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Asymptotics of characters of symmetric groups, genus expansion and free probability. (English) Zbl 1096.20016

The author’s primary goal is to investigate the asymptotic behaviour of a sequence of characters \(\chi_n\) of the symmetric groups \(S_n\) as \(n\to\infty\) on an element \(x\) whose cycles of length \(>1\) are prescribed. It is assumed that the shapes of the Young diagrams to which the characters \(\chi_n\) correspond converge in the transition measure introduced by S. Kerov (see, for example, Interlacing measures, Kirillov’s seminar on representation theory. Providence, RI: AMS. Transl., Ser. 2, Am. Math. Soc. 181(35), 35-83 (1998; Zbl 0890.05074)]).
Under these hypotheses, P. Biane [Adv. Math. 138, No. 1, 126-181 (1998; Zbl 0927.20008)] has shown that the leading term for an asymptotic expression for \(\chi_n(x)\) can be described in terms of a surface of genus \(0\) and can be computed using the free probability theory of D. V. Voiculescu (see [D. V. Voiculescu, K. J. Dykema and A. Nica, Free Random Variables, CRM Monograph Ser. 1. Providence, RI: AMS (1992; Zbl 0795.46049)]). It has been pointed out by A. Okounkov [Int. Math. Res. Not. 2000, No. 20, 1043-1095 (2000; Zbl 1018.15020)] that Biane’s results also provide information about the distribution of the largest eigenvalue in a random matrix in the Gaussian unitary ensemble (GUE).
The present paper is an attempt to simplify some of Biane’s arguments and to refine his results. In particular, the author proves a conjecture of Biane by providing the second term in the asymptotic expansion and a method which, in principal, would permit one to find further terms. More precisely, the author computes the coefficients of the two highest degree terms of the Kerov polynomials.
The transition measure and free probability are related to the algebraic structure through the Jucys-Murphy element \(J\). For any finite set \(A\) and \(*\notin A\), the Jucys-Murphy element is a sum of transpositions in the group algebra \(\mathbb{C}(S_{A\cup\{*\}})\) given by \(J:=\sum_{a\in A}(a,*)\). The \(k\)-th moment of the Jucys-Murphy element is a central element of \(\mathbb{C}(S_A)\) defined by \(M_k^{JM}:=\sum(a_1 ,*)\cdots(a_k,*)\) where the sum is over all \(a_1,\dots,a_k\in A\) for which \((a_1,*)\cdots(a_k,*)\in S_A\).
These moments span the centre of \(\mathbb{C}(S_A)\), and their behaviour under convergence in the transition measure is relatively simple, and so the author turns to the question of how to express the class sums of \(\mathbb{C}(S_A)\) in terms of the moments of \(J\). He does this indirectly. First he associates a class sum with a partition of the underlying ordered set, and then he gives a combinatorial construction of the decomposition into sums of moments of \(J\) via the free cumulants of R. Speicher [Math. Ann. 298, No. 4, 611-628 (1994; Zbl 0791.06010)]. In following the arguments, a reader may find it useful to be able to refer to Biane’s paper mentioned above.

MSC:

20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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References:

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