Weingarten function explained

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 U_d of products of matrix coefficients of the form

\int _{U _{d}} U _{i _{1j} _{1}} … U _{i _{qj} _{q}} U ^{*} _{i ^{\prime} _{1}} … U ^{*} _{i ^{\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 ⊗ … ⊗ U ⊗ U ^{\dagger ⊗ … ⊗} U ^{\dagger} .

This integral is equal to

\sum _{\sigma,\tau\in S _{q}} \delta _{i ^{\prime} _{\sigma( 1 )}} … \delta _{i ^{\prime} _{\sigma(q)}} \delta _{j ^{\prime} _{\tau( 1 )}} … \delta _{j ^{\prime} _{\tau(q)}} Wg(\sigma\tau ^{-1} ,d) where Wg is the Weingarten function, given byWg(\sigma,d) = (1)/(q! ^{2}) \sum _{λ} (\chi ^{λ(1)} ^{2\chi} ^{λ(\sigma)})/(s _{λ,d} ( 1 )) where the sum is over all partitions λ of q . Here χ^λ is the character of 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.

Values of the Weingarten function for simple permutations

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

\displaystyle Wg(d) = 1 (The trivial case where q = 0)\displaystyle Wg( 1 ,d) = (1)/(d) \displaystyle Wg( 2 ,d) = (-1)/(d(d ^{2-1)}) \displaystyle Wg( 1 ^{2,d)} = (1)/(d ^{2-1}) \displaystyle Wg( 3 ,d) = (2)/(d(d ^{2-1)(d} ^{2-4)}) \displaystyle Wg( 21 ,d) = (-1)/((d ^{2-1)(d} ^{2-4)}) \displaystyle Wg( 1 ^{3,d)} = (d ^{2-2})/(d(d ^{2-1)(d} ^{2-4)}) \displaystyle Wg( 4 ,d) = (-5)/(d(d ^{2-1)(d} ^{2-4)(d} ^{2-9)}) \displaystyle Wg( 31 ,d) = (2 d ^{2-3})/(d ^{2(d} ^{2-1)(d} ^{2-4)(d} ^{2-9)}) \displaystyle Wg( 2 ^{2,d)} = (d ^{2+6})/(d ^{2(d} ^{2-1)(d} ^{2-4)(d} ^{2-9)}) \displaystyle Wg( 21 ^{2,d)} = (-1)/(d(d ^{2-1)(d} ^{2-9)}) \displaystyle Wg( 1 ^{4,d)} = (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.^{[1] [2]}

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:$$\backslash int\_\; dU\; U\_\; \backslash bar\; U\_=\; \backslash delta\_\backslash delta\_\; \backslash operatorname(1,d)=\; \backslash frac.$$$$\backslash int\_\; dU\; U\_\; U\_\; \backslash bar\; U\_\backslash bar\; U\_=\; (\backslash delta\_\backslash delta\_\; \backslash delta\_\backslash delta\_\; +\; \backslash delta\_\backslash delta\_\; \backslash delta\_\backslash delta\_)\; \backslash operatorname(1^2,d)\; +(\backslash delta\_\; \backslash delta\_\; \backslash delta\_\backslash delta\_+\; \backslash delta\_\; \backslash delta\_\; \backslash delta\_\backslash delta\_)\; \backslash operatorname(2,d).$$

Asymptotic behavior

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

Wg(\sigma,d) = d ^{-n-|\sigma|} \prod ^{|C _{i|-1}} _{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| 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 . Their theory is similar to the case of the unitary group. They are parameterized by partitions such that all parts have even size.

References

1. Z. Puchała and J.A. Miszczak, Symbolic integration with respect to the Haar measure on the unitary group in Mathematica., arXiv:1109.4244 (2011).
2. M. Fukuda, R. König, and I. Nechita, RTNI - A symbolic integrator for Haar-random tensor networks., arXiv:1902.08539 (2019).
3. P.W. Brouwer and C.W.J. Beenakker, Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems, J. Math. Phys. 37, 4904 (1996), arXiv:cond-mat/9604059.