Asymptotics of higher Lie characters

2025/09/16 by Adin, Ron M., Roichman, Yuval, Tsilevich, Natalia
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2509.12904

Abstract

Higher Lie characters form a distinguished family of symmetric group characters, which appear in many areas of algebra and combinatorics. An old open problem of Thrall is to decompose them into irreducibles. We propose a novel asymptotic approach to this problem, showing that many families of higher Lie characters tend to be proportional to the regular character. In particular, a random higher Lie character tends, in probability, to be proportional to the regular character.

Citations

• Super major index and Thrall's problem
• Koszulity, supersolvability, and Stirling representations
• Invariant theory for the face algebra of the braid arrangement
• What is a combinatorial interpretation?
• Modular Idempotents for the Descent Algebras of Type A and Higher Lie Powers and Modules
• Higher Lie characters and cyclic descent extension on conjugacy classes
• Cyclic sieving, necklaces, and branching rules related to Thrall's\n problem
• Asymptotics for skew standard Young tableaux via bounds for characters
• On the Existence of Tableaux with Given Modular Major Index
• On conjugacy classes of S_n containing all irreducibles
• Character formulas and descents for the hyperoctahedral group
• Spectra of Symmetrized Shuffling Operators

Related

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• The high-dimension limit of characters of compact reductive Lie groups and restrictions on the production of quantum randomness
• Character ratios, representation varieties and random generation of finite groups of Lie type
• Irreducible characters of the symmetric group and exponential growth
• Bounds for asymptotic characters of simple Lie groups