Averages over classical Lie groups, twisted by characters

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:114

Article

Dehaye, Paul-Olivier

关键词: Random matrices;    classical invariant theory;    Schur-Weyl duality;    symmetric functions;   

DOI  :  10.1016/j.jcta.2007.01.008

来源: Elsevier

【 摘 要 】

We compute E-G(Pi(i)tr(g(lambda i))), where g is an element of G = Sp(2n) or SO(m) (m = 2n, 2n + 1) with Haar measure. This was first obtained by Diaconis and Shahshahani [Persi Diaconis, Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994) 49-62. Studies in applied probability], but our proof is more self-contained and gives a combinatorial description for the answer. We also consider how averages of general symmetric functions E-G Phi(n) are affected when we introduce a character chi(G)(lambda) into the integrand. We show that the value of E-G chi(G)(lambda)Phi(n)/E-G Phi(n) approaches a constant for large n. More surprisingly, the ratio we obtain only changes with Phi(n) and lambda and is independent of the Cartan type of G. Even in the unitary case, Bump and Diaconis [Daniel Bump, Persi Diaconis, Toeplitz minors, J. Combin. Theory Ser. A 97 (2) (2002) 252-271. Erratum for the proof of Theorem 4 available at http://sporadic.stanford.edu/bump/coffection.ps and in a third reference in the abstract] have obtained the same ratio. Finally, those ratios can be combined with asymptotics for E-G Phi(n) due to Johansson [Kurt Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (3) (1997) 519-545] and provide asymptotics for E-G chi(G)(lambda)Phi(n). 2007 Elsevier Inc. All rights reserved.

【 授权许可】

Free