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Irreducible Characters of Slk(ℤ/P ℤ)

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Irreducible Characters of Slk(ℤ/P ℤ) University of Alberta n Irreducible Characters of SLk(Z=p Z) by Trevor Lee Pasanen A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Masters of Science in Pure Mathematics Department of Mathematical and Statistical Sciences ⃝c Trevor Lee Pasanen Fall, 2009 Edmonton, Alberta Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. 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Examining Committee Dr. Gerald Cliff, Mathematical and Statistical Sciences Dr. David McNeilly, Mathematical and Statistical Sciences Dr. Lorna Stewart, Computing Science Abstract n In this paper we find irreducible characters of G = SLk(Z=p Z), where n ≥ 2, k = 2; 3 and, p is an odd prime. In the case k = 2 we give a construction for every irreducible character of G without calculating the character values. Our method is based on finding a normal subgroup of G and applying Clifford theory. Acknowledgements I would like to thank Dr. Gerald Cliff, my supervisor, for his constant support during this research. I am also thankful to my fellow student Qian- glong Wen for his comments and suggestions. I should also mention that my graduate studies in Canada were supported in part by NSERC. Of course, I am grateful to my wife and parents for their patience and love. Without them this work would never have come into existence. Contents 1 Introduction 1 n 1.1 Understanding SLk(Z=p Z) ............................... 1 2 Preliminary Results 4 n 2.1 The Size of SLk(Z=p Z) ................................. 4 n 2.2 Quadratic Residues of Z=p Z and the Subgroup S ................... 5 n 2.3 Cubic Residues of Z=p Z and the Subgroup S ...................... 9 2.4 Character Theory . 11 n 3 Irreducible Character Degrees of SL2(Z=p Z) 17 3.0 Introduction . 17 pn−2(p2−1) 3.1 An Irreducible Character of Degree 2 : ..................... 22 3.1.1 When n is Even. 22 3.1.2 When n is Odd. 26 3.2 An Irreducible Character of Degree pn−1(p + 1) . 29 3.2.1 When n is Even. 29 3.2.2 When n is Odd. 32 3.3 An Irreducible Character of Degree pn−1(p − 1) . 37 3.3.1 When n is Even. 37 3.3.2 When n is Odd. 41 3.4 Conclusion about Character Degrees . 52 3.5 Counting Characters . 53 3.5.1 Conjugacy Classes . 53 3.5.2 The Number of Irreducible Characters For Each Degree . 55 n 4 Irreducible Character Degrees of SL3(Z=p Z) 63 4.0 Introduction . 63 4.1 An Irreducible Character of Degree p2n−4(p3 − 1)(p + 1): . 64 4.1.1 When n is Even. 64 4.1.2 When n is Odd. 65 4.2 An Irreducible Character of Degree p2n−2(p2 + p + 1): . 68 4.2.1 When n is Even. 68 4.2.2 When n is Odd. 70 4.3 An Irreducible Character of Degree p3n−3(p − 1)2(p + 1): . 72 4.3.1 When n is Even. 72 4.3.2 When n is Odd. 74 Bibliography . 76 Appendix: Maple Commands . 77 List of Tables n Table 1: Conjugacy Classes of SL2(Z=p Z) page: 53 Table 2: Number of Irreducible Characters of Each Type page: 56 List of Symbols Mn(R) The n × n matrices over R tr(A) The trace of a matrix A th Aij The i; j entry of a matrix A A mod m The matrix A 2 Mn(Z) with each entry reduced mod m I The identity matrix In The n × n identity matrix Cn The cyclic group of order n & Logical AND inj The natural injective map dxe The ceiling of a real number H The restriction of a map to the subgroup H jH The restriction of a map to the subgroup H Chapter 1 Introduction n 1.1 Understanding SLk(Z=p Z) Definition 1.1.1. Given a commutative ring with unity R, the general linear group GLk(R) is the group of invertible k × k matrices with entries in R, under the group operation of matrix multiplication. From linear algebra over a field F we know a matrix M with entries in F is invertible , det M 6= 0. More generally if we consider linear algebra over a commutative ring with unity R there is an analogous result shown below. The determinant of a square matrix M over a commutative ring with unity can be defined exactly as for a square matrix over a field, and the usual elementary properties of determinants carry over. We make use of the following facts carried over from standard linear algebra: (i) A; B 2 Mk(R) ) det AB = det A det B (ii) M 2 Mk(R) ) M ⋅ adj(M) = adj(M) ⋅ M = det M ⋅ Ik, where adj(M) is the transpose of the matrix of cofactors. Proposition 1.1.2. Let R be a commutative ring with unity, and let M 2 Mk(R). Then M is invertible , det M 2 R× 1 n 1.1. Understanding SLk(Z=p Z) Proof. \ ) " Let M 2 Mk(R)be invertible ) M ⋅ N = Ik for some N 2 Mk(R) ) det(M ⋅ N) = det Ik ) det M ⋅ det N = 1 using (i) ) det M 2 R× since (det M)−1 = det N in R: \(" × Let M 2 Mk(R) with det M 2 R ) (det M)−1 ⋅ M ⋅ adj (M) = (det M)−1 ⋅ adj (M) ⋅ M −1 −1 = (det M) ⋅ det M ⋅ Ik multiplying (ii) by (det M) −1 −1 ) M ⋅ (det M) ⋅ adj (M) = (det M) ⋅ adj (M) ⋅ M = Ik ) M is invertible. So, now × GLk(R) = fM 2 Mk(R) j det M 2 R g; and we can define GLk(R)'s subgroup of interest: Definition 1.1.3. Given a commutative ring with unity R, the special linear group SLk(R) is a subgroup of GLk(R) with matrices having determinant 1. Proposition 1.1.4. Let R be a commutative ring with unity, there is a short exact sequence: inj det × 1 ! SLk(R) −! GLk(R) −! R ! 1 Proof. To prove this we show the map det is surjective and image inj = ker det ([3],p.379). The fact image inj = ker det follows directly form the definition of SLk(R), so we are left to show: det 2 n 1.1. Understanding SLk(Z=p Z) is surjective. Let x 2 R× x 0 ::: 0 0 1 ::: 0 ) = x . .. 0 0 ::: 1 2 3 x 0 ::: 0 6 7 6 7 6 0 1 ::: 0 7 ) 6 7 = M 2 GL (R); by Proposition 1.1.2. 6 . 7 k 6 . .. 7 6 . 7 4 5 0 0 ::: 1 So we have det(M) = x and det is surjective. Applying the first isomorphism theorem to this short exact sequence gives the following corol- lary: Corollary 1.1.5. Given a commutative ring with unity R we have: SLk(R) ⊴ GLk(R) and, × GLk(R)=SLk(R) ' R : 3 Chapter 2 Preliminary Results n 2.1 The Size of SLk(Z=p Z) In the following proof we use the fact ([1],p.11): k ! n k2(n−1) Y k (t−1) jGLk(Z=p Z)j = p p − p : t=1 Proposition 2.1.1.
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