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Orthogonal Weingarten functions
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Numbers
$N$
$\mu$ 
$\operatorname{Wg}^{O}_N(\mu)$
1
1:
1
2
1:
1/2
2
2:
-1/8
2
1, 1:
3/8
3
1:
1/3
3
2:
-1/30
3
1, 1:
2/15
3
3:
1/105
3
2, 1:
-1/42
3
1, 1, 1:
8/105
4
1:
1/4
4
2:
-1/72
4
1, 1:
5/72
4
3:
1/576
4
2, 1:
-1/192
4
1, 1, 1:
13/576
4
4:
-13/28800
4
3, 1:
1/900
4
2, 2:
3/3200
4
2, 1, 1:
-83/28800
4
1, 1, 1, 1:
287/28800
5
1:
1/5
5
2:
-1/140
5
1, 1:
3/70
5
3:
1/1890
5
2, 1:
-1/540
5
1, 1, 1:
19/1890
5
4:
-31/498960
5
3, 1:
1/5544
5
2, 2:
17/124740
5
2, 1, 1:
-71/124740
5
1, 1, 1, 1:
4/1485
5
5:
47/3243240
5
4, 1:
-1/28080
5
3, 2:
-4/135135
5
3, 1, 1:
7/77220
5
2, 2, 1:
11/147420
5
2, 1, 1, 1:
-19/77220
5
1, 1, 1, 1, 1:
2/2145
6
1:
1/6
6
2:
-1/240
6
1, 1:
7/240
6
3:
1/4800
6
2, 1:
-1/1200
6
1, 1, 1:
13/2400
6
4:
-1/67200
6
3, 1:
1/20160
6
2, 2:
1/28800
6
2, 1, 1:
-37/201600
6
1, 1, 1, 1:
73/67200
6
5:
1/627200
6
4, 1:
-13/2822400
6
3, 2:
-19/5644800
6
3, 1, 1:
79/5644800
6
2, 2, 1:
29/2822400
6
2, 1, 1, 1:
-11/235200
6
1, 1, 1, 1, 1:
229/940800
6
6:
-409/1219276800
6
5, 1:
431/522547200
6
4, 2:
2497/3657830400
6
4, 1, 1:
-7663/3657830400
6
3, 3:
823/1219276800
6
3, 2, 1:
-59/34836480
6
3, 1, 1, 1:
271/48771072
6
2, 2, 2:
-23/16257024
6
2, 2, 1, 1:
16249/3657830400
6
2, 1, 1, 1, 1:
-6479/406425600
6
1, 1, 1, 1, 1, 1:
81859/1219276800
7
1:
1/7
7
2:
-1/378
7
1, 1:
4/189
7
3:
1/10395
7
2, 1:
-1/2310
7
1, 1, 1:
34/10395
7
4:
-41/8648640
7
3, 1:
1/56160
7
2, 2:
17/1441440
7
2, 1, 1:
-163/2162160
7
1, 1, 1, 1:
115/216216
7
5:
61/194594400
7
4, 1:
-31/29937600
7
3, 2:
-67/97297200
7
3, 1, 1:
353/97297200
7
2, 2, 1:
37/14968800
7
2, 1, 1, 1:
-1387/97297200
7
1, 1, 1, 1, 1:
4489/48648600
7
6:
-2027/66162096000
7
5, 1:
1367/15437822400
7
4, 2:
29671/463134672000
7
4, 1, 1:
-7741/28945917000
7
3, 3:
14453/231567336000
7
3, 2, 1:
-2917/15437822400
7
3, 1, 1, 1:
24763/28945917000
7
2, 2, 2:
-32467/231567336000
7
2, 2, 1, 1:
35263/57891834000
7
2, 1, 1, 1, 1:
-8401/2756754000
7
1, 1, 1, 1, 1, 1:
25379/1447295850
8
1:
1/8
8
2:
-1/560
8
1, 1:
9/560
8
3:
1/20160
8
2, 1:
-1/4032
8
1, 1, 1:
43/20160
8
4:
-23/12700800
8
3, 1:
1/132300
8
2, 2:
61/12700800
8
2, 1, 1:
-457/12700800
8
1, 1, 1, 1:
1243/4233600
8
5:
17/203212800
8
4, 1:
-1/3225600
8
3, 2:
-13/67737600
8
3, 1, 1:
83/67737600
8
2, 2, 1:
23/29030400
8
2, 1, 1, 1:
-373/67737600
8
1, 1, 1, 1, 1:
953/22579200
8
6:
-41/8047226880
8
5, 1:
677/40236134400
8
4, 2:
443/40236134400
8
4, 1, 1:
-2347/40236134400
8
3, 3:
17/1609445376
8
3, 2, 1:
-151/4023613440
8
3, 1, 1, 1:
4339/20118067200
8
2, 2, 2:
-1033/40236134400
8
2, 2, 1, 1:
523/3657830400
8
2, 1, 1, 1, 1:
-36349/40236134400
8
1, 1, 1, 1, 1, 1:
257717/40236134400
9
1:
1/9
9
2:
-1/792
9
1, 1:
5/396
9
3:
1/36036
9
2, 1:
-1/6552
9
1, 1, 1:
53/36036
9
4:
-17/21621600
9
3, 1:
1/277200
9
2, 2:
1/450450
9
2, 1, 1:
-103/5405400
9
1, 1, 1, 1:
17/96525
9
5:
1/36756720
9
4, 1:
-41/367567200
9
3, 2:
-1/15315300
9
3, 1, 1:
1/2042040
9
2, 2, 1:
1/3281850
9
2, 1, 1, 1:
-227/91891800
9
1, 1, 1, 1, 1:
10/459459
9
6:
-587/502831929600
9
5, 1:
23/5324102784
9
4, 2:
269/102851985600
9
4, 1, 1:
-1189/70710740100
9
3, 3:
373/150849578880
9
3, 2, 1:
-1/99034650
9
3, 1, 1, 1:
661/9428098680
9
2, 2, 2:
-2459/377123947200
9
2, 2, 1, 1:
25049/565685920800
9
2, 1, 1, 1, 1:
-3511/10475665200
9
1, 1, 1, 1, 1, 1:
26311/9428098680
10
1:
1/10
10
2:
-1/1080
10
1, 1:
11/1080
10
3:
1/60480
10
2, 1:
-1/10080
10
1, 1, 1:
1/945
10
4:
-1/2661120
10
3, 1:
1/532224
10
2, 2:
1/887040
10
2, 1, 1:
-29/2661120
10
1, 1, 1, 1:
299/2661120
10
5:
41/4023613440
10
4, 1:
-23/502951680
10
3, 2:
-103/4023613440
10
3, 1, 1:
887/4023613440
10
2, 2, 1:
67/502951680
10
2, 1, 1, 1:
-2477/2011806720
10
1, 1, 1, 1, 1:
12293/1005903360
10
6:
-863/2615348736000
10
5, 1:
1/741312000
10
4, 2:
251/326918592000
10
4, 1, 1:
-73/12573792000
10
3, 3:
1873/2615348736000
10
3, 2, 1:
-961/290594304000
10
3, 1, 1, 1:
2809/104613949440
10
2, 2, 2:
-2677/1307674368000
10
2, 2, 1, 1:
4283/261534873600
10
2, 1, 1, 1, 1:
-62633/435891456000
10
1, 1, 1, 1, 1, 1:
1785989/1307674368000
11
1:
1/11
11
2:
-1/1430
11
1, 1:
6/715
11
3:
1/96525
11
2, 1:
-1/14850
11
1, 1, 1:
76/96525
11
4:
-61/315057600
11
3, 1:
1/954720
11
2, 2:
97/157528800
11
2, 1, 1:
-521/78764400
11
1, 1, 1, 1:
329/4375800
11
5:
89/20951330400
11
4, 1:
-17/821620800
11
3, 2:
-1/89535600
11
3, 1, 1:
379/3491888400
11
2, 2, 1:
79/1232431200
11
2, 1, 1, 1:
-773/1163962800
11
1, 1, 1, 1, 1:
4259/581981400
11
6:
-191/1759911753600
11
5, 1:
19/39188318400
11
4, 2:
5081/19359029289600
11
4, 1, 1:
-11041/4839757322400
11
3, 3:
2341/9679514644800
11
3, 2, 1:
-127/101889627840
11
3, 1, 1, 1:
5597/483975732240
11
2, 2, 2:
-1447/1935902928960
11
2, 2, 1, 1:
1039/151242416325
11
2, 1, 1, 1, 1:
-15031/219988969200
11
1, 1, 1, 1, 1, 1:
440053/604969665300
12
1:
1/12
12
2:
-1/1848
12
1, 1:
13/1848
12
3:
1/147840
12
2, 1:
-1/21120
12
1, 1, 1:
89/147840
12
4:
-1/9434880
12
3, 1:
1/1621620
12
2, 2:
37/103783680
12
2, 1, 1:
-437/103783680
12
1, 1, 1, 1:
155/2965248
12
5:
1/518918400
12
4, 1:
-1/98841600
12
3, 2:
-1/188697600
12
3, 1, 1:
17/296524800
12
2, 2, 1:
23/691891200
12
2, 1, 1, 1:
-113/296524800
12
1, 1, 1, 1, 1:
1367/296524800
12
6:
-1153/28768836096000
12
5, 1:
6697/34522603315200
12
4, 2:
1087/10788313536000
12
4, 1, 1:
-667/674269596000
12
3, 3:
1321/14384418048000
12
3, 2, 1:
-3007/5753767219200
12
3, 1, 1, 1:
78343/14384418048000
12
2, 2, 2:
-4399/14384418048000
12
2, 2, 1, 1:
136777/43153254144000
12
2, 1, 1, 1, 1:
-12967/368831232000
12
1, 1, 1, 1, 1, 1:
237901/575376721920
Definition
The orthogonal Weingarten function $\operatorname{Wg}^{O}_N(\mu)$ is the coefficient for coset type $\mu$ in normalized Haar integration over the full orthogonal group $O(N)$ [3]. The spherical expansion (1) fixes its normalization, including singular dimensions.
Parameters
$N$
—   matrix dimension ($N\geq1$)
$\mu$
—   coset type (A nonempty integer partition, written as comma-separated positive parts in nonincreasing order; $k=|\mu|$ is their sum.)
Formulas
(1)
$\displaystyle \operatorname{Wg}^{O}_N(\mu)=\frac{2^k k!}{(2k)!}\sum_{\substack{\lambda\vdash k\\\ell(\lambda)\leq N}}\frac{f^{2\lambda}\,\omega^\lambda(\sigma_\mu)}{D_\lambda(N)},\qquad D_\lambda(N)=\prod_{i=1}^{\ell(\lambda)}\prod_{j=1}^{\lambda_i}(N+2j-i-1).$ Here $\omega^\lambda(\sigma)=(2^k k!)^{-1}\sum_{h\in H_k}\chi^{2\lambda}(\sigma h)$, $\chi^{2\lambda}$ is the irreducible character of $S_{2k}$ indexed by $2\lambda=(2\lambda_1,2\lambda_2,\ldots)$, and $f^{2\lambda}=\chi^{2\lambda}(e)$ is its dimension, where $e$ is the identity permutation [1].
(2)
$\displaystyle \mathbb E\!\left[\prod_{a=1}^{2k}R_{i_a j_a}\right]=\sum_{p,q\in\mathcal P_2(2k)}\Delta_p(\mathbf i)\Delta_q(\mathbf j)\operatorname{Wg}^{O}_N\!\left(\operatorname{ct}(p,q)\right).$ The matrix $R$ has normalized Haar law on $O(N)$, all indices lie in $\{1,\ldots,N\}$, and $\Delta_p(\mathbf i)=\prod_{\{a,b\}\in p}\delta_{i_a,i_b}$, where $\delta$ is the Kronecker delta [1].
(3)
$\displaystyle G_N(p,q)=N^{\ell(\operatorname{ct}(p,q))},\qquad (G_N^{-1})(p,q)=\operatorname{Wg}^{O}_N\!\left(\operatorname{ct}(p,q)\right)\quad(N\geq k).$ The rows and columns are indexed by pairings in $\mathcal P_2(2k)$, and $\ell(\nu)$ is the number of parts of a partition $\nu$ [2].
(4)
$\displaystyle \operatorname{Wg}^{O}_N((1))=\frac1N\quad(N\geq1),\qquad \operatorname{Wg}^{O}_N((1,1))=\frac{N+1}{N(N+2)(N-1)},\qquad \operatorname{Wg}^{O}_N((2))=-\frac1{N(N+2)(N-1)}\quad(N\geq2).$ These are the cases of one and two pairs in [1], Example 2.3.
Comments
(5)
$N$ is the real matrix dimension, and Haar measure on $O(N)$ has total mass one. The coefficients retain their signs and are not multiplied by factorials, powers of $N$, or double-coset sizes.
(6)
Let $\mathcal P_2(2k)$ be the pairings of $\{1,\ldots,2k\}$, and let $p_0=\{\{1,2\},\{3,4\},\ldots,\{2k-1,2k\}\}$. For pairings $p,q$, the partition $\operatorname{ct}(p,q)$ consists of half the sizes of the connected components of their superposition. For a permutation $\sigma\in S_{2k}$ its coset type is $\operatorname{ct}(p_0,\sigma p_0)$, and $H_k$ is the subgroup preserving $p_0$ [1]. Thus the label $3,1,1$ means a partition of $k=5$, with component sizes $6,2,2$. The notation $1^k$ means the partition consisting of $k$ ones.
(7)
A representative $\sigma_\mu$ acts on each consecutive block of $2r$ letters belonging to a part $r$ of $\mu$ with one-line images $(1,2r,2,3,\ldots,2r-1)$ relative to that block; for $r=1$ these are $(1,2)$. Concatenate the blocks in partition order. This is Matsumoto's representative (2.3), although the orthogonal coefficient depends only on the coset type [1].
(8)
When $N\geq k$, the pairing Gram matrix in (3) is invertible. When $N<k$, (1) gives the conventional generalized inverse by retaining only partitions $\lambda$ with $\ell(\lambda)\leq N$ [2]. The entries cover only the invertible range.
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 math import factorial, prod
from collections import Counter


def orthogonal_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
        f = s[tuple(2*r for r in lam)].scalar(p[[1]*(2*k)])
        c = p(J[lam]).coefficient(mu)
        d = prod(N + 2*j - i 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(orthogonal_weingarten(13, (3, 1)))
References
[1]
Sho Matsumoto, Weingarten calculus for matrix ensembles associated with compact symmetric spaces, Random Matrices: Theory and Applications 2 (2013), 1350001. Section 2.2, Theorem 2.2 and Example 2.3. (arXiv) (doi)
[2]
Benoît Collins and Sho Matsumoto, On some properties of orthogonal Weingarten functions, Journal of Mathematical Physics 50 (2009), 113516. Sections 2 and 3, especially equations (3.5)-(3.7) and Theorem 3.1. (arXiv) (doi)
Links
Similar tables
Random-matrix factors in characteristic-polynomial moments —   give leading factors in 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 coset type with $1\leq|\mu|\leq\min(6,N)$ for $1\leq N\leq12$, covering all pairing-contraction patterns through total degree twelve at each selected dimension in the invertible range. The dimension and degree bounds are practical, noncanonical small-matrix and verification budgets; singular cases $|\mu|>N$, larger dimensions and higher degrees are not covered)
How they were obtained:

The attached generator evaluates the spherical expansion over $\mathbb Q$.

more

It obtains zonal spherical coefficients from the expansion in power-sum symmetric polynomials of Jack polynomials $J_\lambda^{(2)}$, with the normalization of [2], equations (3.5)-(3.7), and computes $f^{2\lambda}$ by the hook-length formula. The values are exact fractions, without numerical rounding. All 242 coefficients were independently recovered from pairing Gram inverse equations by rational Gaussian elimination, using up to 10,395 pairings. The checks also verify 57 moments $\mathbb E R_{11}^{2k}=(2k-1)!!/\prod_{j=0}^{k-1}(N+2j)$ from the uniform distribution on the unit sphere, and 64 explicit coefficients through $k=3$ from [2], section 7. Nonzero controls and an altered coefficient test the checking procedure. The reader program agrees with every entry; the zonal specializations and the singular $N=1$ convention are checked separately in the attachments.



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