Spin chain techniques for angular momentum quasicharacters

The present work is closely related to our previous papers [1–3], which are part of a program that aims at developing a non-perturbative quantum theory of gauge fields in the Hamiltonian framework with special emphasis on the role of non-generic gauge orbit types. In this approach, the starting point is a finite-dimensional lattice approximation of the classical Hamiltonian model. This approximation yields a Hamiltonian system endowed with the action of (the lattice counterpart of) the group of local gauge transformations. If the gauge group G is non-Abelian, then this action necessarily has more than one orbit type. Correspondingly, the reduced phase space of the Hamiltonian system, obtained by symplectic reduction, is a stratified symplectic space. The corresponding quantum theory is obtained via canonical quantization. It is best described in the language of $C^*$-algebras (see [4, 5]). To study the influence of the classical orbit type stratification at the quantum level, the concept of costratification of the quantum Hilbert space as developed by Huebschmann [6] is used. The latter is naturally implemented within the framework of holomorphic quantization (a case study is given in [7]). Within this holomorphic picture, the relations characterizing the classical gauge orbit strata may be implemented at the quantum level. Thereby, each element of the stratification corresponds to the zero locus of a finite subset $\{p_1, \ldots, p_r\}$ of the algebra $\mathcal{R}(G)$ of G-invariant representative functions on $G^N_{\mathbb{C}}$, which will be referred to as quasicharacters³. Via this route, we were led to study the algebra $\mathcal{R}(G)$. This was done in [2, 3] for $G = {\mathrm{SU}}(2)$ and, subsequently, for an arbitrary compact Lie group G. For $G = {\mathrm{SU}}(2)$, our analysis boils down to a problem in the combinatorics of angular momentum theory. For the latter we refer to [8–12]. Using this theory, the multiplicative structure constants of $\mathcal{R}(G)$ may be expressed in terms of Clebsch–Gordan coefficients. As a consequence, we have obtained a characterization of the costrata of quantum theory in terms of systems of linear equations with real coefficients built from Wigner's 3nj symbols. The latter may be further expressed in terms of 9j symbols. For these symbols there exist efficient calculators, that is, the above coefficients can be calculated explicitly. Using the same techniques, we have been also able to reduce the eigenvalue problem for the quantum Hamiltonian to a problem in linear algebra.

Although angular momentum theory yields an effective (computer supported) calculus, we believe that it is interesting to relate the latter to the calculus of classical invariant theory. This is the aim of the present paper. The main tools used for accomplishing our goal come from the theory of characteristic identities, see [13–19]. For a given reduction scheme of the tensor product $(D_{\underline{j}}, H_{\underline{j}})$ of N irreps of ${\mathrm{SU}}(2)$, this theory provides a family of operators ${\mathbb{P}}^j_{\underline{j}, \underline{l}}$ projecting on to the irreducible subspaces $H_{\underline{j} , \underline{l}}$ of $H_{\underline{j}}$. It turns out that the quasicharacters of $D_{\underline{j}}$ may be expressed in terms of traces of the form $\mathrm{Tr} \big( {\mathbb{P}}^j_{\underline{j}, \underline{l}^{^{\prime}}} D_{\underline{j}}(\underline{u}) {\mathbb{P}}^j_{\underline{j}, \underline{l} } \big)$. This observation is the starting point of our calculus, which aims at expressing the quasicharacters in terms of ordinary trace invariants of group elements. Moreover, we are also able to analyse the multiplication laws for quasicharacters along these lines. In the present paper, we only analyse certain types of products. A general analysis for quasicharacter products with distinct group elements is left to future work. For that purpose, we will have to combine the methods used in the present paper with recoupling calculus based on different tree labelling. See lemma A.3 for some insight.

An outline of the paper is as follows. In sections 2 and 3 we provide the reader with an introduction to the algebra of quasicharacters and to the theory of characteristic identities. In section 4 we identify the irreducible quasicharacters with traces of representations of group elements (the N-fold Cartesian product $\times^{N}{\mathrm{SU}}(2)$), over totally coupled angular momentum states, labelled by binary coupling trees T with N leaves and associated intermediate labels. The angular momentum states themselves are realized via symmetrized tensor products of the fundamental spin-$\textstyle{\frac 12}$ representation. Explicit evaluations for certain low degree cases for N = 2 and $N = 3,4$ via projection techniques (section 4.1) show that the quasicharacters are simply polynomials in $2 \times 2$ matrix traces of various group element strings. In section 4.2, certain pointwise products, of spin chain realizations for some generic N = 2 irreducible quasicharacters (arbitrary spin j coupled to spin-$\textstyle{\frac 12}$), multiplied by basic spin-$\textstyle{\frac 12}$ quasicharacters, are themselves expanded in the quasicharacter basis, thus recovering certain structure constants of the multiplicative ring.

In section 5, we briefly review the application of quasicharacter theory to Hamiltonian lattice quantum gauge theory, the central focus of our previous work which has motivated the present investigation.

To make the present paper more self-contained, in the appendix we complement the constructive methods approach with a brief review of the abstract formalism of coupling and recoupling for quasicharacters (as developed in our previous work for both ${\mathrm{SU}}(2)$ [2] and compact G [3]). We provide a summary of important properties of the quasicharacter calculus, such as the behaviour under change of coupling tree, and the role of Racah coefficients as structure constants under pointwise multiplication (appendix A.2). In particular, the results of section 4.2 for N = 2 products can be viewed as a de novo explicit evaluation of certain Racah 9j symbols. For completeness, the 9j result is given an independent proof (appendix A.3), where the general result, that the Racah coefficients which are multiplicative structure constants for arbitrary N, are expressible as products of 9j symbols, is also stated [2, 3]. An index of notation used in the paper is given in table 6.

In the concluding section 6, we draw together some further aspects of the concrete calculations of section 4, with the results reviewed in the appendix, in relation to the combinatorial nature of the quasicharacter ring itself, and also formal role of the recoupling calculus in organizing its structure.

For the convenience of the reader, let us recall some basics (we refer to [20] or [21] for more details). First, we will consider the general case of a compact Lie group. Next, we will limit our attention to the special case $G = {\mathrm{SU}}(2)$. We will assume that all representations under consideration are continuous and unitary.

Let G be a compact Lie group and let $\mathfrak{R}(G)$ be the commutative algebra of representative functions of G. Let $\widehat G$ be the set of isomorphism classes of finite-dimensional irreps of G. Given a finite-dimensional unitary representation $(H, \pi)$ of G, let $C(G)_\pi \subset \mathfrak{R}(G)$ denote the subspace of representative functions of π, that is, the subspace spanned by all matrix coefficients $\langle \zeta , \pi(\cdot) v \rangle$ with $v \in H$ and $\zeta \in H^\ast$ of π. Moreover, let ${\chi}_\pi \in C(G)_\pi$ be the character of π, defined by ${\chi}_\pi(a) : = \operatorname{tr}\big(\pi(a)\big)$. The same notation will be used for the Lie group G^N.

The elements of $\widehat G$ will be labelled by the corresponding highest weight λ relative to some chosen Cartan subalgebra and some chosen dominant Weyl chamber. Assume that for every $\lambda \in \widehat G$ a concrete unitary irrep $(H_\lambda,\pi_\lambda)$ of highest weight λ in the Hilbert space H_λ has been chosen. Given $\underline{\lambda} = (\lambda^1,\dots,\lambda^N) \in \widehat G^N$, we define a representation $(H_{\underline{\lambda}},\pi_{\underline{\lambda}})$ of G^N by

$$\begin{align} H_{\underline{\lambda}} = \bigotimes_{i = 1}^N H_{\lambda^i} \,,\quad \pi_{\underline{\lambda}}\left(\underline{a}\right) = \bigotimes_{i = 1}^N \pi_{\lambda^i}\left(a_i\right)\,,\end{align} \tag{ 2.1 }$$

where $\underline{a} = (a_1 , \dots , a_N)$. This representation is irreducible and we have

$$\begin{align*} C\left(G^N\right)_{\pi_{\underline{\lambda}}} \cong \bigotimes_{i = 1}^N C\left(G\right)_{\pi_{\lambda^i}} \,,\end{align*}$$

isometrically with respect to the L²-norms. Using this, together with the Peter–Weyl theorem for G, we obtain that $\bigoplus_{\underline{\lambda} \in \widehat G^N} C(G^N)_{\pi_{\underline{\lambda}}}$ is dense in $L^2 (G^N, {\mathrm{d}}^N a)$. Since $\bigoplus_{\underline{\lambda} \in \widehat G^N} C(G^N)_{\pi_{\underline{\lambda}}} \subset \bigoplus_{\pi \in \widehat{G^N}} C(G^N)_\pi$, this implies

Lemma 2.1. Every irreducible representation of G^N is equivalent to a product representation $(H_{\underline{\lambda}},\pi_{\underline{\lambda}})$ with $\underline{\lambda} \in \widehat G^N$. If $(H_{\underline{\lambda}},\pi_{\underline{\lambda}})$ and $(H_{\underline{\lambda}^{^{\prime}}},\pi_{\underline{\lambda}^{^{\prime}}})$ are isomorphic, then $\underline{\lambda} = \underline{\lambda}^{^{\prime}}$.□

Given $\underline{\lambda} \in \widehat G^N$, let $\pi^d_{\underline{\lambda}}$ denote the representation of G on $H_{\underline{\lambda}}$ defined by

$$\begin{align} \pi^d_{\underline{\lambda}}\left(a\right) : = \pi_{\underline{\lambda}} \left(a, \ldots, a\right) \,.\end{align} \tag{ 2.2 }$$

This representation will be referred to as the diagonal representation induced by $\pi_{\underline{\lambda}}$. It is reducible and has the isotypical decomposition

$$\begin{align*} H_{\underline{\lambda}} = \bigoplus_{\lambda \in \widehat G} H_{\underline{\lambda},\lambda}\end{align*}$$

into uniquely determined subspaces $H_{\underline{\lambda},\lambda}$. Recall that these subspaces may be obtained as the images of the orthogonal projectors

$$\begin{align} {\mathbb{P}}_\lambda : = \dim \left(H_\lambda\right) \int_G \overline{{\chi}_{\pi_\lambda}\left(a\right)} \, \pi_{\underline{\lambda}}\left(a\right) \, \mathrm{d} a\end{align} \tag{ 2.3 }$$

on $H_{\underline{\lambda}}$. These projectors commute with one another and with $\pi^d_{\underline{\lambda}}$. If an isotypical subspace $H_{\underline{\lambda},\lambda}$ is reducible, we can further decompose it in a non-unique way into irreducible subspaces of isomorphism type λ. Let $m_{\underline{\lambda}}(\lambda)$ denote the number of these irreducible subspaces (the multiplicity of π_λ in $\pi^d_{\underline{\lambda}}$) and let $\widehat G_{\underline{\lambda}}$ denote the subset of $\widehat G$ consisting of the highest weights λ such that $m_{\underline{\lambda}}(\lambda) \gt 0$. In this way, we obtain a unitary G-representation isomorphism

$$\begin{align} \varphi_{\underline{\lambda}} : H_{\underline{\lambda}} ~\to~ \bigoplus_{\lambda \in \widehat G_{\underline{\lambda}}} \, \bigoplus_{k = 1}^{m_{\underline{\lambda}}\left(\lambda\right)} H_\lambda \,.\end{align} \tag{ 2.4 }$$

Let

$$\begin{align} \mathrm{pr}^{\underline{\lambda}}_{\lambda,k} : \bigoplus_{\lambda \in \widehat G_{\underline{\lambda}}} \, \bigoplus_{l = 1}^{m_{\underline{\lambda}}\left(\lambda\right)} H_\lambda \to H_\lambda \,,\qquad \mathrm{i}^{\underline{\lambda}}_{\lambda,k} : H_\lambda \to \bigoplus_{\lambda \in \widehat G_{\underline{\lambda}}} \, \bigoplus_{l = 1}^{m_{\underline{\lambda}}\left(\lambda\right)} H_\lambda \, ,\end{align} \tag{ 2.5 }$$

denote the natural projections and injections of the direct sum. For every $\lambda \in \widehat G_{\underline{\lambda}}$ and every $k,l = 1, \dots , m_{\underline{\lambda}}(\lambda)$, define a G-representation endomorphism $A^{\underline{\lambda},\lambda}_{k,l}$ of $\pi^d_{\underline{\lambda}}$ by

$$\begin{align} A^{\underline{\lambda},\lambda}_{k,l} : = \frac{1}{\sqrt{\dim \left(H_\lambda\right)}} ~ \varphi^{-1}_{\underline{\lambda}} \circ \mathrm{i}^{\underline{\lambda}}_{\lambda,k} \circ \mathrm{pr}^{\underline{\lambda}}_{\lambda,l} \circ \varphi_{\underline{\lambda}}.\end{align} \tag{ 2.6 }$$

We define

$$\begin{align} \left({\chi}_{\underline{\lambda}}\right)^{\lambda}_{k,l}\left(\underline{a}\right) : = \sqrt{\dim\left(H_{\underline{\lambda}}\right)} \, \operatorname{tr}\left(\pi_{\underline{\lambda}}\left(\underline{a}\right) A^{\underline{\lambda},\lambda}_{k,l}\right) \,.\end{align} \tag{ 2.7 }$$

Proposition 2.2. The family of functions

$$\begin{align*} \left\{ \left({\chi}_{\underline{\lambda}}\right)^{\lambda}_{k,l} ~:~ \underline{\lambda} \in \widehat G^N ,~ \lambda \in \widehat G_{\underline{\lambda}} \,,~ k,l = 1 , \dots , m_{\underline{\lambda}}\left(\lambda\right) \right\}\end{align*}$$

constitutes an orthonormal basis in $L^2(G^N)^G$.□

For the proof we refer to [2]. By this proposition, the G-invariant representative functions $({\chi}_{\underline{\lambda}})^{\lambda}_{k,l}$, which in the sequel will be referred to as quasicharacters, span the subalgebra

$$\begin{align*} \mathcal{R}\left(G\right) : = \mathfrak{R}\left(G^N\right)^G\end{align*}$$

of G-invariant elements of $\mathfrak{R}(G^N)$.

Next, let us turn to the discussion of the multiplicative structure of $\mathcal{R}(G)$. As the $({\chi}_{\underline{\lambda}})^{\lambda}_{k,l}$ form a basis, it is enough to find the multiplication law for these functions. For that purpose, we assume that a unitary G-representation isomorphism $\varphi_{\underline{\lambda}}$ as given by (2.4) has been chosen for every $\underline{\lambda} \in \widehat G^N$ and every N. Denote $d_{\lambda} : = \dim H_\lambda$ and $d_{\underline{\lambda}} : = \dim H_{\underline{\lambda}}$. Then, using (2.7), we can write

$$\begin{align} \nonumber & \left({\chi}_{\underline{\lambda}_1}\right)^{\lambda_1}_{k_1,l_1}\left(\underline{a}\right) \, \left({\chi}_{\underline{\lambda}_2}\right)^{\lambda_2}_{k_2,l_2}\left(\underline{a}\right)\end{align}$$

$$\begin{align} & \quad = \sqrt{d_{\underline{\lambda}_1} d_{\underline{\lambda}_2}} \, \operatorname{tr} \left( \left( A^{\underline{\lambda}_1,\lambda_1}_{k_1,l_1} \otimes A^{\underline{\lambda}_2,\lambda_2}_{k_2,l_2} \right) \circ \left(\pi_{\underline{\lambda}_1}\left(\underline{a}\right) \otimes \pi_{\underline{\lambda}_2}\left(\underline{a}\right)\right) \right) \,.\end{align} \tag{ 2.8 }$$

If we wish to decompose this product in terms of the $({\chi}_{\underline{\lambda}})^{\lambda}_{k,l}$, we have to decompose the G^N-representation $\pi_{\underline{\lambda}_1} \otimes \pi_{\underline{\lambda}_2}$ into G^N-irreps $\underline{\lambda}$ and then relate the latter to the basis functions using $\varphi_{\underline{\lambda}}$. This procedure is explained in detail in [2] with the result stated there as proposition 3.10.

If we introduce composite indices $I, J\,, \cdots$ representing the label sets $\underline{\lambda}, \lambda, k, l$, the product rule for the quasicharacter ring reads schematically as follows:

$$\begin{align} {\chi}_{I_1} {\chi}_{I_2} = \left.\sum\right._{I_3} C^{I_3}_{I_1I_2} {\chi}_{I_3}\, ,\end{align} \tag{ 2.9 }$$

with the structure constants $C^{I_3}_{I_1I_2}$ depending on two auxiliary unitary G-representation isomorphisms constructed by using the chosen isomorphism $\varphi_{\underline{\lambda}}$ and are given, up to normalization, by group theoretical Racah recoupling coefficients for G. Below, we will see that for $G = {\mathrm{SU}}(2)$ the coefficients boil down to recoupling coefficients of angular momentum theory.

Thus, let us turn to the special case $G = {\mathrm{SU}}(2)$. Here, the highest weights of irreps $\widehat {\mathrm{SU}}(2)$ are in one-to-one correspondence with spins $j = 0 , \frac 1 2 , 1 , \frac 3 2 , \dots$. We will use the common notation D_j for π_j. Thus, $(H_j,D_j)$ is the standard ${\mathrm{SU}}(2)$-irrep of spin j, with dimension $d_j = 2j+1 (\equiv {[}j{]})$ spanned by the orthonormal ladder basis $\{|j,m\rangle : m = -j , -j+1 , \dots , j\}$ which is unique up to a phase. Accordingly, every sequence of highest weights corresponds to a sequence $\underline{j} = j_1,j_2, \cdots, j_N$ of spins. We write $D_{\underline{j}} \equiv \pi_{\underline{j}}$ for the corresponding irrep of $\times^{N}{\mathrm{SU}}(2) = {\mathrm{SU}}(2)^N$ and $D^d_{\underline{j}} \equiv \pi^d_{\underline{j}}$ for the induced diagonal representation of ${\mathrm{SU}}(2)$. To fix the G-representation isomorphisms

$$\begin{align} \varphi_{\underline{j}} : H_{\underline{j}} \to \bigoplus_j \bigoplus_{i = 1}^{m_{\underline{j}}\left(j\right)} H_j \,,\end{align} \tag{ 2.10 }$$

we choose the following reduction scheme for tensor products of N irreps of ${\mathrm{SU}}(2)$. Given nonnegative half integers s₁, s₂, denote

$$\begin{align*} \langle s_1 , s_2 \rangle : = \left\{|s_1-s_2| , |s_1-s_2|+1 , |s_1-s_2|+2 , \dots , s_1+s_2\right\}\end{align*}$$

and recall that the representation space $H_{s_1} \otimes H_{s_2}$ decomposes into unique irreducible subspaces $(H_{s_1} \otimes H_{s_2})_s$ of spin $s \in \langle s_1 , s_2 \rangle$. We start by decomposing $H_{j_1}\otimes H_{j_2}$ into the unique irreducible subspaces $(H_{j_1}\otimes H_{j_2})_{l_2}$ with $l_2 \in \langle j_1 , j_2 \rangle$. Then, we decompose the invariant subspaces

$$\begin{align*} \left(H_{j_1}\otimes H_{j_2}\right)_{l_2} \otimes H_{j_3} \subset H_{j_1}\otimes H_{j_2}\otimes H_{j_3}\end{align*}$$

into unique irreducible subspaces

$$\begin{align*} \left(\left(H_{j_1}\otimes H_{j_2}\right)_{l_2} \otimes H_{j_3}\right)_{l_3} \,,\quad l_3 \in \langle l_2 , j_3 \rangle \,.\end{align*}$$

Iterating this, we end up with a decomposition of $H_{\underline{j}}$ into unique irreducible subspaces

$$\begin{align} H_{\underline{j} , \underline{l}} : = \left( \cdots \left(\left(H_{j_1}\otimes H_{j_2}\right)_{l_2} \otimes H_{j_3}\right)_{l_3} \cdots \otimes H_{j_N}\right)_{l_N} \,,\end{align} \tag{ 2.11 }$$

where $\underline{l} = (l_1 , \dots , l_N)$ is a sequence of nonnegative half integers satisfying $l_1 = j_1$ and $l_i \in \langle l_{i-1} , j_i\rangle$ for $i = 2 , 3 , \dots , N$. Note that for each l_i, the sequence $l_i, j_{i+ 1}, \cdots, j_N$ labels a subspace $H_{l_i}\otimes H_{j_{i+1}}\otimes \cdots\otimes H_{j_N}$ isomorphic to an irreducible representation of member $N- i + 1$ of a descending chain of subgroups

$$\begin{align} \times^N {\mathrm{SU}}\left(2\right) \supset \times^{N-1}{\mathrm{SU}}\left(2\right)\supset\cdots \times^2 {\mathrm{SU}}\left(2\right)\supset {\mathrm{SU}}\left(2\right)^d\,.\end{align} \tag{ 2.12 }$$

Let us denote the totality of such sequences by $R_{\underline{j}}$. Moreover, denote

$$\begin{align} R_{\underline{j}} \left(j\right) = \left\{\underline{l} \in R_{\underline{j}} : l_N = j\right\} \,.\end{align} \tag{ 2.13 }$$

Then, $m_{\underline{j}}(j) = |R_{\underline{j}} (j)|$, and the isotypical component $H_{\underline{j},j}$ of $H_{\underline{j}}$ is given by the direct sum of the subspaces $H_{\underline{j} , \underline{l}}$ with $\underline{l} \in R_{\underline{j}}(j)$.

To define the isomorphism $\varphi_{\underline{j}}$, we choose⁴ ladder bases in the irreducible subspaces $H_{\underline{j} , \underline{l}}$. Denote their elements by $|\underline{j} , \underline{l} , m\rangle$, where $m = -j, -j+1 , \dots , j$. Then,

$$\begin{align} \left\{|\underline{j} , \underline{l} , m\rangle : \underline{l} \in R_{\underline{j}} \left(j\right), m = - j , -j+1 , \dots , j\right\}\end{align} \tag{ 2.14 }$$

is an orthonormal basis in $H_{\underline{j}}$, and we can use the sequences $\underline{l} \in R_{\underline{j}}(j)$ to label the copies of H_j in the direct sum decomposition of the target space of $\varphi_{\underline{j}}$. Thus, the natural injections (2.5) related to this decomposition read $\mathrm{i}^{\underline{j}}_{j,\underline{l}}$, the basis functions read $({\chi}_{\underline{j}})^{j}_{\underline{l},\underline{l}^{^{\prime}}}$ and the endomorphisms appearing in their definition read $A^{\underline{j},j}_{\underline{l},\underline{l}^{^{\prime}}}$. We define $\varphi_{\underline{j}}$ by

$$\begin{align*} \varphi_{\underline{j}}\left(|\underline{j},\underline{l},m\rangle\right) : = \mathrm{i}^{\underline{j}}_{j,\underline{l}}\left(|j,m\rangle\right) \,,\end{align*}$$

where $|j,m\rangle$ denotes the elements of the orthonormal ladder basis in H_j. Under this choice, one obtains

$$\begin{align} A^{\underline{j},j}_{\underline{l},\underline{l}^{^{\prime}}} = \frac{1}{\sqrt{d_j}} \, \sum_{m = -j}^j |\underline{j},\underline{l},m\rangle \langle\,\underline{j},\underline{l}^{^{\prime}},m| \,,\qquad \underline{l},\underline{l}^{^{\prime}} \in R\left(\underline{j},j\right) \, ,\end{align} \tag{ 2.15 }$$

and

$$\begin{align} \left({\chi}_{\underline{j}}\right)^{j}_{\underline{l},\underline{l}^{^{\prime}}}\left(\underline{a}\right) = \sqrt{\frac{d_{\underline{j}}}{d_j}} \, \sum_{m = -j}^j \langle\,\underline{j},\underline{l}^{^{\prime}},m| D^{\underline{j}}\left(\underline{a}\right) |\underline{j},\underline{l},m\rangle \,,\qquad \underline{l},\underline{l}^{^{\prime}} \in R\left(\underline{j},j\right) \,.\end{align} \tag{ 2.16 }$$

Using the matrix elements $D^{j_i}_{m_i,m_i^{^{\prime}}}$, $i = 1,\dots,N$, together with the Clebsch–Gordan coefficients $C^{s_1,s_2,s}_{m_1,m_2,m}$, we obtain the following.

Proposition 2.3. We have

$$\begin{align*} \left({\chi}_{\underline{j}}\right)^{j}_{\underline{l},\underline{l}^{^{\prime}}}\left(\underline{a}\right) = \sqrt{\frac{d_{\underline{j}}}{d_j}} \sum_{m = -j}^j \sum_{\underline{m}} \sum_{\underline{m}^{^{\prime}}} \, C\left(\underline{j},\underline{l},\underline{m}\right) \, C\left(\underline{j},\underline{l}^{^{\prime}},\underline{m}^{^{\prime}}\right) \, D^{j_1}_{m_1^{^{\prime}},m_1}\left(a_1\right) \cdots D^{j_N}_{m_N^{^{\prime}},m_N}\left(a_N\right) \,,\end{align*}$$

where $\sum_{\underline{m}}$ means the sum over all sequences $\underline{m} = (m_1 , \dots , m_N)$ such that

$$\begin{align*} m_i = - j^i , \dots , j^i \ \textrm{for } i = 1 , \dots , N \,,\qquad m_1 + \cdots + m_N = m \,,\end{align*}$$

and where

$$\begin{align*} C\left(\underline{j},\underline{l},\underline{m}\right) = C^{j^1,j^2,l^2}_{m_1,m_2,m_1+m_2} C^{l^2,j^3,l^3}_{m_1+m_2,m_3,m_1+m_2+m_3} \cdots C^{l^{N-1},j^N,l^N}_{m_1+\cdots+m_{N-1},m_N,m} \,.\end{align*}$$

□

For the proof we refer to [2]. Next, one can derive the following multiplication law for the representative functions.

Proposition 2.4. For $G = {\mathrm{SU}}(2)$, the multiplication law for the basis functions $({\chi}_{\underline{j}})^{j}_{\underline{l},\underline{l}^{^{\prime}}}$ reads as follows:

$$\begin{align} \nonumber \left({\chi}_{\underline{j}_1}\right)^{j_1}_{\underline{l}_1,\underline{l}_1^{^{\prime}}} \cdot \left({\chi}_{\underline{j}_2}\right)^{j_2}_{\underline{l}_2,\underline{l}_2^{^{\prime}}} & = \sqrt{\frac{d_{\underline{j_1}}d_{\underline{{j_2}}}}{d_{j_1}d_{j_2}}} ~ \sum_{\underline{j} \in \langle \underline{j}_1,\underline{j}_2 \rangle} ~ \sum_{\underline{l},\underline{l}^{^{\prime}}\in R_{\underline{j}}\left( j\right)} ~ \sqrt{\frac{d_j}{d_{\underline{j}}}} \end{align}$$

$$\begin{align} & \quad \times U_{\underline{j}_1,\underline{j}_2}\left(\underline{j},\underline{l};\underline{l}_1,\underline{l}_2\right) \, U_{\underline{j}_1,\underline{j}_2}\left(\underline{j},\underline{l}^{^{\prime}};\underline{l}_1^{^{\prime}},\underline{l}_2^{^{\prime}}\right) \, \left({\chi}_{\underline{j}}\right)^{j}_{\underline{l},\underline{l}^{^{\prime}}} \,,\end{align}$$

where

$$\begin{align} U_{\underline{j}_1,\underline{j}_2}\left(\underline{j},\underline{l};\underline{l}_1,\underline{l}_2\right) = \langle\,\underline{j}_1,\underline{j}_2;\underline{j},\underline{l},m|\underline{j}_1,\underline{j}_2;\underline{l}_1,\underline{l}_2;j,m\rangle\end{align} \tag{ 2.17 }$$

for every $\underline{j} \in \langle \underline{j}_1,\underline{j}_2 \rangle$, $j \in \langle j_1,j_2 \rangle$ and $\underline{l} \in R(\underline{j}, j)$, and for any admissible m.□

For the proof see [2]. It can be seen from the proof that the recoupling coefficients U are (up to some phase choices) uniquely determined by the above choice of the isomorphism $\varphi_{\underline{\lambda}}$. Up to normalization, they are given by what is known as $3(2N-1)j$ symbols. Finally, it turns out that the recoupling coefficients can be expressed in terms of the recoupling coefficients for N = 2.

Lemma 2.5. The recoupling coefficients $U_{\underline{j}_1,\underline{j}_2}(\underline{j},\underline{l};\underline{l}_1,\underline{l}_2)$ are given by

$$\begin{align*} U_{\underline{j}_1,\underline{j}_2}\left(\underline{j},\underline{l};\underline{l}_1,\underline{l}_2\right) = \prod_{i = 2}^N \begin{pmatrix} l_1^{i-1} & l_2^{i-1} & l^{i-1} \\ j_1^i & j_2^i & j^i \\ l_1^i & l_2^i & l^i \end{pmatrix} ,\end{align*}$$

where $l_1^1 : = j_1^1$, $l_2^1 : = j_2^1$ and $l^1 : = j^1$. Here,

$$\begin{align*} \begin{pmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \\ j_7 & j_8 & j_9 \end{pmatrix} = \sqrt{d_{j_3}d_{j_6}d_{j_7}d_{j_8}} \begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \\ j_7 & j_8 & j_9 \end{Bmatrix},\end{align*}$$

with the bracket on the right hand side denoting the 9j Racah symbols.

To summarize, the above results relate the structure of the algebra $\mathcal R({\mathrm{SU}}(2))$ to the combinatorics of recoupling theory of angular momentum as provided in [8–12]. Moreover, as there exist efficient calculators for 9j symbols, provided e.g. by the Python library SymPy [22] or online by Anthony Stone's Wigner coefficient calculator⁵, lemma 2.5 provides an explicit knowledge of the multiplication law in the commutative algebra $\mathcal{R}({\mathrm{SU}}(2))$.

Recall that the analysis of the algebra $\mathcal{R}({\mathrm{SU}}(2))$ in terms of generalized Clebsch–Gordan coefficients provided in the previous section relies on the sequence of irreducible modules displayed in (2.11). Clearly, associated with that decomposition of $H_{\underline{j}}$, there is an ordered composition of projection operators from pairwise tensor products of irreducible spin representations on to specific irreducible constituents. This observation provides the algorithm that we are going to propose as an alternative calculational method for dealing with the algebra $\mathcal{R}({\mathrm{SU}}(2))$. It will be based on the general theory of characteristic identities for Lie algebras (see [13–19]).

For the convenience of the reader, let us give a short introduction to the subject. Our presentation is along the lines of [15, 16] (see also [23]). We also refer to [24] for the basics on Lie algebras.

In the standard notation of [24], let L be a semi-simple Lie algebra (over some field $\mathbb{F}$), let H be a fixed Cartan subalgebra with dual $H^\ast$. Let Φ be the set of roots relative to H and let $\Phi^+$ be the set of positive roots. Let us choose a basis $\{x_1, \ldots,x_l \}$ of L and let $\{x^1, \ldots,x^l \}$ be the dual basis with respect to the Killing form of L. In particular, we can choose a basis consisting of root space elements, together with a basis $\{h_1, \ldots, h_m \}$ of H. Moreover, let us denote $\delta = \tfrac{1}{2} \sum_{\alpha \in \Phi^+} \alpha$, let $(\cdot, \cdot)$ be the inner product on $H^\ast$ induced by the symmetric Killing form and Λ be the labelling operator, that is, the operator which by acting on an irrep of L with highest weight µ coincides with µ. For a chosen basis $\{h_1, \ldots, h_m \}$ of H it is represented by a vector operator with components $\Lambda(h_i)$ taking constant values $\mu(h_i)$.

Now, let π be a finite-dimensional representation of L and let us consider a matrix operator A with entries

$$\begin{align} A^i{}_j = -\frac{1}{2} \sum_{r = 1}^l \left( \pi\left(x_r\right)^i{}_j x^r + \pi\left(x^r\right)^i{}_j x_r \right)\, ,\end{align} \tag{ 3.1 }$$

together with its adjoint $\overline{A}^i{}j = - A_j{}^i$. Let T be a contragredient tensor operator relative to π. Then one can show the following identities:

$$\begin{align} A T - T \overline{A} = \left(2A -\pi\left(c_L\right) \right) T = \left[c_L, T\right] \, ,\end{align} \tag{ 3.2 }$$

where c_L is the universal Casimir element of L.

Let $(V_\lambda, \pi_\lambda)$ be a finite-dimensional irreducible L-module with highest weight λ, and for this section let $\{\lambda_i\}_{i = 1}^k$ $= \{\lambda_1, \ldots, \lambda_k\}$ be the list of the distinct weights in $V(\lambda)$. Let A^λ be the corresponding matrix operator given by (3.1) with π replaced by π_λ. Then, any irreducible contragredient tensor operator T with highest weight λ may be decomposed into shift tensors $T_{[i]}$, $i = 1,\cdots, k$, which decrease the eigenvalue of the labelling operator Λ on an irrep by the weight λ_i,

$$\begin{align*} \Lambda\left(h_j\right) T_{\left[i\right]} = T_{\left[i\right]} \left(\Lambda - \lambda_i\right) \left(h_j\right) \, .\end{align*}$$

In terms of the shift operators, the identities (3.2) take the following form:

$$\begin{align} \left( A^\lambda - \Lambda_i\right) T_{\left[i\right]} = 0 \, , \quad T_{\left[i\right]} \left(\overline{A}^\lambda - \overline{\Lambda}_i \right) = 0\, ,\end{align} \tag{ 3.3 }$$

where $\Lambda_i$ and $\overline{\Lambda}_i$ are given by

$$\begin{align} \Lambda_i & = \tfrac{1}{2} \left(\lambda, \lambda + 2 \delta\right) - \tfrac{1}{2} \left(\lambda_i, 2\left(\Lambda + \delta\right) + \lambda_i\right) \, ,\end{align} \tag{ 3.4 }$$

$$\begin{align} \overline{\Lambda}_i & = \tfrac{1}{2} \left(\lambda, \lambda + 2 \delta\right) + \tfrac{1}{2} \left(\lambda_i, 2\left(\Lambda + \delta\right) - \lambda_i\right) \, .\end{align} \tag{ 3.5 }$$

Moreover, we have the following.

Proposition 3.1. The matrix operators A^λ and $\overline A^\lambda$ satisfy the following characteristic identities:

$$\begin{align} \prod_{i = 1}^k \left( A^\lambda -\Lambda_i \right) = 0 \, , \quad \prod_{i = 1}^k \left( \overline{A}^\lambda - \overline{\Lambda}_i \right) = 0 \, .\end{align} \tag{ 3.6 }$$

Remark 3.1. If $V(\mu)$ is an irreducible module, then A^λ acting on $V(\mu)$ yields

$$\begin{align} A^{\lambda, \mu} = \tfrac{1}{2} \left(\pi_\lambda \otimes \pi_\mu \left(c_L\right) - \pi_\lambda\left( c_L\right) \otimes 1 - 1 \otimes \pi_\mu\left( c_L\right)\right) \, .\end{align} \tag{ 3.7 }$$

Thus, for every highest weight module V_µ, A^λ yields an operator $A^{\lambda, \mu}$ acting on the tensor product $V(\lambda) \otimes V(\mu)$. If we decompose this product into irreps,

$$\begin{align} V\left(\lambda\right) \otimes V\left(\mu\right) = \bigoplus_{i = 1}^k m_{\lambda \mu}^{\mu + \lambda_i} \, V\left(\mu + \lambda_i \right) \, ,\end{align} \tag{ 3.8 }$$

then $A^{\lambda, \mu}$ yields the constant value

$$\begin{align} \alpha_i = \tfrac{1}{2} \left(\lambda, \lambda + 2 \delta\right) - \tfrac{1}{2} \left(\lambda_i, 2\left(\mu + \delta\right) + \lambda_i\right)\end{align} \tag{ 3.9 }$$

on each subspace $V(\mu + \lambda_i )$. This, together with the observation that a diagonal matrix B with distinct eigenvalues b_i satisfies $\Pi_{i = 1}^k (B - b_i) = 0$, essentially yields the proof of the characteristic identities (3.6).□

By extending the relation $A^\lambda T_{[i]} = \Lambda_i T_{[i]}$ in (3.3) to any polynomial $p(A^\lambda)$ and choosing, in particular, $p(A^\lambda) = \Pi_{l \neq j } (A^\lambda - \Lambda_l)$, we obtain

$$\begin{align} \Pi_{l \neq j } \left(A^\lambda - \Lambda_l\right) T_{\left[i\right]} = \Pi_{l \neq j } \left(\Lambda_i - \Lambda_l\right) T_{\left[i\right]} \, .\end{align} \tag{ 3.10 }$$

Thus, we have

$$\begin{align} \mathbb{P}_j^\lambda T_{\left[i\right]} = \delta_{ij} T_{\left[i\right]} \, ,\end{align} \tag{ 3.11 }$$

where $\mathbb{P}_j^\lambda$ is a projection operator given by

$$\begin{align} \mathbb{P}_j^\lambda = \prod_{l \neq j} \frac{A^\lambda- \Lambda_l}{\Lambda_j - \Lambda_l} \, .\end{align} \tag{ 3.12 }$$

Correspondingly, for $\overline{A}^\lambda$, we obtain

$$\begin{align} T_{\left[i\right]} \overline{\mathbb{P}}_j^\lambda = \delta_{ij} T_{\left[i\right]} \, , \quad \overline{\mathbb{P}}_j^\lambda = \prod_{l \neq j} \frac{\overline{A}^\lambda - \overline{\Lambda}_l}{\overline{\Lambda}_j - \overline{\Lambda}_l} \, .\end{align} \tag{ 3.13 }$$

If we decompose the tensor operator T into its shift components, $T = \sum_{i = 1}^k T_{[i]}$, then we clearly have

$$\begin{align} T_{\left[i\right]} = \mathbb{P}_i^\lambda T \, , \quad T_{\left[i\right]} = T \overline{\mathbb{P}}_i^\lambda \, .\end{align} \tag{ 3.14 }$$

Now, equations (3.6), (3.12)–(3.14) immediately imply the following.

Proposition 3.2. The projection operators $\mathbb{P}_j^\lambda$ and $\overline{\mathbb{P}}_j^\lambda$ fulfil

$$\begin{align} A^\lambda \mathbb{P}_j^\lambda = \Lambda_j \mathbb{P}_j^\lambda \, , \quad \overline{A}^\lambda \overline{\mathbb{P}}_j^\lambda = \overline{\Lambda}_j \overline {\mathbb{P}}_j \, ,\end{align} \tag{ 3.15 }$$

$$\begin{align} \mathbb{P}_i^\lambda \mathbb{P}_j^\lambda = \delta_{ij} \mathbb{P}_j^\lambda \, , \quad \overline{\mathbb{P}}_i^\lambda \overline{\mathbb{P}}_j^\lambda = \delta_{ij} \overline{\mathbb{P}}_j^\lambda \, ,\end{align} \tag{ 3.16 }$$

$$\begin{align} \sum_{i = 1}^k \mathbb{P}_i^\lambda = \unicode{x1D7D9} \, , \quad \sum_{i = 1}^k \overline{\mathbb{P}}_i^\lambda = \unicode{x1D7D9} \, .\end{align} \tag{ 3.17 }$$

Note that (3.15) yields the spectral decompositions of A^λ and $\overline{A}^\lambda$. That is, given the Clebsch–Gordan decomposition (3.8), the operator $\mathbb{P}_j^\lambda$ projects onto the irreducible component $V(\mu + \lambda_j)$ and, thus, the action of $A^\lambda \mathbb{P}_j^\lambda$ onto an element of $V(\lambda) \otimes V(\mu)$ yields a vector of $V(\mu + \lambda_j)$ multiplied by the eigenvalue α_j given by (3.9). In the sequel, if an irreducible module V_µ is chosen, then in accordance with the notation $A^{\lambda, \mu}$, the corresponding projectors will be denoted by

$$\begin{align} \mathbb{P}_{\lambda, \mu}^{\mu + \lambda_j} \, .\end{align} \tag{ 3.18 }$$

Remark 3.2. In general, for a given irrep $V(\mu)$, not all of the representations $V(\mu + \lambda_i)$ occur in the Clebsch–Gordan decomposition (3.8). This fact leads to a reduction of the characteristic identities (3.6). As was shown in [16], the product over all i in (3.6) gets reduced to the following subset:

$$\begin{align*} \overline{I} \left(\lambda, \mu\right) = \left\{i : \left(\mu + \delta - \lambda_i, \alpha \right) \neq 0 \, \, \ \textrm{for all } \, \, \alpha \in \Phi^+ \right\} \, .\end{align*}$$

□

Turning to the formalism for the case $G = {\mathrm{SU}}(2)$, we denote the matrix operators $A^{\lambda, \mu}$ by $\mathbb{J}^{j,k}$ and the corresponding projectors by ${\mathbb{P}}_{j,k}^{k + \lambda_j}$. For convenience, from now on, we will use the Dynkin labelling, that is, we will write $j = 2 \tfrac 12 j$ for spin $\tfrac 12 j$. In this notation, the Casimir invariant reads $\textstyle{\frac 14}j(j+2)$ ) (cf equation (5.3)).

From (3.7) we have

$$\begin{align} \mathbb{J}^{j,k} = \tfrac{1}{2} \left(\pi_j \otimes \pi_k \left(c_L\right) - \pi_j\left( c_L\right) \otimes 1 - 1 \otimes \pi_k\left( c_L\right)\right) \, .\end{align} \tag{ 3.19 }$$

Note that the angular momentum generators for arbitrary spin are $\mathbf{J}^j = \pi_j(\tfrac 12 \boldsymbol{\sigma})$, so that

$$\begin{align} \mathbb{J}^{j,k} = \left({\mathbf{J}^j}\otimes \textsf{Id}\right)\cdot \left(\textsf{Id}\otimes {\mathbf{J}^k}\right) \equiv {\mathbf{J}^j} \cdot {\mathbf{J}^k} \, ,\end{align} \tag{ 3.20 }$$

with eigenvalues

$$\begin{align} \alpha_{\ell} = \textstyle{\frac 12}\left(\textstyle{\frac 12}\ell\left(\textstyle{\frac 12}\ell+1\right) - \textstyle{\frac 12}j\left(\textstyle{\frac 12}j+1\right)-\textstyle{\frac 12}k\left(\textstyle{\frac 12}k+1\right)\right)\,,\quad \mbox{for} \quad\ell \in \langle j,k\rangle\,.\end{align} \tag{ 3.21 }$$

In the sequel, the following projectors will be useful.

Lemma 3.3 (Angular momentum projectors for $H_1 \otimes H_j$ and $H_2 \otimes H_j$). 

$$\begin{align} {\mathbb{P}}^{j\!+\!1}_{\left(1,j\right)} : = & \, \frac{ 2{\mathbb{J}}^{\left(1,j\right)} \!+\! \left(\textstyle{\frac 12}j\!+\!1\right) \unicode{x1D7D9} \otimes \unicode{x1D7D9}^j}{j\!+\!1} \, , \quad {\mathbb{P}}^{j-1}_{\left(1,j\right)} : = \frac{\textstyle{\frac 12}j\unicode{x1D7D9}\otimes \unicode{x1D7D9}^j -2{\mathbb{J}}^{\left(1,j\right)}}{j\!+\!1}\, .\end{align} \tag{ 3.22 }$$

$$\begin{align} {\mathbb{P}}^{j\!+\!2}_{\left(2,j\right)} : = &\, \frac{\left(\mathbb{J}^{2,j}\right)^2\!+\!\left(\textstyle{\frac 12}j\!+\!2\right)\left(\mathbb{J}^{2,j}\right) \!+\! \left(\textstyle{\frac 12}j\!+\!1\right)\unicode{x1D7D9}^{2}\otimes \unicode{x1D7D9}^j}{\left(\textstyle{\frac 12}j\!+\!1\right)\left(j\!+\!1\right)}\, , \nonumber \\ {\mathbb{P}}^{j}_{\left(2,j\right)} : = &\, \frac{-\left(\mathbb{J}^{2,j}\right)^2-\left(\mathbb{J}^{2,j}\right) \!+\! \textstyle{\frac 12}j\left(\textstyle{\frac 12}j\!+\!1\right) \unicode{x1D7D9}^{2}\otimes \unicode{x1D7D9}^j}{\left(\textstyle{\frac 12}j\!+\!1\right)\textstyle{\frac 12}j}\, , \nonumber \\ {\mathbb{P}}^{j-2}_{\left(2,j\right)} : = &\, \frac{\left(\mathbb{J}^{2,j}\right)^2\!+\!\left(1-\textstyle{\frac 12}j\right)\left(\mathbb{J}^{2,j}\right) -\textstyle{\frac 12}j \unicode{x1D7D9}^{2}\otimes \unicode{x1D7D9}^j}{\left(j\!+\!1\right)\textstyle{\frac 12}j}\, .\end{align} \tag{ 3.23 }$$

□

Proof. We will prove the first formula, to check the remaining ones is left to the reader. By (3.12), we have

$$\begin{align*} \mathbb{P}^{j+1}_{\left(1,j\right)} = \frac{\mathbb{J}^{\left(1,j\right)} -\alpha_{j-1} \unicode{x1D7D9}}{\alpha_{j+1} - \alpha_{j-1}}\, .\end{align*}$$

Next, by (3.20), we have the eigenvalues $\alpha_{j+1} = \tfrac{1}{4}j$ and $\alpha_{j-1} = -\tfrac{1}{4}j - \frac{1}{2}$. Inserting them into the above formula yields the assertion. □

In this section we are going to study the algebra of ${\mathrm{SU}}(2)$ quasicharacters in terms of the characteristic projectors introduced above:

• 1.  
  We will develop an algebraic calculus which allows to determine the quasicharacters in terms of trace invariants.
• 2.  
  We will apply our method to the explicit analysis of the multiplication law in terms of trace invariants.

Clearly, one cannot hope to produce general formulae, so in this paper we will concentrate on the cases N = 2 as well as examples from $N = 3, N = 4$. Our presentation will be expository rather than formal, with the algorithmic steps being presented by way of examples.

To start with, recall from (2.11) that in terms of the chosen reduction scheme we have a decomposition of $H_{\underline{j}}$ into irreducible subspaces given by

$$\begin{align} H_{\underline{j} , \underline{l}} : = \left( \cdots \left(\left(H_{j^1}\otimes H_{j^2}\right)_{l^2} \otimes H_{j_3}\right)_{l^3} \cdots \otimes H_{j^N}\right)_{l^N} \, .\end{align} \tag{ 4.1 }$$

There is an associated family of operators projecting onto these irreps given by

$$\begin{align} {\mathbb{P}}^j_{\underline{j}, \underline{l}} = {\mathbb{P}}^j_{\left(l_{N-1} , j_N\right)}\circ\left({\mathbb{P}}^{l_{N-1}}_{l_{N-2}, j_{N-1}} \otimes \unicode{x1D7D9} \right) \cdots \left({\mathbb{P}}^{l_3}_{l_2, j_3} \otimes \unicode{x1D7D9} \right) \circ \left({\mathbb{P}}^{l_2}_{j_1, j_2} \otimes \unicode{x1D7D9} \right) \, ,\end{align} \tag{ 4.2 }$$

with $l_N = j$. If we choose ladder bases $|\underline{j},\underline{l},m\rangle$ in the irreducible subspaces $H_{\underline{j} , \underline{l}}$, then we obtain

$$\begin{align} {\mathbb{P}}^j_{\underline{j}, \underline{l}} = \sum_{m = -j}^j |\underline{j},\underline{l},m\rangle \langle\,\underline{j},\underline{l},m| \, .\end{align} \tag{ 4.3 }$$

Thus, we can rewrite (2.16) as follows:

$$\begin{align*} \left({\chi}_{\underline{j}}\right)^{j}_{\underline{l},\underline{l}^{^{\prime}}}\left(\underline{u}\right) & = \sum_{m = -j}^j \langle\,\underline{j},\underline{l}^{^{\prime}},m| D^{\underline{j}}\left(\underline{u}\right) |\underline{j},\underline{l},m\rangle \\ & = \sum_{m = -j}^j \langle\,\underline{j},\underline{l}^{^{\prime}},m| \left( \left.\bigoplus\right._{j,\tilde {\underline{l}}} {\mathbb{P}}^j_{\underline{j}, \tilde {\underline{l}}} \right) D^{\underline{j}}\left(\underline{u}\right) \left( \left. \bigoplus \right._{j,{\tilde {\underline{l}}}^{\prime}} {\mathbb{P}}^j_{\underline{j}, {\tilde {\underline{l}}}^{\prime} } \right) |\underline{j},\underline{l},m\rangle \\ & = \sum_{m = -j}^j \langle\,\underline{j},\underline{l}^{^{\prime}},m| {\mathbb{P}}^j_{\underline{j}, \underline{l}^{^{\prime}}} D^{\underline{j}}\left(\underline{u}\right) {\mathbb{P}}^j_{\underline{j}, \underline{l} } |\underline{j},\underline{l},m\rangle \,.\end{align*}$$

Here, $|\underline{j},\underline{l},m\rangle \in H_{\underline{j} , \underline{l}}$ and $|\underline{j},\underline{l}^{\prime},m\rangle \in H_{\underline{j} , \underline{l}^{\prime}}$, where $l_N = l^{\prime}_N = j$. Thus, $H_{\underline{j} , \underline{l}} \cong H_{\underline{j} , \underline{l}^{\prime}}$ and both spaces may be identified with the abstract irrep H_j with ladder basis $| \,j m\rangle$. Under this isomorphism the sum in the above formula may be interpreted as the trace over H_j and we obtain

$$\begin{align} \left({\chi}_{\underline{j}}\right)^{j}_{\underline{l},\underline{l}^{^{\prime}}}\left(\underline{u}\right) = \mathrm{Tr} \left( {\mathbb{P}}^j_{\underline{j}, \underline{l}^{^{\prime}}} D^{\underline{j}}\left(\underline{u}\right) {\mathbb{P}}^j_{\underline{j}, \underline{l} } \right) \, .\end{align} \tag{ 4.4 }$$

Remark 4.1 (Notation). In the sequel, we use the following simplified notation. First, recall that we adopt Dynkin notation and from now we will write $\pi_j (\cdot)$ for an irrep with spin $\tfrac{1}{2} j$ and dimension j + 1. Moreover, we will write

$$\begin{align*} \left({\chi}_{\underline{j}}\right)^{j}_{\underline{l},\underline{l}^{^{\prime}}} = {\chi}^j_{\left(\underline{j}\right), \underline{l},\underline{l}^{^{\prime}}} \, .\end{align*}$$

Group elements $\underline{u}$, and generically local operators (on single members of tensor product spaces) are denoted u, v, a, $\unicode{x1D7D9}$, or $\sf x, \sf y, \ldots$; nonlocal operators are capitalized in blackboard font, e.g. ${\mathbb{J}}^{j,k}$, ${\mathbb{P}}^{j,1}$ or ${\mathbf{J}}^1 \cdot {\mathbf{J}}^2$ where the separate spaces are indicated explicitly. Repeated tensor products are written as $u^{\otimes_j}$, $\unicode{x1D7D9}^{\otimes_j}\equiv \unicode{x1D7D9}^j$ (grouplike), $\textsf{x}^{\otimes_j}$ (primitive). The forms $\left.\sum\right._\textsf{x} \!\left.\sum\right._\textsf{y} \textsf{x} \cdots \textsf{y}\cdots \textsf{y}^{^{\prime}}\cdots \textsf{x}^{^{\prime}}$ will signify correlated sums over independent basis sets, inserted at various positions in tensor products.□

In this notation, for the angular momentum cases $N = 2,3$, we have

$$\begin{align} {{\chi}}^l_{jk}\left(u,v\right) = &\, \operatorname{tr}\left( {\mathbb{P}}^l_{j,k} \pi_j\left(u\right)\otimes \pi_k\left(v\right)\right)\,;\end{align} \tag{ 4.5 }$$

$$\begin{align} {{\chi}}^j_{\left(j_1 j_2\right) j_3,k k^{^{\prime}}}\left(u,v,w\right) = &\, \operatorname{tr}\left( {\mathbb{P}}^j_{\left(j_1j_2\right)j_3,k} \pi_{j_1}\left(u\right)\otimes \pi_{j_2}\left(v\right)\otimes \pi_{j_3}\left(v\right) {\mathbb{P}}^j_{\left(j_1j_2\right)j_3,k^{^{\prime}}} \right)\, ,\end{align} \tag{ 4.6 }$$

with the understanding that in the lists $\underline{l}, \underline{l}^{\prime}$ the final j is omitted.

Now, having fixed the notation, let us explain the basic idea of our calculus. For that purpose, recall the following classical result of Cayley and Sylvester. Let $H_1 \cong {\mathbb{C}}^2$ be the basic 2-dimensional representation space of ${\mathrm{SU}}(2)$. Then, every irreducible representation of ${\mathrm{SU}}(2)$ is isomorphic to $S^n(H_1)$ for some n, where $S^n(H_1)$ is the symmetric power of H₁. Thus, in order to realize the irreducible representation $\pi_j(u)$ we use the symmetrized tensor product of $\otimes^j \pi_1(u)\equiv \otimes^j u$, with generator

$$\begin{align*} \pi_j\left({\mathbf{J}}\right) = \textstyle{\frac 12}{\mathbb{S}}^j \boldsymbol{\sigma}^{\otimes_j}{\mathbb{S}}^j \, ,\end{align*}$$

where $\boldsymbol{\sigma}^{\otimes_j}$ is the $(j-1)$-fold iterated coproduct

$$\begin{align*} \boldsymbol{\sigma}^{\otimes_j} : = \boldsymbol{\sigma} \otimes \unicode{x1D7D9} \otimes \cdots \otimes \unicode{x1D7D9} + \unicode{x1D7D9} \otimes \boldsymbol{\sigma} \otimes \unicode{x1D7D9} \otimes \cdots \otimes \unicode{x1D7D9} \cdots + \unicode{x1D7D9} \otimes \unicode{x1D7D9} \otimes \cdots \otimes \boldsymbol{\sigma}\end{align*}$$

and ${\mathbb{S}}^j$ is the symmetrization operator (a sum over all permutations on $\otimes^j {\mathbb{C}}^2$), given by

$$\begin{align} {\mathbb{S}}^j = &\, \frac{1}{j!}\sum_{\sigma \in {\mathfrak S}_j} {\mathbb{K}}_\sigma\,,\end{align} \tag{ 4.7 }$$

$$\begin{align} {\mathbb{S}}^2 = &\,\frac{1}{2}\left( 1 + {\mathbb{K}}_{\left(12\right)}\right)\,,\quad {\mathbb{S}}^3 = \frac{1}{6} \left( 1 + {\mathbb{K}}_{\left(123\right)}+ {\mathbb{K}}_{\left(132\right)}+ {\mathbb{K}}_{\left(12\right)}+ {\mathbb{K}}_{\left(23\right)}+ {\mathbb{K}}_{\left(31\right)}\right)\,.\end{align} \tag{ 4.8 }$$

Here, ${\mathbb{K}}_{(12)}$ is the elementary transposition operator (the switch operator) and ${\mathbb{K}}_{(123)} = {\mathbb{K}}_{(23)} \circ {\mathbb{K}}_{(12)}$. In the sequel, we will use the following relation

$$\begin{align*} {\mathbb{K}}_{\left(12\right)} = \textstyle{\frac 12} \boldsymbol{\sigma}\cdot \boldsymbol{\sigma} + \textstyle{\frac 12} \unicode{x1D7D9} \otimes \unicode{x1D7D9}\, .\end{align*}$$

If we choose the $(2\times 2)$-elementary matrices $\left\{e_{pq} \right\}$, we have ${\mathbb{K}}_{(12)} = \sum_{p,q = 1}^2 e_{pq}\otimes e_{qp}$, which in the above introduced notation reads

$$\begin{align*} {\mathbb{K}}_{\left(12\right)} = \left.\sum\right._\textsf{x} \textsf{x}\otimes \textsf{x}^{^{\prime}} \,.\end{align*}$$

Using this transcription we have

$$\begin{align} 2{\mathbb{J}}^{j,k} = \left.\sum\right._\textsf{x} \textsf{x}^{\otimes_j}\otimes\textsf{x}^{\otimes_k} - \textstyle{\frac 12}jk\unicode{x1D7D9}^{\otimes_j}\otimes\unicode{x1D7D9}^{\otimes_k}\, ,\end{align} \tag{ 4.9 }$$

and finally, using (3.22), we obtain the following.

Proposition 4.1. In the above calculus the projectors (3.22) read⁶

$$\begin{align} {\mathbb{P}}_{j\!+\!1}^{j,1} = &\, \frac{1}{j\!+\!1}\left.\sum\right._{\textsf{x}} {\mathbb{S}}^j\textsf{x}^{\otimes_j}{\mathbb{S}}^j \otimes \textsf{x}^{^{\prime}} \!+\! \frac{1}{j\!+\!1}\unicode{x1D7D9}^{j}\otimes \unicode{x1D7D9}\,,\end{align} \tag{ 4.10 }$$

$$\begin{align} {\mathbb{P}}_{j-1}^{j,1} = &\, -\frac{1}{j\!+\!1}\left.\sum\right._\textsf{x} {\mathbb{S}}^j\textsf{x}^{\otimes_j} {\mathbb{S}}^j\otimes \textsf{x}^{^{\prime}} \!+\! \frac{j}{j\!+\!1}\unicode{x1D7D9}^{j}\otimes \unicode{x1D7D9}\,;\end{align} \tag{ 4.11 }$$

with inverses

$$\begin{align} \left.\sum\right._\textsf{x} {\mathbb{S}}^j\textsf{x}^{\otimes_j}{\mathbb{S}}^j \otimes \textsf{x}^{^{\prime}} = &\, j {\mathbb{P}}_{j\!+\!1}^{j,1} - {\mathbb{P}}_{j-1}^{j,1}\,, \qquad \unicode{x1D7D9}^{j}\otimes \unicode{x1D7D9} \equiv {\mathbb{P}}_{j\!+\!1}^{j,1} + {\mathbb{P}}_{j-1}^{j,1}\,.\end{align} \tag{ 4.12 }$$

4.1. Evaluation of quasicharacters for $N = 2, 3$ and 4

To demonstrate the use of the projection formalism, we now show some illustrative steps leading to the evaluation of some N = 2 and $N = 3,4$ characters. To start with, let us consider the following example.

Example 4.2 (The quasicharacter ${\chi}^4_{(31)}(u,v)$). Noting that $\mathbb{K}_{(132)} = \mathbb{K}_{(23)} \circ \mathbb{K}_{(12)}$, the first term of equation (4.10) will include the contribution

$$\begin{align*} \textstyle{\frac 16} \operatorname{tr}\left(\left(\left(\left.\sum\right._\textsf{x} \textsf{x} \otimes \unicode{x1D7D9} \otimes \unicode{x1D7D9} + \unicode{x1D7D9} \otimes \textsf{x} \otimes \unicode{x1D7D9} + \unicode{x1D7D9} \otimes \unicode{x1D7D9} \otimes \textsf{x}\right)\otimes \textsf{x}^{^{\prime}} \right) \cdot \left(\left(\textsf{y}u \otimes \textsf{z}\textsf{y}^{^{\prime}} u \otimes \textsf{z}^{^{\prime}}u\right)\otimes v \right)\right)\end{align*}$$

which entails three terms with triple sums over elementary matrices representing the pairwise switches. Using the following properties

$$\begin{align*} \left.\sum\right._\textsf{x} \operatorname{tr}\left(\textsf{x} a \textsf{x}^{^{\prime}} b\right) = &\, \operatorname{tr}\left(a\right) \operatorname{tr}\left(b\right)\,, \qquad \left.\sum\right._\textsf{x} \operatorname{tr}\left(\textsf{x}a\right)\operatorname{tr}\left(\textsf{x}^{^{\prime}}b\right) = \operatorname{tr}\left(ab\right), \\ \left.\sum\right._\textsf{x}\! \left.\sum\right._\textsf{y} {\operatorname{tr}\left({\sf xy}a\right)\operatorname{tr}\left({\sf x^{^{\prime}}y^{^{\prime}}} b\right) } = &\, {\operatorname{tr}\left(a\right)\operatorname{tr}\left(b\right)}\,,\qquad \left.\sum\right._\textsf{x} \!\left.\sum\right._\textsf{y} \operatorname{tr}\left(\textsf{xy}a\right)\operatorname{tr}\left(\textsf{y}^{^{\prime}}\textsf{x}^{^{\prime}} b\right) = \operatorname{tr}\left(\unicode{x1D7D9}\right)\operatorname{tr}\left(ab\right)\,,\end{align*}$$

with similar contributions from the remaining permutations contained in the symmetrizer, and using the Cayley-Hamilton identity $u^2 = \langle u\operatorname{tr}(u)- \unicode{x1D7D9}$ for elements of ${\mathrm{SU}}(2)$, the traces $\operatorname{tr}(u^2) = \operatorname{tr}(u)^2 - 2$ and $\operatorname{tr}(\unicode{x1D7D9}) = 2$ yield the inhomogeneous form⁷

$$\begin{align*} {\chi}^4_{\left(31\right)}\left(u,v\right) = \,- {\frac{1}{4}}\operatorname{tr}\left(uv\right)- {\frac{1}{4}}\operatorname{tr}\left(u\right)\operatorname{tr}\left(v\right) + {\frac{3}{4}}\operatorname{tr}\left(u\right)^2\operatorname{tr}\left(uv\right)- {\frac{3}{8}}\operatorname{tr}\left(u\right)^2\operatorname{tr}\left(v\right)+ {\frac{1}{16}}\operatorname{tr}\left(u\right)^3\operatorname{tr}\left(v\right).\end{align*}$$

□

For N = 3 the relevant projections are developed pairwise according to the order of couplings. As an example we consider the character with $\underline{j} = (j,1,1)$, diagonal in internal intermediate labels $k = j+1 = k^{^{\prime}}$:

$$\begin{align*} {\chi}^{j+2}_{\left(j11\right),j\!+\!1,j\!+\!1}\left(u,v,w\right) & = \operatorname{tr}\left({\mathbb{P}}^{1,j\!+\!1}_{j\!+\!2}\!\cdot\! \left( {\mathbb{P}}^{1,j}_{j\!+\!1} \!\cdot\! \pi_j\left(u\right)\otimes v\right) \otimes w \right)\nonumber\\ & = \operatorname{tr}\left({\mathbb{P}}^{1,j\!+\!1}_{j\!+\!2}\!\cdot\! \left( {\mathbb{P}}^{1,j}_{j\!+\!1} \otimes \unicode{x1D7D9} \!\cdot\! \pi_j\left(u\right)\otimes v \otimes w \right)\right)\,.\end{align*}$$

As a final example, we consider the extension of the method to the N = 4 quasicharacter ${\chi}^{j\!+\!3\,j\!+\!2}_{(j111),j\!+\!1\,j\!+\!2,j\!+\!1\,j\!+\!2}$. This requires a composition of three projectors,

$$\begin{align*} {\mathbb{P}}^{1,j\!+\!2}_{j\!+\!3}\!\cdot\!\left( {\mathbb{P}}^{1,j\!+\!1}_{j\!+\!2}\!\otimes \unicode{x1D7D9}\! \right)\!\cdot\! \left( {\mathbb{P}}^{1,j}_{j\!+\!1} \!\otimes\! \unicode{x1D7D9}\!\otimes\! \unicode{x1D7D9}\right)\,,\end{align*}$$

whose switch operator form will entail symmetrizations of the form ${\mathbb{S}}^{j\!+\!2} \otimes \unicode{x1D7D9}$ composed with ${\mathbb{S}}^{j\!+\!1} \otimes \unicode{x1D7D9} \otimes \unicode{x1D7D9}$ on the appropriate subspaces.

Example 4.3 (The quasicharacter ${\chi}^4_{(1111)2\,3,2\,3}(r,s,t,u)$). For the case j = 1, the switch operator form is⁸

$$\begin{align*} 24{\chi}^4_{(1111)2\,3,2\,3}(r,s,t,u) & = \sum \operatorname{tr} \left\{\rule{0pt}{10pt}\big({\textsf{z}} \!\cdot \! 1 \!\cdot \! 1 \!\cdot \! {\textsf{z}}^{^{\prime}} + 1 \!\cdot \! {\textsf{z}} \!\cdot \! 1 \!\cdot \! {\textsf{z}}^{^{\prime}}+1 \!\cdot \! 1 \!\cdot \! {\textsf{z}} \!\cdot \! {\textsf{z}}^{^{\prime}}) +1\!\cdot \! 1\!\cdot \! 1\!\cdot \! 1\big)\right.\\ &\quad \left. \cdot \big(\tfrac 23({\textsf{y}} \!\cdot \! 1 \!\cdot \! {\textsf{y}}^{^{\prime}} \!\cdot \! 1 + 1 \!\cdot \! {\textsf{y}} \!\cdot \! {\textsf{y}}^{^{\prime}} \!\cdot \! 1 + {\textsf{y}} \!\cdot \! {\textsf{y}}^{^{\prime}} \!\cdot \! 1 \!\cdot \! 1) +1\!\cdot \! 1\!\cdot \! 1\!\cdot \! 1\big)\cdot\right.\\ &\quad \left. \rule{0pt}{10pt}\cdot \big(\tfrac 13({\textsf{x}} \!\cdot \! {\textsf{x}}^{^{\prime}} \!\cdot \! 1 \!\cdot \! 1+ {\textsf{x}}^{^{\prime}} \!\cdot \! 1 \!\cdot \! {\textsf{x}}\!\cdot \! 1+1 \!\cdot \! {\textsf{x}} \!\cdot \! {\textsf{x}}^{^{\prime}}\!\cdot \! 1)+1\!\cdot \! 1\!\cdot \! 1\!\cdot \! 1\big)\cdot (r\!\cdot \! s\!\cdot \! t\!\cdot \! u)\,\right\}\,.\end{align*}$$

Here the overall symmetrization

$$\begin{align*} \left({\mathbb{S}}^{3} \otimes \unicode{x1D7D9}\right)\cdot \left({\mathbb{S}}^{2} \otimes \unicode{x1D7D9} \otimes \unicode{x1D7D9}\right) \equiv \left({\mathbb{S}}^{3} \otimes \unicode{x1D7D9}\right)\end{align*}$$

has been implemented to replace contributions from $\mathbb{J}^{1,1}$ and $\mathbb{J}^{2,1}$ (proposition 4.1 above) with the appropriately re-weighted forms, for example

$$\begin{align*} {\textsf{x}} \!\cdot \! {\textsf{x}}^{^{\prime}} \!\cdot \! 1 \!\cdot \! 1 \Rightarrow \tfrac 13 \left({\textsf{x}} \!\cdot \! {\textsf{x}}^{^{\prime}} \!\cdot \! 1 \!\cdot \! 1+ {\textsf{x}}^{^{\prime}} \!\cdot \! 1 \!\cdot \! {\textsf{x}}\!\cdot \! 1+1 \!\cdot \! {\textsf{x}} \!\cdot \! {\textsf{x}}^{^{\prime}}\!\cdot \! 1\right)\,.\end{align*}$$

The computation entails accumulating all the contributions from the 64 terms and attributing them to trace types with the appropriate strengths. An example is (with overall trace understood):

$$\begin{align*} {\textsf{z}} r \cdot {\textsf{y}} {\textsf{x}} s \cdot {\textsf{y}}^{^{\prime}} {\textsf{x}}^{^{\prime}} t \cdot {\textsf{z}}^{^{\prime}} u \Rightarrow&\, \operatorname{tr} \left( r u \right) \operatorname{tr} \left( {\textsf{y}} {\textsf{x}} s \cdot {\textsf{y}}^{^{\prime}} {\textsf{x}}^{^{\prime}} t \right) \Rightarrow \operatorname{tr} \left( r u \right) \operatorname{tr} \left( {\textsf{x}} s {\textsf{x}}^{^{\prime}} t \right) \Rightarrow \operatorname{tr} \left( r u \right) \operatorname{tr} \left( s \right)\operatorname{tr} \left( t \right)\,.\end{align*}$$

The final result is⁹

$$\begin{align*} 24{\chi}_{\left(1111\right) 23,23}^4 = &\, \left(\frac 23 \operatorname{tr}\left( rstu +rsut+ rust\right) + \frac 89 \operatorname{tr}\left( rtsu+ rtus + ruts\right) \right) \\ &\, + \left(\operatorname{tr}\left( s \right)\operatorname{tr}\left( rtu+rut\right) +\operatorname{tr}\left( r \right)\operatorname{tr}\left( stu+sut\right) \operatorname{tr}\left( t \right)\operatorname{tr}\left( rsu+rus\right) +\frac 23 \operatorname{tr}\left( u \right)\operatorname{tr}\left( rst+rts\right) \right)\\ &\, +\left(\operatorname{tr}\left( rs \right) \operatorname{tr}\left( t\right) \operatorname{tr}\left( u \right) +\frac{13}{9}\left(\operatorname{tr}\left( ru \right) \operatorname{tr}\left( s\right) \operatorname{tr}\left( t \right)+\operatorname{tr}\left( su \right) \operatorname{tr}\left( r\right) \operatorname{tr}\left( t \right)\right) + \frac 49 \operatorname{tr}( ut \right) \operatorname{tr}( r) \operatorname{tr}( s )\big)\\ &\, + 2\big(\operatorname{tr}( rs ) \operatorname{tr}( ut) + \operatorname{tr}( ru ) \operatorname{tr}( st) + \operatorname{tr}( rt ) \operatorname{tr}( su)\big) +\frac{13}{9}\operatorname{tr}( r ) \operatorname{tr}( s) \operatorname{tr}( t ) \operatorname{tr}( u )\,.\end{align*}$$

□

In tables 1 and 2 we give further explicit values of some irreducible χ's, for N = 2 and $N = 3, N = 4$, respectively, computed via projection techniques, as illustrated in the above examples.

Table 1. Tabulation of some N = 2 quasicharacters ${\chi}_{(j_1 j_2)}^{j} \big(u,v\big)$ up to degree 4.

$\left.{\strut}\right.^j_{(j_1j_2)}$	$\frac{1}{\sqrt{[j_1][j_2]}}{\chi}^j_{(j_1j_2)} (u,v)$
$\left.{\strut}\right.{(\textstyle{\frac12}0)}^{\textstyle{\frac12}}$	${\mathrm{Tr}}(u)$
$\left.{\strut}\right._{(\textstyle{\frac12}\textstyle{\frac12})}^{1}$	${\frac 12}\big({\mathrm{Tr}}(uv)+{\mathrm{Tr}}(u){\mathrm{Tr}}(v) \big)$
$\left.{\strut}\right._{(\textstyle{\frac12}\textstyle{\frac12})}^{0}$	${\frac 12}\big(-{\mathrm{Tr}}(uv)+{\mathrm{Tr}}(u){\mathrm{Tr}}(v) \big)$
$\left.{\strut}\right._{(10)}^{1}$	$-1+{\mathrm{Tr}}(u)^2$
$\left.{\strut}\right._{(\textstyle{\frac12}1)}^{\textstyle{\frac12}}$	${\frac 13}\big(-{\mathrm{Tr}}(u)-2{\mathrm{Tr}}(v){\mathrm{Tr}}(uv) +2{\mathrm{Tr}}(u){\mathrm{Tr}}(v)^2\big)$
$\left.{\strut}\right._{(\textstyle{\frac12}1)}^{\textstyle{\frac32}}$	${\frac 13}\big(-2{\mathrm{Tr}}(u)+ 2{\mathrm{Tr}}(v){\mathrm{Tr}}(uv) +{\mathrm{Tr}}(u){\mathrm{Tr}}(v)^2 \big)$
$\left.{\strut}\right._{(\textstyle{\frac32}0)}^{\textstyle{\frac12}}$	$-2{\mathrm{Tr}}(u)+ {\mathrm{Tr}}(u)^3$
$\left.{\strut}\right._{(11)}^{0}$	${\frac 13}\big({\mathrm{Tr}}(uv)^2 -2{\mathrm{Tr}}(u){\mathrm{Tr}}(v){\mathrm{Tr}}(uv)+{\mathrm{Tr}}(u)^2{\mathrm{Tr}}(v)^2 -1\big)$
$\left.{\strut}\right._{(11)}^{1}$	${\frac 12}\big(-{\mathrm{Tr}}(u)^2-{\mathrm{Tr}}(v)^2-{\mathrm{Tr}}(uv)^2+{\mathrm{Tr}}(u)^2{\mathrm{Tr}}(v)^2 +2 \big)$
$\left.{\strut}\right._{(11)}^{2}$	${\frac 13} - {\frac 12}{\mathrm{Tr}}(u)^2 - {\frac 12}{\mathrm{Tr}}(v)^2 + {\frac 16}{\mathrm{Tr}}(uv)^2 + {\frac 23}{\mathrm{Tr}}(u){\mathrm{Tr}}(v){\mathrm{Tr}}(uv) + {\frac 16}{\mathrm{Tr}}(u)^2{\mathrm{Tr}}(v)^2$
$\left.{\strut}\right._{(\textstyle{\frac32}\textstyle{\frac12})}^{2}$	$- {\frac{1}{4}}{\mathrm{Tr}}(uv)- {\frac{1}{4}}{\mathrm{Tr}}(u){\mathrm{Tr}}(v) + {\frac{3}{4}}{\mathrm{Tr}}(u)^2{\mathrm{Tr}}(uv)- {\frac{3}{8}}{\mathrm{Tr}}(u)^2{\mathrm{Tr}}(v)+ {\frac{1}{16}}{\mathrm{Tr}}(u)^3{\mathrm{Tr}}(v)$
$\left.{\strut}\right._{(\textstyle{\frac32}\textstyle{\frac12})}^{1}$	$+ {\frac{1}{4}}{\mathrm{Tr}}(uv)+ {\frac{7}{4}}{\mathrm{Tr}}(u){\mathrm{Tr}}(v) - {\frac{3}{4}}{\mathrm{Tr}}(u)^2{\mathrm{Tr}}(uv)+ {\frac{3}{8}}{\mathrm{Tr}}(u)^2{\mathrm{Tr}}(v)+ {\frac{15}{16}}{\mathrm{Tr}}(u)^3{\mathrm{Tr}}(v)$
$\left.{\strut}\right._{(20)}^{2}$	$1-3{\mathrm{Tr}}(u)^2 + {\mathrm{Tr}}(u)^4$

Table 2. Tabulation of some N = 3 and N = 4 quasicharacters.

$\left.{\strut}\right.^j_{(j_1j_2j_3)l_2l_2^{^{\prime}}}$	$\frac{1}{\sqrt{[j_1][j_2][j_3]}}{\chi}^j_{(j_1j_2j_3)l_2l_2^{^{\prime}}} (u,v,w)$
${\strut}_{(\textstyle{\frac12}\textstyle{\frac12}\textstyle{\frac12})11}^{\textstyle{\frac32}}$	$\tfrac 16 \Big(\operatorname{tr} ( uwv ) + \operatorname{tr} ( vwu ) + \operatorname{tr} ( uv )\operatorname{tr} ( w ) +\operatorname{tr} ( uw )\operatorname{tr} ( v ) + \operatorname{tr} ( vw )\operatorname{tr} ( u ) +\operatorname{tr} ( u )\operatorname{tr} ( v )\operatorname{tr} ( w )\Big)$
${\strut}_{(\textstyle{\frac12}\textstyle{\frac12}\textstyle{\frac12})11}^{\textstyle{\frac12}}$	$\tfrac 16\Big(\!\!-\operatorname{tr} ( uwv )\! -\! \operatorname{tr} ( vwu ) \!-\!\operatorname{tr} ( uw )\operatorname{tr} ( v )\! +\! 2\operatorname{tr} ( uv )\operatorname{tr} ( w )\!-\! \operatorname{tr} ( vw )\operatorname{tr} ( u )\!+\! 2\operatorname{tr} ( u )\operatorname{tr} ( v )\operatorname{tr} ( w )\!\Big)$
${\strut}_{(\textstyle{\frac12}\textstyle{\frac12}\textstyle{\frac12})00}^{\textstyle{\frac12}}$	$\tfrac 12\big(\operatorname{tr} ( u )\operatorname{tr} ( v ) \operatorname{tr} ( w )- \operatorname{tr} ( uv )\operatorname{tr} ( w )\big)$
$\left.{\strut}\right.^j_{(j_1j_2j_3)l_2l_3,l_2^{^{\prime}}l_3^{^{\prime}}}$	$\frac{1}{\sqrt{[j_1][j_2][j_3][j_4]}}{\chi}^j_{(j_1j_2j_3j_4)l_2l_3,l_2^{^{\prime}}l_3^{^{\prime}}} (r,s,t,u)$
$\left.{\strut}\right._{(\textstyle{\frac12}\textstyle{\frac12}\textstyle{\frac12}\textstyle{\frac12})1\tfrac32,1\tfrac32}^{2}$	$\frac{1}{24}\left\{\big(\frac 23 \operatorname{tr}( rstu +rsut+ rust) + \frac 89\operatorname{tr}( rtsu+ rtus + ruts)\big)\right.$
	$+ \big(\operatorname{tr}( s )\operatorname{tr}( rtu+rut) +\operatorname{tr}( r )\operatorname{tr}( stu+sut) \operatorname{tr}( t )\operatorname{tr}( rsu+rus) +\frac 23 \operatorname{tr}( u )\operatorname{tr}( rst+rts) \big)$
	$+\big(\operatorname{tr}( rs ) \operatorname{tr}( t) \operatorname{tr}( u )+\frac{13}{9}(\operatorname{tr}( ru ) \operatorname{tr}( s) \operatorname{tr}( t )+\operatorname{tr}( su ) \operatorname{tr}( r) \operatorname{tr}( t )) + \frac 49 \operatorname{tr}( ut ) \operatorname{tr}( r) \operatorname{tr}( s )\big)$
	$\left.+ 2\big(\operatorname{tr}( rs ) \operatorname{tr}( ut) + \operatorname{tr}( ru ) \operatorname{tr}( st) +\operatorname{tr}( rt ) \operatorname{tr}( su)\big)+\frac{13}{9}\operatorname{tr}( r ) \operatorname{tr}( s) \operatorname{tr}( t ) \operatorname{tr}( u )\right\}.$

4.2. Products of N = 2 quasicharacters and 9j symbols

It turns out that, for N = 2, one can explicitly derive the general multiplication law for quasicharacters via a corresponding manipulation of the tensor products of the characteristic projectors themselves. We will see that, given this law, the general multiplication law for the quasicharacters given by proposition 2.4 and lemma 2.5 follows immediately. We have

$$\begin{align*} \mathbb{P}_{j_1j_2}^k \otimes\mathbb{P}_{j^{^{\prime}}_1j^{^{\prime}}_2}^{k^{^{\prime}}} = &\, \sum | k m, j_1j_2\rangle \langle k m, j_1j_2 |\otimes | k^{^{\prime}} m^{^{\prime}}, j^{^{\prime}}_1j^{^{\prime}}_2\rangle \langle k^{^{\prime}} m^{^{\prime}}, j^{^{\prime}}_1j^{^{\prime}}_2 |\,,\end{align*}$$

leading, after uncoupling of the paired angular momenta, to

$$\begin{align*} \mathbb{P}_{j_1j_2}^k \otimes\mathbb{P}_{j^{^{\prime}}_1j^{^{\prime}}_2}^{k^{^{\prime}}} = &\, \sum |j_1 m_1, j_2 m_2, j^{^{\prime}}_1 m^{^{\prime}}_1, j^{^{\prime}}_2 m^{^{\prime}}_2\rangle \langle j_1 m_1, j_2 m_2 | k m, j_1j_2 \rangle \langle j^{^{\prime}}_1 m^{^{\prime}}_1, j^{^{\prime}}_2 m^{^{\prime}}_2 | k^{^{\prime}} m^{^{\prime}}, j^{^{\prime}}_1j^{^{\prime}}_2 \rangle\\ &\,{\boldsymbol{\cdot}}\langle k m, j_1j_2 |j_1 n_1, j_2 n_2 \rangle \langle k^{^{\prime}} m^{^{\prime}}, j^{^{\prime}}_1j^{^{\prime}}_2 |j^{^{\prime}}_1 n^{^{\prime}}_1 ,j^{^{\prime}}_2 n^{^{\prime}}_2 \rangle \langle j_1 n_1, j_2 n_2, j^{^{\prime}}_1 n^{^{\prime}}_1, j^{^{\prime}}_2 n^{^{\prime}}_2 |\,.\end{align*}$$

In the following manipulations we assume summation over repeated indices, including coupled angular momenta. In the last step, the summations

$$\begin{align*} \left(\cdots\right)_{m,m^{^{\prime}}} \left(\cdots\right)_{m,m^{^{\prime}}} \equiv \left(\cdots\right)_{m,m^{^{\prime}}} \left(\cdots\right)_{m^*,m^{^{\prime}}{}^*} \delta_{m,m^*}\delta_{m^{^{\prime}},m^{^{\prime}}{}^*}\end{align*}$$

are re-written as

$$\begin{align*} \left(\cdots\right)_{m,m^{^{\prime}}} \left(\cdots\right)_{m^*,m^{^{\prime}}{}^*} C^{k k^{^{\prime}} K^*}_{m m^* M^*}C^{k k^{^{\prime}} K^*}_{m^{^{\prime}} m^{^{\prime}}{}^* M^*}\,,\end{align*}$$

with the δ-constraint being implemented by the unitarity of the corresponding Clebsch–Gordan coefficient matrix (summation over $K^*,M^*$ with $K^* \in \langle k,k^{^{\prime}}\rangle$). We have then

$$\begin{align*} &\, |j_1 m_1, j_2 m_2, j^{^{\prime}}_1 m^{^{\prime}}_1, j^{^{\prime}}_2 m^{^{\prime}}_2\rangle C^{j_1j_2k}_{m_1m_2m}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{m^{^{\prime}}_1m^{^{\prime}}_2m^{^{\prime}}}C^{j_1j_2k}_{n_1n_2n}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{n^{^{\prime}}_1n^{^{\prime}}_2n^{^{\prime}}} \langle j_1 n_1, j_2 n_2, j^{^{\prime}}_1 n^{^{\prime}}_1, j^{^{\prime}}_2 n^{^{\prime}}_2 |\\ &\quad = \, |J_1M_1,J_2M_2\rangle \langle J_1M_1, j_1j^{^{\prime}}_1|j_1 m_1 ,j^{^{\prime}}_1 m^{^{\prime}}_1\rangle \langle J_2M_2, j_2j^{^{\prime}}_2|j_2 m_2, j^{^{\prime}}_2 n^{^{\prime}}_2\rangle \\ &\qquad {\boldsymbol{\cdot}} C^{j_1j_2k}_{m_1m_2m}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{m^{^{\prime}}_1m^{^{\prime}}_2m^{^{\prime}}}C^{j_1j_2k}_{n_1n_2n}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{n^{^{\prime}}_1n^{^{\prime}}_2n^{^{\prime}}} \langle j_1n_1, j^{^{\prime}}_1n^{^{\prime}}_1|J_1M^{^{\prime}}_1,j_1j^{^{\prime}}_1\rangle \langle j_2n_2, j^{^{\prime}}_2n^{^{\prime}}_2|J_2M^{^{\prime}}_2,j_2j^{^{\prime}}_2\rangle \langle J_1M^{^{\prime}}_1,J_1M^{^{\prime}}_2|\\ &\quad = \, |J_1M_1,J_2M_2\rangle C^{j_1j^{^{\prime}}_1 J_1}_{m_1m^{^{\prime}}_1M_1}C^{j_2j^{^{\prime}}_2 J_2}_{m_2m^{^{\prime}}_2M_2}{\boldsymbol{\cdot}} C^{j_1j_2k}_{m_1m_2m}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{m^{^{\prime}}_1m^{^{\prime}}_2m^{^{\prime}}}C^{j_1j_2k}_{n_1n_2m}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{n^{^{\prime}}_1n^{^{\prime}}_2m^{^{\prime}}}{\boldsymbol{\cdot}} C^{j_1j^{^{\prime}}_1 J_1}_{n_1n^{^{\prime}}_1M^{^{\prime}}_1}C^{j_2j^{^{\prime}}_2 J_2}_{n_2n^{^{\prime}}_2M^{^{\prime}}_2}\langle J_1M^{^{\prime}}_1,J_1M^{^{\prime}}_2|\,\\ &\quad = |KM\rangle C^{J_1J_2 K}_{M_1M_2M}C^{j_1j^{^{\prime}}_1 J_1}_{m_1m^{^{\prime}}_1M_1}C^{j_2j^{^{\prime}}_2 J_2}_{m_2m^{^{\prime}}_2M_2}{\boldsymbol{\cdot}} C^{j_1j_2k}_{m_1m_2m}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{m^{^{\prime}}_1m^{^{\prime}}_2m^{^{\prime}}}C^{j_1j_2k}_{n_1n_2m}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{n^{^{\prime}}_1n^{^{\prime}}_2m^{^{\prime}}}{\boldsymbol{\cdot}} C^{j_1j^{^{\prime}}_1 J_1}_{n_1n^{^{\prime}}_1M^{^{\prime}}_1}C^{j_2j^{^{\prime}}_2 J_2}_{n_2n^{^{\prime}}_2M^{^{\prime}}_2}C^{J_1J_2K^{^{\prime}}}_{M^{^{\prime}}_1M^{^{\prime}}_2M^{^{\prime}}}\langle K^{^{\prime}}M^{^{\prime}}|\,\\ &\quad \equiv |KM\rangle C^{J_1J_2 K}_{M_1M_2M}C^{j_1j^{^{\prime}}_1 J_1}_{m_1m^{^{\prime}}_1M_1}C^{j_2j^{^{\prime}}_2 J_2}_{m_2m^{^{\prime}}_2M_2}{\boldsymbol{\cdot}} C^{j_1j_2k}_{m_1m_2m}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{m^{^{\prime}}_1m^{^{\prime}}_2m^*}C^{j_1j_2k}_{n_1n_2m^*}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{n^{^{\prime}}_1n^{^{\prime}}_2m^{^{\prime}}{}^*} \\ &\qquad {\boldsymbol{\cdot}}\delta_{mm^*}\delta_{m^{^{\prime}}m^{^{\prime}}{}^*} {\boldsymbol{\cdot}}C^{j_1j^{^{\prime}}_1 J_1}_{n_1n^{^{\prime}}_1M^{^{\prime}}_1}C^{j_2j^{^{\prime}}_2 J_2}_{n_2n^{^{\prime}}_2M^{^{\prime}}_2}C^{J_1J_2K^{^{\prime}}}_{M^{^{\prime}}_1M^{^{\prime}}_2M^{^{\prime}}}\langle K^{^{\prime}}M^{^{\prime}}|\,\\ &\quad = |KM\rangle C^{J_1J_2 K}_{M_1M_2M}C^{j_1j^{^{\prime}}_1 J_1}_{m_1m^{^{\prime}}_1M_1}C^{j_2j^{^{\prime}}_2 J_2}_{m_2m^{^{\prime}}_2M_2}{\boldsymbol{\cdot}} C^{j_1j_2k}_{m_1m_2m}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{m^{^{\prime}}_1m^{^{\prime}}_2m^*}C^{j_1j_2k}_{n_1n_2m^*}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{n^{^{\prime}}_1n^{^{\prime}}_2m^{^{\prime}}{}^*} \\ &\qquad {\boldsymbol{\cdot}} C^{k k^{^{\prime}} K^*}_{m m^{^{\prime}} M^*}C^{k k^{^{\prime}} K^*}_{m^* m^{^{\prime}}{}^* M^*} {\boldsymbol{\cdot}} C^{j_1j^{^{\prime}}_1 J_1}_{n_1n^{^{\prime}}_1M^{^{\prime}}_1}C^{j_2j^{^{\prime}}_2 J_2}_{n_2n^{^{\prime}}_2M^{^{\prime}}_2}C^{J_1J_2K^{^{\prime}}}_{M^{^{\prime}}_1M^{^{\prime}}_2M^{^{\prime}}}\langle K^{^{\prime}}M^{^{\prime}}|\,.\end{align*}$$

Finally regrouping the 12 Clebsch–Gordan factors we have

$$\begin{align} \mathbb{P}_{j_1j_2}^k \otimes\mathbb{P}_{j^{^{\prime}}_1j^{^{\prime}}_2}^{k^{^{\prime}}} = &\,|KM,J_1J_2\rangle \langle K^{^{\prime}}M^{^{\prime}},J_1J_2| \nonumber \\ & {\boldsymbol{\cdot}}\left[ C^{j_1j_2k}_{m_1m_2m}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{m^{^{\prime}}_1m^{^{\prime}}_2m^{^{\prime}}}C^{k k^{^{\prime}} K^*}_{m m^{^{\prime}} M^*}{\boldsymbol{\cdot}} C^{j_1j^{^{\prime}}_1 J_1}_{m_1m^{^{\prime}}_1M_1}C^{j_2j^{^{\prime}}_2 J_2}_{m_2m^{^{\prime}}_2M_2}C^{J_1J_2 K}_{M_1M_2M}\right] \end{align} \tag{ 4.13 }$$

$$\begin{align} &{\boldsymbol{\cdot}} \left[ C^{j_1j_2k}_{n_1n_2m^*}C^{j^{^{\prime}}_1j^{^{\prime}}_2k^{^{\prime}}}_{n^{^{\prime}}_1n^{^{\prime}}_2m^{^{\prime}}{}^*} C^{k k^{^{\prime}} K^*}_{m^* m^{^{\prime}}{}^* M^*} {\boldsymbol{\cdot}} C^{j_1j^{^{\prime}}_1 J_1}_{n_1n^{^{\prime}}_1M^{^{\prime}}_1}C^{j_2j^{^{\prime}}_2 J_2}_{n_2n^{^{\prime}}_2M^{^{\prime}}_2}C^{J_1J_2K^{^{\prime}}}_{M^{^{\prime}}_1M^{^{\prime}}_2M^{^{\prime}}}\right]\,.\end{align} \tag{ 4.14 }$$

Each bracket contains independent summations (over magnetic states $m_1,m_2, m^{^{\prime}}_1,m^{^{\prime}}_2$ and $n_1,n_2, n^{^{\prime}}_1,n^{^{\prime}}_2$ respectively) and represents the overlap of totally coupled 4 spin states on 2 different 4 leaf trees. This implements $\delta_{K^*K}\delta_{K^*K^{^{\prime}}}\delta_{M^*M}\delta_{M^*M^{^{\prime}}}$ and identifies the brackets as the square of the corresponding 9j symbol, independent of label M, and the summation becomes as before¹⁰

$$\begin{align} \mathbb{P}_{j_1j_2}^k \otimes \mathbb{P}_{j^{^{\prime}}_1j^{^{\prime}}_2}^{k^{^{\prime}}} = &\, \sum_{J_1, J_2, K} \left[J_1\right]\left[J_2\right] \left[k\right]\left[k^{^{\prime}}\right]\left.\left\{\begin{array}{ccc} j_1 & j_2 & k\\j^{^{\prime}}_1 & j^{^{\prime}}_2 & k^{^{\prime}} \\ J_1 & J_2 & K \end{array}\right\}\right.^{\!\!2} \mathbb{P}^{K}_{J_1J_2} .\end{align} \tag{ 4.15 }$$

Note that each grand projector $\mathbb{P}^{K}_{J_1J_2}$ is in fact a projection over $H_{J_1}\otimes H_{J_2}$ which is a $[J_1]\cdot[J_2]$-dimensional subspace of the full four-fold factor space $H_{j_1}\otimes H_{j_2}\otimes H_{j^{^{\prime}}_1}\otimes H_{j^{^{\prime}}_2}$, that is, the spaces $H_{J_1}\otimes H_{J_2}$ are isomorphic copies of the corresponding spin irreps here realized in the total space. The formula for pointwise product of quasicharacters now follows directly from that for the (tensor) product of the associated projectors:

$$\begin{align*} {\chi}^{j_1, j_2}_{J_1}\left(u,v\right){\boldsymbol{\cdot}} {\chi}^{j^{^{\prime}}_1, j^{^{\prime}}_2}_{J_2}\left(u,v\right) = &\, \operatorname{tr} \left( \mathbb{P}_{j_1j_2}^{J_1} {\boldsymbol{\cdot}} D_{j_1}\left(u\right) \otimes D_{j_2}\left(v\right) \right){\boldsymbol{\cdot}} \operatorname{tr} \left( \mathbb{P}_{j^{^{\prime}}_1j^{^{\prime}}_2}^{J_2}{\boldsymbol{\cdot}} D_{j^{^{\prime}}_1}\left(u\right) \otimes D_{j_2}\left(v\right) \right)\\ = &\, \operatorname{tr}\left(\left(\mathbb{P}_{j_1j_2}^{J_1}\otimes\mathbb{P}_{j^{^{\prime}}_1j^{^{\prime}}_2}^{J_2}\right) {\boldsymbol{\cdot}} \left(D_{j_1}\left(u\right) \otimes D_{j_2}\left(v\right) \otimes D_{j^{^{\prime}}_1}\left(u\right) \otimes D_{j^{^{\prime}}_2}\left(v\right)\right)\right).\end{align*}$$

Thus finally, we obtain the following.

Lemma 4.4 (Pointwise product of N = 2 irreducible quasicharacters). 

$$\begin{align} {\chi}_{j_1, j_2}^{k}\left(u,v\right){\boldsymbol{\cdot}} {\chi}_{j^{^{\prime}}_1, j^{^{\prime}}_2}^{k^{^{\prime}}}\left(u,v\right) = &\, \sum_{J_1, J_2, K} \left[J_1\right]\left[J_2\right] \left[k\right]\left[k^{^{\prime}}\right]\left.\left\{\begin{array}{ccc} j_1 & j_2 & k\\j^{^{\prime}}_1 & j^{^{\prime}}_2 & k^{^{\prime}} \\ J_1 & J_2 & K \end{array}\right\}\right.^{\!\!2} {\chi}_{J_1, J_2}^{K}\left(u,v\right)\,.\end{align} \tag{ 4.16 }$$

□

Table 3 lists the derived 9j symbols (after normalization) for cases evaluated explicitly in section 4.2 (with standard angular momentum labelling).

Table 3. 9j symbols derived from the expansion of N = 2 quasicharacter products.

${\chi}_{\textstyle{\frac 12} 0}^{\textstyle{\frac 12}}\cdot \chi_{j \textstyle{\frac 12}}^{j\!+\!\textstyle{\frac 12}}$	$1\cdot \chi_{j\!+\!\textstyle{\frac 12} \textstyle{\frac 12}}^{ j\!+\! 1}$	$\displaystyle{\frac{1}{(2j\!+\! 1)^2}}\cdot \chi_{j\!+\!\textstyle{\frac 12} \textstyle{\frac 12}}^{ j}$	$\displaystyle{\frac{2j(2j\!+\! 2)}{(2j\!+\! 1)^2}}\cdot \chi_{j\!-\!\textstyle{\frac 12} \textstyle{\frac 12}}^{j}$	$0\cdot \chi_{j\!-\!\textstyle{\frac 12} \textstyle{\frac 12}}^{j\!-\! 1}$
$\left\{\begin{array}{ccc}{\cdot}&{\cdot}&{\cdot}\\ {\cdot}&{\cdot}&{\cdot}\\ {\cdot}&{\cdot}&{\cdot}\end{array}\right\}$	$\displaystyle{\frac{1}{2(2j\!+\! 2)}}$	$\displaystyle{\frac{1}{2(2j\!+\! 1)(2j\!+\! 2)}}$	$\displaystyle{\frac{1}{2(2j\!+\! 1)}}$	0
	$\left\{\begin{array}{ccc}\textstyle{\frac 12}&0&\textstyle{\frac 12}\\ j&\textstyle{\frac 12}&j\!+\!\textstyle{\frac 12}\\ j\!+\! \textstyle{\frac 12} &\textstyle{\frac 12} &j \!+\! 1\end{array}\right\}$	$\left\{\begin{array}{ccc}\textstyle{\frac 12}&0&\textstyle{\frac 12}\\ j&\textstyle{\frac 12}&j\!+\!\textstyle{\frac 12}\\ j\!+\! \textstyle{\frac 12} &\textstyle{\frac 12} &j\end{array}\right\}$	$\left\{\begin{array}{ccc}\textstyle{\frac 12}&0&\textstyle{\frac 12}\\ j&\textstyle{\frac 12}&j\!-\!\textstyle{\frac 12}\\ j\!-\! \textstyle{\frac 12} &\textstyle{\frac 12} &j\end{array}\right\}$	$\left\{\!\!\begin{array}{ccc}\textstyle{\frac 12}&0&\textstyle{\frac 12}\\ j&\textstyle{\frac 12}&j\!-\!\textstyle{\frac 12}\\ j\!-\! \textstyle{\frac 12} &\textstyle{\frac 12} &j\!-\! 1\end{array}\!\!\right\}$

Compiling the set of structure constants for pointwise multiplication of quasicharacters would of course be immediate if the entire character table were at hand. As a further application of the projection operator method of construction, we treat the computation of certain N = 2 products directly, exploiting the formalism described above (sections 4 and 4.1), to develop explicit product formulae for some special cases (multiplication by fundamental group characters). As is known [2, 3] from the formal viewpoint (section 1 above, and sections A.2, A.3 below), the multiplicative structure constants are (squares of) Racah 9j coefficients, so that the present computations amount to de novo computation of some special 9j Racah symbols, and serve to verify the overall formalism.

To explain our method, in the sequel, we are going to establish the multiplication rules for the following special types:

$$\begin{align*} \operatorname{tr}\left(u\right)\cdot {\chi}_{j,1}^\ell\left(u,v\right), \quad \operatorname{tr}\left(v\right)\cdot{\chi}_{j,1}^\ell\left(u,v\right)\,, \quad \ell = j \pm 1 \, ,\end{align*}$$

for arbitrary j. This amounts to evaluating pointwise products, of some instances of χ constructed formally above in terms of the switch operator calculus. These basic multiplicands of course happen to be standard irreducible, ${\mathrm{SU}}(2)$ spin-$\textstyle{\frac 12}$ characters: $\operatorname{tr}(u)\equiv {\chi}_{10}^1(u,v)$, $\operatorname{tr}(v)\equiv {\chi}_{01}^{1}(u,v)$. We proceed with the calculation for the maximal $\ell$ cases $\ell = j\!+\! 1$; the expansions for $\ell = j\!-\! 1$ can be inferred using similar methods.

As a preliminary remark, note that from equation (2.8), the quasicharacter product necessitates decoupling the constituent paired angular momenta, followed by subsequent rearrangement and recoupling. In the case of the pointwise product with $\operatorname{tr}(u)$, this latter step will entrain the tensor product reduction $H_{1} \otimes H_j \cong H_{j+1} \oplus H_{j-1}$ within the $(j+1)$-fold tensor power $\otimes^{j+1}{\mathbb{C}}^2$. This is consistent with the standard diagrammatic Littlewood–Richardson rule

which corresponds to the sum of the totally symmetric plus mixed symmetry Young projectors $\unicode{x1D7D9}^{j+1} = {\mathbb{S}}^{j+1}+{\mathbb{M}}^{j+1}$. Noting the definition (4.7), these projectors (and the corresponding resolution of the identity) are given recursively as

$$\begin{align} {\mathbb{S}}^{j\!+\!1} = &\, \frac{1}{\left(j\!+\!1\right)}\left({\mathbb{I}} + {\mathbb{K}}_{\left(j\!+\!1 1\right)} +{\mathbb{K}}_{\left(j\!+\!1 2\right)} +\cdots + {\mathbb{K}}_{\left(j\!+\!1 j\right)}\right)\left({\mathbb{S}}^j\otimes\unicode{x1D7D9}\right)\,,\end{align} \tag{ 4.17 }$$

$$\begin{align} {\mathbb{M}}^{j\!+\!1} = &\, \frac{1}{\left(j\!+\!1\right)}\left(j {\mathbb{I}} - {\mathbb{K}}_{\left(j\!+\!1 1\right)} -{\mathbb{K}}_{\left(j\!+\!1 2\right)} -\cdots - {\mathbb{K}}_{\left(j\!+\!1 j\right)}\right)\left({\mathbb{S}}^j\otimes\unicode{x1D7D9}\right)\,.\end{align} \tag{ 4.18 }$$

In the case of the pointwise product with $\operatorname{tr}(v)$, the corresponding tensor product is simply $H_{1} \otimes H_1 \cong H_{2} \oplus H_{0}$, and the mixed symmetry projector coincides with the antisymmetrizer, $\unicode{x1D7D9}^{2} = {\mathbb{S}}^{2}+{\mathbb{A}}^{2}$.

Evaluation of $\operatorname{tr}(u)\cdot {\chi}_{j,1}^{j\!+\! 1}(u,v)$:

Recall the explicit projection operator (3.22), and the switch operator equivalent¹¹, (4.10). As above, we note the isomorphic description of spin-$(\textstyle{\frac 12}j\!-\!\textstyle{\frac 12})$ (Dynkin label $(j\!-\!1)$) recovered from the $(j\!+\!1)$-fold tensor product of the fundamental representation (rather than the $(j-1)$-fold tensor power) using the mixed symmetry projection,

$$\begin{align} \widetilde{\pi}_{j-1}\left(a\right) : = {\mathbb{M}}^{j+1} u^{\otimes_{j+1}} {\mathbb{M}}^{j+1}\,.\end{align} \tag{ 4.19 }$$

The invariant ${\mathbb{J}}^{j-1,1}$ must satisfy the appropriate characteristic equation, but when rewritten in terms of switch operator insertions, equation (4.9) leads to a different coefficient of the identity operator. Using equations (4.19), (4.10) and (4.11) for the case in hand (tensoring spin-$(\textstyle{\frac 12}j -\textstyle{\frac 12})$ (but working with its realization in $\otimes^{j+1}{\mathbb{C}}^2$)  with the fundamental, we have for the corresponding projection operators

$$\begin{align} {\mathbb{P}}_{j}^{\left(j\!-\!1,1\right)} = &\, \frac{2\widetilde{\mathbb{J}}^{\left(j-1,1\right)} + \left(\textstyle{\frac 12}j+\textstyle{\frac 12}\right)}{j}\, ,\quad {\mathbb{P}}_{j\!-\!2}^{\left(j\!-\!1,1\right)} = \frac{-2\widetilde{\mathbb{J}}^{\left(j-1,1\right)} +\left(\textstyle{\frac 12}j-\textstyle{\frac 12}\right)}{j}\,, \nonumber\\ \mbox{whence} \quad \widetilde{\mathbb{P}}^{j\!-\!1, 1}_j = &\, \frac{1}{j}\left.\sum\right._\textsf{x} {\mathbb{M}}^{j+1}\textsf{x}^{\otimes_{j+1}}{\mathbb{M}}^{j+1} \otimes \textsf{x}^{^{\prime}} \,,\\ \mbox{and} \quad\widetilde{\mathbb{P}}^{j\!-\!1, 1}_{j\!-\!2} = &\, -\frac{1}{j}\left.\sum\right._\textsf{x}{\mathbb{M}}^{j+1} \textsf{x}^{\otimes_{j+1}}{\mathbb{M}} \otimes \textsf{x}^{^{\prime}} + \unicode{x1D7D9}^{j+1}\otimes \unicode{x1D7D9}\,.\nonumber\end{align} \tag{ 4.20 }$$

For the product $\operatorname{tr}(u)\cdot {\chi}_{j,1}^{j\!+\!1}(u,v)$ we have from equation (4.10),

$$\begin{align} \operatorname{tr}\left(u\right){\chi}_{j,1}^{j+1} = & \,\frac{1}{j+1} \left.\sum\right._\textsf{x} \operatorname{tr}\left( \left(\textsf{x}^{\otimes_j}\cdot u^{\otimes_j} \cdot{\mathbb{S}}^j\right)\otimes u\right)\operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right) + \frac{1}{j+1}\operatorname{tr}\left(\left(u^{\otimes_j} \cdot{\mathbb{S}}^j\right)\otimes u\right)\operatorname{tr}\left(v\right)\,\\ \equiv &\, \frac{1}{j+1}{[}A{]} + \frac{1}{j+1}{[}B{]}\, . \nonumber\end{align} \tag{ 4.21 }$$

Term ${[}B{]}$ entails a product of fundamental characters, and by the composition rule for angular momenta we have

$$\begin{align} {[}B{]} = \operatorname{tr}\left(\pi_j\left(u\right)\right) {\mathrm{Tr}}\left(u\right){\mathrm{Tr}}\left(v\right) = &\, {\mathrm{Tr}}\left(\pi_{j\!+\!1}\left(u\right)\right){\mathrm{Tr}}\left(v\right) + {\mathrm{Tr}}\left(\pi_{j\!-\!1}\left(u\right)\right){\mathrm{Tr}}\left(v\right)\,.\end{align} \tag{ 4.22 }$$

Using (4.17), (4.18), term ${[}A{]}$ is on the other hand

$$\begin{align} {[}A{]} = &\,\operatorname{tr}\left( \left(\textsf{x}^{\otimes_j}\cdot u^{\otimes_j} \cdot{\mathbb{S}}^j\right)\otimes u\right) \operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right) = \operatorname{tr}\left( \left(\textsf{x}^{\otimes_j}\otimes \unicode{x1D7D9}\right)\cdot u^{\otimes_{j\!+\!1}} \cdot\left({\mathbb{S}}^j\otimes \unicode{x1D7D9}\right)\right)\operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right) \nonumber \\ = &\,\operatorname{tr}\left( \left(\textsf{x}^{\otimes_j}\otimes \unicode{x1D7D9}\right)\cdot u^{\otimes_{j\!+\!1}} \cdot{\mathbb{S}}^{j\!+\!1}\right)\operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right) + \operatorname{tr}\left( \left(\textsf{x}^{\otimes_j}\otimes \unicode{x1D7D9}\right)\cdot u^{\otimes_{j\!+\!1}} \cdot{\mathbb{M}}^{j\!+\!1}\right)\operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right)\end{align} \tag{ 4.23 }$$

$$\begin{align} \equiv &\, {[}C{]} + {[}D{]}\,.\end{align} \tag{ 4.24 }$$

Using the cyclicity of the trace, the coproduct $\sf x^j \otimes \unicode{x1D7D9}$ in [C] may be smeared across the $(j\!+\!1)$-fold tensor product, so that

$$\begin{align} {[}C{]} = &\, \frac{j}{\left(j\!+\!1\right)} \operatorname{tr}\left( \textsf{x}^{\otimes_{j\!+\!1}}\cdot u^{\otimes_{j\!+\!1}} \cdot{\mathbb{S}}^{j\!+\!1}\right)\operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right)\,.\end{align} \tag{ 4.25 }$$

Proceeding with the second, mixed term, we may use (from Young diagram )

$$\begin{align*} {\mathbb{M}}^{j\!+\!1} = \frac{j}{\left(j\!+\!1\right)}\left( {\mathbb{I}} - {\mathbb{K}}_{\left(j\!+\!1 1\right)}\right)\left({\mathbb{S}}^j\otimes\unicode{x1D7D9}\right)\,,\end{align*}$$

as proxy for (4.18), in the presence of the symmetrization over positions $1,2,\cdots, j$ implemented by $\big({\mathbb{S}}^j\otimes\unicode{x1D7D9}\big)$ across the trace:

$$\begin{align} {[}D{]} = &\, \frac{j}{j\!+\!1} \operatorname{tr}\left( \left(\textsf{x}^{\otimes_j}\otimes \unicode{x1D7D9}\right)\cdot u^{\otimes_{j\!+\!1}} \cdot\left({\mathbb{S}}^j\otimes \unicode{x1D7D9}\right)\right)\operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right) \end{align} \tag{ 4.26 }$$

$$\begin{align} \nonumber &\, - \frac{j}{j\!+\!1}\left.\sum\right._\textsf{y}\operatorname{tr}\left( \left(\textsf{x}^{\otimes_j}\otimes \unicode{x1D7D9}\right)\cdot u^{\otimes_{j\!+\!1}} \cdot\left(\textsf{y}\otimes \unicode{x1D7D9} \cdots \otimes \textsf{y}^{^{\prime}}\right) \cdot\left({\mathbb{S}}^j\otimes \unicode{x1D7D9}\right)\right)\operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right)\end{align}$$

$$\begin{align} = & \, \frac{j}{j\!+\!1}\left.\sum\right._\textsf{y}\operatorname{tr}\left( \left(\textsf{x}^{\otimes_j}\right)\cdot u^{\otimes_{j}} \cdot\left({\mathbb{S}}^j\right)\right)\operatorname{tr}\left(u\right)\operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right)\end{align} \tag{ 4.27 }$$

$$\begin{align} \nonumber &\, - \frac{j}{j\!+\!1}\left.\sum\right._\textsf{x, y}\operatorname{tr}\left( \left(\textsf{x}^{\otimes_j}\right)\cdot \left(u\textsf{y}\otimes u^{\otimes_{j-1}}\right) \cdot {\mathbb{S}}^j\right)\operatorname{tr}\left(u\textsf{y}^{^{\prime}}\right)\operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right)\,.\end{align}$$

Now note that $\left.\sum\right._\textsf{y} \operatorname{tr}(\cdots u \textsf{y} \cdots )\operatorname{tr}(u\textsf{y}^{^{\prime}}) \equiv \operatorname{tr}(\cdots u^2 \cdots)$ which can be replaced by $u^2 = u \operatorname{tr}(u) -\unicode{x1D7D9}$; thus the first two contributions cancel leaving

$$\begin{align*} {[}D{]} = \frac{j}{j\!+\!1}{\mathrm{Tr}}\left( \left(\textsf{x}^{\otimes_j}\right)\cdot \left(\unicode{x1D7D9} \otimes u^{\otimes_{j-1}}\right) \cdot {\mathbb{S}}^j\right){\mathrm{Tr}}\left(\textsf{x}^{^{\prime}}v\right)\,.\end{align*}$$

By similar techniques, writing now $\textsf{x}^{\otimes_j} = \textsf{x}\otimes \unicode{x1D7D9}^{j-1} + \unicode{x1D7D9} \otimes \textsf{x}^{\otimes_{j-1}}$, and implementing (4.17) for ${\mathbb{S}}^j$, further consideration of tensor product structure across the trace leads to

$$\begin{align} {[}D{]} = &\,\frac{1}{j\!+\!1}\operatorname{tr}\left(u^{\otimes_{j-1}}{\mathbb{S}}^{j-1}\right) \operatorname{tr}\left(v\right) + \frac{j\!+\!2}{j\!+\!1} \left.\sum\right._\textsf{x} \operatorname{tr}\left(\textsf{x}^{\otimes_{j-1}}\cdot u^{\otimes_{j-1}}\cdot {\mathbb{S}}^{j-1}\right) \operatorname{tr}\left(\textsf{x}^{^{\prime}}v\right) \,.\end{align} \tag{ 4.28 }$$

With equations (4.25), (4.28) for the first and second terms of ${[}A{]}$, together with the switch independent part ${[}B{]}$, we are in a position to rewrite contributions to (4.21) as combinations of projection operators, using equation (4.12), and hence to infer the expansion of the pointwise product in question. Collecting terms we arrive at the following final form.

Lemma 4.5 (Pointwise product ${\chi}_{1,0}^{1}(u,v){\chi}_{j,1}^{j\!+\!1}(u,v)$). 

$$\begin{align} {\chi}_{1,0}^{1}\left(u,v\right){\chi}_{j,1}^{j\!+\!1}\left(u,v\right)& = {\chi}_{j\!+\!1,1}^{j\!+\!2}\left(u,v\right) + \frac{1}{\left(j\!+\!1\right)^2}{\chi}_{j\!+\!1,1}^{j}\left(u,v\right) + \frac{j\left(j\!+\!2\right)}{\left(j\!+\!1\right)^2}{\chi}_{j\!-\!1,1}^{j}\left(u,v\right)\nonumber\\& \quad + 0\cdot {\chi}_{j\!-\!1,1}^{j\!-\!2}\left(u,v\right) .\end{align} \tag{ 4.29 }$$

□

Evaluation of ${\mathrm{Tr}}(v)\cdot {\chi}_{j,1;j\!+\! 1}(u,v)$:

Turning to the corresponding calculation of the pointwise product with ${\mathrm{Tr}}(v)$, we have

$$\begin{align*} \operatorname{tr}(v){\chi}_{j,1}^{j+1}(u,v)& = \operatorname{tr}(v)\operatorname{tr}\big({\mathbb{P}}_{j+1}^{(j,1)} \cdot \pi_j(u)\otimes v\big) \\ & = \, \frac{1}{j+1}\operatorname{tr}\Big(2{\mathbb{J}}^{j,1}\otimes \unicode{x1D7D9} \cdot \big(u^{\otimes_j} {\mathbb{S}}^j \otimes v\big)\otimes v \Big)\nonumber\\& \quad + \frac{\textstyle{\frac 12}j+1}{j+1}\operatorname{tr}\Big(\unicode{x1D7D9}^{j+1} \otimes \unicode{x1D7D9} \cdot \big(\big(u^{\otimes_j} {\mathbb{S}}^j\otimes v\big)\otimes v \Big)\end{align*}$$

which becomes in terms of the correlated sums (see (4.9) above) and projection operators ${\mathbb{I}}^2 = {\mathbb{S}}^2+{\mathbb{A}}^2$,

$$\begin{align*} \operatorname{tr}(v){\chi}_{j,1}^{j+1}(u,v) = &\, \frac{1}{j+1}\left.\sum\right._\textsf{x} \operatorname{tr}\Big(\big(\textsf{x}^{\otimes_j} \otimes \textstyle{\frac 12}\big(\textsf{x}^{\otimes_2}\big)\! \cdot\!\big(u^{\otimes_j} {\mathbb{S}}^j\big) \otimes {\mathbb{S}}^2\big(v\otimes v \big)\Big) \\ &\,+ \frac{1}{j+1}\left.\sum\right._\textsf{x} \operatorname{tr}\Big(\big(\textsf{x}^{\otimes_j}\otimes\textstyle{\frac 12}\big(\textsf{x}^{\otimes_2}\big)\big)\! \cdot\! \big(\big(u^{\otimes_j} {\mathbb{S}}^j\big) \otimes {\mathbb{A}}^2\big(v\otimes v \big)\Big)\\ &\, + \frac{1}{j+1} \operatorname{tr}\Big( \big(\big(u^{\otimes_j} {\mathbb{S}}^j\big) \otimes {\mathbb{S}}^2\big(v\otimes v \big)\Big)+ \frac{1}{j+1}\operatorname{tr}\Big(\big(u^{\otimes_j} {\mathbb{S}}^j\big) \otimes {\mathbb{A}}^2\big(v\otimes v \big)\Big)\,,\end{align*}$$

where the insertion $\textsf{x}^{^{\prime}} \otimes \unicode{x1D7D9}$ has been averaged as $\textstyle{\frac 12} \textsf{x}^{\otimes_2}$ in the presence of overall (anti)symmetrization. Thus

$$\begin{align} \operatorname{tr}\left(v\right){\chi}_{j,1}^{j+1} = &\, \frac{1}{j+1}\sum \operatorname{tr}\left(\left({\mathbb{J}}^{j,2}+ \left(\textstyle{\frac 12}j+1\right)\right)\! \cdot\! \left(u^{\otimes_j} {\mathbb{S}}^j\right) \otimes {\mathbb{S}}^2\left(v\otimes v \right)\right) \nonumber \\ &\,+ \frac{\left(\textstyle{\frac 12}j+1\right)}{j+1}\sum \operatorname{tr}\left(\left(u^{\otimes_j} {\mathbb{S}}^j\right) \otimes {\mathbb{A}}^2\left(v\otimes v \right)\right)\,,\end{align} \tag{ 4.30 }$$

again using equation ( (4.9) and recognising that $\big({\mathbb{J}}^{j,2}\cdot \big(u^{\otimes_j} {\mathbb{S}}^j\big) \cdot \big(v^{\otimes_2} {\mathbb{A}}^2\big)$ vanishes as $\big(v^{\otimes_2} {\mathbb{A}}^2\big)$ is one-dimensional. Introducing the projection operators ${\mathbb{P}}_{\ell}^{(j,2)}$, $\ell = j, j\!\pm\!2$ (lemma 3.3, equations (3.23)), we reorganize the contributions to (4.30) above leading to the final form¹²

Lemma 4.6 (Pointwise product ${\chi}_{0,1}^{1}(u,v){\chi}_{j,1}^{j\!+\!1}(u,v)$). 

$$\begin{align} {\chi}_{0,1}^{1}\left(u,v\right)\, {\chi}_{j,1}^{j+1}\left(u,v\right) = {{\chi}_{j,2}^{j+2}\left(u,v\right) + \frac{\textstyle{\frac 12}j}{j+1}{\chi}_{j,2}^{j}\left(u,v\right) + \frac{\textstyle{\frac 12}j+1}{j+1}{\chi}_{j,0}^{j}}\left(u,v\right).\end{align} \tag{ 4.31 }$$

□

Remark 4.7 (Ab initio 9j Racah coefficients). In the expansions (4.29), (4.31) resulting from the pointwise product computations the coefficients arising (tables 3 and 4), after normalization, are squares of Racah 9j symbols (equations (4.14), (4.15) and lemma 4.16), which are functions of the three spin and internal labels of the two multiplicands, and the three labels of each summand.□

Table 4. 9j symbols derived from the expansion of N = 2 quasicharacter products (ctd.).

$\chi_{0 \textstyle{\frac 12}}^{\textstyle{\frac 12}}\cdot \chi_{j \textstyle{\frac 12}}^{j\!+\!\textstyle{\frac 12}}$	$1\cdot \chi_{j 1}^{j\!+\! 1}$	$\displaystyle{\frac{j}{(2j\!+\! 1)}}\cdot \chi_{j 1}^{ j\!-\! 1}$	$\displaystyle{\frac{(j\!+\! 1)}{(2j\!+\! 1)}}\cdot \chi_{j 0}^{j}$
$\left\{\begin{array}{ccc}\cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot\end{array}\right\}$	$\displaystyle{\frac{1}{2\sqrt{3(j\!+\! 1)(2j\!+\! 1)}}}$	$\displaystyle{\frac{1}{2(2j\!+\! 1)}\sqrt{\frac{j}{3(j\!+\! 1)} } }$	$\displaystyle{\frac{1}{2(2j\!+\! 1)}\sqrt{\frac{1}{2j\!+\! 1}}}$
	$\left\{\begin{array}{ccc}0&\textstyle{\frac 12}&\textstyle{\frac 12}\\ j&\textstyle{\frac 12}&j\!+\!\textstyle{\frac 12}\\ j &1 &j \!+\! 1\end{array}\right\}$	$\left\{\begin{array}{ccc}0&\textstyle{\frac 12}&\textstyle{\frac 12}\\ j&\textstyle{\frac 12}&j\!+\!\textstyle{\frac 12}\\ j &1 &j\end{array}\right\}$	$\left\{\begin{array}{ccc}0&\textstyle{\frac 12}&\textstyle{\frac 12}\\ j&\textstyle{\frac 12}&j\!+\!\textstyle{\frac 12}\\ j &0 &j\end{array}\right\}$

The results are tabulated in conventional angular momentum labelling rather than Dynkin labelling; specific numerical instances can be verified directly by using known tabulations, or an on-line calculator¹³. Note that further pointwise products for other low degree cases are in principle calculable using these spin chain methods, although the combinatorial manipulations will increase in complexity. For example, the product with $\operatorname{tr}(u)\operatorname{tr}(v)\equiv \operatorname{tr}(u\otimes v)$ is available from lemmata 4.5 and 4.6, while that for $\operatorname{tr}(uv) = \sum \operatorname{tr}(u\textsf{z}\otimes v\textsf{z}^{^{\prime}})$ would necessitate rearranging the switch terms into combinations of ${\mathbb{J}}_{j\pm 1,1}$ and ${\mathbb{J}}_{j\pm 1,0}$, and extracting the corresponding projectors, finally yielding (cf table 1) the expansions of ${\chi}^1_{(\textstyle{\frac12}\textstyle{\frac12})}\cdot {\chi}{}_{j,\textstyle{\frac12}}^{j\pm 1/2}$ and ${\chi}^0_{(\textstyle{\frac12}\textstyle{\frac12})}\cdot {\chi}{}_{j,\textstyle{\frac12}}^{j\pm 1/2}$. $\Box$

In this section we shall demonstrate how the knowledge of the multiplicative structure of $\mathcal R(G)$ is fundamental in Hamiltonian lattice gauge theory (with gauge group $G = {\mathrm{SU}}(2)\,$ for this case). The material in the section is based on [2, 3]. Here, we limit our attention to the study of the spectral problem of the Hamiltonian. In [2], the reader may also find a detailed discussion of the role of the classical gauge orbit type structure [1] at the quantum level, and the associated implications for quasicharacters.

Let Λ be a finite spatial lattice with lattice spacing δ, and let Λ⁰, Λ¹ and Λ² denote, respectively, the sets of lattice sites, lattice links and lattice plaquettes. For the links and plaquettes, let there be chosen an arbitrary orientation. In lattice gauge theory, gauge potentials (the variables) are approximated by their parallel transporters along links and gauge transformations (the symmetries) are approximated by their values at the lattice sites. Thus, the classical configuration space is the space $G^{\Lambda^1}$ of mappings $\Lambda^1 \to G$, the classical symmetry group is the group $G^{\Lambda^0}$ of mappings $\Lambda^0 \to G$ with pointwise multiplication and the action of $g \in G^{\Lambda^0}$ on $\underline{a} \in G^{\Lambda^1}$ is given by

$$\begin{align} \left(g \cdot \underline{a}\right)\left(\ell\right) : = g\left(x\right) \underline{a}\left(\ell\right) g\left(y\right)^{-1}\,,\end{align} \tag{ 5.1 }$$

where $\ell = (x,y) \in \Lambda^1$ where x, y denote the starting point and the endpoint of $\ell$, respectively. The classical phase space is given by the associated Hamiltonian G-manifold, and the reduced classical phase space is obtained from that by symplectic reduction (details are given in [2, 3]). Dynamics is ruled by the classical counterpart of the Kogut–Susskind lattice Hamiltonian. If we identify $\mathrm{T}^\ast G$ with $G \times \mathfrak{g}$, and, thus, $\mathrm{T}^\ast G^{\Lambda^1}$ with $G^{\Lambda^1} \times \mathfrak{g}^{\Lambda^1}$, by means of left-invariant vector fields, the classical Hamiltonian is given by

$$\begin{align*} H\left(a,E\right) = \frac{g^2}{2 \delta} \sum_{\ell \in \Lambda^1} \|E\left(\ell\right)\|^2 - \frac{1}{g^2 \delta} \sum_{p \in \Lambda^2} \left(\operatorname{tr} a\left(p\right) + \overline{\operatorname{tr} a\left(p\right)}\right) \,.\end{align*}$$

Here, g is the coupling constant, and for plaquette $p = (\ell_1,\ell_2,\ell_3,\ell_4)$, $a(p) = a(\ell_1)a(\ell_2)a(\ell_3)a(\ell_4)$, where the links $\ell_1,\ell_2,\ell_3,\ell_4$, in this order, form the boundary of p and are endowed with the boundary orientation. Finally, $\| \cdot \|$ denotes the norm given by the scalar product (Killing form) on $\mathfrak{g} = \mathfrak{su}(2)$.

We now turn to the quantum Kogut–Susskind Hamiltonian H, obtained via canonical quantization in the tree gauge (see below). The pure gauge part, treated here, reads:

$$\begin{align} H = \frac{g^2}{2\delta} \, \mathfrak{C} - \frac{1}{g^2 \delta} \, \mathfrak{W} \,.\end{align} \tag{ 5.2 }$$

Here,

$$\begin{align*} \mathfrak{C} : = \sum_{\ell \in \Lambda^1} E_{ij}\left(\ell\right) E_{ji}\left(\ell\right)\end{align*}$$

is the Casimir operator (negative of the group Laplacian) of ${\mathrm{SU}}(2)^N$ and

$$\begin{align*} \mathfrak{W} : = \sum_{p \in \Lambda^2} \left( W \left(p\right) + W\left(p\right)^*\right) \,,\end{align*}$$

where the so-called Wilson loop operator W(p) is the quantum counterpart of $\operatorname{tr} a(p)$ (the corresponding multiplication operator on Hilbert space $\mathcal{H}$). For details, see [4, 5, 25]. Recall that the representative functions of spin j on ${\mathrm{SU}}(2)$ are eigenfunctions of the Casimir operator of ${\mathrm{SU}}(2)$ corresponding to the eigenvalue¹⁴

$$\begin{align} \epsilon_j = 4 j\left(j+1\right) \,.\end{align} \tag{ 5.3 }$$

It follows that the invariant representative functions $({\chi}_{\underline{j}})^{j}_{\underline{l},\underline{l}^{^{\prime}}}$ are eigenfunctions of ${\mathfrak C}$ corresponding to the eigenvalues

$$\begin{align} \epsilon_{\underline{j}} = \epsilon_{j^1} + \cdots + \epsilon_{j^N} \,.\end{align} \tag{ 5.4 }$$

Let us analyse $\mathfrak{W}$. For that purpose, for our regular cubic lattice, we define a standard gauge fixing tree as follows. By a line, we mean a maximal straight line consisting of lattice links. First, choose a lattice site x₀ and a line L₁ through x₀. Next, choose a second line L₂ through x₀ perpendicular to L₁ and add all lines parallel to L₂ in the plane spanned by L₁ and L₂. Finally, add all lines perpendicular to that plane. Let B be such a standard tree. Since $a(\ell) = \unicode{x1D7D9}$ for every $\ell \in B^1$, we can decompose $\mathfrak{W}$ into the sum of three pieces¹⁵. It is easy to check that there exists an orientation and a labelling of the off-tree links such that for every plaquette with four off-tree links (all of these plaquettes are parallel to the plane spanned by the lines L₁ and L₂), the boundary links are labelled and oriented consistently. This means that for one of the two possible orientations of the plaquette, they carry the induced boundary orientation, and that their numbers increase in that direction. Then,

$$\begin{align*} \mathfrak{W} = ~ & \sum_{\left\{p:\,p \cap B = \emptyset\right\}} \operatorname{tr}\left(a_{r_p} a_{s_p} a_{t_p} a_{u_p}\right) + \overline{\operatorname{tr}\left(a_{r_p} a_{s_p} a_{t_p} a_{u_p}\right)} \\ & + \sum_{\left\{p:\,|p \cap B| = 2\right\}}\operatorname{tr}\left(a_{r_p} a_{s_p}\right) + \overline{\operatorname{tr}\left(a_{r_p} a_{s_p}\right)} \\ & + \sum_{\left\{p:\,|p \cap B| = 3\right\}}\operatorname{tr}\left(a_{r_p}\right)+\overline{\operatorname{tr}\left(a_{r_p}\right)}.\end{align*}$$

To find the matrix elements of H with respect to the basis functions $\{({\chi}_{\underline{j}})^{j}_{\underline{l},\underline{l}^{^{\prime}}}\}$, we have to find the corresponding expansion of $\mathfrak{W}$. The sequences $\underline{j}$ occurring here will have at most four nonzero entries, so we can write $\underline{j} = (j_1r_1,\dots,j_kr_k)$ if $\underline{j}$ has entries 0 except for j_i at the places r_i, $i = 1,\dots,k$. The function $T_r(\underline{a}) = \operatorname{tr}(a_r)$ coincides with the basis function $({\chi}_{\underline{j}})^{j}_{\underline{l},\underline{l}^{^{\prime}}}$ with $\underline{j} = (\frac12 r)$. If we omit the irrelevant indices $j , \underline{l} , \underline{l}^{^{\prime}}$, we thus have

$$\begin{align*} T_r = \chi_{\left(\frac12 r\right)} \,.\end{align*}$$

The function $T_{rs}(\underline{a}) : = \operatorname{tr}(a_r a_s)$ is a linear combination of the basis functions $({\chi}_{\underline{j}})^{j}_{\underline{l} \underline{l}^{^{\prime}}}$ with $\underline{j} = (\frac12r,\frac12s)$. Again, we may omit the irrelevant labels $\underline{l}$, $\underline{l}^{^{\prime}}$. Now, using proposition 2.3, we can write

$$\begin{align} \left({\chi}_{\left(j_rr,j_ss\right)}\right)^{j}_{} = \sqrt{\frac{d_{j_r} d_{j_s}}{d_j}} ~ \sum\limits_{m = -j}^j ~ \sum\limits_{n_r + n_s = m \atop n_r^{^{\prime}} + n_s^{^{\prime}} = m} ~ C^{j_r j_s j}_{n_r n_s m} C^{j_r j_s j}_{n_r^{^{\prime}} n_s^{^{\prime}} m} \, D^{j_r}_{n_r n_r^{^{\prime}}}\left(a_r\right) \, D^{j_s}_{n_s n_s^{^{\prime}}}\left(a_s\right) \, .\end{align} \tag{ 5.5 }$$

Using this, together with the orthogonality relation

$$\begin{align} \langle D^j_{m_1 m_2}|D^{j^{^{\prime}}}_{m_1^{^{\prime}} m_2^{^{\prime}}}\rangle = \frac{1}{d_j} \, \delta_{jj^{^{\prime}}} \delta_{m_1 m_1^{^{\prime}}} \delta_{m_2 m_2^{^{\prime}}}\end{align} \tag{ 5.6 }$$

and the normalization condition $\sum_{m_1,m_2 = \pm\frac12} (C^{\frac12 \frac12 j}_{m_1 m_2 m})^2 = 1$, we obtain

$$\begin{align*} \langle\left({\chi}_{\left(\frac12r,\frac12s\right)}\right)^{j}_{}|T_{rs}\rangle = \left(-1\right)^{1-j} \, \frac{\sqrt{d_j}}{2}\,.\end{align*}$$

Thus,

$$\begin{align} T_{rs} = \frac{\sqrt 3}{2} \, \left({\chi}_{\left(\frac12 r,\frac12 s\right)}\right)^{1}_{} - \frac 1 2 \, \left({\chi}_{\left(\frac12 r,\frac12 s\right)}\right)^{0}_{} \,.\end{align} \tag{ 5.7 }$$

Finally, the function $T_{rstu}(\underline{a}) : = \operatorname{tr}(a_r a_s a_t a_u)$ is a linear combination of the basis functions $({\chi}_{\underline{j}})^{j}_{\underline{l} \underline{l}^{^{\prime}}}$ with $\underline{j} = (\frac12r,\frac12s,\frac12t,\frac12u)$. Here, the sequences $\underline{l} \in R(\underline{j},j)$ have entries $l^1 = \cdots = l^{r-1} = 0$, $l^r = \cdots = l^{s-1} = \frac12$, $l^s = \cdots = l^{t-1} = l$, $l^t = \cdots = l^{u-1} = k$ and $l^u = \cdots = l^N = j$, where $l = 0,1$ and $k \in \langle l , \frac12 \rangle$ so that $j \in \langle k , \frac12 \rangle$. This means that they are labelled by two intermediate spins $l,k$, so that in our notation we may replace the labels $\underline{l}$ and $\underline{l}^{^{\prime}}$ by (l, k) and $(l^{^{\prime}},k^{^{\prime}})$, respectively. Expressing these basis functions in terms of matrix elements according to proposition 2.3, and using once again the orthogonality relation (5.6), we obtain

$$\begin{align*} \langle\left({\chi}_{\left(\frac12r,\dots,\frac12u\right)}\right)^{j}_{\left(l,k\right),\left(l^{^{\prime}},k^{^{\prime}}\right)}|T_{rstu}\rangle & = \frac{1}{4 \sqrt{d_j}} \, \sum_{m = -j}^j ~ \sum_{m_r+\cdots+m_u = m} ~ \\ & \quad \times C^{\frac12 \frac12 l}_{m_r,m_s,m_r+m_s} C^{l \frac12 k}_{m_r+m_s,m_t,m_r+m_s+m_t} C^{k \frac12 j}_{m_r+m_s+m_t,m_u,m} \\ & \quad \times C^{\frac12 \frac12 l^{^{\prime}}}_{m_u,m_r,m_r+m_u} C^{l^{^{\prime}} \frac12 k^{^{\prime}}}_{m_r+m_u,m_s,m_r+m_s+m_u} C^{k^{^{\prime}} \frac12 j}_{m_r+m_s+m_u,m_t,m} \,.\end{align*}$$

A detailed calculation (see [3]) leads to

$$\begin{align} T_{rstu} = &\, \textstyle \frac 1 8 \left({\chi}_{\left(\frac12r,\cdots,\frac12u\right)}\right)^{0}_{\left(0\frac12\right)\left(0\frac12\right)} -\frac{\sqrt{3}}{8} \left({\chi}_{\left(\frac12r,\dots,\frac12u\right)}\right)^{0}_{\left(0\frac12\right)\left(1\frac12\right)}\nonumber \\ & -\frac{\sqrt{3}}{8} \left({\chi}_{\left(\frac12r,\dots,\frac12u\right)}\right)^{0}_{\left(1\frac12\right)\left(0\frac12\right)} -\frac{1}{8} \left({\chi}_{\left(\frac12r,\dots,\frac12u\right)}\right)^{0}_{\left(1\frac12\right)\left(1\frac12\right)}\nonumber \\ &\,\textstyle-\frac{\sqrt{3}}{8} \left({\chi}_{\left(\frac12r,\dots,\frac12u\right)}\right)^{1}_{\left(0\frac12\right)\left(0\frac12\right)} +\frac 3 8 \left({\chi}_{\left(\frac12r,\dots,\frac12u\right)}\right)^{1}_{\left(0\frac12\right)\left(1\frac12\right)} + 0 \, ({\chi}_{(\frac12r,\dots,\frac12u)})^{1}_{(0\frac12)(1\frac32)}\nonumber \\ &\, \textstyle-\frac 1 8 ({\chi}_{(\frac12r,\dots,\frac12u)})^{1}_{(1\frac12)(0\frac12)} -\frac{1}{8\sqrt{3}} ({\chi}_{(\frac12r,\dots,\frac12u)})^{1}_{(1\frac12)(1\frac12)} +\frac1{\sqrt{6}} ({\chi}_{(\frac12r,\dots,\frac12u)})^{1}_{(1\frac12)(1\frac32)}\nonumber \\ &\,\textstyle-\frac{1}{\sqrt{8}} ({\chi}_{(\frac12r,\dots,\frac12u)})^{1}_{(1\frac32)(0\frac12)} -\frac{1}{2\sqrt{6}} ({\chi}_{(\frac12r,\dots,\frac12u)})^{1}_{(1\frac32)(1\frac12)} -\frac{1}{4\sqrt{3}} ({\chi}_{(\frac12r,\dots,\frac12u)})^{1}_{(1\frac32)(1\frac32)}\nonumber \\ &\,\textstyle+\frac{\sqrt{5}}{4} ({\chi}_{(\frac12r,\dots,\frac12u)})^{2}_{(1\frac32)(1\frac32)} \,.\end{align} \tag{ 5.8 }$$

Remark 5.1. The relation between the quasicharacters contributing in equations (5.7) and (5.8), and the various trace types occurring, is seen by reference to the evaluations for N = 2 and N = 4 carried out in section 4 above. For example, (5.7) follows directly from the N = 2 table 1 after normalization; see example 4.3 for explicit evaluation of the N = 4 quasicharacter in (5.8) with maximal spin and internal labels, namely ${}^j(l,k)(l^{^{\prime}},k^{^{\prime}}) = {}^2(1,\tfrac 32) (1,\tfrac 32)$.

An alternative strategy for the T_rstu expansion would be to exploit lemma A.3 (for quasicharacter products with distinct group elements), to build iteratively up to N = 4. While such products rely only on the angular momentum Clebsch–Gordan series (with unit 3j coefficients), the resultant expansion would then contain quasicharacters based on different tree labelling, necessitating further rearrangement (lemma A.1) to arrive at an expansion such as (5.8) over an orthonormal basis.□

Now, consider the eigenvalue problem for H. For that purpose, we simplify the notation by collecting the labels $\underline{j}$, j, $\underline{l}$ and $\underline{l}^{^{\prime}}$ of the basis functions in a multi-index

$$\begin{align*} I : = \left(\underline{j} ; j ; \underline{l} ; \underline{l}^{^{\prime}}\right).\end{align*}$$

Let $\mathcal{I}$ denote the totality of all these multi-indices. According to proposition 2.4, the structure constants of multiplication, defined by

$$\begin{align} \chi_{I_1}^{{\mathbb{C}}} \cdot \chi_{I_2}^{{\mathbb{C}}} = \sum_{I \in \mathcal{I}} ~ C^I_{I_1 I_2} \, \chi_{I}^{{\mathbb{C}}} \,,\end{align} \tag{ 5.9 }$$

are given by

$$\begin{align} C^{I}_{I_1 I_2} ~ = ~ \sqrt{\frac{d_{\underline{j}_1} d_{\underline{j}_2} d_j}{d_{j_1} d_{j_2} d_{\underline{j}}}} ~ U_{\underline{j}_1,\underline{j}_2}\left(\underline{j},\underline{l};\underline{l}_1,\underline{l}_2\right) ~ U_{\underline{j}_1,\underline{j}_2}\left(\underline{j},\underline{l}^{^{\prime}};\underline{l}_1^{^{\prime}},\underline{l}_2^{^{\prime}}\right) \,,\end{align} \tag{ 5.10 }$$

where $I_i = \left(\underline{j}_i;j_i;\underline{l}_i;\underline{l}_i^{^{\prime}}\right)$ and $I = \left(\underline{j};j;\underline{l};\underline{l}^{^{\prime}}\right)$. Expanding

$$\begin{align} \psi = \sum_J \psi^J \chi_{J} \,,\qquad {\mathfrak W} = \sum_I W^I \left(\chi_{I} + \overline{\chi_{I}}\right) \,,\end{align} \tag{ 5.11 }$$

and using (5.9), as well as the fact that $\langle\chi_{K}|\overline{\chi_{I}}\chi_{J}\rangle = \langle\chi_{I} \chi_{K}|\chi_{J}\rangle = C_{IK}^J$ implies

$$\begin{align*} \overline{\chi_{I}} \chi_{J} = C_{IK}^J \chi_{K} \,.\end{align*}$$

Finally, the eigenvalue problem reads

$$\begin{align} \sum_{J \in \mathcal{I}} \left\{\left( 2{g^2}{\delta} \epsilon_J - {\cal E} \right) \delta_J^K - \frac{1}{g^2 \delta} \sum_{I \in \mathcal{I}} W^I \left(C_{IJ}^K + C_{IK}^J\right)\right\} \psi^J = 0 \,,\end{align} \tag{ 5.12 }$$

for all $K \in \mathcal{I}$. Here, ε_J stands for the eigenvalue of the Casimir operator $\mathfrak{C}$ corresponding to the eigenfunction χ_J, given by (5.4). Thus, we have obtained a homogeneous system of linear equations for the eigenfunction coefficients ψ^J. The eigenvalues $\cal E$ are determined by the requirement that the determinant of this system must vanish. Note that the sum over I in (5.12) is finite, because there are only finitely many nonvanishing W^I. Moreover, by remark 4.12 in [2], also the sum over J is finite for every fixed K. Thus, we have reduced the eigenvalue problem for the Hamiltonian to a problem in linear algebra. Combining this with well-known asymptotic properties of 3nj symbols (see [8] (Topic 9), [28], and further references therein), we obtain an algebraic setting which allows for a computer algebra supported study of the spectral properties of H.

In the present study we have developed an analysis of the algebra ${\mathcal R}(G)$ of G-invariant representative functions over N copies of a group G, adapted to the case of angular momentum, $G = {\mathrm{SU}}(2)$. The property of invariance under simultaneous conjugation by a fixed group element is a generalization of the N = 1 case, which characterizes the standard theory of group characters. We have introduced the corresponding generalized irreducible quasicharacters, and for $G = {\mathrm{SU}}(2)$ we have provided concrete methods for computations in examples of low degree—both explicit evaluations, and product expansions (sections 4.1 and 4.2). The existence of orthonormal basis sets guaranteed by the group invariant measure, leads to a characterization of equivalent bases, arising via traces of group element representatives over totally coupled states. As discussed above, the expansion of pointwise products, as well as changes of coupling scheme, or tree (see appendix), lead to a distinguished role for Racah recoupling coefficients as overlap and multiplicative structure constants. In particular, the latter are shown to be $3(2N-1)j$ Racah symbols, generalizing the N = 1 case where the 3j symbols simply specify the angular momentum triangle condition (sections A.2 and A.3).

We refer to table 5 for an examination of the combinatorial structure of the invariant ring ${\mathcal R}(G)$ in the ${\chi}^T(\underline{u})$ basis in more detail. The column sums (the Catalan sequence) reflect the overall multiplicity of total angular momenta obtained in the tensor product of N copies of the fundamental spin-$\textstyle{\frac 12}$ representation (and coincide with the squared sum $\sum_j m_{\underline{j}}(j)^2$ of multiplicities in the reduction of each module $V_{\underline{j}}$, by bracketing the tensor products of spin-$\textstyle{\frac 12}$ modules (as in equation (2.11)) and assembling the resulting $V_{\underline{j},j}$ modules). The list entries refer to monomials in trace strings which are in bijection with the total count. For example, for N = 2 we have $\{{\mathrm{Tr}}(uv), {\mathrm{Tr}}(u){\mathrm{Tr}}(v)\}$, as well as $\{{\mathrm{Tr}}(u){\mathrm{Tr}}(v){\mathrm{Tr}}(w), {\mathrm{Tr}}(uv) {\mathrm{Tr}}(w),{\mathrm{Tr}}(vw) {\mathrm{Tr}}(u),{\mathrm{Tr}}(wu) {\mathrm{Tr}}(v)$, ${\mathrm{Tr}}(u[v,w])\}$ for N = 3. For degree higher than 5, however, the entries marked ${}^*$ represent reduced counts modulo additional relations which hold (including the restriction to partitions with parts of length $\unicode{x2A7D}$3), pertaining to the generating sets of various matrix invariant rings (see [29] and for example [30, 31]). In general, however, it is evident that the entries in tables 1 and 2 represent Kostka-type transformation coefficients between the orthonormal basis of irreducible quasicharacters, and selected monomial basis representatives.

Table 5. The Catalan number C(n) of monomials in linear, quadratic and cubic traces at degree $n = 3p\!+\!2q\!+\!r$.

n	Monomial & number	Total
2	$\begin{array}{llr}(1^2)&1\\(2)&1 \end{array}$	2
3	$\begin{array}{llr}(1^3)&1\\(21)&3 \\3&1 \end{array}$	5
4	$\begin{array}{llr}(1^4)&1\\(21^4)&6 \\(2^2)&3 \\(31) & 4\end{array}$	14

n	Monomial & number	Total
5	$\begin{array}{llr} (1^5)&1\\(21^3)&10\\(31^2)&10\\(2^21)&15\\(32)& 6^* \end{array}$	42
6	$\begin{array}{llr}(1^6)&1\\(21^4)&15\\(31^3)&20\\(2^21^2)&45\\(2^3)&15\\(321)& 36^* \\(33)&0^* \end{array}$	132

Table 6. Index of notation.

$H_{\underline{j}}$	irreducible $\times^N{{\mathrm{SU}}(2)}$ module ($= H_{j_1}\otimes H_{j_2}\otimes \cdots \otimes H_{j_N}$)
$d_j = 2j+1 \equiv{[}j{]}$	dimension of irreducible ${{\mathrm{SU}}(2)}$ module (spin j, Dynkin label 2j)
$H_{j, \underline{j}}$	isocopy space, $= H_j^{(1)} \! + \! H_j^{(2)} \! + \! \cdots \! + \! H_j^{(r)}$, $H_j^{(a)} \cong H_j\,, a = 1,2,\cdots, r$
$m_{\underline{j}}(j)$	multiplicity of Hj within $H_{\underline{j}, j}$ $( = r)$
$\underline{j}, \underline{l}, j$	spin labels (standard coupling; $l_1\equiv j_1$, $l_N \simeq j$ )
$T, \underline{j}, \underline{k}, j$	spin labels (tree, leaf edges, internal edges, root edge)
$\mid(j_1 j_2) j_{12},m\rangle$	two spin coupled state
$\mid {\underline{j}}, {\underline{l}}, j,m\rangle$	multi-spin coupled state (standard coupling)
$\mid T({\underline{j}}, {\underline{k}}), j,m\rangle$	multi-spin coupled state (general T)
$({\chi}_{\underline{j}})^{j}_{\underline{l},\underline{l}^{^{\prime}}}(\underline{u})$	${\mathrm{SU}}(2)$ quasicharacter (standard coupling)
${{\chi}}^{T,j}_{(\underline{j}),\underline{k},\underline{k}^{^{\prime}}}({\underline{u}})$	${\mathrm{SU}}(2)$ quasicharacter (general T)
${{\chi}}^j_{k l}(u,v)$	N = 2 quasicharacter
${{\chi}}^{j}_{(j_1j_2j_3)k, k^{^{\prime}}}(u,v,w)$	N = 3 quasicharacter
$C^{j_1,j_2, j_{12}}_{m_1,m_2,m_{12}}$	Clebsch–Gordan coefficient $( = \langle j_1m_1,j_2m_2\mid (j_1j_2) j_{12},m\rangle$)
$C(T)^{\underline{j}}_{{\underline{m}}; {\underline{k}}, j, m }$	Clebsch–Gordan coefficient $( = \langle \underline{j}, \underline{m} |T(\underline{j} ,\underline{k}) jm \rangle)$
$R\big(T^{^{\prime}}\mid T\big)^{\underline{j},j}_{\underline{k}^{^{\prime}}, \underline{k}}$	Racah recoupling coefficient ($= \langle T^{^{\prime}}(\underline{j}, \underline{k}^{^{\prime}}); j m | T(\underline{j} ,\underline{k}); jm \rangle$)
$\left.\left\{\begin{array}{ccc} j_1 & j_2 & k\\j^{^{\prime}}_1 & j^{^{\prime}}_2 & k^{^{\prime}} \\ J_1 & J_2 & K \end{array}\right\}\right.$	Racah 9j symbol
${\mathbb{P}}^j_{j_1,j_2}$	two spin projection operator
${\mathbb{P}}(T)^j_{\underline{j},\underline{k}}$	multi-spin projection operator
${\mathbf{J}} = D_{j}(\textstyle {\frac 12}\boldsymbol{\sigma})$	spin-j generator
${\mathbb{K}}_{(123)}$	permutation operator ($= {\mathbb{K}}_{(23)} \circ {\mathbb{K}}_{(12)}$)
${\mathbb{S}}^j$	j-fold symmetrization operator (for $\pi_j(u)$)
${\mathbb{A}}^2, {\mathbb{M}}^{j+1}$	antisymmetrization, mixed symmetrization operator (for $\pi_{(j\!-\!1)}(u)$)

In the present context, the well known association between the recoupling calculus, and tensor categories of group representations, thus becomes mirrored in the role of the recoupling calculus in supporting the invariant ring of generalized group characters. Even for the N = 1 case and $G = {\mathrm{SU}}(2)$ (angular momentum), this can be seen in the associativity $\langle \langle j_1, j_2\rangle, j_3\rangle = \langle j_1,\langle j_2, j_3\rangle\rangle$ of the set-theoretical operator specifying the angular momentum combination rules (equivalently, associativity of the 3j structure constants governing pointwise products of irreducible characters). On the other hand, from the pointwise product Lemma, while not manifest, the presence of recoupling coefficients as the $N\unicode{x2A7E} 2$ structure constants ensures that associativity (and of course commutativity) is still consistent. This is clearly a concomitant of the tensor-categorical origins of the recoupling calculus itself [32].

It should further be noted that the general representation-theoretical setting for quasicharacters, namely, an N-fold tensor product of G-modules, is precisely the arena in which the so-called quadratic Racah algebra R(n) is defined (see for example [33]). In this case the algebraic generators are the partial Casimir operators ${C}_{ij}: = (\mathbf{J}_i\!+\! \mathbf{J}_j)^2$ rather than the projection operators $\mathbb{P}_{j_ij_j}^j$ which arise in the spectral decomposition ${C}_{ij} = {\sum}_j j(j\!+\!1)\mathbb{P}_{j_ij_j}^j$. The rule (4.15) for expanding projection operator products, with the expansion coefficients being 9j Racah symbols, is thus an aspect of fine-grained structure wherein, in this instance, underlying the general R(n) quadratic bracket algebra, certain operator products themselves can be evaluated explicitly in closed form.

There appears moreover to be a functorial association between the pointwise algebraic product, and the combinatorial tree operations of join and multiply. For example, while products of more than two quasicharacters can by definition be evaluated pairwise, it is natural to conjecture that the resultant expansions can be recombined into a final summation reflecting the tree structure. In this way, the binary multiplication, schematically,

$$\begin{align*} {\chi}^T \!\cdot\! {\chi}^T = \sum R\left(\left(T\!\cdot\! T\right)| T^{\boldsymbol{\vee}}\right)^2 {\chi}^T\,,\end{align*}$$

would have a ternary generalization

$$\begin{align*} \left({\chi}^T \!\cdot\! {\chi}^T\right)\!\cdot\! {\chi}^T = \sum R\left(\left(T\!\cdot\! T\right)\!\cdot\!T| {T^{{\boldsymbol{\vee}}}\!\!\!\!\! \boldsymbol{\vee}}\right)^2 {\chi}^T\,,\end{align*}$$

(with a similar expression for the opposite bracketing given that the associator must vanish), and also possibly for higher order products.

Additional structure in the case of the angular momentum recoupling calculus, includes the classification of fundamental recouplings according to equivalence classes of certain maximally connected trivalent graphs [10, 11] associated with the coupling trees involved¹⁶. It is a natural question to investigate how these distinctions influence specific quasicharacters, in their behaviour as multiplicands. It can also be expected that the functorial correspondences between algebraic operations on characters, and combinatorial moves on trees, emerging from our preliminary analysis, can be given more formal foundations. All of these considerations have relevance to the motivation of the applications in Hamiltonian lattice gauge theory and geometric quantization, and will form topics for ongoing study. In the case of physical gauge group $G = SU(3)$, the recoupling theory must take account of the non-multiplicity free nature of the tensor products (see [3]). We hope to return to these questions in future work.

The authors thank Ronald King for discussions and valuable suggestions on the work.

No new data were created or analysed in this study.

In the foregoing the ring ${\mathcal R}({\mathrm{SU}}(2))$ of ${\mathrm{SU}}(2)$-invariant representative functions associated to the irreducible quasicharacters has been analysed in the context of related combinatorial results in invariant theory, especially as regards counts and gradings. It is natural also to review the algebraic structure in more detail. This is the focus of the present section, bringing together various results that have been provided in [2, 3]. While the $G = {\mathrm{SU}}(2)$ case is our primary concern here, some of the discussion nonetheless applies more generally, and this will be indicated at the appropriate places.

In this appendix we replace the label set $\underline{j}, \underline{\ell}$ by the data of a binary coupling tree. The choice of (planted), semi-labelled binary coupling tree and the composite data $T, \underline{j}, \underline{k}, j$ (the notation will be explained presently), will turn out to be key to unravelling the algebraic structure of the group invariant representative functions ${\mathcal R}({\mathrm{SU}}(2))$ in terms of concrete manipulations with the associated irreducible quasicharacters χ, via the dependence on T and transformations between different coupling types. We therefore preface the discussion here with some notational conventions regarding tree and list manipulation. In the following subsections, the behaviour of the χ characters under change of T, and also their structure as a commutative algebra (pointwise multiplication), will be described.

A.1. Notation

A.2. Properties: change of coupling tree $\langle \chi^{\widetilde{T}}_{j} |\chi^T_j\rangle$, and pointwise multiplication $\langle \chi^{T}_{j^{^{\prime\prime}}}|\chi^T_j\!\cdot \! \chi^{T}_{j^{^{\prime}}}\rangle$

Recall that the irreducible quasicharacters are specified by sums over total angular momentum matrix elements between coupled states constructed using a given coupling tree, but possibly different internal labels:

$$\begin{align*} {{\chi}}^T_{\left(\underline{j}\right)\underline{k} \underline{k}^{^{\prime}}; j}\left(\underline{u}\right) = \sum_m \langle T\left(\underline{j}\right) \underline{k}; jm| D_{\underline{j}}\left(\underline{u}\right) | T\left(\underline{j}\right) \underline{k}^{^{\prime}}; jm \rangle\,.\end{align*}$$

Inserting complete sets of coupled states for a second coupling tree $\widetilde{T}$, and rearranging, leads to the

Lemma A.1 (Transformation rule under change of coupling tree). 

$$\begin{align*} {\chi}^T_{\left(\underline{j}\right)\underline{k} \underline{k}^{^{\prime}};j}\left(\underline{u}\right) = &\, \sum_{J,\underline{K},\underline{K}^{^{\prime}}} R\left(T|\widetilde{T}\right)^{\underline{j}}_{\underline{k} j \underline{K} j} \, {\chi}^{\widetilde{T}}_{\left(\underline{j}\right)\underline{K} \underline{K}^{^{\prime}};j}\left(\underline{u}\right)\, R\left(\widetilde{T}|{T}\right)^{\underline{j}}_{\underline{K}^{^{\prime}}j \underline{k}^{^{\prime}}j} \, .\end{align*}$$

Evaluation of pointwise products of general quasicharacters proceeds via a schematic coupling and uncoupling manipulation. On the one hand, the product of traces can be viewed as a trace over a tensor product, for which totally coupled states are associated with the join $T\cdot T$ of the individual coupling trees. On the other hand, the coupling tree T is a subtree of the thread $T \ast {\boldsymbol{\vee}}$, where the 2N angular momentum labels $\underline{j}$ and $\underline{j}^{^{\prime}}$ must also be reordered as $\underline{j} \ast \underline{j}^{^{\prime}}$ leaf labels, with the coupled N-component $\underline{J}$ in turn providing the leaf labels of the subtree T:

Lemma A.2 (Pointwise product of quasicharacters). 

$$\begin{align*} {\chi}^{T,j_1}_{\underline{j},\underline{k}_1\underline{k}^{^{\prime}}_1}\left(\underline{u}\right) & \cdot {\chi}^{T,j_2}_{\underline{j},\underline{k}_2,\underline{k}^{^{\prime}}_2}\left(\underline{u}\right) \nonumber\\ & \, = \sum_{j, \underline{J}\underline{K} ,\underline{K}^{^{\prime}} } \left(R^{\underline{j}\ast \underline{j}}_{T\cdot T,T\ast {\boldsymbol{\vee}}}\right)_{\underline{k}_{1\cdot2},j_1 j_2,j; \underline{J},\underline{K},j } \left(R^{\underline{j}\ast \underline{j}}_{T\ast {\boldsymbol{\vee}},T\cdot T}\right)_{\underline{J},\underline{K}^{^{\prime}},j ; \underline{k}_{1^{^{\prime}}\cdot2^{^{\prime}}},j_1 j_2,j} \cdot {\chi}^{T,j}_{\underline{J}, \underline{K},\underline{K}^{^{\prime}}}\left(\underline{u}\right)\,.\end{align*}$$

In this case the recoupling coefficients measure the overlap of two 2N-leaf coupling trees, and so each entail $2(2N)-1$ labels; discounting for each the N repeated spin labels within $\underline{j} \ast \underline{j}$, and the overall common j, they are thus of $(6N-3)j$- type, as appropriate for structure constants arising from expanding the pointwise product of two χ characters into a sum, $3(2N-1) = 6N-3$. The structure constants reflect the commutativity of the pointwise product because of their symmetry (under interchange of $\underline{k}_1, \underline{k}^{^{\prime}}_1$ and $\underline{k}_2, \underline{k}^{^{\prime}}_2$ and $\ell_1, \ell_2$). (As shown in lemma A.4 (see [2, 3]), the recoupling coefficients factorize as a product of 9j symbols).

For pointwise multiplication ${\chi}^{T_1}{\large \cdot} {\chi}^{T_2}$ of quasicharacters from two different coupling trees, the above formalism also goes through, and indeed the result can be expressed in the ${\chi}^{T_3}$ basis provided by a third tree (either directly, or via the transformation rule under change of coupling tree).

An important instance of pointwise multiplication is for a quasicharacter product

$$\begin{align*} {\chi}\left(T_1\right)_{\left(\underline{j}^{\left(1\right)}\right)\underline{k}^{\left(1\right)}\underline{k}^{^{\prime}}{}^{\left(1\right)}}\left(\underline{u}^{\left(1\right)}\right)\cdot {\chi}\left(T_2\right)_{\left(\underline{j}^{\left(2\right)}\right)\underline{k}^{\left(2\right)}\underline{k}^{^{\prime}}{}^{\left(2\right)}}\left(\underline{u}^{\left(2\right)}\right)\end{align*}$$

of type N₁ with type N₂, with distinct group elements $(u^{(1)}_1, u^{(1)}_2, \cdots, u^{(1)}_{N_1})$ and $(u^{(2)}_1, u^{(2)}_2, \cdots, u^{(2)}_{N_2})$, respectively. This can be treated using lemma A.2 above, regarding each quasicharacter to be of type $N_1+N_2$ by expanding the spin labels of each multiplicand, namely $(\underline{j}^{(1)}, \underline{0})$, and $(\underline{0}, \underline{j}^{(2)})$, respectively. However, as usual, quasicharacter products may be written as the trace over the tensor product of the appropriate projectors in the total space. This tensor product is the penultimate pairing in the sequence of projectors based on the tree $T = T_1 \cdot T_2$, whose previous couplings operate entirely separately on each subspace $H_{\underline{j}^{(1)}}\otimes H_{\underline{j}^{(2)}}$, corresponding to the left and right subtrees (see section¹⁸ A.1 and equations (2.11), (2.12)). Thus we can write, for the final pairing

$$\begin{align*} H_{\underline{j}^{\left(1\right)}}\otimes H_{\underline{j}^{\left(2\right)}} \cong \sum_{J \in \langle{j_1,j_2}\rangle }\!\! \!{\oplus}\, H_J\end{align*}$$

and correspondingly

$$\begin{align*} \mathbb{P}\left(T_1\right)_{\left(\underline{j}^{\left(1\right)}\right)\underline{k}^{\left(1\right)}}\otimes\mathbb{P}\left(T_2\right)_{\left(\underline{j}^{\left(2\right)}\right)\underline{k}^{\left(2\right)}} = \sum_{J \in \langle{j_1,j_2}\rangle }\mathbb{P}^J_{j_1,j_2} \left(\mathbb{P}\left(T_1\right)_{\left(\underline{j}^{\left(1\right)}\right)\underline{k}^{\left(1\right)}}\otimes\mathbb{P}\left(T_2\right)_{\left(\underline{j}^{\left(2\right)}\right)\underline{k}^{\left(2\right)}}\right)\,.\end{align*}$$

The manipulation on each pair $\underline{k}^{(1)}, \underline{k}^{^{\prime}}{}^{(1)}, j^{(1)}$, $\underline{k}{}^{(2)}, \underline{k}^{^{\prime}}{}^{(2)}, j^{(2)}$ of degenerately labelled subspaces leads to

Lemma A.3 (Pointwise product with independent group elements). 

$$\begin{align*} {\chi}\left(T_1\right)^{j^{\left(1\right)}}_{\left(\underline{j}^{\left(1\right)}\right)\underline{k}^{\left(1\right)}\underline{k}^{^{\prime}}{}^{\left(1\right)}}\left(\underline{u}^{\left(1\right)}\right)\cdot {\chi}\left(T_2\right)^{j^{\left(2\right)}}_{\left(\underline{j}^{\left(2\right)}\right)\underline{k}^{\left(2\right)}\underline{k}^{^{\prime}}{}^{\left(2\right)}}\left(\underline{u}^{\left(2\right)}\right) = \sum_{J \in \langle j^{\left(1\right)},j^{\left(2\right)}\rangle} {\chi}\left(T_1\cdot T_2\right)^J_{\left(\underline{j}^{\left(1\right)}\underline{j}^{\left(2\right)}\right) \underline{K}, \underline{K}^{^{\prime}} }\end{align*}$$

with internal edge labels $\underline{K} = (\underline{k}^{(1)},\underline{k}{}^{(2)}, j^{(1)},j^{(2)} )$ and $\underline{K}^{^{\prime}} = (\underline{k}{}^{(2)},\underline{k}^{^{\prime}}{}^{(2)}, j^{(1)} j^{(2)})$.

A.3. Products of N = 2 quasicharacters and 9j symbols

In the N = 2 case both the tree join, and multiplication (leaf duplication), of two 2-leaf trees (planted cherries), ${\boldsymbol{\vee}}\cdot {\boldsymbol{\vee}}$ and ${\boldsymbol{\vee}}^{{\boldsymbol{\vee}}}$, have the same shape (the balanced, 4-leaf tree (figure 1)), but of course with rearranged leaf labels according to $\underline{j}\cdot \underline{j}^{^{\prime}} = (j_1,j_2,j_1^{^{\prime}},j_2^{^{\prime}})$, $\underline{j}\ast\underline{j}^{^{\prime}} = (j_1,j_1^{^{\prime}},j_2,j_2^{^{\prime}})$. The recoupling coefficients between totally coupled angular momentum states according to the respective schemes are thus dependent on nine quantities: four leaf labels, two pairs of internal labels and the total angular momentum.

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In section 4.2, in the N = 2 case, rather than a direct calculation of the quasicharacter product ${\chi}_{j_1 j_2;k}(u,v) {\chi}_{j^{^{\prime}}_1 j^{^{\prime}}_2;k^{^{\prime}}}(u,v)$, it was shown that the tensor product of the associated projectors $\mathbb{P}_{j_1j_2}^k \otimes\mathbb{P}_{j^{^{\prime}}_1j^{^{\prime}}_2}^{k^{^{\prime}}}$ can be manipulated in such a way that the quasicharacter product expansion follows immediately by taking an overall trace (see (4.16)).

As mentioned above, for specific evaluations it is convenient to refer all products to a fixed coupling scheme. Consider for definiteness the specific choice based on the totally unbalanced or so-called 'caterpillar' tree $T^{\texttt{cat}}$; that is, with $N\!-\!2$ internal angular momenta assigned sequentially, viz. $j_{(12)}$, $j_{((12)3)}, \cdots j_{(\cdots ((12)3)\cdots N\!-\!1})$, (labelled $k_{1i}\,,k_{2i}$ below with $j : = j_{(\cdots ((12)3)\cdots N)}$). This is of course identical to the standard Clebsch–Gordan $\underline{j}, \underline{l}$ scheme of equation (2.11) with, in this case, $k_1 = l_2\,, k_2 = l_3\,, \cdots$, and $k_{N\!-\!2} = l_{N\!-\!1}$. The statement of lemma 2.5, in the composite tree labelling notation, becomes (see [2, 3])

Lemma A.4 (Pointwise product and 9j factorization). The multiplicative structure constants for products of standard caterpillar tree quasicharacters are given (for $N \unicode{x2A7E} 3$) by

$$\begin{align*} \,{\chi}^{\texttt{cat},j_1}_{\underline{j},\underline{k}_1\underline{k}^{^{\prime}}_1}\left(\underline{u}\right) \cdot {\chi}^{\texttt{cat},j_2}_{\underline{j},\underline{k}_2,\underline{k}^{^{\prime}}_2}\left(\underline{u}\right)& = \sum_{j, \underline{J}\underline{K} ,\underline{K}^{^{\prime}} }{\chi}^{\texttt{cat},J}_{\underline{J}, \underline{K},\underline{K}^{^{\prime}}}\left(\underline{u}\right) \cdot \left[\left(\!\begin{array}{ccc} j_{11}& j_{21}&J_1 \\j_{12}& j_{22}&J_2\\ k_{11}& k_{21}& K_1\end{array}\!\right) \!\cdot\!\prod_{i = 1}^{N\!-\!3} \left(\!\begin{array}{ccc}k_{1i}& k_{2i}& K_i \\ j_{1i\!+\!2}& j_{2i\!+\!2}& J_{i\!+\!2}\\ j_{1i\!+\!1}& j_{2i\!+\!1}& K_{i\!+\!1} \end{array}\!\right) \right. \nonumber\\& \quad \left. \!\cdot\! \left(\!\begin{array}{ccc}k_{1N\!-\!2}& k_{2N\!-\!2}& K_{N\!-\!2} \\ j_{1N\!-\!1}& j_{2N\!-\!2}& J_{N\!-\!1} \\ j_1& j_2 & J \end{array}\!\right)\right] \!\cdot\! \left[\left(\!\begin{array}{c} \cdots \\ \cdots \\ \cdots \end{array}\!\right)\right] \,,\end{align*}$$

where the symbol $\left[(\!\begin{array}{c} \cdots \end{array}\!)\right]$ repeats the product of 9j symbols (see lemma 2.5), with $\underline{k}_1, \underline{k}_2, \underline{K}$ replaced by $\underline{k}^{^{\prime}}_1, \underline{k}^{^{\prime}}_2, \underline{K}^{^{\prime}}$. For N = 2, each product collapses to a single term yielding the coefficient (as in lemma 4.4)¹⁹)

$$\begin{align*} \left(\!\begin{array}{ccc} j_{11}& j_{21}&J_1 \\j_{12}& j_{22}&J_2\\ j_1& j_2 & J\end{array}\!\right)^{\!\!2}\,.\end{align*}$$