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Unitary Weingarten functions
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Numbers
$N$
$\mu$ 
$\operatorname{Wg}^{U}_N(\mu)$
1
1:
1
2
1:
1/2
2
2:
-1/6
2
1, 1:
1/3
3
1:
1/3
3
2:
-1/24
3
1, 1:
1/8
3
3:
1/60
3
2, 1:
-1/40
3
1, 1, 1:
7/120
4
1:
1/4
4
2:
-1/60
4
1, 1:
1/15
4
3:
1/360
4
2, 1:
-1/180
4
1, 1, 1:
7/360
4
4:
-1/1008
4
3, 1:
29/20160
4
2, 2:
11/10080
4
2, 1, 1:
-1/420
4
1, 1, 1, 1:
67/10080
5
1:
1/5
5
2:
-1/120
5
1, 1:
1/24
5
3:
1/1260
5
2, 1:
-1/504
5
1, 1, 1:
23/2520
5
4:
-1/8064
5
3, 1:
47/201600
5
2, 2:
31/201600
5
2, 1, 1:
-1/1920
5
1, 1, 1, 1:
431/201600
5
5:
1/25920
5
4, 1:
-101/1814400
5
3, 2:
-37/907200
5
3, 1, 1:
1/11340
5
2, 2, 1:
23/362880
5
2, 1, 1, 1:
-299/1814400
5
1, 1, 1, 1, 1:
29/51840
6
1:
1/6
6
2:
-1/210
6
1, 1:
1/35
6
3:
1/3360
6
2, 1:
-1/1120
6
1, 1, 1:
17/3360
6
4:
-1/36288
6
3, 1:
23/362880
6
2, 2:
1/25920
6
2, 1, 1:
-1/5670
6
1, 1, 1, 1:
169/181440
6
5:
1/259200
6
4, 1:
-13/1814400
6
3, 2:
-1/226800
6
3, 1, 1:
1/67200
6
2, 2, 1:
17/1814400
6
2, 1, 1, 1:
-17/453600
6
1, 1, 1, 1, 1:
109/604800
6
6:
-1/950400
6
5, 1:
13/8553600
6
4, 2:
461/419126400
6
4, 1, 1:
-167/69854400
6
3, 3:
1/931392
6
3, 2, 1:
-17/10348800
6
3, 1, 1, 1:
59/13970880
6
2, 2, 2:
-169/139708800
6
2, 2, 1, 1:
599/209563200
6
2, 1, 1, 1, 1:
-47/5174400
6
1, 1, 1, 1, 1, 1:
2627/69854400
7
1:
1/7
7
2:
-1/336
7
1, 1:
1/48
7
3:
1/7560
7
2, 1:
-1/2160
7
1, 1, 1:
47/15120
7
4:
-1/120960
7
3, 1:
19/846720
7
2, 2:
11/846720
7
2, 1, 1:
-1/13440
7
1, 1, 1, 1:
403/846720
7
5:
1/1425600
7
4, 1:
-221/139708800
7
3, 2:
-61/69854400
7
3, 1, 1:
1/249480
7
2, 2, 1:
47/19958400
7
2, 1, 1, 1:
-1739/139708800
7
1, 1, 1, 1, 1:
1499/19958400
7
6:
-1/11404800
7
5, 1:
13/79833600
7
4, 2:
131/1341204480
7
4, 1, 1:
-29/87091200
7
3, 3:
311/3353011200
7
3, 2, 1:
-37/191600640
7
3, 1, 1, 1:
5167/6706022400
7
2, 2, 2:
-13/106444800
7
2, 2, 1, 1:
283/609638400
7
2, 1, 1, 1, 1:
-421/191600640
7
1, 1, 1, 1, 1, 1:
82477/6706022400
8
1:
1/8
8
2:
-1/504
8
1, 1:
1/63
8
3:
1/15120
8
2, 1:
-1/3780
8
1, 1, 1:
31/15120
8
4:
-1/332640
8
3, 1:
5/532224
8
2, 2:
1/190080
8
2, 1, 1:
-1/27720
8
1, 1, 1, 1:
359/1330560
8
5:
1/5702400
8
4, 1:
-37/79833600
8
3, 2:
-19/79833600
8
3, 1, 1:
1/725760
8
2, 2, 1:
31/39916800
8
2, 1, 1, 1:
-403/79833600
8
1, 1, 1, 1, 1:
1447/39916800
8
6:
-1/74131200
8
5, 1:
1/32947200
8
4, 2:
211/13076743680
8
4, 1, 1:
-307/4086482400
8
3, 3:
977/65383718400
8
3, 2, 1:
-47/1188794880
8
3, 1, 1, 1:
27499/130767436800
8
2, 2, 2:
-251/10897286400
8
2, 2, 1, 1:
1957/16345929600
8
2, 1, 1, 1, 1:
-433/594397440
8
1, 1, 1, 1, 1, 1:
163021/32691859200
9
1:
1/9
9
2:
-1/720
9
1, 1:
1/80
9
3:
1/27720
9
2, 1:
-1/6160
9
1, 1, 1:
79/55440
9
4:
-1/798336
9
3, 1:
53/11975040
9
2, 2:
29/11975040
9
2, 1, 1:
-1/51840
9
1, 1, 1, 1:
1973/11975040
9
5:
1/18532800
9
4, 1:
-127/778377600
9
3, 2:
-31/389188800
9
3, 1, 1:
1/1801800
9
2, 2, 1:
79/259459200
9
2, 1, 1, 1:
-1817/778377600
9
1, 1, 1, 1, 1:
239/12355200
9
6:
-1/345945600
9
5, 1:
71/9340531200
9
4, 2:
139/37362124800
9
4, 1, 1:
-1/45619200
9
3, 3:
1/296524800
9
3, 2, 1:
-1/92252160
9
3, 1, 1, 1:
127/1779148800
9
2, 2, 2:
-1/166053888
9
2, 2, 1, 1:
113/2874009600
9
2, 1, 1, 1, 1:
-19/65894400
9
1, 1, 1, 1, 1, 1:
4109/1779148800
10
1:
1/10
10
2:
-1/990
10
1, 1:
1/99
10
3:
1/47520
10
2, 1:
-1/9504
10
1, 1, 1:
49/47520
10
4:
-1/1729728
10
3, 1:
197/86486400
10
2, 2:
53/43243200
10
2, 1, 1:
-1/90090
10
1, 1, 1, 1:
4603/43243200
10
5:
1/51891840
10
4, 1:
-17/259459200
10
3, 2:
-1/32432400
10
3, 1, 1:
1/3991680
10
2, 2, 1:
1/7413120
10
2, 1, 1, 1:
-1/842400
10
1, 1, 1, 1, 1:
577/51891840
10
6:
-1/1297296000
10
5, 1:
1/432432000
10
4, 2:
137/128432304000
10
4, 1, 1:
-487/64216152000
10
3, 3:
61/64216152000
10
3, 2, 1:
-71/19758816000
10
3, 1, 1, 1:
1801/64216152000
10
2, 2, 2:
-83/42810768000
10
2, 2, 1, 1:
971/64216152000
10
2, 1, 1, 1, 1:
-1273/9879408000
10
1, 1, 1, 1, 1, 1:
75541/64216152000
11
1:
1/11
11
2:
-1/1320
11
1, 1:
1/120
11
3:
1/77220
11
2, 1:
-1/14040
11
1, 1, 1:
119/154440
11
4:
-1/3459456
11
3, 1:
239/190270080
11
2, 2:
127/190270080
11
2, 1, 1:
-1/147840
11
1, 1, 1, 1:
13679/190270080
11
5:
1/129729600
11
4, 1:
-83/2854051200
11
3, 2:
-19/1427025600
11
3, 1, 1:
1/8108100
11
2, 2, 1:
17/259459200
11
2, 1, 1, 1:
-1853/2854051200
11
1, 1, 1, 1, 1:
251/37065600
11
6:
-1/4151347200
11
5, 1:
37/45664819200
11
4, 2:
7/19570636800
11
4, 1, 1:
-37/12454041600
11
3, 3:
43/136994457600
11
3, 2, 1:
-17/12454041600
11
3, 1, 1, 1:
1681/136994457600
11
2, 2, 2:
-1/1383782400
11
2, 2, 1, 1:
893/136994457600
11
2, 1, 1, 1, 1:
-157/2490808320
11
1, 1, 1, 1, 1, 1:
88261/136994457600
12
1:
1/12
12
2:
-1/1716
12
1, 1:
1/143
12
3:
1/120120
12
2, 1:
-1/20020
12
1, 1, 1:
71/120120
12
4:
-1/6486480
12
3, 1:
19/25945920
12
2, 2:
1/2594592
12
2, 1, 1:
-1/231660
12
1, 1, 1, 1:
653/12972960
12
5:
1/296524800
12
4, 1:
-29/2075673600
12
3, 2:
-1/159667200
12
3, 1, 1:
1/15375360
12
2, 2, 1:
71/2075673600
12
2, 1, 1, 1:
-71/188697600
12
1, 1, 1, 1, 1:
427/98841600
12
6:
-1/11762150400
12
5, 1:
67/211718707200
12
4, 2:
409/3027577512960
12
4, 1, 1:
-101/78843164400
12
3, 3:
197/1681987507200
12
3, 2, 1:
-43/74755000320
12
3, 1, 1, 1:
58897/10091925043200
12
2, 2, 2:
-1517/5045962521600
12
2, 2, 1, 1:
1451/473058986400
12
2, 1, 1, 1, 1:
-1237/37377500160
12
1, 1, 1, 1, 1, 1:
943073/2522981260800
Definition
The unitary Weingarten function $\operatorname{Wg}^{U}_N(\mu)$ is the coefficient for cycle type $\mu$ in normalized Haar integration over $U(N)$ [3]. Its normalization, including singular dimensions, is fixed by the character sum (1).
Parameters
$N$
—   matrix dimension ($N\geq 1$)
$\mu$
—   cycle type (A nonempty integer partition, written as comma-separated positive parts in nonincreasing order; $k=|\mu|$ is their sum.)
Formulas
(1)
$\displaystyle \operatorname{Wg}^{U}_N(\mu)=\frac{1}{k!}\sum_{\substack{\lambda\vdash k\\\ell(\lambda)\leq N}}\frac{f^\lambda\chi^\lambda(\mu)}{C_\lambda(N)},\qquad C_\lambda(N)=\prod_{i=1}^{\ell(\lambda)}\prod_{j=1}^{\lambda_i}(N+j-i).$ Here $\chi^\lambda$ is the irreducible character of $S_k$ indexed by $\lambda$, and $f^\lambda=\chi^\lambda(1^k)$ is its dimension [1].
(2)
$\displaystyle \mathbb E\!\left[\prod_{r=1}^{k}U_{i_rj_r}\,\overline{U_{i'_rj'_r}}\right]=\sum_{\sigma,\tau\in S_k}\delta_\sigma(\mathbf i,\mathbf i')\delta_\tau(\mathbf j,\mathbf j')\operatorname{Wg}^{U}_N\!\left(\operatorname{type}(\sigma^{-1}\tau)\right).$ Here $U$ has normalized Haar law on $U(N)$, all matrix indices are in $\{1,\ldots,N\}$, $\delta_\sigma(\mathbf i,\mathbf i')=\prod_{r=1}^{k}\delta_{i_{\sigma(r)},i'_r}$, and $\operatorname{type}$ denotes cycle type [1].
(3)
$\displaystyle \sum_{\tau\in S_k}N^{c(\tau)}\operatorname{Wg}^{U}_N\!\left(\operatorname{type}(\tau^{-1}\sigma)\right)=\delta_{\sigma,e}\quad(N\geq k).$ The integer $c(\tau)$ is the number of cycles of $\tau$, and $e$ is the identity permutation in $S_k$ [1].
(4)
$\displaystyle \operatorname{Wg}^{U}_N((1))=\frac1N\quad(N\geq1),\qquad \operatorname{Wg}^{U}_N((1,1))=\frac{1}{N^2-1},\qquad \operatorname{Wg}^{U}_N((2))=-\frac{1}{N(N^2-1)}\quad(N\geq2).$ These are the degree-one and degree-two cases in [1].
Comments
(5)
$N$ is the complex matrix dimension, and Haar measure on the full unitary group $U(N)$ has total mass one. The coefficients retain their signs and are not multiplied by factorials, powers of $N$, or conjugacy-class sizes.
(6)
The partition $\mu=(\mu_1,\ldots,\mu_\ell)$ records the cycle lengths of a permutation in $S_k$, where $k=\mu_1+\cdots+\mu_\ell$. For example, the entry label $3,1,1$ means cycle type $(3,1,1)$ in $S_5$. The partition $1^k$ denotes the identity cycle type.
(7)
When $N\geq k$, the coefficients form the ordinary convolution inverse in (3). When $N<k$, the character sum defines the conventional generalized inverse by retaining only partitions $\lambda$ with $\ell(\lambda)\leq N$ [1]. The stored entries cover only $N\geq k$.
(8)
Schur polynomials are the characters of the polynomial representations of $U(N)$ used in the representation-theoretic derivation of these coefficients [2].
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.symbolic.ring import SR
from sage.combinat.partition import Partitions
from sage.combinat.sf.sf import SymmetricFunctions
from math import factorial, prod


def unitary_weingarten(N, mu):
    k = sum(mu)
    SF = SymmetricFunctions(QQ)
    s, p = SF.schur(), SF.power()
    answer = QQ(0)
    for lam in Partitions(k):
        if len(lam) > N:
            continue
        f = s[lam].scalar(p[[1] * k])
        chi = s[lam].scalar(p[list(mu)])
        c = prod(N + j - i for i, row in enumerate(lam) for j in range(row))
        answer += QQ(f * chi) / QQ(c)
    return answer / QQ(factorial(k))


print(unitary_weingarten(13, (3, 1)))  # 67/149909760
References
[1]
Sho Matsumoto, Weingarten calculus for matrix ensembles associated with compact symmetric spaces, Random Matrices: Theory and Applications 2 (2013), 1350001. Section 2.1, equation (2.1) and Theorem 2.1. (arXiv)
[2]
Benoît Collins and Piotr Śniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Communications in Mathematical Physics 264 (2006), 773–795. Section 2, equation (9) and Corollary 2.4. (arXiv) (doi)
Links
Similar tables
Schur polynomials —   give the characters of polynomial unitary representations used in the derivation
Random-matrix factors in characteristic-polynomial moments —   give leading factors of characteristic-polynomial moments of the classical compact groups
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds every cycle type with $1\le |\mu|\le\min(6,N)$ for $1\le N\le12$, covering all contraction patterns through total degree twelve at every selected dimension in the stable range. The dimension and degree bounds are practical, noncanonical limits; the singular range $|\mu|>N$ is not covered)
How they were obtained:

The attached generator evaluates the character sum over $\mathbb Q$, using the Murnaghan–Nakayama rule for symmetric-group characters and the hook-length formula for their dimensions. Every value is an exact rational; no numerical precision or rounding is used.

more

All 242 coefficients were independently recovered by enumerating permutations and solving the conjugacy-class Gram inverse equations with rational Gaussian elimination. The checks also compare all coefficients with Sage's symmetric-function character expansion, all 57 admissible pairs $(N,k)$ with the Haar moment $\mathbb E|U_{11}|^{2k}=k!/[N(N+1)\cdots(N+k-1)]$, and all 34 degree-one and degree-two coefficients with (4). Nonzero controls and a deliberately corrupted inverse coefficient test the checking procedure.



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