(1)
where for
(2)
Schur polynomials
are implemeted e.g. in SageMath, and characters
can be obtained either using charater_table() or via Frobenius characteristic map from Schurs.
(3)
for because the Haar measure is invariant under the central ,
(4)
while under this phase shift
(5)
the integrand picks up a factor . Hence for
(6)
References
[CŚ06]
B. Collins and P. Śniady, “Integration with respect to the haar measure on unitary, orthogonal and symplectic group,” Comm. Math. Phys., vol. 264, no. 3, pp. 773–795, 2006, doi: 10.1007/s00220-006-1554-3. arXiv: math-ph/0402073.
[Col03]
B. Collins, “[],” Int. Math. Res. Notices, vol. 2003, no. 17, p. 953, 2003, doi: 10.1155/s107379280320917x.
[Wei78]
D. Weingarten, “Asymptotic behavior of group integrals in the limit of infinite rank,” J. Math. Phys., vol. 19, no. 5, pp. 999–1001, 1978, doi: 10.1063/1.523807.