We present a simple technique to compute moments of derivatives of unitary characteristic polynomials. The first part of the technique relies on an idea of Bump and Gamburd: it uses orthonormality of Schur functions over unitary groups to compute matrix averages of characteristic polynomials. In order to consider derivatives of those polynomials, we here need the added strength of the Generalized Binomial Theorem of Okounkov and Olshanski. This result is very natural as it provides coefficients for the Taylor expansions of Schur functions, in terms of shifted Schur functions. The answer is finally given as a sum over partitions of functions of the contents. One can also obtain alternative expressions involving hypergeometric functions of matrix arguments.Résumé. Nous introduisons une nouvelle technique, en deux parties, pour calculer les moments de dérivées de polynômes caractéristiques. La première étape repose sur une idée de Bump et Gamburd et utilise l'orthonormalité des fonctions de Schur sur les groupes unitaires pour calculer des moyennes de polynômes caractéristiques de matrices aléatoires. La deuxième étape, qui est nécessaire pour passer aux dérivées, utilise une généralisation du théorème binomial due à Okounkov et Olshanski. Ce théorème livre les coefficients des séries de Taylor pour les fonctions de Schur sous la forme de "shifted Schur functions". La réponse finale est donnée sous forme de somme sur les partitions de fonctions des contenus. Nous obtenons aussi d'autres expressions en terme de fonctions hypergéométriques d'argument matriciel.
The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices (n points). In this paper, we generalize many aspects of this situation. We introduce random shifts of the eigenvalues of the permutation matrices, in two different ways: independently or not for each subset of eigenvalues associated to the same cycle. We also consider vastly more general functions than the characteristic polynomial of a permutation matrix, by first finding an equivalent definition in terms of cycle-type of the permutation. We consider other groups than the symmetric group, for instance the alternating group and other Weyl groups. Finally, we compute some asymptotics results when n tends to infinity. This last result requires additional ideas: it exploits properties of the Feller coupling, which gives asymptotics for the lengths of cycles in permutations of many points.
We compute E G ( i tr(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 G Φ n are affected when we introduce a character χ G λ into the integrand. We show that the value of E G χ G λ Φ n /E G Φ 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 G Φ 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 χ G λ Φ n .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.