论文标题
代数组合和表示理论中的Gini指数
The Gini Index in Algebraic Combinatorics and Representation Theory
论文作者
论文摘要
Gini指数是一个试图衡量资源在整个人群中分配的数字，并且通常在经济学中用作衡量财富或收入不平等的衡量。 Gini指数通常定义为分布的“ Lorenz曲线”与平等线之间的区域，归一化为零和一个之间。以这种方式，我们将在整个整数分区中定义一个Gini索引，并证明与之相关的一些组合结果；最终在GINI指数的预期值的身份证明中达到最终形式。然后，我们将讨论对称多项式，并表明Gini指数可以理解为某些Kostka-Foulkes多项式的程度。这种识别产生了概括，我们可以在复杂反射组的不可还原表示或连接的还原线性代数组上定义GINI指数。
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.
