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Weingarten function
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<h2 id="unitary-groups">Unitary groups</h2> <p>Weingarten functions are used for evaluating integrals over the <a href="/facts/Unitary_group/lEvudIhU">unitary group</a> <i>U</i><i>d</i> of products of matrix coefficients of the form </p> ∫ U d U i 1 j 1 ⋯ U i q j q U i 1 ′ j 1 ′ ∗ ⋯ U i q ′ j q ′ ∗ d U , {\displaystyle \int _{U_{d}}U_{i_{1}j_{1}}\cdots U_{i_{q}j_{q}}U_{i_{1}^{\prime }j_{1}^{\prime }}^{*}\cdots U_{i_{q}^{\prime }j_{q}^{\prime }}^{*}dU,} <p>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 q j q j 1 ′ i 1 ′ … j q ′ i q ′ {\displaystyle i_{1}j_{1}\ldots i_{q}j_{q}j'_{1}i'_{1}\ldots j'_{q}i'_{q}} matrix element of U ⊗ ⋯ ⊗ U ⊗ U † ⊗ ⋯ ⊗ U † {\displaystyle U\otimes \cdots \otimes U\otimes U^{\dagger }\otimes \cdots \otimes U^{\dagger }} . </p><p>This integral is equal to </p> ∑ σ , τ ∈ S q δ i 1 i σ ( 1 ) ′ ⋯ δ i q i σ ( q ) ′ δ j 1 j τ ( 1 ) ′ ⋯ δ j q j τ ( q ) ′ W g ( σ τ − 1 , d ) {\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)} <p>where <i>Wg</i> is the Weingarten function, given by </p> W g ( σ , d ) = 1 q ! 2 ∑ λ χ λ ( 1 ) 2 χ λ ( σ ) s λ , d ( 1 ) {\displaystyle W\!g(\sigma ,d)={\frac {1}{q!^{2}}}\sum _{\lambda }{\frac {\chi ^{\lambda }(1)^{2}\chi ^{\lambda }(\sigma )}{s_{\lambda ,d}(1)}}} <p>where the sum is over all partitions λ of <i>q</i> (Collins 2003). Here χλ is the character of <i>S</i><i>q</i> corresponding to the partition λ and <i>s</i> is the <a href="/facts/Schur_polynomial/mBzF2Mdr">Schur polynomial</a> of λ, so that <i>s</i>λ<i>d</i>(1) is the dimension of the representation of <i>U</i><i>d</i> corresponding to λ. </p><p>The Weingarten functions are rational functions in <i>d</i>. They can have poles for small values of <i>d</i>, 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 <i>d</i> parts. This is no longer a rational function of <i>d</i>, but is finite for all positive integers <i>d</i>. The two sorts of Weingarten functions coincide for <i>d</i> larger than <i>q</i>, and either can be used in the formula for the integral. </p> <h3>Values of the Weingarten function for simple permutations</h3> <p>The first few Weingarten functions <i>Wg</i>(σ, <i>d</i>) are </p> W g ( , d ) = 1 {\displaystyle \displaystyle W\!g(,d)=1} (The trivial case where <i>q</i> = 0) W g ( 1 , d ) = 1 d {\displaystyle \displaystyle W\!g(1,d)={\frac {1}{d}}} W g ( 2 , d ) = − 1 d ( d 2 − 1 ) {\displaystyle \displaystyle W\!g(2,d)={\frac {-1}{d(d^{2}-1)}}} W g ( 1 2 , d ) = 1 d 2 − 1 {\displaystyle \displaystyle W\!g(1^{2},d)={\frac {1}{d^{2}-1}}} W g ( 3 , d ) = 2 d ( d 2 − 1 ) ( d 2 − 4 ) {\displaystyle \displaystyle W\!g(3,d)={\frac {2}{d(d^{2}-1)(d^{2}-4)}}} W g ( 21 , d ) = − 1 ( d 2 − 1 ) ( d 2 − 4 ) {\displaystyle \displaystyle W\!g(21,d)={\frac {-1}{(d^{2}-1)(d^{2}-4)}}} W g ( 1 3 , d ) = d 2 − 2 d ( d 2 − 1 ) ( d 2 − 4 ) {\displaystyle \displaystyle W\!g(1^{3},d)={\frac {d^{2}-2}{d(d^{2}-1)(d^{2}-4)}}} W g ( 4 , d ) = − 5 d ( d 2 − 1 ) ( d 2 − 4 ) ( d 2 − 9 ) {\displaystyle \displaystyle W\!g(4,d)={\frac {-5}{d(d^{2}-1)(d^{2}-4)(d^{2}-9)}}} W g ( 31 , d ) = 2 d 2 − 3 d 2 ( d 2 − 1 ) ( d 2 − 4 ) ( d 2 − 9 ) {\displaystyle \displaystyle W\!g(31,d)={\frac {2d^{2}-3}{d^{2}(d^{2}-1)(d^{2}-4)(d^{2}-9)}}} W g ( 2 2 , d ) = d 2 + 6 d 2 ( d 2 − 1 ) ( d 2 − 4 ) ( d 2 − 9 ) {\displaystyle \displaystyle W\!g(2^{2},d)={\frac {d^{2}+6}{d^{2}(d^{2}-1)(d^{2}-4)(d^{2}-9)}}} W g ( 21 2 , d ) = − 1 d ( d 2 − 1 ) ( d 2 − 9 ) {\displaystyle \displaystyle W\!g(21^{2},d)={\frac {-1}{d(d^{2}-1)(d^{2}-9)}}} W g ( 1 4 , d ) = d 4 − 8 d 2 + 6 d 2 ( d 2 − 1 ) ( d 2 − 4 ) ( d 2 − 9 ) {\displaystyle \displaystyle W\!g(1^{4},d)={\frac {d^{4}-8d^{2}+6}{d^{2}(d^{2}-1)(d^{2}-4)(d^{2}-9)}}} <p>where permutations σ are denoted by their cycle shapes. </p><p>There exist computer algebra programs to produce these expressions.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h3>Explicit expressions for the integrals in the first cases</h3> <p>The explicit expressions for the integrals of first- and second-degree polynomials, obtained via the formula above, are: ∫ 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 . {\displaystyle \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}}.} ∫ 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 ) . {\displaystyle \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).} </p> <h3>Asymptotic behavior</h3> <p>For large <i>d</i>, the Weingarten function <i>Wg</i> has the asymptotic behavior </p> W g ( σ , d ) = d − n − | σ | ∏ i ( − 1 ) | C i | − 1 c | C i | − 1 + O ( d − n − | σ | − 2 ) {\displaystyle W\!g(\sigma ,d)=d^{-n-|\sigma |}\prod _{i}(-1)^{|C_{i}|-1}c_{|C_{i}|-1}+O(d^{-n-|\sigma |-2})} <p>where the permutation σ is a product of cycles of lengths <i>C</i><i>i</i>, and <i>c</i><i>n</i> = (2<i>n</i>)!/<i>n</i>!(<i>n</i> + 1)! is a <a href="/facts/Catalan_number/R9vd4D3V">Catalan number</a>, and |σ| is the smallest number of transpositions that σ is a product of. There exists a diagrammatic method<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> to systematically calculate the integrals over the unitary group as a <a href="/facts/Power_series/vYh6sTps">power series</a> in <i>1/d</i>. </p> <h2 id="orthogonal-and-symplectic-groups">Orthogonal and symplectic groups</h2> <p>For <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal</a> and <a href="/facts/Symplectic_group/G4CKBc7z">symplectic groups</a> 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. </p> <h2 id="external-links">External links</h2> <ul><li>Collins, Benoît (2003), "Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability", <i>International Mathematics Research Notices</i>, 2003 (17): 953–982, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math-ph/0205010">math-ph/0205010</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1155%2FS107379280320917X">10.1155/S107379280320917X</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1959915">1959915</a></li> <li>Collins, Benoît; Śniady, Piotr (2006), "Integration with respect to the Haar measure on unitary, orthogonal and symplectic group", <i>Communications in Mathematical Physics</i>, 264 (3): 773–795, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math-ph/0402073">math-ph/0402073</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2006CMaPh.264..773C">2006CMaPh.264..773C</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00220-006-1554-3">10.1007/s00220-006-1554-3</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2217291">2217291</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:16122807">16122807</a></li> <li>Weingarten, Don (1978), "Asymptotic behavior of group integrals in the limit of infinite rank", <i><a href="/facts/Journal_of_Mathematical_Physics/hfGFBt5M">Journal of Mathematical Physics</a></i>, 19 (5): 999–1001, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1978JMP....19..999W">1978JMP....19..999W</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1063%2F1.523807">10.1063/1.523807</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0471696">0471696</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Z. Puchała and J.A. Miszczak, Symbolic integration with respect to the Haar measure on the unitary group in Mathematica., arXiv:1109.4244 (2011). <a href="https://arxiv.org/abs/1109.4244" target="_blank">https://arxiv.org/abs/1109.4244</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>M. Fukuda, R. König, and I. Nechita, RTNI - A symbolic integrator for Haar-random tensor networks., arXiv:1902.08539 (2019). <a href="https://arxiv.org/abs/1902.08539" target="_blank">https://arxiv.org/abs/1902.08539</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="https://arxiv.org/abs/cond-mat/9604059" target="_blank">https://arxiv.org/abs/cond-mat/9604059</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>