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Representations of symmetric groups and free probability. (English) Zbl 0927.20008
Close connections are established between asymptotic properties of representations of symmetric groups \(S_q\) when \(q\to\infty\) and the combinatorial treatment, due to R. Speicher, or free probability theory. This investigation, inspired by the works of A. M. Vershik and S. V. Kerov [see, e.g. A. M. Vershik in: Proc. Int. Congr. Math. Zürich 1994, Vol. 2, 1384-1394 (1995; Zbl 0843.05003), and especially S. V. Kerov, Funkts. Anal. Prilozh. 27, No. 2, 32-49 (1993; Zbl 0808.05098), Tr. St-Peterbg. Mat. Obshch. 4, 165-192 (1996)], develops some previous results of the author [cf. e.g., Pac. J. Math. 171, No. 2, 373-387 (1995; Zbl 0854.60070), Discrete Math. 175, No. 1-3, 41-53 (1997; Zbl 0892.05006)].
The representations considered mainly correspond to “\(A\)-balanced” Young diagrams \(\lambda_q\), with \(q\) boxes, whose largest column and line do not exceed \(Aq^{1/2}\), \(A>1\). If \(\lambda_q\) (viewed as continuous piecewise linear functions on \(\mathbb{R}\)) upon rescaling by a factor \(q^{-1/2}\) converge to a certain limit shape \(\omega\), then the exact asymptotics of the normalized characters corresponding to \(\lambda_q\) is given in terms of the free cumulants of a unique probability measure \(m_\omega\) (associated with \(\omega\) by equating the generating function of the continuous Young diagram \(\omega\) to the Cauchy transform of compactly supported \(m_\omega\) on \(\mathbb{R}\)). Moreover, specific asymptotic patterns are found to be inherent to operations on the set of equivalence classes of representations, such as tensor product, restriction to a subgroup, and induction (outer product). It is shown that, for large enough initial diagrams, most Young diagrams appearing in the decomposition of the resulting representations are close to a specific shape which can be computed using the methods of free probability theory.

MSC:
20C30 Representations of finite symmetric groups
46L53 Noncommutative probability and statistics
20C32 Representations of infinite symmetric groups
05E10 Combinatorial aspects of representation theory
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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