Averages over classical Lie groups, twisted by characters

Journal article

We compute E_G(∏_itr(g^λ_i)), where g ∈ 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_GΦ_n are affected when we introduce a character χ^G_λ into the integrand. We show that the value of E_Gχ^G_λΦ_n/E_GΦ_n approaches a constant for large n. More surprisingly, the ratio we obtain only changes with Φ_n and λ 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/correction.ps and in a third reference in the abstract] have obtained the same ratio. Finally, those ratios can be combined with asymptotics for E_GΦ_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_Gχ^G_λΦ_n. © 2007 Elsevier Inc. All rights reserved.

Authors: Dehaye, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Journal of Combinatorial Theory, Series A
• Volume: 114
• Issue: 7
• Pages: 1278-1292
• Publication date: 2007-10-01
• DOI: 10.1016/j.jcta.2007.01.008
• EISSN: 1096-0899
• ISSN: 0097-3165
• Language: English
• Keywords: Random matrices; classical invariant theory; symmetric functions; Schur-Weyl duality