Cyclic Actions in Combinatorial Invariant Theory

Abstract

The major original contributions of this thesis are as follows: Theorem 3.3.1 and Proposition 3.3.3 together show that a natural q-analogue of the rational Schr\"oder polynomial is (separately) unimodal in both its even and odd coefficient sequences. Theorem 4.1.2 which, for certain parameters, defines an elementary (WxC)-action on the classical parking space for a Weyl group. When this action is defined, it agrees with the more technical algebraic construction of Armstrong, Reiner, and Rhoades. Theorem 5.1.3 is a general cyclic sieving result which in particular recovers the q=-1 phenomenon for Catalan necklaces, as well as higher-order sieving for a more general family of necklaces.