Class Note for MATH 796 at KU
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Date Created: 02/06/15
Monday 414 Let G be a nite group All representations are nitedimensional and over C Here s the machinery we developed on Friday Character formulas We proved that 1 may 9 Xpy m y 2 Xm y m 3 Mumy XpyXpy 4 momma y m M a An important missing piece is a formula for the character of Homg p p Fixed spaces We de ned the xed space of a representation G as VG v E V l 91 h V9 6 G and observed that Homg V W HomcV WGi The inner product For X41 6 CZG we de ned 1 7 ltx7 1 Zxg y gEG and proved the formulas 1 G 7 7 5 611ch 7 G EXAM gEG 6 ltXm XpgtG dimC HOmG 7 Pl Schur7s Lemma and the Orthogonality Relations What happens when p and p are irreducible representations Proposition 1 Schurls Lemma Let G be a group and let p V and p V be nitedimensional repre sentations of C over a eld lF l fp and p are irreducible then every Gequivariant 45 V A V is either zero or an isomorphism 2 If in addition lF is algebraically closed then g HomGV7 V g lF Zf 7 p 0 otherwise Proof For 1 recall that kerqb and imqb are Ginvariant subspaces But since p rho are simple there are not many possibilities Either kerqb 0 and im 45 W when 45 is an isomorphismi Otherwise ker 45 V or imqb 0 either of which implies that 45 0 For 2 let 45 E Homg V i If p 2 p then 45 0 by l and we re done Otherwise we may as well assume that V V i Since lF is algebraically closed 45 has an eigenvalue A Then 45 7 AI is Gequivariant and singular hence zero by So 45 All We7ve just shown that the only G equivariant maps from V to itself are scalar multiplication by some A D Theorem 2 Let p V and p V be nitedimensional representations of C over C i fp and p are irreducible then ltX X gt 1 if P E 77 p G 0 otherwise ii If phi l l pin are distinct irreducible representations and n n p mEBem GB EW i1 m1 i1 then n 09 szgtc mi ltXp7 XpgtG Em i1 In particular ltXp XpgtG 1 if and only ifp is irreducible iii If Xp Xp then p E p iv If phi l l n is a complete list of irreducible representations of G then n N d39 7 Preg 69 P Imp i1 and consequently n Zltdimm2 101 i1 V The irreducible characters e characters of irreducible representations form an orthonormal basis for CZG In particular the number of irreducible characters equals the number of conjugacy classes of G Example 1 Find all the irreducible characters of 64 There are ve conjugacy classes in 64 corresponding to the cycleshapes 1111 211 22 31 and 4 The squares of their dimensions must add up to 164 24 The only list of ve positive integers with that property is 1 1 2 3 3 We start by writing down some characters that we knowi Cycle shape 1111 211 22 31 4 Size of conjugacy class 1 6 3 8 6 X1 Xtriv 1 1 1 1 1 X2 Xsign 1 1 1 1 1 Xdef 4 2 0 1 0 Xre 24 0 0 0 0 Of course va and Xsign are irreducible since they are 1dimensionali On the other hand Xdef can t be irreducible because 64 doesnlt have a 4dimensional irrepl lndeed ltXdef7 XdefgtG 2 which means that pdef must be a direct sum of two distinct irrepsi If it were the direct sum of two copies of the unique 2dimensional irrep then ltXdef Xdefgtc would be 4 not 2 by ii of Theorem 2 We calculate ltXdef7 XtrivgtG 17 ltXdef7 XsigngtG 0 Therefore X3 Xdef 7 va is an irreducible character The other 3dimensional irreducible character is X4 X3 8 Xsign we can check that X4 X4gtG 1i The other irreducible character X5 has dimension 2 We can calculate it from the regular character and the other four irreducibles7 because Xreg X1 X2 3X3 X4 2X5 So here is the complete character table of G4 Cycle shape 1111 211 22 31 4 Size of conjugacy class 1 6 3 8 6 X1 1 1 1 1 1 X2 1 71 1 1 71 X3 3 1 71 0 71 X4 3 71 71 0 1 X5 2 0 2 71 0 Now7 the proof of Theorem 2 Assertion follows from part 2 of Schur s Lemma together with Proposition 67 and ii follows because the inner product is bilinear on direct sumsi For iii7 Maschke s Lemma says that every complex representation p can be written as a direct sum of irreduciblesi Their multiplicities determine p up to isomorphism7 and can be recovered from Xp by assertion ii For iv7 recall that Xng 1g 1G1 and Xregg 0 for g f 19 Therefore 1 7 1 ltXreg7 igtG Zxreg9 i9 lmHOG Chm z gEG so pi appears in preg with multiplicity equal to its dimensioni That the irreducible characters are orthonormal hence linearly independent in CZG follows from Schur7s Lemma together with assertion The hard part is to show that they in fact span CZGi We will do this next time
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