Kerov’s Central Limit Theorem for the Plancherel Measure on Young Diagrams
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In this article, a reconstruction of Kerov's proof of Gaussian fluctuations around the limit shape of a curve is presented, largely based on the unpublished notes of the original proof.Abstract:
Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure M n. That is, the weight M n(λ) of a diagram λ equals dim2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (Logan-Shepp 1977, Vershik-Kerov 1977). In 1993, Kerov announced a remarkable theorem describing Gaussian fluctuations around the limit shape Ω. Here we propose a reconstruction of his proof. It is largely based on Kerov’s unpublished work notes, 1999read more
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References
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Symmetric functions and Hall polynomials
TL;DR: In this paper, the characters of GLn over a finite field and the Hecke ring of GLs over finite fields have been investigated and shown to be symmetric functions with two parameters.