The Gini Index in Algebraic Combinatorics and Representation Theory

TL;DR

This paper introduces a Gini index for integer partitions, explores its combinatorial properties, and links it to symmetric polynomials and representation theory, providing new insights into algebraic combinatorics.

Contribution

It defines a Gini index on partitions, proves related combinatorial identities, and connects it to Kostka-Foulkes polynomials and representation theory.

Findings

Derived an identity for the expected Gini index on partitions

Linked the Gini index to degrees of Kostka-Foulkes polynomials

Extended the Gini index concept to representations of complex reflection groups

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. 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 complex reflection group, or connected reductive linear algebraic group.