#### Date of Award

May 2021

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Jeb F Willenbring

#### Committee Members

Allen D Bell, Kevin B McLeod, Boris L Okun, Yi Ming Zou

#### Keywords

Algebra, Combinatorics, Generating Functions, Gini index, Lorenz curve, Representation Theory

#### Abstract

The Gini index is a number that attempts to measure how equitably a resource is distributed throughout a population, and is commonly used in economics as a measurement of inequality of wealth or income. The Gini index is often defined as the area between the "Lorenz curve" of a distribution and the line of equality, normalized to be between zero and one. In this fashion, we will define a Gini index on the set of integer partitions and prove some combinatorial results related to it; culminating in the proof of an identity for the expected value of the Gini index. These results comprise the principle contributions of the author. We will then discuss symmetric polynomials, and show that the Gini index can be understood as the degrees of certain Kostka-foulkes polynomials. This identification yields a generalization whereby we may define a Gini index on the irreducible representations of a finite group generated by reflections, or a connected reductive linear algebraic group.

#### Recommended Citation

Kopitzke, Grant Joseph, "The Gini Index in Algebraic Combinatorics and Representation Theory" (2021). *Theses and Dissertations*. 2680.

https://dc.uwm.edu/etd/2680