Symplectic Weingarten functions

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Numbers

$N$

$\mu$

$\operatorname{Wg}^{Sp}_N(\mu)$

1/2

1/4

1/40

1, 1:

3/40

1/6

1/168

1, 1:

5/168

1/1344

2, 1:

1/672

1, 1, 1:

1/168

1/8

1/432

1, 1:

7/432

1/8640

2, 1:

1/2880

1, 1, 1:

19/8640

17/1330560

3, 1:

1/41580

2, 2:

1/63360

2, 1, 1:

79/1330560

1, 1, 1, 1:

85/266112

1/10

1/880

1, 1:

9/880

1/31680

2, 1:

1/7920

1, 1, 1:

17/15840

1/673920

3, 1:

1/247104

2, 2:

17/7413120

2, 1, 1:

109/7413120

1, 1, 1, 1:

287/2471040

29/207567360

4, 1:

1/3843840

3, 2:

1/6289920

3, 1, 1:

41/69189120

2, 2, 1:

37/103783680

2, 1, 1, 1:

1/540540

1, 1, 1, 1, 1:

151/11531520

1/12

1/1560

1, 1:

11/1560

1/87360

2, 1:

1/17472

1, 1, 1:

53/87360

1/3203200

3, 1:

1/900900

2, 2:

17/28828800

2, 1, 1:

151/28828800

1, 1, 1, 1:

73/1372800

1/76876800

4, 1:

1/28828800

3, 2:

1/57657600

3, 1, 1:

1/8870400

2, 2, 1:

1/16473600

2, 1, 1, 1:

19/38438400

1, 1, 1, 1, 1:

61/12812800

89/84687482880

5, 1:

383/195432652800

4, 2:

739/635156121600

4, 1, 1:

2789/635156121600

3, 3:

47/42343741440

3, 2, 1:

29/12098211840

3, 1, 1, 1:

2623/211718707200

2, 2, 2:

1/678585600

2, 2, 1, 1:

137/19848628800

2, 1, 1, 1, 1:

1721/35286451200

1, 1, 1, 1, 1, 1:

46301/105859353600

Definition

The symplectic Weingarten coefficient $\operatorname{Wg}^{Sp}_N(\mu)$ is Matsumoto's coefficient at the even coset representative $\sigma_\mu$ for normalized Haar integration on $\operatorname{USp}(2N)$ [3].

Parameters

$N$

— symplectic rank ($N\geq1$)

$\mu$

— coset type (A nonempty integer partition, written as comma-separated positive parts in nonincreasing order; $k=|\mu|$ is their sum.)

Formulas

(1)

$\displaystyle \operatorname{Wg}^{Sp}_N(\mu)=\frac{2^k k!}{(2k)!}\sum_{\substack{\lambda\vdash k\\\ell(\lambda)\leq N}}\frac{f^{\lambda\cup\lambda}\pi^\lambda(\sigma_\mu)}{D'_\lambda(N)},\qquad D'_\lambda(N)=\prod_{i=1}^{\ell(\lambda)}\prod_{j=1}^{\lambda_i}(2N-2i+j+1).$ Here $\lambda\cup\lambda=(\lambda_1,\lambda_1,\lambda_2,\lambda_2,\ldots)$, $\chi^{\lambda\cup\lambda}$ is the corresponding irreducible character of $S_{2k}$, $f^{\lambda\cup\lambda}=\chi^{\lambda\cup\lambda}(e)$, where $e$ is the identity permutation, and $\pi^\lambda(\sigma)=(2^k k!)^{-1}\sum_{h\in H_k}\operatorname{sgn}(h)\chi^{\lambda\cup\lambda}(\sigma h)$ [1], equation (2.8).

(2)

$\displaystyle \operatorname{Wg}^{Sp}(\sigma;N)=\operatorname{sgn}(\sigma)\operatorname{Wg}^{Sp}_N\!\left(\operatorname{ct}(p_0,\sigma p_0)\right)\quad(\sigma\in S_{2k}).$ The left side is the function on permutations; the stored coefficient uses the even representative of its coset type [1], section 2.3.1.

(3)

$\displaystyle \mathbb E\!\left[\prod_{a=1}^{2k}S_{i_a j_a}\right]=\sum_{\sigma,\tau\in M_{2k}}\Delta'_\sigma(\mathbf i)\Delta'_\tau(\mathbf j)\operatorname{Wg}^{Sp}(\sigma^{-1}\tau;N),\qquad \Delta'_\sigma(\mathbf i)=\prod_{r=1}^{k}(J_N)_{i_{\sigma(2r-1)},i_{\sigma(2r)}}.$ Here $S$ has Haar law on $\operatorname{USp}(2N)$, all matrix indices lie in $\{1,\ldots,2N\}$, and $M_{2k}$ consists of the permutations satisfying $\sigma(2r-1)<\sigma(2r)$ and $\sigma(1)<\sigma(3)<\cdots<\sigma(2k-1)$ [1], Theorem 2.4.

(4)

$\displaystyle \operatorname{Wg}^{Sp}_N((1))=\frac1{2N}\quad(N\geq1),\qquad \operatorname{Wg}^{Sp}_N((1,1))=\frac{2N-1}{4N(N-1)(2N+1)},\qquad \operatorname{Wg}^{Sp}_N((2))=\frac1{4N(N-1)(2N+1)}\quad(N\geq2).$ These are the cases of one and two pairs in [1], Example 2.4.

Comments

(5)

The rank is $N$, the complex matrix dimension is $2N$, and $\operatorname{USp}(2N)=\{S\in U(2N):S^T J_NS=J_N\}$ with $J_N=\begin{pmatrix}0&I_N\\-I_N&0\end{pmatrix}$. Haar measure has total mass one. The coefficients retain their signs without factorial, dimension, or class-size rescaling [1].

(6)

Let $\mathcal P_2(2k)$ be the pairings of $\{1,\ldots,2k\}$ and $p_0=\{\{1,2\},\ldots,\{2k-1,2k\}\}$. For pairings $p,q$, write $\operatorname{ct}(p,q)$ for the partition of half the component sizes of their superposition. The coset type of $\sigma\in S_{2k}$ is $\operatorname{ct}(p_0,\sigma p_0)$, and $H_k$ is the subgroup preserving $p_0$. Thus the label $3,1,1$ means component sizes $6,2,2$, and $1^k$ means $k$ parts equal to one.

(7)

For a part $r$ of $\mu$, the representative $\sigma_\mu$ has one-line images $(1,2r,2,3,\ldots,2r-1)$ on its consecutive block of $2r$ letters; for $r=1$ the images are $(1,2)$. Concatenate these blocks in partition order. Each representative is even, as in [1], equation (2.3). For example, $\sigma_{(2)}=(1,4,2,3)$ in one-line notation.

(8)

When $N\geq k$, every denominator in (1) is nonzero and the pairing Gram matrix is invertible. At smaller ranks, the sum restricted to $\ell(\lambda)\leq N$ specifies the usual generalized inverse. The stored entries cover only $k\leq N$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.symbolic.ring import SR
import sage.rings.polynomial.laurent_polynomial_ring
from sage.combinat.sf.sf import SymmetricFunctions
from sage.combinat.partition import Partitions
from collections import Counter
from math import factorial, prod


def symplectic_weingarten(N, mu):
    k = sum(mu)
    sf = SymmetricFunctions(QQ)
    p, s, J = sf.power(), sf.schur(), sf.jack(t=QQ(2)).J()
    z = prod(r**m * factorial(m) for r, m in Counter(mu).items())
    total = QQ(0)
    for lam in Partitions(k):
        if len(lam) > N:
            continue
        doubled_rows = tuple(r for r in lam for _ in (0, 1))
        f = s[doubled_rows].scalar(p[[1]*(2*k)])
        c = p(J[lam.conjugate()]).coefficient(mu)
        d = prod(2*N - 2*i + j for i, row in enumerate(lam) for j in range(row))
        total += QQ(f*c) / QQ(d)
    return QQ(2**len(mu)*z) * total / QQ(factorial(2*k))


print(symplectic_weingarten(2, (2,)))  # 1/40

References

[1]

Sho Matsumoto, Weingarten calculus for matrix ensembles associated with compact symmetric spaces, Random Matrices: Theory and Applications 2 (2013), 1350001. Sections 2.2.1 and 2.3, especially equations (2.3), (2.8), and Theorem 2.4. (arXiv) (doi)

[2]

Benoît Collins and Sho Matsumoto, On some properties of orthogonal Weingarten functions, Journal of Mathematical Physics 50 (2009), 113516. Section 3, equations (3.5) and (3.6). (arXiv) (doi)

Links

[3]

Matsumoto: Weingarten calculus for compact symmetric spaces

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Data properties

Entries are of type: rational number

How well the digits are known: exact

Table is complete: no (it holds every coset type with $1\leq|\mu|\leq N$ for $1\leq N\leq6$, covering all stable pairing-contraction patterns for complex matrix dimensions two through twelve. The rank and degree bounds are practical, noncanonical small-matrix and verification budgets. Larger ranks, higher degrees, and singular cases $|\mu|>N$ are not covered)

How they were obtained:

The attached generator evaluates the twisted spherical expansion over $\mathbb Q$. At the even representatives, $\pi^\lambda(\sigma_\mu)=\omega^{\lambda'}(\sigma_\mu)$, where $\lambda'$ is the conjugate partition and $\omega^{\lambda'}$ is the zonal spherical function. It extracts $\omega^{\lambda'}$ from the expansion of the Jack polynomial $J_{\lambda'}^{(2)}$ in power-sum symmetric polynomials, using [2], equation (3.5), and obtains $f^{\lambda\cup\lambda}$ by the hook-length formula. Every stored value is an exact fraction, without rounding.

All 68 coefficients were independently recovered by rational elimination of the pairing Gram equations, using up to 10,395 pairings. Tensor contractions with $J_N$ determine the Gram signs. The checks recover all 21 admissible moments $\mathbb E|S_{11}|^{2k}=k!/\prod_{j=0}^{k-1}(2N+j)$ from the uniform complex sphere, and compare all 16 coefficients with $k\leq2$ to (4). Known nonzero controls, 38 direct tensor contractions, and an intentionally changed coefficient test the checker. The reader program agrees with every entry.