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. Consider in general an integral of the form ∫ U d U i 1 j 1 ⋯ U i n j n U i 1 ′ j 1 ′ ∗ ⋯ U i n ′ ′ j n ′ ′ ∗ d U , {\displaystyle \int _{U_{d}}U_{i_{1}j_{1}}\cdots U_{i_{n}j_{n}}U_{i_{1}^{\prime }j_{1}^{\prime }}^{*}\cdots U_{i_{n'}^{\prime }j_{n'}^{\prime }}^{*}dU,} It can be shown that the integral would be zero, unless n = n ′ {\displaystyle n=n'} . Thus, consider only integrals of the form
∫ U d U i 1 j 1 ⋯ U i n j n U i 1 ′ j 1 ′ ∗ ⋯ U i n ′ j n ′ ∗ d U , {\displaystyle \int _{U_{d}}U_{i_{1}j_{1}}\cdots U_{i_{n}j_{n}}U_{i_{1}^{\prime }j_{1}^{\prime }}^{*}\cdots U_{i_{n}^{\prime }j_{n}^{\prime }}^{*}dU,}
where ∗ {\displaystyle *} denotes complex conjugation. Note that U j i ∗ = ( U † ) i j {\displaystyle U_{ji}^{*}=(U^{\dagger })_{ij}} where U † {\displaystyle U^{\dagger }} is the conjugate transpose of U {\displaystyle U} , so one can interpret the above expression as being for the i 1 j 1 … i n j n j 1 ′ i 1 ′ … j n ′ i n ′ {\displaystyle i_{1}j_{1}\ldots i_{n}j_{n}j'_{1}i'_{1}\ldots j'_{n}i'_{n}} matrix element of U ⊗ ⋯ ⊗ U ⊗ U † ⊗ ⋯ ⊗ U † {\displaystyle U\otimes \cdots \otimes U\otimes U^{\dagger }\otimes \cdots \otimes U^{\dagger }} . This integral is equal to ∑ σ , τ ∈ S n δ i 1 i σ ( 1 ) ′ ⋯ δ i n i σ ( n ) ′ δ j 1 j τ ( 1 ) ′ ⋯ δ j n j τ ( n ) ′ W σ , τ ( d ) {\displaystyle \sum _{\sigma ,\tau \in S_{n}}\delta _{i_{1}i_{\sigma (1)}^{\prime }}\cdots \delta _{i_{n}i_{\sigma (n)}^{\prime }}\delta _{j_{1}j_{\tau (1)}^{\prime }}\cdots \delta _{j_{n}j_{\tau (n)}^{\prime }}W_{\sigma ,\tau }(d)} where W σ , τ ( d ) = Wg ( σ τ − 1 , d ) {\displaystyle W_{\sigma ,\tau }(d)=\operatorname {Wg} (\sigma \tau ^{-1},d)} , and Wg {\displaystyle \operatorname {Wg} } is the Weingarten function, given by
Wg ( σ , d ) = 1 n ! 2 ∑ λ χ λ ( 1 ) 2 χ λ ( σ ) s λ , d ( 1 ) {\displaystyle \operatorname {Wg} (\sigma ,d)={\frac {1}{n!^{2}}}\sum _{\lambda }{\frac {\chi ^{\lambda }(1)^{2}\chi ^{\lambda }(\sigma )}{s_{\lambda ,d}(1)}}}
where the sum is over all partitions λ of n (Collins 2003). Here χλ is the character of Sn corresponding to the partition λ and s is the Schur polynomial of λ, so that sλ,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 n, and either can be used in the formula for the integral. The integrals are called the link integrals in U ( N ) {\displaystyle U(N)} lattice gauge theory. See for a graphical method for evaluating these integrals, inspired by quantum string diagrams.
Examples of Weingarten functions The first few Weingarten functions Wg ( , d ) = 1 Wg ( 1 , d ) = 1 d Wg ( 2 , d ) = − 1 d ( d 2 − 1 ) Wg ( 1 2 , d ) = 1 d 2 − 1 Wg ( 3 , d ) = 2 d ( d 2 − 1 ) ( d 2 − 4 ) Wg ( 21 , d ) = − 1 ( d 2 − 1 ) ( d 2 − 4 ) Wg ( 1 3 , d ) = d 2 − 2 d ( d 2 − 1 ) ( d 2 − 4 ) Wg ( 4 , d ) = − 5 d ( d 2 − 1 ) ( d 2 − 4 ) ( d 2 − 9 ) Wg ( 31 , d ) = 2 d 2 − 3 d 2 ( d 2 − 1 ) ( d 2 − 4 ) ( d 2 − 9 ) Wg ( 2 2 , d ) = d 2 + 6 d 2 ( d 2 − 1 ) ( d 2 − 4 ) ( d 2 − 9 ) Wg ( 21 2 , d ) = − 1 d ( d 2 − 1 ) ( d 2 − 9 ) Wg ( 1 4 , d ) = d 4 − 8 d 2 + 6 d 2 ( d 2 − 1 ) ( d 2 − 4 ) ( d 2 − 9 ) {\displaystyle {\begin{aligned}\operatorname {Wg} (,d)&=1\\\operatorname {Wg} (1,d)&={\frac {1}{d}}\\\operatorname {Wg} (2,d)&={\frac {-1}{d(d^{2}-1)}}\\\operatorname {Wg} (1^{2},d)&={\frac {1}{d^{2}-1}}\\\operatorname {Wg} (3,d)&={\frac {2}{d(d^{2}-1)(d^{2}-4)}}\\\operatorname {Wg} (21,d)&={\frac {-1}{(d^{2}-1)(d^{2}-4)}}\\\operatorname {Wg} (1^{3},d)&={\frac {d^{2}-2}{d(d^{2}-1)(d^{2}-4)}}\\\operatorname {Wg} (4,d)&={\frac {-5}{d(d^{2}-1)(d^{2}-4)(d^{2}-9)}}\\\operatorname {Wg} (31,d)&={\frac {2d^{2}-3}{d^{2}(d^{2}-1)(d^{2}-4)(d^{2}-9)}}\\\operatorname {Wg} (2^{2},d)&={\frac {d^{2}+6}{d^{2}(d^{2}-1)(d^{2}-4)(d^{2}-9)}}\\\operatorname {Wg} (21^{2},d)&={\frac {-1}{d(d^{2}-1)(d^{2}-9)}}\\\operatorname {Wg} (1^{4},d)&={\frac {d^{4}-8d^{2}+6}{d^{2}(d^{2}-1)(d^{2}-4)(d^{2}-9)}}\end{aligned}}}
where permutations σ are denoted by their cycle shapes. For example, in Wg ( , d ) = 1 {\displaystyle \operatorname {Wg} (,d)=1} , the permutation is the trivial permutation on 0 elements. In Wg ( 21 2 , d ) = − 1 d ( d 2 − 1 ) ( d 2 − 9 ) {\displaystyle \operatorname {Wg} (21^{2},d)={\frac {-1}{d(d^{2}-1)(d^{2}-9)}}} , the permutation is the permutation on 4 elements that preserves 2 elements, and exchanges 2 elements. There exist computer algebra programs to produce these expressions.
Examples of link integrals
∫ U d d U U i j U ¯ k ℓ = δ i k δ j ℓ Wg ( 1 , d ) = δ i k δ j ℓ d ∫ U d d U U i j U k ℓ U ¯ m n U ¯ p q = ( δ i m δ j n δ k p δ ℓ q + δ i p δ j q δ k m δ ℓ n ) Wg ( 1 2 , d ) + ( δ i m δ j q δ k p δ ℓ n + δ i p δ j n δ k m δ ℓ q ) Wg ( 2 , d ) ∫ U d d U U 11 U 22 U 33 ¯ U 12 U 23 U 31 = Wg ( 3 , d ) = 2 d ( d 2 − 1 ) ( d 2 − 4 ) {\displaystyle {\begin{aligned}\int _{U_{d}}dUU_{ij}{\bar {U}}_{k\ell }&=\delta _{ik}\delta _{j\ell }\operatorname {Wg} (1,d)={\frac {\delta _{ik}\delta _{j\ell }}{d}}\\\int _{U_{d}}dUU_{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)\\\int _{U_{d}}dU\;{\overline {U_{11}U_{22}U_{33}}}U_{12}U_{23}U_{31}&=\operatorname {Wg} (3,d)={\frac {2}{d(d^{2}-1)(d^{2}-4)}}\end{aligned}}}
Asymptotics Given a permutation σ {\displaystyle \sigma } , there is a trivial part and a nontrivial part. The trivial part fixes elements, and the nontrivial part permutes the elements in cycles. This can be written out in the cycle notation for permutations. For example, the following permutation ( 123 ) ( 45 ) ( 6 ) ( 7 ) ( 8 ) {\displaystyle (123)(45)(6)(7)(8)} has the nontrivial part ( 123 ) ( 45 ) {\displaystyle (123)(45)} . Given a fixed permutation σ {\displaystyle \sigma } on n {\displaystyle n} elements, as the size of the unitary group grows to d → ∞ {\displaystyle d\to \infty } , we have the asymptotic formula Wg ( σ , d ) = d − n − | σ | [ ∏ C i ∈ σ ( − 1 ) | C i | − 1 c | C i | − 1 + O ( d − 2 ) ] {\displaystyle \operatorname {Wg} (\sigma ,d)=d^{-n-|\sigma |}\left[\prod _{C_{i}\in \sigma }(-1)^{|C_{i}|-1}c_{|C_{i}|-1}+O(d^{-2})\right]} where C 1 , C 2 , … {\displaystyle C_{1},C_{2},\dots } are the cycles of σ {\displaystyle \sigma } , | C i | {\displaystyle |C_{i}|} is the length of cycle C i {\displaystyle C_{i}} , c n := ( 2 n ) ! n ! ( n + 1 ) ! {\displaystyle c_{n}:={\frac {(2n)!}{n!(n+1)!}}} is a Catalan number, and | σ | {\displaystyle |\sigma |} is the smallest number of transpositions (pairwise exchange) that σ {\displaystyle \sigma } is composed of. This formula can be used to prove that the algebra of the gaussian unitary ensemble converges to the free probability algebra. Higher order expansions exist, of the form d − n − | σ | ( a 0 + a 2 d − 2 + a 4 d − 4 + ⋯ ) {\displaystyle d^{-n-|\sigma |}(a_{0}+a_{2}d^{-2}+a_{4}d^{-4}+\cdots )} where a 0 {\displaystyle a_{0}} is given above, and a 2 , a 4 , … {\displaystyle a_{2},a_{4},\dots } are real numbers that depend on σ {\displaystyle \sigma } but not d {\displaystyle d} . The full expansion is: W ρ σ ( d ) = ( − 1 ) | ρ − 1 σ | d n + | ρ − 1 σ |