Cyclic Sieving, Necklaces, and Branching Rules Related to Thrall's Problem

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 Kraśkiewicz-Weyman, Stembridge, and Schocker related to the so-called higher Lie modules and branching rules for inclusions Ca ≀ S b ֒→ S ab . Extending the approach gives monomial expansions for certain graded Frobenius series arising from a generalization of Thrall's problem. Lemma 1.4. [AS18, Lemma 8.3] Let W be a finite set of length n words closed under the C n -action, where C n acts by cyclic rotations. Then, the triple W, C n , W flex (q) exhibits the CSP.

Higher Lie characters and cyclic descent extension on conjugacy classes

“…The best result so far is Schocker's expansion [29,Theorem 3.1], which however involves signs and rational coefficients. For recent discussions see, e.g., [25,5,36].…”