Available at: https://digitalcommons.calpoly.edu/theses/3085
Date of Award
6-2025
Degree Name
MS in Mathematics
Department/Program
Mathematics
College
College of Science and Mathematics
Advisor
Anthony Mendes
Advisor Department
Mathematics
Advisor College
College of Science and Mathematics
Abstract
In this thesis, we explore the relationship between the degrees of irreducible matrix representations of the symmetric group and prime numbers. We start by building up the required background necessary to construct our results. We dive into topics of discrete mathematics including matrix representations, characters of representations, integer partitions, conjugacy classes, class functions, and tableaux. Some other key ingredients in our work include using character tables to classify the irreducible rep- resentations of the symmetric group and the hook-length formula to easily compute the degrees of these representations. Once we have laid the groundwork for our in- vestigation, we begin exploring the relationship between degrees of representations and primes. The bulk of the work in this thesis hinges upon proving the claim that the total number of irreducible representations of the symmetric group of size n with degrees not divisible by a prime p is itself divisible by the same prime p. To do so, we utilize additional topics including abaci, cores and quotients of integer partitions, and facts regarding the factorization of prime numbers.