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An Exposition on Group Characters

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AN EXPOSITION ON GROUP CHARACTERS THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Mathematical Science in the Graduate School of the Ohio State University By Aaron Margraff, BS Graduate Program in Mathematics The Ohio State University 2014 Master's Examination Committee: Dr. James Cogdell, Advisor Dr. Warren Sinnott c Copyright by Aaron Margraff 2014 ABSTRACT This paper is an educational approach to group characters through examples which introduces the beginner algebraist to representations and characters of finite groups. My hope is that this exploration might help the advanced undergraduate student discover some of the foundational tools of Character Theory. The prerequisite material for this paper includes some elementary Abstract and Linear Algebra. The basic groups used in the examples are intended to excited a student into exploration of groups they understand from their undergraduate studies. Throughout the section of examples there are exercises used to check understanding and give the reader opportunity to explore further. After taking a course in Abstract Algebra one might find that groups are not concrete objects. Groups model actions, rotations, reflections, movements, and permutations. Group representations turn these abstract sets of objects into sets of n × n matrices with real or complex entries, which can be easily handled by a computer for any number of calculations. ii ACKNOWLEDGMENTS The paper that is before you is not only evidence of my hard work and under- standing of characters and representations but also evidence of the patience of many teachers that have poured their time and knowledge into me. It was my honor to work with an advisor as accomplished as Dr. James Cogdell. As much as he instructed me in mathematics he also instructed me with his patience. As a future educator I greatly esteem the self-restraint he showed when I presented him with questions that he had previously answered. His calm approach to education removed the barriers that would otherwise impede learning. Thank you for investing your valuable time into a growing student. I would also like to thank Dr. Warren Sinnott for his willingness to participate as a member on my thesis committee and spending his time to read and review my work. I would be amiss not to pour out my gratitude to my family and church for their support of my schooling and education over the past 20 years of my life. I attribute my position spiritually, mentally, bodily, and financially to your faithfulness in caring for me. You all are truly a gift from God. iii VITA 1989 . Born in Columbus, Ohio 2012 . Bachelor of Science in Education at Youngstown State University 2012-Present . Graduate Teaching Associate, The Ohio State University FIELDS OF STUDY Major Field: Mathematics Specialization: Education iv TABLE OF CONTENTS Abstract . ii Acknowledgments . iii Vita......................................... iv CHAPTER PAGE 1 Introduction . .1 1.1 Introductory Definitions and Facts for Representations and Char- acters of Finite Groups . .1 1.2 Helpful Facts for Finding Representations . .4 2 Examples . .8 2.1 Characters of Abelian Groups . .8 2.2 Characters of Non-Abelian Groups . 16 3 Proofs . 32 Bibliography . 52 v CHAPTER 1 INTRODUCTION 1.1 Introductory Definitions and Facts for Representations and Characters of Finite Groups The structure of this paper is modeled after the method in which I was taught to find group characters. I was given the definitions and major results of representations and characters, taking them as truth, and used them to discover the character tables of various groups. In formal mathematical developments it is unsettling to use a result before the result is proven to be true, so this method is perhaps an educational approach to character theory. I begin this paper with an introduction to definitions and results I used in my discovery process. In the second chapter I use the definitions and results to derive the character tables for some abelian and non-abelian groups. In the last chapter of the paper I give formal definitions and prove the results given in this section. Let's look at some of the basic definitions and results to begin the study. Words in italics indicate a new definition. A representation of a group G is a homomorphism ρ : G ! GL(V ) where V is a vector space over the field C and ρ(g) is the image of an element g. No- tice each element ρ(g) is a element of the group of invertible matrices. Also, since ρ is 1 a homomorphism these matrices preserve the group action, that is ρ(gh) = ρ(g)·ρ(h) [1]. Many groups have a geometric interpretation and have representations that reflect their geometric meaning. One can find such a representation by choosing a suitable basis for the vector space for which the geometric interpretation is understood. The dimension of the representation is given by the dimension of the vector space chosen, V . In some sense it would be nice to know when a representation captures the com- plete behavior of a group, which leads us to defining the next term. A representation is called faithful when it maps G isomorphically to its image. Every finite group has a finite number of irreducible representations. These are the building blocks for all other representations of the group. For this section of the paper irreducible representation will mean a representation that cannot be written as the direct sum of two or more representations. In Chapter 3 we will develop a more rigorous definition of irreducible representations. Also in Chapter 3 we find that the number of irreducible representations is the number of conjugacy classes of the group, hence a finite number of them. It is natural at this point to ask, what are the dimensions of the irreducible representations? With the number of irreducible representations known one can specify the dimension of the irreducible representations by using the property that the sum of the squares of the dimensions of the irreducible representations is equal to the order of the group. This result is written as, 2 2 2 jGj = d1 + d2 + ··· + dr where di is the dimension of the irreducible representation ρi and r is the number of conjugacy classes of G. Let G be a group and ρ : G ! GL(V ) be a representation on a vector space V over the field C. The character of ρ is the mapping χρ : G ! C 2 defined by χρ(g) = tr ρ(g). The degree of the character is the dimension of its cor- responding representation. In general the trace is not a homomorphism, hence the character is not always a homomorphism either. The character χo assumes the value of 1 at each element of G, this is called the trivial character. Sometimes the trivial character is written as 1, 1G or simply as 1. The mathematical term \character" was first introduced by Carl Friedrich Gauss in Disquisitiones Arithmeticae. He used characters to assign numerical information to classes of binary quadratic forms, in order to separate classes of forms with the same determinant [3]. The characters in representation theory are associated with a respective representation where χρ(g) is the trace of ρ(g). An irreducible character is the character of an irreducible represen- tation. Hence there are r irreducible characters and representations. Later we will find that characters are class functions, which means they are constant over conjugacy classes, −1 χρ(hgh ) = χρ(g): A character χ can be thought of as vectors where elements of a group G are listed as G = fg1; g2; : : : ; gng and the vector representation of χ is t χ = (χ(g1);:::; χ(gn)) ; where t denotes the transpose and n is the order of the group [1]. Since characters are constant over conjugacy classes it is natural to list the group elements according to their conjugacy class. The most valuable result in relating group characters to vectors is found by constructing a hermitian dot product as: 1 X hχ; χ0i = χ(g)χ0(g): jGj g2G With this inner product we have that the irreducible characters are orthonormal. This fact is extremely valuable when one needs to determine the reducibility of a 3 representation. If taking the inner product of the character of the representation with itself results in unity then the character came from an irreducible representation, if not the representation is reducible. If we let χ be the trivial character and χ0 is any non-trivial irreducible character then X χ0(g) = 0: g2G This fact is useful when finding the rows in a character table. A character table of a group is a two dimensional table whose rows correspond to the irreducible characters and columns correspond to conjugacy classes. The sum of the rows of the table weighted by the size of the conjugacy class is zero. A similar statement about the columns can be found in Theorem 3.0.16. Its result is that, X deg(χ0) · χ0(g) = 0; for g 6= 1: χ0 irreducible Another way that the above relation can be stated is that the sum of the irreducible characters, weighted by their dimension, evaluated at a non-trivial conjugacy class is zero. If you sum the irreducible characters, weighted by their dimensions, over the trivial conjugacy class you will receive the order of the group. 1.2 Helpful Facts for Finding Representations Finding representations of groups can be difficult, and sometimes not necessary to complete the character table. But when it is helpful to find the representation there are some standard methods which can be used to isolate potential representations. One-dimensional representations can be found by using parity, the tensor product of two other one-dimensional representations, or perhaps the determinant of a higher dimension representation.
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