Chiral unitary Weingarten functions

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Numbers

$a$

$b$

$\mu$

$\operatorname{Wg}^{AIII}_{a,b}(\mu)$

2/3

1, 1:

-1/3

1/3

1, 1:

-2/15

2, 1:

1/5

1, 1, 1:

-2/15

4/15

1, 1:

-1/15

2, 1:

1, 1, 1:

-4/105

3, 1:

1/35

2, 2:

2/21

2, 1, 1:

-4/105

1, 1, 1, 1:

1/35

1/2

1/5

1, 1:

1/5

-1/15

2, 1:

2/15

1, 1, 1:

1/30

2/105

3, 1:

-1/21

2, 2:

2/105

2, 1, 1:

3/35

1, 1, 1, 1:

-1/21

1/5

1, 1:

-2/105

2, 1:

1/21

1, 1, 1:

-2/105

-1/105

3, 1:

1/525

2, 2:

23/525

2, 1, 1:

1, 1, 1, 1:

-2/525

6/35

1, 1:

-1/35

2, 1:

1, 1, 1:

-2/315

3, 1:

1/315

2, 2:

2/63

2, 1, 1:

-2/315

1, 1, 1, 1:

1/315

3/5

2/15

1, 1:

1/3

-4/105

2, 1:

2/21

1, 1, 1:

17/105

1/105

3, 1:

-1/35

2, 2:

1/105

2, 1, 1:

1/15

1, 1, 1, 1:

2/35

1/3

16/105

1, 1:

3/35

-2/105

2, 1:

2/35

1, 1, 1:

1/105

-2/945

3, 1:

-1/189

2, 2:

22/945

2, 1, 1:

16/945

1, 1, 1, 1:

-1/189

1/7

1, 1:

-2/315

2, 1:

1/45

1, 1, 1:

-2/315

-1/315

3, 1:

1/2205

2, 2:

47/2205

2, 1, 1:

1, 1, 1, 1:

-2/2205

8/63

1, 1:

-1/63

2, 1:

1, 1, 1:

-8/3465

3, 1:

1/1155

2, 2:

58/3465

2, 1, 1:

-8/3465

1, 1, 1, 1:

1/1155

2/3

2/21

1, 1:

3/7

-1/42

2, 1:

1/14

1, 1, 1:

11/42

1/189

3, 1:

-1/54

2, 2:

1/189

2, 1, 1:

10/189

1, 1, 1, 1:

4/27

3/7

5/42

1, 1:

1/6

-1/63

2, 1:

1/18

1, 1, 1:

1/18

3, 1:

-1/147

2, 2:

2/147

2, 1, 1:

1/42

1, 1, 1, 1:

2/147

1/4

5/42

1, 1:

1/21

-1/126

2, 1:

2/63

1, 1, 1:

1/252

-1/693

3, 1:

-1/693

2, 2:

10/693

2, 1, 1:

1/154

1, 1, 1, 1:

-1/693

5/7

1/14

1, 1:

1/2

-1/63

2, 1:

1/18

1, 1, 1:

43/126

1/315

3, 1:

-4/315

2, 2:

1/315

2, 1, 1:

3/70

1, 1, 1, 1:

71/315

1/2

2/21

1, 1:

5/21

-4/315

2, 1:

16/315

1, 1, 1:

67/630

2/3465

3, 1:

-13/1980

2, 2:

59/6930

2, 1, 1:

2/77

1, 1, 1, 1:

43/990

3/4

1/18

1, 1:

5/9

-1/90

2, 1:

2/45

1, 1, 1:

73/180

1/495

3, 1:

-1/110

2, 2:

1/495

2, 1, 1:

7/198

1, 1, 1, 1:

16/55

Definition

The chiral unitary Weingarten coefficient $\operatorname{Wg}^{AIII}_{a,b}(\mu)$ is defined by (1) for $W=U^*\operatorname{diag}(I_a,-I_b)U$, with $U$ Haar distributed on $U(a+b)$ [2].

Parameters

$a$

— multiplicity of the eigenvalue $+1$ ($a\ge b\ge1$)

$b$

— multiplicity of the eigenvalue $-1$ ($1\le b\le a$)

$\mu$

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

Formulas

(1)

$\displaystyle \operatorname{Wg}^{AIII}_{a,b}(\mu)=\sum_{\substack{\lambda\vdash k\\\ell(\lambda)\le N}}\frac{s_\lambda(1^a,(-1)^b)\,\chi^\lambda(\mu)}{C_\lambda(N)},\qquad C_\lambda(N)=\prod_{i=1}^{\ell(\lambda)}\prod_{j=1}^{\lambda_i}(N+j-i).$ Here $N=a+b$, $k=|\mu|$, $s_\lambda$ is a Schur polynomial, and $\chi^\lambda$ is the irreducible character of $S_k$ indexed by $\lambda$. The notation $1^a,(-1)^b$ means $a$ arguments equal to $+1$ followed by $b$ arguments equal to $-1$ [1].

(2)

$\displaystyle \mathbb E\!\left[\prod_{r=1}^{k}W_{i_rj_r}\right]=\sum_{\sigma\in S_k}\left(\prod_{r=1}^{k}\delta_{i_{\sigma(r)},j_r}\right)\operatorname{Wg}^{AIII}_{a,b}(\operatorname{type}(\sigma)).$ All matrix indices lie in $\{1,\ldots,N\}$; $\delta_{ij}$ is the Kronecker delta and $\operatorname{type}(\sigma)$ is the cycle type [1].

(3)

$\displaystyle \operatorname{Wg}^{AIII}_{a,b}(\operatorname{type}(\sigma))=\sum_{\tau\in S_k}N^{e(\tau)}(a-b)^{o(\tau)}\operatorname{Wg}^{U}_{N}(\operatorname{type}(\tau^{-1}\sigma)).$ Here $\operatorname{Wg}^{U}_{N}$ is the unitary Weingarten function, and $e(\tau)$ and $o(\tau)$ count the even-length and odd-length cycles of $\tau$, respectively; a factor with exponent zero is $1$ [1].

(4)

$\displaystyle \operatorname{Wg}^{AIII}_{a,b}((1))=\frac{a-b}{N},\qquad \operatorname{Wg}^{AIII}_{a,b}((1,1))=\frac{(a-b)^2-1}{N^2-1},\qquad \operatorname{Wg}^{AIII}_{a,b}((2))=\frac{4ab}{N(N^2-1)}.$ These are the degree-one and degree-two cases, with $N=a+b$ [1].

(5)

$\displaystyle \operatorname{Wg}^{AIII}_{b,a}(\mu)=(-1)^{|\mu|}\operatorname{Wg}^{AIII}_{a,b}(\mu).$ The symbol $\operatorname{Wg}^{AIII}_{b,a}$ uses the same definition with the two eigenvalue multiplicities interchanged. In particular, if $a=b$ and $|\mu|$ is odd, the coefficient is zero.

Comments

(6)

Write $N=a+b$ and $D=\operatorname{diag}(I_a,-I_b)$. The matrix $W=U^*DU$ is Hermitian and unitary, with $a$ eigenvalues $+1$ and $b$ eigenvalues $-1$. It is the matrix used in Matsumoto's class AIII integration theorem; the symmetric-space realization $V=DW$ has different matrix entries [1].

(7)

The partition $\mu$ records ordinary cycle lengths in the symmetric group $S_k$, where $k=|\mu|$ is the degree of the matrix-entry moment. Thus $2,1,1$ means cycle type $(2,1,1)$ in $S_4$, and $1^k$ denotes the identity cycle type. The stored coefficients retain their signs and have no factorial, class-size or dimension multiplier.

(8)

The eigenspace of $W$ for the eigenvalue $+1$ is a uniformly distributed $a$-dimensional complex subspace of $\mathbb C^N$. Its orthogonal projection is $(I_N+W)/2$, and the corresponding complex Grassmannian is $U(N)/(U(a)\times U(b))$.

(9)

When $k\le N$, the permutation Gram matrix is invertible. For $k>N$, (1) retains only partitions $\lambda$ of length at most $N$, as in the generalized-inverse convention for unitary Weingarten functions [1]. The stored entries are all in the range $k\le N$.

Programs

(P1)

Sage

# Download the attached generate.py into the working directory.
from generate import weingarten

# One signature beyond the default dimension bound.
print(weingarten(8, 1, (2, 1)))  # 2/55

References

[1]

Sho Matsumoto, Weingarten calculus for matrix ensembles associated with compact symmetric spaces, Random Matrices: Theory and Applications 2 (2013), 1350001. Sections 2.1 and 4.1, especially Theorem 4.1, Example 4.1 and the character expansion on page 18. (arXiv) (doi)

Links

[2]

Matsumoto: Weingarten calculus for compact symmetric spaces

Similar tables

Unitary Weingarten functions — give the Haar integration kernel in the convolution defining the class AIII coefficients

Schur polynomials — give the symmetric polynomials whose signature specializations occur in the character expansion

Data properties

Entries are of type: rational number

How well the digits are known: exact

Table is complete: no (it holds every signature $a\ge b\ge1$ with $a+b\le8$ and every cycle type $\mu$ with $1\le |\mu|\le\min(4,a+b)$, covering balanced eigenspaces, rank-one negative complements and intermediate ranks together with all contraction patterns through degree four in the invertible range. The dimension and degree bounds are practical, noncanonical limits chosen to allow full exact permutation checks; larger dimensions and degrees are not covered)

How they were obtained:

The attached generator evaluates the character expansion over $\mathbb Q$. It obtains Schur specializations from the Jacobi-Trudi determinant and symmetric-group characters from the Murnaghan-Nakayama rule. Every stored value is an exact reduced rational, with no decimal rounding.

All 163 coefficients were independently recovered by rational Gaussian elimination on conjugacy-class Gram equations. The stored values were then checked against the full permutation equations, all 61 diagonal moments from the beta integral for $W_{11}$, all 48 degree-one and degree-two cases in (4), signature reversal, and the convolution with all corresponding coefficients read from the unitary Weingarten table. Nonzero known controls and a deliberately corrupted coefficient test the checking procedure. The attached checks.py reproduces these tests, and argument-selection.md explains the complete finite selection.