Mixed symmetries of S_n: immanants in the sampling of U(d) submatrices

TL;DR

This paper investigates the statistical properties of immanants of submatrices from Haar-distributed unitary matrices, focusing on their mean and higher moments, with implications for understanding symmetries in random matrix sampling.

Contribution

It provides new results on the moments of immanants of Haar-random submatrices, expanding the understanding of their distributional properties without relying on detailed proofs.

Findings

Mean and higher moments of immanants are characterized.

Results apply to Haar-distributed unitary matrix submatrices.

Insights into symmetries of S_n in random matrix sampling.

Abstract

We provide results on the mean and higher moments of immanants of submatrices of ensembles of Haar-distributed unitary matrices, mostly without proofs. This paper is based on a talk presented at ISQS29 in Prague in July 2025 by Trevor Welsh.

Tables

Table 1. Table 1: Evaluations of the 1st and 2nd moments

$\lambda$	${\int }_{U\left(d\right)}𝑑U{\left|{\mathrm{Imm}}^{\lambda }M\right|}^2$	${\int }_{U\left(d\right)}𝑑U{\left|{\mathrm{Imm}}^{\lambda }M\right|}^4$
(2)	$\frac{2}{d\left(d+1\right)}$	$4\frac{3d^2-d+2}{d^2\left(d^2-1\right)\left(d+2\right)\left(d+3\right)}$
(3)	$\frac{6}{d\left(d+1\right)\left(d+2\right)}$	$144\frac{d^2+d+4}{d^2\left(d^2-1\right)\left(d+2\right)\left(d+3\right)\left(d+4\right)\left(d+5\right)}$
(2,1)	$\frac{6}{d\left(d^2-1\right)}$	$36\frac{5d^3-3d^2-8d+12}{d^2{\left(d^2-1\right)}^2\left(d^2-4\right)\left(d+3\right)}$
(4)	$\frac{24}{d\left(d+1\right)\left(d+2\right)\left(d+3\right)}$	$576\frac{5d^4+30d^3+127d^2+294d+264}{d^2\left(d^2-1\right)\left(d+1\right){\left(d+2\right)}^2\left(d+3\right)\left(d+4\right)\left(d+5\right)\left(d+6\right)\left(d+7\right)}$
(3,1)	$\frac{24}{d\left(d^2-1\right)\left(d+2\right)}$	$48\frac{73d^5+27d^4-585d^3-421d^2+742d-1240}{d^2{\left(d^2-1\right)}^2\left(d^2-4\right)\left(d+2\right)\left(d^2-9\right)\left(d+4\right)\left(d+5\right)}$
($2^2$)	$\frac{24}{d^2\left(d^2-1\right)}$	$144\frac{19d^4-112d^3+239d^2-224d+132}{d^2{\left(d^2-1\right)}^2{\left(d^2-4\right)}^2\left(d^2-9\right)}$
(2,$1^2$)	$\frac{24}{d\left(d^2-1\right)\left(d-2\right)}$	$48\frac{73d^3-170d^2-151d+542}{2d^2{\left(d^2-1\right)}^2\left(d-2\right)\left(d^2-4\right)\left(d^2-9\right)}$
(5)	$\frac{120}{d\left(d+1\right)\left(d+2\right)\left(d+3\right)\left(d+4\right)}$	$28800\frac{3d^4+26d^3+173d^2+598d+880}{d^2\left(d^2-1\right)\left(d+1\right){\left(d+2\right)}^2\left(d+3\right)\left(d+4\right)\left(d+5\right)\left(d+6\right)\left(d+7\right)\left(d+8\right)\left(d+9\right)}$
(4,1)	$\frac{120}{d\left(d^2-1\right)\left(d+2\right)\left(d+3\right)}$	$960\frac{100d^6+581d^5+611d^4-3373d^3-9643d^2-7304d-12204}{d^2\left(d^2-1\right)\left(d^2-4\right)\left(d^2-9\right)\left(d+1\right)\left(d+2\right)\left(d+3\right)\left(d+4\right)\left(d+5\right)\left(d+6\right)\left(d+7\right)}$
(3,2)	$\frac{120}{d^2\left(d^2-1\right)\left(d+2\right)}$	$240\frac{394d^6-367d^5-3331d^4+7568d^3-10747d^2+17575d+1428}{d^3{\left(d^2-1\right)}^2{\left(d^2-4\right)}^2\left(d^2-9\right)\left(d+3\right)\left(d+4\right)\left(d+5\right)}$
(3,$1^2$)	$\frac{120}{d\left(d^2-1\right)\left(d^2-4\right)}$	$240\frac{418d^5-813d^4-5424d^3+7276d^2+18977d-35968}{d^2{\left(d^2-1\right)}^2{\left(d^2-4\right)}^2\left(d^2-9\right)\left(d^2-16\right)\left(d+5\right)}$
($2^2$,1)	$\frac{120}{d^2\left(d^2-1\right)\left(d-2\right)}$	$240\frac{394d^5-3725d^4+12404d^3-15268d^2+609d+9540}{d^3{\left(d^2-1\right)}^2{\left(d^2-4\right)}^2\left(d^2-9\right)\left(d-3\right)\left(d-4\right)}$
(2,$1^3$)	$\frac{120}{d\left(d^2-1\right)\left(d-2\right)\left(d-3\right)}$	$960\frac{100d^3-433d^2-47d+1638}{d^2{\left(d^2-1\right)}^2\left(d^2-4\right)\left(d-2\right)\left(d^2-9\right)\left(d-3\right)\left(d-4\right)}$

Table 2. Table 2: Evaluations of the leading term coefficient J λ J^{\lambda}

$\lambda$	$J^{\lambda }$
(1)	2
(2)	12
(3)	144
(2,1)	180
(4)	2880
(3,1)	3504
(2,2)	2736
(5)	86400
(4,1)	96000
(3,2)	94560
(3,$1^2$)	100320

Equations

$$\mathrm{Det}M$$

$$\mathrm{Perm}M$$

$${\mathrm{Imm}}^{\lambda }M=\sum _{\pi \in {\mathfrak{S}}_n}{\chi }^{\lambda }(\pi )M_{1\pi (1)}{M}_{2\pi (2)}\cdots M_{n\pi (n)}.$$

$${\mathrm{Imm}}^{(2,1)}\left(\begin{array}{ccc}{M}_{11}& M_{12}& M_{13}\\ M_{21}& M_{22}& M_{23}\\ M_{31}& M_{32}& M_{33}\end{array}\right)=\{\begin{array}{l}(2)M_{11}{M}_{22}{M}_{33}+(0)M_{11}{M}_{23}{M}_{32}+(-1)M_{12}{M}_{23}{M}_{31}\\ +(0)M_{13}{M}_{22}{M}_{31}+(-1)M_{13}{M}_{21}{M}_{32}+(0)M_{12}{M}_{21}{M}_{33}.\end{array}$$

$${\int }_{U(d)}dU|{\mathrm{Imm}}^{\lambda }M{|}^2=\frac{\dim (\lambda ,{\mathfrak{S}}_n)}{\dim (\lambda ,U(d))}.$$

$${\mathcal{Y}}^{\lambda }=\{(i,j):1\le i\le \mathrm{len}(\lambda ),1\le j\le {\lambda }_i\}.$$

$$H^{\lambda }=\prod _{(i,j)\in {\mathcal{Y}}^{\lambda }}({\lambda }_i+{\lambda }_j^{\prime }-i-j+1).$$

$$\dim (\lambda ,{\mathfrak{S}}_n)=\frac{n!}{H^{\lambda }}.$$

$$\dim ((6,4,1),{\mathfrak{S}}_{11})=\frac{11!}{8\times 6\times 5\times 4\times 2\times 5\times 3\times 2}=\frac{11!}{57600}=693.$$

$$N^{\lambda }(d)=\prod _{(i,j)\in {\mathcal{Y}}^{\lambda }}(d+j-i)$$

$$\dim (\lambda ,U(d))=\frac{N^{\lambda }(d)}{H^{\lambda }}.$$

$$N^{(6,4,1)}(d)=d^2(d+1{)}^2(d+2{)}^2(d+3)(d+4)(d+5)(d-1)(d-2)$$

$$\dim ((6,4,1),U(d))$$

$${\int }_{U(d)}dU|{\mathrm{Imm}}^{\lambda }M{|}^2=\frac{n!}{N^{\lambda }(d)}.$$

$${\int }_{U(d)}dU|\mathrm{Det}M{|}^2=\frac{n!}{d(d-1)(d-2)\cdots (d-n+1)}={\left(\genfrac{}{}{0ex}{}{d}{n}\right)}^{-1}$$

$${\int }_{U(d)}dU|\mathrm{Perm}M{|}^2=\frac{n!}{d(d+1)(d+2)\cdots (d+n-1)}={\left(\genfrac{}{}{0ex}{}{d+n-1}{n}\right)}^{-1}.$$

$${\int }_{U(d)}dU|{\mathrm{Imm}}^{\lambda }M{|}^2=\frac{n!}{d^n}+O\left(\frac{1}{d^{n+1}}\right).$$

$${\int }_{U(d)}dU|{\mathrm{Imm}}^{\lambda }M{|}^2>{\int }_{U(d)}dU|{\mathrm{Imm}}^{\mu }M{|}^2.$$

$${\int }_{U(d)}dU{U}_{i_1{k}_1}\cdots U_{i_m{k}_m}{U}_{j_1{\ell }_1}^{\ast }\cdots U_{j_m{\ell }_m}^{\ast }=\sum _{\sigma ,\tau \in {\mathfrak{S}}_m}{\delta }_{\sigma }(\mathbf{i},\mathbf{j}){\delta }_{\tau }(\mathbf{k},\mathbf{\ell })W(\sigma {\tau }^{-1},d)$$

$${\delta }_{\sigma }(\mathbf{i},\mathbf{j})=\prod _{s=1}^m{\delta }_{i_{\sigma (s)},j_s}$$

$$W(\rho ,d)=\frac{1}{(m!{)}^2}\sum _{\xi ⊢m}\frac{\dim (\xi ,{\mathfrak{S}}_m{)}^2}{\dim (\xi ,U(d))}{\chi }^{\xi }(\rho ).$$

$$\begin{array}{rl}{\int }_{U(d)}dU|{\mathrm{Imm}}^{\lambda }M{|}^2& =\sum _{\pi ,\gamma \in {\mathfrak{S}}_n}{\chi }^{\lambda }(\pi ){\chi }^{\lambda }(\gamma )U_{1\pi (1)}\cdots U_{n\pi (n)}{U}_{1\gamma (1)}^{\ast }\cdots U_{n\gamma (n)}^{\ast }\\ & =\sum _{\pi ,\gamma \in {\mathfrak{S}}_n}{\chi }^{\lambda }(\pi ){\chi }^{\lambda }(\gamma )\sum _{\sigma ,\tau \in {\mathfrak{S}}_n}{\delta }_{\sigma }(\mathbf{i},\mathbf{j}){\delta }_{\tau }(\mathbf{k},\mathbf{\ell })W(\sigma {\tau }^{-1},d)\end{array}$$

$$\begin{array}{rl}{\int }_{U(d)}dU|{\mathrm{Imm}}^{\lambda }M{|}^2& =\sum _{\pi ,\gamma \in {\mathfrak{S}}_n}{\chi }^{\lambda }(\pi ){\chi }^{\lambda }(\gamma )W(\pi {\gamma }^{-1},d)\\ & =\frac{1}{(n!{)}^2}\sum _{\xi ⊢n}\frac{\dim (\xi ,{\mathfrak{S}}_n{)}^2}{\dim (\xi ,U(d))}\sum _{\pi ,\gamma \in {\mathfrak{S}}_n}{\chi }^{\lambda }(\pi ){\chi }^{\lambda }(\gamma ){\chi }^{\xi }(\pi {\gamma }^{-1}).\end{array}$$

$$\begin{array}{r}\sum _{\pi ,\gamma \in {\mathfrak{S}}_n}{\chi }^{\lambda }(\pi ){\chi }^{\lambda }(\gamma ){\chi }^{\xi }(\pi {\gamma }^{-1})={\delta }_{\xi \lambda }\frac{n!}{\dim (\xi ,{\mathfrak{S}}_n)}\sum _{\pi \in {\mathfrak{S}}_n}{\chi }^{\lambda }(\pi ){\chi }^{\xi }(\pi )={\delta }_{\xi \lambda }\frac{(n!{)}^2}{\dim (\xi ,{\mathfrak{S}}_n)}.\end{array}$$

$${\int }_{U(d)}dU|{\mathrm{Imm}}^{\lambda }M{|}^{2t}$$

$${\int }_{U(d)}dU|{\mathrm{Imm}}^{\lambda }M{|}^4=\sum _{\xi ⊢2n}\frac{1}{H^{\xi }{N}^{\xi }(d)}\sum _{\begin{array}{c}\sigma ,\rho \in {\mathfrak{H}}_n\end{array}}{T}^{\lambda ,\xi }(\sigma ,\rho )$$

$$T^{\lambda,\xi}(\sigma,\rho)=\sum_{\begin{subarray}{c}\pi,\gamma\in\mathfrak{V}_{n}\end{subarray}}\!\hat{\chi}^{\lambda}(\pi)\,\hat{\chi}^{\lambda}(\gamma)\,\chi^{\xi}\bigl{(}\sigma\pi\rho\gamma)$$

$${\int }_{U(d)}dU|\mathrm{Det}M{|}^4=\frac{1}{\dim ((2^n),U(d))}.$$

$${\int }_{U(d)}dU|\mathrm{Perm}M{|}^4=(n!{)}^2\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }{2}^{2n-4k}{\left((2n-4k+1)\frac{(n-k)!}{k!(2n-2k+1)!}\right)}^2\frac{1}{\dim ((2n-2k,2k),U(d))}.$$

$$\begin{array}{rl}{\int }_{U(d)}dU|{\mathrm{Imm}}^{\lambda }M{|}^4& =\sum _{\xi ⊢2n}\frac{1}{H^{\xi }{N}^{\xi }(d)}\\ & \times \sum _{\ell =0}^n\left(\genfrac{}{}{0ex}{}{n}{\ell }\right)\sum _{j=0}^{\ell }\left(\genfrac{}{}{0ex}{}{\ell }{j}\right)\sum _{k=0}^{\min \left\{\begin{array}{c}n-\ell -j\\ \ell -j\end{array}\right\}}{\zeta }_{n,\ell ,k,j}\left(\genfrac{}{}{0ex}{}{n-\ell }{k}\right)T^{\lambda ,\xi }\left({ϵ}_{[\ell ]},{ϵ}_{[\ell +k\backslash \ell -j]}\right)\end{array}$$

$${\int }_{U(d)}dU|{\mathrm{Imm}}^{\lambda }M{|}^4=\frac{1}{d^{2n}}{J}^{\lambda }+O\left(\frac{1}{d^{2n+1}}\right)$$

Full text

,

1 Introduction

The quantum state of $n$ indistinguishable particles (bosons or fermions) must transform by a one-dimensional representation of $\mathfrak{S}_{n}$, the symmetric group on $n$ symbols [1, 2]. This, in turn, naturally introduces determinant and permanent states for fermions and bosons respectively (see for instance [3, 4, 5]). Applications of permanents in physics have been re-energized by the BosonSampling problem [6], which connects the computational complexity of sampling permanents of $n\times n$ submatrices in the $d$-dimensional unitary group $U(d)$ ($n\ll d$) to linear optics experiments.

For $n$ partially distinguishable particles, states with mixed permutation symmetries (i.e. states transforming under irreducible representations (irreps) of $\mathfrak{S}_{n}$ that are not fully symmetric or fully antisymmetric) may appear. It is then natural to investigate immanants either in their own right [7, 8], for applications in chemistry [8], as a gateway to transition probabilities in $n$-body problems [9], or because they are sums of group functions which appear in transition probabilities [10].

In this contribution, we investigate moments of immanants of $n\times n$ submatrices of random matrices in $U(d)$, with particular emphasis on the first (the mean) and second moments. They are of interest because they can be used to understand the (anti)concentration property of immanants of submatrices and thus provide a pathway to prove average case complexity of these matrix functions, thereby demonstrating a possible computational advantage [11, 12, 13]. Using Weingarten Calculus allows us to produce, for the full range of immanants, explicit $d$-dependent formulae for the mean, and an algorithmic formula for the second moment. We evaluate this latter formula for all immanants for $n\leq 5$, giving explicit rational polynomials in $d$, with only the growth in computational requirements preventing us for going further. Nonetheless, our analysis also yields a formula for the leading term, which is evaluated here for $n\leq 9$. In the extremal cases of the immanant — the permanent and determinant — our approach cogently reproduces results of Nezami [14]. Full proofs of our results will be given elsewhere.

2 Determinants, permanents and immanants

The determinant $\operatorname{Det}M$ and permanent $\operatorname{Perm}M$ of an $n\times n$ matrix $M$ may be defined by

[TABLE]

where $\operatorname{sgn}(\pi)=\pm 1$ is the sign of the permutation $\pi\in\mathfrak{S}_{n}$.

In the definition (1) for the determinant, the coefficient $\operatorname{sgn}(\pi)$ of each summand is the character of the alternating representation of $\mathfrak{S}_{n}$; in the definition (2), the coefficients are all one and so may be regarded as the values of the character of the trivial representation of $\mathfrak{S}_{n}$.

The immanants interpolate this construction by taking the coefficients to be values of the characters $\chi^{\lambda}$ of irreps of $\mathfrak{S}_{n}$. The irreps of $\mathfrak{S}_{n}$ are labelled by partitions $\lambda$ of $n$. Such partitions, denoted $\lambda\vdash n$, are non-increasing sequences $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{p})$ of positive integer parts for which $\lambda_{1}+\lambda_{2}+\cdots+\lambda_{p}=n$. For later convenience, define $\operatorname{len}(\lambda)=p$. Often, repeated parts of a partition are denoted using exponents: for example, $(3,2^{3},1^{2})\equiv(3,2,2,2,1,1)$. For $\lambda\vdash n$, the immanant $\operatorname{Imm}^{\lambda}M$ of an $n\times n$ matrix $M$ is defined by

[TABLE]

The partition $\lambda=(n)$ labels the trivial irrep of $\mathfrak{S}_{n}$, and so $\operatorname{Imm}^{(n)}M=\operatorname{Perm}M$. The partition $\lambda=(1^{n})$ labels the alternating irrep of $\mathfrak{S}_{n}$, and so $\operatorname{Imm}^{(1^{n})}M=\operatorname{Det}M$.

For $n=3$, the only partition other than $(1^{3})$ or $(3)$ is $\lambda=(2,1)$, with corresponding immanant

[TABLE]

Here, for clarity, the values of the character $\chi^{(2,1)}$ have been placed in parentheses.

3 The mean

In this work, for fixed $n\leq d$, we consider $n\times n$ submatrices $M$ of unitary matrices $U\in U(d)$. We aim to understand the distribution of $|{\operatorname{Imm}}^{\lambda}M|^{2}$ as $U$ is sampled with respect to the Haar measure of $U(d)$.

The following result giving the mean of $|{\operatorname{Imm}}^{\lambda}M|^{2}$ is proved below in Section 5.

Theorem 3.1

For $\lambda\vdash n$ and $d\geq n$,

[TABLE]

Eq. (5) is the ratio of dimensions of irreps of $\mathfrak{S}_{n}$ and $U(d)$, both labelled by the same partition $\lambda$. By using known expressions for these dimensions, we can obtain a rational polynomial expression for $\int_{U(d)}dU\,|\operatorname{Imm}^{\lambda}M|^{2}$ for any $\lambda\vdash n$.

To give these dimension formulae, first define the Young diagram ${\mathcal{Y}}^{\lambda}$ of $\lambda\vdash n$ to be the following subset of ${\mathbb{Z}}_{+}^{2}$:

[TABLE]

The Young diagram ${\mathcal{Y}}^{\lambda}$ can be represented graphically by placing a box at position $(i,j)$ for each $(i,j)\in{\mathcal{Y}}^{\lambda}$. In the example $\lambda=(6,4,1)$, this gives:

\yng

(6,4,1)

For a partition $\lambda$, its conjugate partition $\lambda^{\prime}$ is defined so that ${\mathcal{Y}}^{\lambda^{\prime}}=\{(j,i):(i,j)\in{\mathcal{Y}}^{\lambda}\}$. Thereupon, $\lambda^{\prime}$ is obtained by reading the column lengths of the diagram for ${\mathcal{Y}}^{\lambda}$. For example, $(6,4,1)^{\prime}=(3,2^{3},1^{2})$.

The hook-product $H^{\lambda}$ of a partition $\lambda$ is defined by

[TABLE]

Often, the hook-length $\lambda_{i}+\lambda^{\prime}_{j}-i-j+1$ is placed at position $(i,j)$ for each box in ${\mathcal{Y}}^{\lambda}$. In the following diagram, we have done this for $\lambda=(6,4,1)$. $H^{\lambda}$ is simply the product of these values.

\young

(865421,5321,1)

The dimension of the representation of $\mathfrak{S}_{n}$ labelled by $\lambda$ is then [15, §2.37]

[TABLE]

For example, the dimension of the irrep of $\mathfrak{S}_{11}$ labelled by $\lambda=(6,4,1)$ is

[TABLE]

After defining the polynomial $N^{\lambda}(d)$ by

[TABLE]

the dimension of the representation of $U(d)$ labelled by the partition $\lambda$ is given by [15, §3.282]

[TABLE]

Then, in our ongoing example,

[TABLE]

so that the dimension of the irrep of $U(d)$ labelled by $\lambda=(6,4,1)$ is

[TABLE]

Applying (8) and (11) to (5) then yields

[TABLE]

In particular, the $\lambda=(1^{n})$ and $\lambda=(n)$ cases give222Nezami [14] also obtains these (see his eqs. (106) and (97)).

[TABLE]

Some other cases are given in the second column of Table 1.

The following asymptotic result follows immediately from (14):

Corollary 3.2

For $\lambda\vdash n$,

[TABLE]

4 Mean dominance

Given partitions $\lambda,\mu\vdash n$, write $\lambda\trianglelefteq\mu$ if $\sum_{i=1}^{k}\lambda_{i}\leq\sum_{i=1}^{k}\mu_{i}$ for $1\leq k\leq\min\{\operatorname{len}(\lambda),\operatorname{len}(\mu)\}$. This defines a partial order on the set of partitions of $n$ known as the dominance order. Note that, generally, it is not a total order, as illustrated in [16, §1.4.7]. Define $\lambda\lhd\mu$ if $\lambda\trianglelefteq\mu$ and $\lambda\neq\mu$. It is easy to see [16, §1.4.9] that if $\lambda\lhd\mu$, then the diagram of $Y^{\mu}$ is obtained from that of $Y^{\lambda}$ by moving one or more boxes up and to the right. Then, with $d$ fixed, (10) gives $N^{\lambda}(d)<N^{\mu}(d)$. Consequently, (14) implies that:

Proposition 4.1

If $\lambda,\mu\vdash n$ with $\lambda\lhd\mu$ then for each $d\geq n$,

[TABLE]

For the three partitions of $n=3$, this result is illustrated in Fig. 4.

5 Weingarten Calculus

The proof of Theorem 3.1 makes use of the following result, a cornerstone of Weingarten Calculus [17]

Theorem 5.1

[18, 19*]**

For fixed sequences $\bm{i}=(i_{1},\ldots,i_{m})$, $\bm{j}=(j_{1},\ldots,j_{m})$, $\bm{k}=(k_{1},\ldots,k_{m})$, $\bm{\ell}=(\ell_{1},\ldots,\ell_{m})$ of elements of the set $\{1,2,\ldots,d\}$,*

[TABLE]

where

[TABLE]

and for each $\rho\in\mathfrak{S}_{m}$, $W(\rho,d)$ is defined by

[TABLE]

*Proof of Theorem 3.1: * Using (3) and the $m=n$ case of (19) yields:

[TABLE]

where $\bm{i}=\bm{j}=(1,2,\ldots,n)$, \bm{k}=\bigl{(}\pi(1),\pi(2),\ldots,\pi(n)\bigr{)} and \bm{\ell}=\bigl{(}\gamma(1),\gamma(2),\ldots,\gamma(n)\bigr{)}. Then, only $\sigma=\operatorname{id}$ and $\tau=\gamma\pi^{-1}$ contribute to the second sum above, resulting in

[TABLE]

The sum over $\gamma\in\mathfrak{S}_{n}$ can be performed using an extension of the first orthogonality property of finite group characters (see [20, eq. (31.16)]). This gives

[TABLE]

Applying this to the previous expression then yields (5), as desired. $\Box$

6 Higher Moments

Moments, and in particular the second moment, are required in applications to demonstrate approximate average-case hardness of sampling through the anticoncentration of a output distribution [21]. Thus, it is desirable to be able to calculate

[TABLE]

for all $t\in{\mathbb{Z}_{+}}$. Of course, (5) is the $t=1$ case. However, evaluating (25) for $t>1$ is much trickier. Here, we concentrate on the $t=2$ cases, for which we have partial results. To obtain these, we are led to consider Young subgroups of $\mathfrak{S}_{2n}$.

First, let $\mathfrak{S}_{2n}$ act to permute the set of symbols ${\mathcal{I}}_{n}=\{{1}^{+},{2}^{+},\ldots,{n}^{+},{1}^{-},{2}^{-},\ldots,{n}^{-}\}$, and consider these symbols arranged in a $n\times 2$ array as in Fig. 2. Then, let $\mathfrak{V}_{n}$ denote the subgroup of $\mathfrak{S}_{2n}$ which permutes the symbols within their columns in this array. Each $\pi\in\mathfrak{V}_{n}$ is then composed from the two permutations $\pi^{+},\pi^{-}\in\mathfrak{S}_{n}$ of the columns. Denote this $\pi=\pi^{+}\oplus\pi^{-}$. Note that $\lvert{\mathfrak{V}_{n}}\rvert=(n!)^{2}$.

Similarly, let $\mathfrak{H}_{n}$ denote the subgroup of $\mathfrak{S}_{2n}$ which keeps the symbols ${\mathcal{I}}_{n}$ in their rows — it simply swaps the signs of some from the set $\{1,2,\ldots,n\}$. Then, for each ${\mathcal{A}}\subset\{1,2,\ldots,n\}$, let $\epsilon_{{\mathcal{A}}}\in\mathfrak{H}_{n}$ swap ${i}^{+}$ and ${i}^{-}$ for all $i\in{\mathcal{A}}$. For example, if $n=6$ and ${\mathcal{A}}=\{2,3,5\}$, the action of $\epsilon_{{\mathcal{A}}}$ is depicted in Fig. 2. Note that $\lvert{\mathfrak{H}_{n}}\rvert=2^{n}$.

Using the definition (3) and then applying the $m=2n$ case of the Weingarten result (19) gives

[TABLE]

with

[TABLE]

for $\sigma,\rho\in\mathfrak{H}_{n}$ and $\hat{\chi}^{\lambda}({\pi}^{+}\oplus{\pi}^{-})=\chi^{\lambda}({\pi}^{+})\chi^{\lambda}({\pi}^{-})$ for ${\pi}^{+},{\pi}^{-}\in\mathfrak{S}_{n}$.

In the determinant case (where $\lambda=(1^{n})$), the double sum over $\mathfrak{H}_{n}$ and $\mathfrak{V}_{n}$ may be recognised as the (square of the) expression for a Young symmetriser, familiar from the representation theory of $\mathfrak{S}_{n}$ [16, §1.5.10]. This leads to:333In fact, the same reasoning gives \int_{U(d)}dU\,|\operatorname{Det}M|^{2t}=\bigl{(}\dim({(t^{n})},U(d))\bigr{)}^{-1}. Nezami [14] also gives this result (see his eq. (106)).

Proposition 6.1

[TABLE]

For the permanent case (where $\lambda=(n)$), we have the following conjecture:444Nezami [14] makes a similar statement (see his eq. (98) which is derived from his conjectured eq. (43)).

Conjecture 6.2

[TABLE]

However, we have no such general expression for other $\lambda$, and so we have resorted to evaluating our expression (26) for $\int_{U(d)}dU\,|\operatorname{Imm}^{\lambda}M|^{4}$ using a computer. Nonetheless, even for small $n$, this is temporally very expensive. Indeed, to obtain the $n=5$ results, we have to be a bit smarter, and use the fact that many of the $T^{\lambda,\xi}(\sigma,\rho)$ are equal to one another.

First, for $\ell\geq 0$, define the set $[\ell]=\{1,2,\ldots,\ell\}$. Then, for ${\mathcal{A}}\subset[\ell]$, define $[\ell]\backslash{\mathcal{A}}=\{i\in[\ell],i\notin{\mathcal{A}}\}$. In addition, for later use, for $\ell\geq m\geq 0$, define $[\ell\backslash m]=[\ell]\backslash[m]\equiv\{m+1,m+2,\ldots,\ell\}$.

To express the symmetries of $T^{\lambda,\xi}(\sigma,\rho)$, let ${\mathcal{A}},{\mathcal{B}}\subset[n]$ be such that $\sigma=\epsilon_{{\mathcal{A}}}$ and $\rho=\epsilon_{{\mathcal{B}}}$. Then:

$T^{\lambda,\xi}(\epsilon_{\alpha({\mathcal{A}})},\epsilon_{\alpha({\mathcal{B}})})=T^{\lambda,\xi}(\epsilon_{{\mathcal{A}}},\epsilon_{{\mathcal{B}}})$ for all $\alpha\in\mathfrak{S}_{n}$; 2. 2.

$T^{\lambda,\xi}(\epsilon_{{\mathcal{B}}},\epsilon_{{\mathcal{A}}})=T^{\lambda,\xi}(\epsilon_{{\mathcal{A}}},\epsilon_{{\mathcal{B}}})$; 3. 3.

$T^{\lambda,\xi}(\epsilon_{[n]\backslash{\mathcal{A}}},\epsilon_{[n]\backslash{\mathcal{B}}})=T^{\lambda,\xi}(\epsilon_{{\mathcal{A}}},\epsilon_{{\mathcal{B}}})$.

Applying these three symmetries to the expression (26) leads to:555Rewriting our original naive expression in this way has made an enormous efficiency improvement — the original one involved evaluating $2^{2n}$ terms $T^{\lambda,\xi}(\sigma,\rho)$ for each $\xi$, whereas (30) makes approximately $(n+3)^{3}/24$ such evaluations.

Proposition 6.3

For $\lambda\vdash n$,

[TABLE]

where $\zeta_{n,\ell,k,j}={4}/{(1+\delta_{k,n-\ell-j})(1+\delta_{k,\ell-j})}$.

The third column of Table 1 gives evaluations of this result for all $\lambda\vdash n$ with $n\leq 5$, excluding those cases covered by Prop. 6.1.

7 Taking it to the limit

From (26), we obtain:

[TABLE]

where the leading term coefficient $J^{\lambda}\in{\mathbb{Z}}$ is given by

[TABLE]

Using this, or Prop. 6.1, we obtain

[TABLE]

Moreover, (32) may be used to show that $J^{\lambda}=J^{\lambda^{\prime}}$, where $\lambda^{\prime}$ is the conjugate of the partition $\lambda$. Therefore, (33) also gives

[TABLE]

For other cases of $\lambda$, we again resort to using a computer, but not before we extract the symmetries of $J^{\lambda}(\sigma,\rho)$. To express these, again let ${\mathcal{A}},{\mathcal{B}}\subset[n]$ be such that $\sigma=\epsilon_{{\mathcal{A}}}$ and $\rho=\epsilon_{{\mathcal{B}}}$. Then:

$J^{\lambda}(\epsilon_{\alpha({\mathcal{A}})},\epsilon_{\alpha({\mathcal{B}})})=J^{\lambda}(\epsilon_{{\mathcal{A}}},\epsilon_{{\mathcal{B}}})$ for all $\alpha\in\mathfrak{S}_{n}$; 2. 2.

$J^{\lambda}(\epsilon_{{\mathcal{A}}},\epsilon_{{\mathcal{B}}})=0$ if $\lvert{{\mathcal{A}}}\rvert\neq\lvert{{\mathcal{B}}}\rvert$; 3. 3.

$J^{\lambda}(\epsilon_{[n]\backslash{\mathcal{A}}},\epsilon_{[n]\backslash{\mathcal{B}}})=J^{\lambda}(\epsilon_{{\mathcal{A}}},\epsilon_{{\mathcal{B}}})$.

The second symmetry of $T^{\lambda,\xi}(\sigma,\rho)$ also extends to this case, but that given here is more powerful.

Applying the above three symmetries to the expression (32) leads to the first part of the following:

Proposition 7.1

For $\lambda\vdash n$,

[TABLE]

In addition,

[TABLE]

where $\theta_{\ell,k}\in\mathfrak{S}_{n}$ is the involution defined by

[TABLE]

Using this result gives the values in Table 2. Because $J^{\lambda}=J^{\lambda^{\prime}}$, we only list one value from each conjugate pair.

Acknowlegement

The work of HdG is supported by NSERC of Canada. Part of this work was completed by JD under the NSERC USRA program.