Cyclic sieving, necklaces, and branching rules related to Thrall's problem

TL;DR

This paper unifies and extends the understanding of cyclic sieving, necklaces, and branching rules in representation theory, providing new combinatorial and algebraic insights into higher Lie modules and Thrall's problem.

Contribution

It offers a unified, bijective approach to branching rules and Frobenius series related to higher Lie modules, extending Thrall's problem.

Findings

Derived monomial expansions for graded Frobenius series.

Unified approach to cyclic sieving and necklace generating functions.

Extended Thrall's problem to new algebraic contexts.

Abstract

We show that the cyclic sieving phenomenon of Reiner--Stanton--White together with necklace generating functions arising from work of Klyachko offer a remarkably unified, direct, and largely bijective approach to a series of results due to Kraskiewicz--Weyman, Stembridge, and Schocker related to the so-called higher Lie modules and branching rules for inclusions $ C_a \wr S_b \hookrightarrow S_{ab} $. Extending the approach gives monomial expansions for certain graded Frobenius series arising from a generalization of Thrall's problem.