Publication Date

2016-06-18

Availability

Open access

Embargo Period

2016-06-18

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PHD)

Department

Mathematics (Arts and Sciences)

Date of Defense

2016-05-10

First Committee Member

Drew Armstrong

Second Committee Member

Ken Baker

Third Committee Member

Alexander Dvorsky

Fourth Committee Member

Victor Milenkovic

Abstract

The idea of the lattice of non-crossing partitions, NC(n), is inspired by early work of Kreweras. In this thesis we study the action of dihedral group D_2n on NC(n), especially the sublattice NC(n)^F in which all the elements are fixed by a reflection F, and then we extend our work to the characters of the dihedral group acting on NC(n). We start from enumerative properties of the lattice NC(n)^F . Next we investigate the recursive structure on the lattice NC(n)^F related to central binomial coefficients and the Catalan numbers. We proceed to look into combinatorial structure of a graded sublattice which is named "the pruned sublattice". Two characters Alpha_S and Beta_S introduced by Stanley of dihedral groups acting on NC(n) are computed with respect to certain rank-selected subposet NC(n)_S in NC(n). We first recall Montenegro's computation of Beta_[n-2] from his unpublished manuscript. Based on the cyclic sieving phenomenon of Reiner, Stanton and White, we obtain a general result for all Alpha_S's, where S is a subset of [n] of size 1 or n-2.

Keywords

non-crossing partition; dihedral group; characters; group action

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