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Polynomial Representations of the General Linear Group

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Polynomial Representations of the General Linear Group Imperial College London MSci Thesis Polynomial Representations of the General Linear Group Author: Supervisor: Misja F.A. Steinmetz Dr. John R. Britnell A thesis submitted in fulfilment of the requirements for the degree of Master in Science in the Algebra Section Department of Mathematics \This is my own work unless otherwise stated." Name: Date: June 2014 Misja F.A. Steinmetz CID: 00643423 KNSM-laan 50 1019 LL Amsterdam the Netherlands [email protected] [email protected] i Abstract The main goal of this project will be to describe and classify all irreducible characters of polynomial representations of GLn(K) for an infinite field K. We achieve this goal in Theorem 4.5.1 at the end of Chapter 4. Our journey towards this big theorem takes us past many interesting topics in algebra and representation theory. In Chapter 1 we will do some of the necessary groundwork: we will introduce the concepts of coalgebras and bialgebras. In Chapter 2 we will introduce finitary functions and coefficient functions (following Green's nomenclature [9]). We will use results from Chapter 1 to deduce some initial consequences from these definitions. In Chapter 3 we will introduce the category MK (n) of finite-dimensional left KGLn(K)-modules which ‘afford’ polynomial representations. This category will be the main object of study in this and the next chapter. Next we introduce the Schur algebra SK (n) and prove that left SK -modules are equivalent to left modules in MK . In Chapter 4 we introduce weights, weight spaces and formal characters. We use these results to prove our big theorem. Finally, in Chapter 5 we will look at the rather long and explicit example of the ir- reducible characters of GL2(Fq) to give the reader some feeling for dealing with the characters of GLn(K), when K is a finite field rather than an infinite one. We will construct a complete character table for the aforementioned groups. ii Contents Contents ii Introduction1 1 Elementary Coalgebra Theory3 1.1 The Definition of a Coalgebra.........................3 1.1.1 Examples of Coalgebras........................4 1.2 The Dual Algebra to a Coalgebra.......................5 1.3 Homomorphisms of Coalgebras........................6 1.4 Subcoalgebras..................................8 1.5 Comodules...................................8 1.6 Bialgebras.................................... 10 1.7 Definitions in Module Theory......................... 11 1.7.1 Extension of the Ground Field.................... 12 1.7.2 Absolute Irreducibility......................... 12 2 Finitary Functions and Coefficient Functions 13 2.1 Basic Representation Theory......................... 13 2.2 Finitary functions............................... 14 2.2.1 F is a K-coalgebra........................... 14 2.3 Coefficient Functions.............................. 17 2.4 The category modA(KΓ)............................ 18 3 Polynomial Representations and the Schur Algebra 19 3.1 The Definition of MK (n) and MK (n; r)................... 21 3.2 Examples of Polynomial Representations................... 22 3.3 The Schur Algebra............................... 23 3.4 The map e : KΓ ! SK (n; r).......................... 25 3.5 The Module E⊗r ................................ 27 4 Weights and Characters 30 4.1 Weights..................................... 30 4.2 Weight Spaces................................. 31 4.2.1 Examples of Weight Spaces...................... 33 4.3 First Results on Weight Spaces........................ 34 4.4 Characters.................................... 35 4.5 Irreducible modules in MK (n; r)....................... 40 iii Contents iv 5 The irreducible characters of GL2 43 5.1 Conjugacy Classes of GL2(Fq)......................... 43 5.2 Irreducible Characters of V; Uα and Vα .................... 45 5.2.1 The Characters of Uα and Vα ..................... 47 5.3 The Characters of Wα,β ............................ 48 5.4 The Characters of Ind ' ............................ 51 Conclusion 56 Acknowledgements.................................. 57 Introduction \What fascinated me so extraordinarily in these investigations [represen- tations of groups] was the fact that here, in the midst of a standstill that prevailed in other areas in the theory of forms, there arose a new and fertile chapter of algebra which is also important for geometry and analysis and which is distinguished by great beauty and perfection." { Issai Schur ([4, p. xii]). Issai Schur was an astoundingly brilliant 20th century German-Jewish mathematician, whose life did not end well. He was born in Russia in 1875, but having most of his life been educated in German he moved to the University of Berlin to study Mathematics in 1894. Under the supervision of Frobenius and Fuchs, he obtained his PhD here in 1901 with a dissertation titled \Uber¨ eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen"[14], which translates loosely as \about a class of matrices, which can be categorised by a given matrix". This dissertation contains an unusually large amount of original ideas and almost all important results in Chapter 3 and Chapter 4 of this project were first found in this dissertation. Schur became a professor in Bonn in 1919, but was forced to step down from his position under the Nazi regime. He died in 1941 in Tel Aviv, Palestine, after having lived the last years of his life here in poverty. I decided to start this project with a short biography of Schur's life, because my main goal in this project is to present, prove and explain some of the most important results from Schur's dissertation. More specifically the main goal of this project is to describe and classify all irreducible characters of GLn(K) for an infinite field K. The approach I have taken here is inspired by, but not completely identical to Schur's approach. As the quotation above might suggest we will come across some beautifully elegant parts of mathematics on our journey towards this goal, but may the reader be warned that elegant is not always a synonym for easy! In Chapter 1 we do some of the algebraic groundwork needed in later chapters. We introduce the concepts of a coalgebra and a bialgebra from scratch. Thereafter we look at some of the theory surrounding these objects, involving concepts such as homomorphisms of coalgebras, subcoalgebras and comodules. In Chapter 2 we introduce finitary functions. It should be noted that finitary functions appear under many different names in the literature (most notably as representative functions), but I have chosen to follow Green's nomenclature here (see [9, p. 3]). We proceed in this chapter by using results from Chapter 1 to prove that the set of finitary functions is a K-bialgebra and we show that coefficient functions form a subcoalgebra. 1 Introduction 2 In Chapter 3 we let Γ = GLn(K) (again following Green's notation) and we define polynomial functions on Γ. We introduce the categories MK (n) and MK (n; r) of ‘finite- dimensional left KΓ-modules, which afford polynomial representations', i.e. represen- tations whose coefficient functions are polynomial functions. Next we introduce the Schur algebra SK (n; r) - the attentive reader may be able to make a wild guess who this has been named after - and we show that left SK (n; r)-modules are, in fact, equivalent to the modules in MK (n; r): We conclude the chapter by considering the evaluation ⊗r map e : KΓ ! SK (n; r) and the module E and use these to show that any module V 2 MK (n; r) is completely reducible. In Chapter 4 we introduce weights and weight spaces. After having looked at some initial consequences of these definitions such as the weight space decomposition of a module in MK (n; r), we shift our focus to the study of formal characters. We study these characters using some of the theory of symmetric polynomials and show that formal characters are very naturally linked to the normal characters we know from representation theory. At the end of this chapter we use our results to find all irreducible modules in MK (n; r): In Chapter 5 we move away from the heavy theory and adopt a more hands-on approach in constructing all irreducible characters of GL2(Fq). This is intended to give the reader a feeling of how to deal with characters of GLn(K), where K is a finite field rather than an infinite one. We construct the character table of GL2(Fq), by first looking at some fairly standard characters and then by inducing characters from (large) subgroups. It should be noted that in 1955 Green([7]) found abstract formulae for the characters of GLn(K), where K is a finite field. However, explicitly constructing these characters often remains challenging. Many different approaches to this subject are possible. The most famous approach is probably through the representation theory of the symmetric group Sn, which is less complicated to understand. Using the Schur functor and the Schur-Weyl duality it is possible to establish a link between the irreducible representations of the symmetric and general linear groups (see e.g. [6]). It has always been my intention to write a project on the representation theory of GLn(K), which is why I decided to take a more direct approach here. It is interesting to note, however, that the link between the representation theory of Sn and GL works two ways, so we could use the results of this project and the Schur functor to deduce many interesting results about the representation theory of the symmetric group (see e.g. [9, Chapter 6]). The main references I have used for this project are Sweedler ([15]) for Chapter 1, Green ([9]) for Chapters 2,3 and 4 and Fulton & Harris ([6]) for Chapter 5. In many places I have included my own proofs and examples, or I have adapted proofs to make the argument more lucid. I have always tried to indicate this clearly at the beginning of the proof. Any result, which I have not explicitly stated to be my own working, was taken from a source that should be clear from context.
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