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 who found their asymptotic behavior, and named by, 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
 * $$\int_{U_d} U_{i_1j_1}\cdots U_{i_qj_q}U^*_{i^\prime_1j^\prime_1}\cdots U^*_{i^\prime_qj^\prime_q}dU,$$

where $$*$$ denotes complex conjugation. Note that $$U^*_{ji}=(U^\dagger)_{ij}$$ where $$U^\dagger$$ is the conjugate transpose of $$U$$, so one can interpret the above expression as being for the $$i_1j_1\ldots i_qj_qj'_1i'_1\ldots j'_qi'_q$$ matrix element of $$U\otimes\cdots\otimes U\otimes U^\dagger\otimes\cdots\otimes U^\dagger$$.

This integral is equal to
 * $$\sum_{\sigma,\tau\in S_q}\delta_{i_1i^\prime_{\sigma(1)}}\cdots\delta_{i_qi^\prime_{\sigma(q)}} \delta_{j_1j^\prime_{\tau(1)}}\cdots\delta_{j_qj^\prime_{\tau(q)}}W\!g(\sigma\tau^{-1},d)$$

where Wg is the Weingarten function, given by

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. Here &chi;&lambda; is the character of Sq corresponding to the partition &lambda; and s is the Schur polynomial of &lambda;, so that s&lambda;d(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.

Values of the Weingarten function for simple permutations
The first few Weingarten functions Wg(σ, d) are
 * $$\displaystyle W\!g(,d) = 1$$ (The trivial case where q = 0)
 * $$\displaystyle W\!g(1,d) = \frac{1}{d}$$
 * $$\displaystyle W\!g(2,d) = \frac{-1}{d(d^2-1)}$$
 * $$\displaystyle W\!g(1^2,d) = \frac{1}{d^2-1}$$
 * $$\displaystyle W\!g(3,d) = \frac{2}{d(d^2-1)(d^2-4)}$$
 * $$\displaystyle W\!g(21,d) = \frac{-1}{(d^2-1)(d^2-4)}$$
 * $$\displaystyle W\!g(1^3,d) = \frac{d^2-2}{d(d^2-1)(d^2-4)}$$
 * $$\displaystyle W\!g(4,d) = \frac{-5}{d(d^2-1)(d^2-4)(d^2-9)} $$
 * $$\displaystyle W\!g(31,d) = \frac{2d^2-3}{d^2(d^2-1)(d^2-4)(d^2-9)} $$
 * $$\displaystyle W\!g(2^2,d) = \frac{d^2+6}{d^2(d^2-1)(d^2-4)(d^2-9)} $$
 * $$\displaystyle W\!g(21^2,d) = \frac{-1}{d(d^2-1)(d^2-9)} $$
 * $$\displaystyle W\!g(1^4,d) = \frac{d^4-8d^2+6}{d^2(d^2-1)(d^2-4)(d^2-9)} $$

where permutations σ are denoted by their cycle shapes.

There exist computer algebra programs to produce these expressions.

Explicit expressions for the integrals in the first cases
The explicit expressions for the integrals of first- and second-degree polynomials, obtained via the formula above, are:$$\int_{U_d} dU U_{ij} \bar U_{k\ell} = \delta_{ik}\delta_{j\ell} \operatorname{Wg}(1,d) = \frac{\delta_{ik}\delta_{j\ell}}{d}.$$$$\int_{U_d} dU U_{ij} U_{k\ell} \bar U_{mn} \bar U_{pq}= (\delta_{im}\delta_{jn} \delta_{kp}\delta_{\ell q} + \delta_{ip}\delta_{jq} \delta_{km}\delta_{\ell n} ) \operatorname{Wg}(1^2,d) + (\delta_{im} \delta_{jq} \delta_{kp}\delta_{\ell n} + \delta_{ip} \delta_{jn} \delta_{km}\delta_{\ell q}) \operatorname{Wg}(2,d).$$

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


 * $$W\!g(\sigma,d) = d^{-n-|\sigma|}\prod_i(-1)^{|C_i|-1}c_{|C_i|-1} +O(d^{-n-|\sigma|-2})$$

where the permutation σ is a product of cycles of lengths Ci, and cn = (2n)!/n!(n + 1)! is a Catalan number, and |&sigma;| is the smallest number of transpositions that &sigma; is a product of. There exists a diagrammatic method 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. Their theory is similar to the case of the unitary group. They are parameterized by partitions such that all parts have even size.