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 U_d of products of matrix coefficients of the form

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

This integral is equal to

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

where Wg is the Weingarten function, given by

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

where the sum is over all partitions λ of q (Collins 2003). Here χ^λ is the character if S_q corresponding to the partition λ and s is the Schur polynomial of λ, so that s_λd(1) is the dimension of the representation of U_d 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(σd) are

$\displaystyle Wg(,d)=1$ (The trivial case where q=0)

$\displaystyle Wg(1,d)={\frac {1}{d}}$

$\displaystyle Wg(2,d)={\frac {-1}{d(d^{2}-1)}}$

$\displaystyle Wg(1^{2},d)={\frac {1}{d^{2}-1}}$

$\displaystyle Wg(3,d)={\frac {2}{d(d^{2}-1)(d^{2}-4)}}$

$\displaystyle Wg(21,d)={\frac {-1}{(d^{2}-1)(d^{2}-4)}}$

$\displaystyle Wg(1^{3},d)={\frac {d^{2}-2}{d(d^{2}-1)(d^{2}-4)}}$

where permutations σ are denoted by their cycle shapes.

Asymptotic behavior

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

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

where the permutation σ is a product of cycles of lengths C_i, and c_n = (2n)!/n!(n + 1)! is a Catalan number, and |σ| iss the smallest number of transpositions that σ is a product of.

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.

References

• Collins, Benoît (2003), "Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability", International Mathematics Research Notices (17): 953–982, doi:10.1155/S107379280320917X, ISSN 1073-7928, MR1959915{{citation}}: CS1 maint: unflagged free DOI (link)
• Collins, Benoît; Śniady, Piotr (2006), "Integration with respect to the Haar measure on unitary, orthogonal and symplectic group", Communications in Mathematical Physics, 264 (3): 773–795, doi:10.1007/s00220-006-1554-3, ISSN 0010-3616, MR2217291
• Weingarten, Don (1978), "Asymptotic behavior of group integrals in the limit of infinite rank", Journal of Mathematical Physics, 19 (5): 999–1001, doi:10.1063/1.523807, ISSN 0022-2488, MR0471696