Asymptotics of the maximal and the typical dimensions of isotypic\n components of tensor representations of the symmetric group

2010/08/23 by Sevak Mkrtchyan, Mkrtchyan, Sevak
Mathematics · #05D40 #05E10 #20C30 #60C05 #Advanced Algebra and Geometry #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Probability (math.PR) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1008.3854

openalex publication_date 2010/08/23 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

Vershik and Kerov gave asymptotical bounds for the maximal and the typical\ndimensions of irreducible representations of symmetric groups S_n. It was\nconjectured by G. Olshanski that the maximal and the typical dimensions of the\nisotypic components of tensor representations of the symmetric group admit\nsimilar asymptotical bounds. The main result of this article is the proof of\nthis conjecture. Consider the natural representation of S_n on\n(\ℂ^N)^{\⊗ n}. Its isotypic components are parametrized by Young\ndiagrams with n cells and at most N rows. P. Biane found the limit shape of\nYoung diagrams when n\→\∞, \√(n)/N\→ c. By showing\nthat this limit shape is the unique solution to a variational problem, it is\nproven here, that after scaling, the maximal and the typical dimensions of\nisotypic components lie between positive constants. A new proof of Biane's\nlimit-shape theorem is obtained.\n

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