Representations of Finite Groups 29/12/2008 Andrew Baker Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland. E-mail address: [email protected] URL: http://www.maths.gla.ac.uk/»ajb Contents Chapter 1. Linear and multilinear algebra 1 1.1. Basic linear algebra 1 1.2. Class functions and the Cayley-Hamilton Theorem 5 1.3. Separability 8 1.4. Basic notions of multilinear algebra 10 Exercises on Chapter 1 12 Chapter 2. Representations of ¯nite groups 15 2.1. Linear representations 15 2.2. G-homomorphisms and irreducible representations 17 2.3. New representations from old 21 2.4. Permutation representations 23 2.5. Properties of permutation representations 25 2.6. Calculating in permutation representations 27 2.7. Generalized permutation representations 28 Exercises on Chapter 2 30 Chapter 3. Character theory 33 3.1. Characters and class functions on a ¯nite group 33 3.2. Properties of characters 35 3.3. Inner products of characters 36 3.4. Character tables 39 3.5. Examples of character tables 43 3.6. Reciprocity formul½ 48 3.7. Representations of semi-direct products 49 3.8. Real representations 51 Exercises on Chapter 3 51 Chapter 4. Some applications to group theory 55 4.1. Characters and the structure of groups 55 4.2. A result on representations of simple groups 57 4.3. A Theorem of Frobenius 58 Exercises on Chapter 4 60 Appendix A. Background information on groups 63 A.1. The Isomorphism and Correspondence Theorems 63 A.2. Some de¯nitions and notation 64 A.3. Group actions 64 A.4. The Sylow theorems 67 A.5. Solvable groups 67 A.6. Product and semi-direct product groups 68 i ii CONTENTS A.7. Some useful groups 68 A.8. Some useful Number Theory 69 Bibliography 71 Solutions 73 Chapter 1 73 Chapter 2 75 Chapter 3 77 Chapter 4 81 CHAPTER 1 Linear and multilinear algebra In this chapter we will study the linear algebra required in representation theory. Some of this will be familiar but there will also be new material, especially that on `multilinear' algebra. 1.1. Basic linear algebra Throughout the remainder of these notes | will denote a ¯eld, i.e., a commutative ring with unity 1 in which every non-zero element has an inverse. Most of the time in representation theory we will work with the ¯eld of complex numbers C and occasionally the ¯eld of real numbers R. However, a lot of what we discuss will work over more general ¯elds, including those of ¯nite characteristic such as Fp = Z=p for a prime p. Here, the characteristic of the ¯eld | is de¯ned to be the smallest natural number p 2 N such that p1 = 1| + ¢{z ¢ ¢ + 1} = 0; p summands provided such a number exists, in which case | is said to have ¯nite or positive characteristic, otherwise | is said to have characteristic 0. When the characteristic of | is ¯nite it is actually a prime number. 1.1.1. Bases, linear transformations and matrices. Let V be a ¯nite dimensional vector space over |, i.e., a |-vector space. Recall that a basis for V is a linearly independent spanning set for V . The dimension of V (over |) is the number of elements in any basis, and is denoted dim| V . We will often view | itself as a 1-dimensional |-vector space with basis f1g or indeed any set fxg with x 6= 0. Given two |-vector spaces V; W , a linear transformation (or linear mapping) from V to W is a function ': V ¡! W such that '(v1 + v2) = '(v1) + '(v2)(v1; v2; v 2 V ); '(tv) = t'(v)(t 2 |): We will be denote the set of all linear transformations V ¡! W by Hom|(V; W ). This is a |-vector space with the operations of addition and scalar multiplication given by (' + θ)(u) = '(u) + θ(u)('; θ 2 Hom|(V; W )); (t')(u) = t('(u)) = '(tu)(t 2 |): The extension property in the next result is an important property of a basis. Proposition 1.1. Let V; W be |-vector spaces with V ¯nite dimensional, and fv1; : : : ; vmg a basis for V where m = dim| V . Given a function ': fv1; : : : ; vmg ¡! W , there is a unique linear transformation '~: V ¡! W such that '~(vj) = '(vj) (1 6 j 6 m): 1 2 1. LINEAR AND MULTILINEAR ALGEBRA We can express this with the aid of the commutative diagram inclusion fv ; : : : ; v g / V 1 Lm LL LL LL ' LL 9!' ~ L% Ä W in which the dotted arrow is supposed to indicate a (unique) solution to the problem of ¯lling in the diagram inclusion fv ; : : : ; v g / V 1 Lm LL LL LL ' LL L% W with a linear transformation so that composing the functions corresponding to the horizontal and right hand sides agrees with the functions corresponding to left hand side. Proof. The formula for' ~ is 0 1 Xm Xm '~ @ ¸jvjA = ¸j'(vj): ¤ j=1 j=1 The linear transformation' ~ is known as the linear extension of ' and is often just denoted by '. Let V; W be ¯nite dimensional |-vector spaces with the bases fv1; : : : ; vmg and fw1; : : : ; wng respectively, where m = dim| V and n = dim| W . By Proposition 1.1, for 1 6 i 6 m and 1 6 j 6 n, the function 'ij : fv1; : : : ; vmg ¡! W ; 'ij(vk) = ±ikwj (1 6 k 6 m) has a unique extension to a linear transformation 'ij : V ¡! W . Proposition 1.2. The collection of functions 'ij : V ¡! W (1 6 i 6 m, 1 6 j 6 n) forms a basis for Hom|(V; W ). Hence dim| Hom|(V; W ) = dim| V dim| W = mn: A particular and important case of this is the dual space of V , V ¤ = Hom(V; |): ¤ ¤ ¤ Notice that dim| V = dim| V . Given any basis fv1; : : : ; vmg of V , de¯ne elements vi 2 V (i = 1; : : : ; m) by ¤ vi (vk) = ±ik; where ±ij is the Kronecker ±-function for which ( 1 if i = j; ±ij = 0 otherwise: ¤ ¤ ¤ Then the set of functions fv1; : : : ; vmg forms a basis of V . There is an associated isomorphism V ¡! V ¤ under which ¤ vj Ã! vj : If we set V ¤¤ = (V ¤)¤, the double dual of V , then there is an isomorphism V ¤ ¡! V ¤¤ under which ¤ ¤ ¤ vj Ã! (vj ) : 1.1. BASIC LINEAR ALGEBRA 3 ¤ ¤ Here we use the fact that the vj form a basis for V . Composing these two isomorphisms we obtain a third V ¡! V ¤¤ given by ¤ ¤ vj Ã! (vj ) : In fact, this does not depend on the basis of V used, although the factors do! This is sometimes called the canonical isomorphism V ¡! V ¤¤. The set of all (|-)endomorphisms of V is End|(V ) = Hom|(V; V ): This is a ring (actually a |-algebra, and also non-commutative if dim| V > 1) with addition as above, and composition of functions as its multiplication. There is a ring monomorphism | ¡! End|(V ); t 7¡! t IdV ; which embeds | into End|(V ) as the subring of scalars. We also have 2 dim| End|(V ) = (dim| V ) : Let GL|(V ) denote the group of all invertible |-linear transformations V ¡! V , i.e., the group of units in End|(V ). This is usually called the general linear group of V or the group of linear automorphisms of V and denoted GL|(V ) or Aut|(V ). Now let v = fv1; : : : ; vmg and w = fw1; : : : ; wng be bases for V and W . Then given a linear transformation ': V ¡! W we may de¯ne the matrix of ' with respect to the bases v and w to be the n £ m matrix with coe±cients in |, w[']v = [aij]; where Xn '(vj) = akjwk: k=1 0 0 0 0 Now suppose we have a second pair of bases for V and W , say v = fv1; : : : ; vmg and w = 0 0 fw1; : : : ; wng. Then we can write Xm Xn 0 0 vj = prjvr; wj = qsjws; r=1 s=1 for some pij; qij 2 |. If we form the m £ m and n £ n matrices P = [pij] and Q = [qij], then we have the following standard result. Proposition 1.3. The matrices w[']v and w0 [']v0 are related by the formul½ ¡1 ¡1 w0 [']v0 = Qw[']vP = Q[aij]P : In particular, if W = V , w = v and w0 = v0, then ¡1 ¡1 v0 [']v0 = P v[']vP = P [aij]P : 1.1.2. Quotients and complements. Let W ⊆ V be a vector subspace. Then we de¯ne the quotient space V=W to be the set of equivalence classes under the equivalence relation » on V de¯ned by u » v if and only if v ¡ u 2 W: We denote the class of v by v + W . This set V=W becomes a vector space with operations (u + W ) + (v + W ) = (u + v) + W; ¸(v + W ) = (¸v) + W 4 1. LINEAR AND MULTILINEAR ALGEBRA and zero element 0 + W . There is a linear transformation, usually called the quotient map q : V ¡! V=W , de¯ned by q(v) = v + W: Then q is surjective, has kernel ker q = W and has the following universal property. Theorem 1.4. Let f : V ¡! U be a linear transformation with W ⊆ ker f. Then there is a unique linear transformation f : V=W ¡! U for which f = f ± q. This can be expressed in the diagram q / V ? V=W ?? ?? ?? f ? | 9! f U in which all the sides represent linear transformations. Proof. We de¯ne f by f(v + W ) = f(v); which makes sense since if v0 » v, then v0 ¡ v 2 W , hence f(v0) = f((v0 ¡ v) + v) = f(v0 ¡ v) + f(v) = f(v): The uniqueness follows from the fact that q is surjective.