REPRESENTATION THEORY OF PSL2(q)

Following are notes from book [1]. The aim is to show the quasirandomness of PSL2(q), i.e., the group has no low dimensional representation. 1. Representation Theory of Finite Groups Let G be a nite group, in the following we recall some elementary representation theory for G. Say (; V) is a representation of G we mean V is a ( nite-dimensional) complex vector space and : G! GL(V) is a group homomorphism. The dimension of the representation (; V) is just the dimension of V over C, we consider only nite dimensional representation in this note. Say a representation (; V) is irreducible if V has no nontrivial G-invariant subspace, for example 1-dimensional representations are all irreducible. Say the representation is faithful if is injective, for example all nontrivial representations for simple groups are faithful. A rst result is the Schur's Lemma. Let (; V) and (;W) be two representations of G, let HomG(; ) be the vector space of interwiners T, that is, T: V! W is a linear map and (g)◦T = T ◦(g) holds for every g 2 G. Say = if there exists an invertible T 2 HomG(; ). Lemma 1.1 (Schur). Let (; V) and (;W) be two nite-dimensional and irreducible represen-tations of G, then dimHomG(; )