# Weingarten function

In mathematics, Weingarten functions are rational functions indexed by partitions of integers that can be used to calculate integrals of products of matrix coefficients over classical groups. They were first studied by Weingarten (1978) who found their asymptotic behavior, and named by Collins (2003), who evaluated them explicitly for the unitary group.

## Unitary groups

Weingarten functions are used for evaluating integrals over the unitary group Ud of products of matrix coefficients of the form

${\displaystyle \int _{U_{d}}U_{i_{1}j_{1}}\cdots U_{i_{q}j_{q}}U_{j_{1}^{\prime }i_{1}^{\prime }}^{*}\cdots U_{j_{q}^{\prime }i_{q}^{\prime }}^{*}dU.}$

(Here ${\displaystyle U^{*}}$ denotes the conjugate transpose of ${\displaystyle U}$, alternatively denoted as ${\displaystyle U^{\dagger }}$.)

This integral is equal to

${\displaystyle \sum _{\sigma ,\tau \in S_{q}}\delta _{i_{1}i_{\sigma (1)}^{\prime }}\cdots \delta _{i_{q}i_{\sigma (q)}^{\prime }}\delta _{j_{1}j_{\tau (1)}^{\prime }}\cdots \delta _{j_{q}j_{\tau (q)}^{\prime }}W\!g(\sigma \tau ^{-1},d)}$

where Wg is the Weingarten function, given by

${\displaystyle W\!g(\sigma ,d)={\frac {1}{q!^{2}}}\sum _{\lambda }{\frac {\chi ^{\lambda }(1)^{2}\chi ^{\lambda }(\sigma )}{s_{\lambda ,d}(1)}}}$

where the sum is over all partitions λ of q (Collins 2003). Here χλ is the character of Sq corresponding to the partition λ and s is the Schur polynomial of λ, so that sλd(1) is the dimension of the representation of Ud corresponding to λ.

The Weingarten functions are rational functions in d. They can have poles for small values of d, which cancel out in the formula above. There is an alternative inequivalent definition of Weingarten functions, where one only sums over partitions with at most d parts. This is no longer a rational function of d, but is finite for all positive integers d. The two sorts of Weingarten functions coincide for d larger than q, and either can be used in the formula for the integral.

### Examples

The first few Weingarten functions Wg(σ, d) are

${\displaystyle \displaystyle W\!g(,d)=1}$ (The trivial case where q = 0)
${\displaystyle \displaystyle W\!g(1,d)={\frac {1}{d}}}$
${\displaystyle \displaystyle W\!g(2,d)={\frac {-1}{d(d^{2}-1)}}}$
${\displaystyle \displaystyle W\!g(1^{2},d)={\frac {1}{d^{2}-1}}}$
${\displaystyle \displaystyle W\!g(3,d)={\frac {2}{d(d^{2}-1)(d^{2}-4)}}}$
${\displaystyle \displaystyle W\!g(21,d)={\frac {-1}{(d^{2}-1)(d^{2}-4)}}}$
${\displaystyle \displaystyle W\!g(1^{3},d)={\frac {d^{2}-2}{d(d^{2}-1)(d^{2}-4)}}}$

where permutations σ are denoted by their cycle shapes.

There exist computer algebra programs to produce these expressions.[1][2]

### Asymptotic behavior

For large d, the Weingarten function Wg has the asymptotic behavior

${\displaystyle W\!g(\sigma ,d)=d^{-n-|\sigma |}\prod _{i}(-1)^{|C_{i}|-1}c_{|C_{i}|-1}+O(d^{-n-|\sigma |-2})}$

where the permutation σ is a product of cycles of lengths Ci, and cn = (2n)!/n!(n + 1)! is a Catalan number, and |σ| is the smallest number of transpositions that σ is a product of. There exists a diagrammatic method[3] to systematically calculate the integrals over the unitary group as a power series in 1/d.

## Orthogonal and symplectic groups

For orthogonal and symplectic groups the Weingarten functions were evaluated by Collins & Śniady (2006). Their theory is similar to the case of the unitary group. They are parameterized by partitions such that all parts have even size.