Applications of the Cyclic Sieving Phenomenon to Words, Branching Rules, and Tableaux

Abstract

Since Reiner-Stanton-White defined the cyclic sieving phenomenon associated to a finite cyclic group action and a polynomial, many compelling examples of cyclic sieving have been found. In this thesis, we focus on what we can accomplish using these known cyclic sieving phenomena as tools. We seek to show that the cyclic sieving phenomenon is more than just a coincidence. There is a natural notion of refinement for many CSP's. We formulate and prove a refinement of a famous CSP on words of fixed content by also fixing the \textit{cyclic descent type}. The argument presented is completely different from Reiner-Stanton-White's representation-theoretic approach. Instead, we build up the words of fixed content and cyclic descent type through a branching algorithm and use it to derive major index generating functions. We also prove a refinement of cyclic sieving on shifted subset sums by extending some ideas of Wagon-Wilf. 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 and direct approach to a series of results due to Kra{\'s}kiewicz--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} $. Our arguments are bijective except for the use of the cyclic sieving phenomenon. Extending this approach gives monomial expansions for certain graded Frobenius series arising from a generalization of Thrall's problem. Following a conjecture of Dennis White, Brendon Rhoades proved an interesting cyclic sieving phenomenon for the action of promotion on rectangular standard Young tableaux \cite[Theorem~1.3]{MR2557880}, which gives an effective enumeration formula for the fixed point sizes of the powers on promotion of rectangular Young tableaux. However, currently unknown are the actual fixed point sets counted by Rhoades's formula. In this thesis, we construct all of the sufficiently large rectangular tableaux fixed by powers of promotion. To do so, we introduce tableau stabilization and prove some of its interesting properties. We also present open problems related to tableau stabilization.