Date of Award
2025
Document Type
Open Access Dissertation
Degree Name
Doctor of Philosophy in Mathematical Sciences (PhD)
Administrative Home Department
Department of Mathematical Sciences
Advisor 1
David Hemmer
Committee Member 1
William Keith
Committee Member 2
Robert Schneider
Committee Member 3
Ramy El-Ganainy
Abstract
Finding the decomposition numbers for the symmetric group is a difficult problem and has led to many different research directions. For a given Specht module, the Schaper sum formula can be refined with knowledge of the Schaper number for the particular partition to give more precise information about the decomposition numbers. In Chapter 2, we give a combinatorial formula for the Schaper number when the partition has at most two columns. At the end, we provide a conjecture for the Schaper number for the partition of shaper (3^n).
Chapter 3 covers the joint work with my advisor, David Hemmer, and is published in the Ramanujan Journal. There is a well-known bijection between finite binary sequences and integer partitions. Sequences of length r correspond to partitions of perimeter r+1. Motivated by work on rational numbers in the Calkin-Wilf tree, we classify partitions whose corresponding binary sequence is a palindrome. We give a generating function that counts these partitions and describe how to efficiently generate all of them. Atypically for partition generating functions, we find an unusual significance to prime degrees. Specifically, we prove there are nontrivial palindrome partitions of n except when n=3 or n+1 is prime. We find an interesting new ``branching diagram" for partitions, similar to Young's lattice, with an action of the Klein four group corresponding to natural operations on the binary sequences.
Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory. Tewodros Amdeberhan conjectures equalities of symmetric group characters sums over a new set called Ev(\lambda). In Chapter 4, I share my results from investigating an alternating sum of characters for Ev(\lambda) written in terms of the inner product of Schur functions and power sum symmetric functions. I found an equality between the alternating sum of power-sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of symmetric group characters over the set Ev(\lambda) equals 0.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
Recommended Citation
Westrem, Karlee J., "Schaper numbers, palindrome partitions, and symmetric functions, with applications to characters of the symmetric group", Open Access Dissertation, Michigan Technological University, 2025.