Control over the KKR bijection with respect to the nesting structure on rigged configurations and a CSP instance involving Motzkin numbers

Abstract

There are two disjoint main projects that this thesis covers. The Motzkin numbers are a sort of “re- laxed version” of the Catalan numbers. For example, Catalan numbers count perfect non-crossing matchings, while Motzkin numbers count not necessarily perfect non-crossing matchings. The first project deals with instances of the cyclic sieving phenomenon involving Motzkin numbers and their standard q−analogue. We also show that the standard q−analogue for Motzkin numbers satisfy a similar generating series interpretation to that of the q−Catalan numbers. The second project deals with understanding the Kerov-Kirillov-Reshetikhin (KKR) bijection between semistandard tableaux and rigged configurations with a particular emphasis on the standard case. In partic- ular, we understand inducing perturbations on the corresponding rigged configuration via direct operations on the tableau. We develop a technique to take a standard tableau T, and output a new standard tableau T′ which has the same corresponding rigged configuration up to a rigging on any desired row of the first rigged partition. We also develop an alternate technique to Kuniba, Okado, Sakamoto, Takagi, and Yamada that “unwraps” the natural nesting structure on rigged configurations. The primary operation from which all the above follows from is “raise” which we introduce and give various combinatorial models for. The operation raise induces a very simple, controlled perturbation on the corresponding rigged configuration. Our results are formulated in terms of paths or (classically) highest weight elements of tensor products of Kirillov-Reshetikhin crystals.