The Gini index in the representation theory of the general linear group

The Gini index is a function that attempts to measure the amount of inequality in the distribution of a finite resource throughout a population. It is commonly used in economics as a measure of inequality of income or wealth. We define a discrete Gini index on the set of integer partitions with at most $n$ parts and show how this function emerges in the representation theory of the complex general linear group.Comment: 13 pages, 4 figure

The Gini Index in Algebraic Combinatorics and Representation Theory

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

Self-Similarity of the 11-Regular Partition Function

The partition function counts the number of ways a positive integer can be written as the sum of a non-increasing sequence of positive integers. These sums are known as partitions. The famous mathematician Srinivasa Ramanujan proved the partition function has beautiful divisibility properties. We will consider the k-regular partition function, which counts the partitions where no part is divisible by k. Results on the arithmetic of k-regular partition functions have been proven by several authors. In this paper we establish self-similarity results for the 11-regular partition function

Oshkosh Scholar, Volume XII

Oshkosh Scholar, the UW Oshkosh undergraduate research journal, is a collaborative effort between students and faculty at UW Oshkosh. Students are involved in editing and production coordination, and faculty review articles through a blind review process.Welcome to Volume XII of Oshkosh Scholar, the undergraduate research journal of the University of Wisconsin Oshkosh. In the following pages students from mathematics, music, economics, and history present original research and engage important scholarly debates. A collaborative effort between students and faculty, Oshkosh Scholar aims to follow the process of academic journals as much as possible. All submissions go through a double-blind review before the student authors and their faculty mentors have the opportunity to revise and resubmit the articles. An interdisciplinary committee of faculty then selects the strongest papers for publication. Student editors are involved in the editing and production process. A student artist designs the cover. By publishing the best research and writing of our undergraduate students, Oshkosh Scholar celebrates the centrality of intellectual curiosity and the search for truth to the university’s mission. The six articles selected for publication in this volume demonstrate both the quality and the diversity of student research on campus. Caitlin Kirchner, from the Department of Music, introduces us to the rich history of traditional Appalachian folk ballads and the trope of the “murdered girl” ballad in particular. Focusing on the song “Shotgun Blues” by Abigail Washburn, Caitlin demonstrates how a modern artist has sought to invert the traditional gender stereotypes of the ballads, and make a statement against misogyny through a reinterpretation of the tradition. From the Department of Mathematics, Grant Kopitzke presents exciting new results on partition function research. Building on the work published in a previous volume of Oshkosh Scholar by UW Oshkosh alumnus Eric Boll, Grant identifies for the first time ever the self-similarity of the 11-regular partition function. This paper makes a truly original contribution to the field of partition function theory. Christopher Smith, from the Economics Department, also presents highly original research into the feasibility of using virtual economies as a tool for studying economic theory. Using the video game EVE: Online, one of the largest virtual economies in the world, he argues that the game’s economy does function according to principles of microeconomic theory and can be used to predict and measure microeconomic behaviors. The second set of articles comes from three history majors, who highlight the value of both historiographical questions and original archival research. Christopher Gauger re-engages an important historiographical debate in South African history on the relationship between the African National Congress and the South African Communist Party (SACP). Countering recent, revisionist claims about the influence of the SACP in the alliance, he argues that the relationship was strategic rather than ideological. David Wieczorek examines the history of the antiabortion movement in Wisconsin, illustrating aspects of the movement that differed from broader national trends. Finally, William Wasielewski advances our understanding of HIV/AIDS activism in Wisconsin through his research on the BEST Clinic, a Milwaukee clinic that played a critical role in the development of HIV/AIDS prevention programs during the 1980s and ’90s. I invite you to enjoy this excellent collection of undergraduate exploration and discovery. I would like to thank all of the student authors, the faculty mentors, the faculty reviewers, and the Selection Committee who contributed their time and efforts to this volume. Additional thanks go to Managing Editor Susan Surendonk, as well as the student editors, Sydney Nelson and Natalie Dillon. Their hard work and dedication have been essential to making this edition of the journal possible. Oshkosh Scholar is a testament to the intellectual vitality of this university. I am proud to share this edition with you.Office of Student Research and Creative Activit