Averages over classical Lie groups, twisted by characters

Abstract

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 .

Toeplitz minors and specializations of skew Schur polynomials

“…It is well known that a Schur polynomial specialized to two variables is equal to a Chebyshev polynomial [24]. We also obtain from formula (14)…”

“…where e j , e k are elementary symmetric polynomials (2) (we assume in the three last identities that N ≥ 1 and 0 ≤ j, k ≤ N ). Moreover, formula (7) gives the asymptotic behaviour (14) as N → ∞, where the h j are complete homogeneous symmetric polynomials (2) (note that the partitions indexing the sum in (7) are now conjugated). In the following, we recall some known explicit inverses of Toeplitz matrices and compute another two in order to obtain evaluations for this Toeplitz minor.…”

“…Finally, taking into account that only one set of variables in the specialization of f needs to be finite in equations (11)- (14), we can study the principal specialization of the above skew Schur polynomials with an infinite number of variables. To do so, we consider the specialization f (z) = Θ δ (z) = E(1, q −1 , .…”

“…= q (d−1)j−dk+d(d−1)N/2 (1 − q) d(N +1)G q (N + 2)G q (d + 1) G q (d + N + 2) (q; q) d+k (q; q) j N r=max (j,k) q r r r − k q d + r − j − 1 r − j qOnce again, this identity coincides with the one given by the hook-content formula for k = 0. It follows from(14) and the Cauchy identity that as N → ∞s (N,...,N d ,j)/(k) (1, q, . .…”

Joint moments of derivatives of characteristic polynomials

“…For a more thorough discussion of why a similar approach should always be attempted and other examples of its applications, please see the author's thesis and the results in [Dehaye 2007b]. …”