Basics of representation theory Clause Samples

Basics of representation theory for finite groups Theorem 2.3.1 (▇▇▇▇▇▇▇’▇ Theorem). Let G be a finite group and let F be a field in which |G| is invertible. Then every finite dimensional FG-module is semisimple. It is easy to find examples of representations of a finite group G over a field of characteristic dividing the order G which are not semisimple. This result therefore partitions the field of representation theory into the study of representations of a finite group G over a field F in which |G| is invertible, which we call ordinary representation theory and the study of representations over a field whose characteristic divides |G|, which we call modular representation theory. An important technique in representation theory is the study of the endomorphism rings of representations. The first key result here is the following. Theorem 2.3.2 (▇▇▇▇▇’▇ Lemma). Let A be a ring with a unit and let S1 and S2 be simple A-modules. Then Hom(S1, S2) = 0 unless S1 ∼= S2 in which case the endomorphism ring EndA(S1) is a division ring. In particular if A is finite dimensional over an algebraically closed field F then EndA(S1) ∼= F. If A is a finite dimensional algebra over a field F we can consider it as an A-module via the algebra multiplication which we call the regular A module and denote AA. In the ordinary case we have the following result. Theorem 2.3.3 (▇▇▇▇▇-▇▇▇▇▇▇▇▇▇▇). Let A be a finite dimensional algebra over a field F such that every finite dimensional A-module is semisimple. Then A is a direct sum of matrix algebras over division rings. That is, if AA ∼= Sn1 ⊕ · · · ⊕ Snr where S1, . . . , Sr are the non-isomorphic simple modules occurring with multiplicities n1, . . . , nr in the regular representation AA, then A ∼= Mn1 (D1) ⊕ · · · ⊕ Mnr (Dr) where Di = EndA(Si)op. Furthermore , if F is algebraically closed then Di = F for all i. 2.3.1 Complex characters