Séminaire Lotharingien de Combinatoire, 82B.37 (2019), 12 pp.

Connor Ahlbach and Joshua P. Swanson

Thrall's problem: cyclic sieving, necklaces, and branching rules

Abstract. The cyclic sieving phenomenon (CSP) of Reiner-Stanton-White offers an explanation for why many polynomials evaluate to non-negative integers at roots of unity. We reverse the usual paradigm and show how certain cyclic sieving results can be used to give new proofs of Schur expansions due to Kra\'skiewicz-Weyman, Stembridge, and Schocker. These results concern the so-called higher Lie modules and branching rules for inclusions C_a wreath S_b -> S_ab. Extending the approach gives monomial expansions for certain Frobenius series arising from a generalization of Thrall's problem.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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