Class Note for MATH 796 at KU 4
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Date Created: 02/06/15
Friday 411 Until further notice G is still a nite group and all representations are nitedimensional over C New Characters from Old In order to investigate characters we need to know how standard vector space or in fact Gmodule functors such as EB and 8 a ect the corresponding characters Throughout let p V p V be representations of G with V O V ll 1 Direct sum To construct a basis for V EB V we can take the union of a basis for V and a bais for V Equivalently we can Write the vectors in V EB V as column block vectors Vex2H3 lvEV Mew Accordingly de ne p EB p V EB V by 7 90739 0 pep gtlthgt fl 0 W t From this it is clear that 1 ngph Xph Xphl 2 Duality Recall that the dual space V of V consists of all lFlinear transformations lt1 V A F Given a representation p V there is a natural action of G on V de ned by MW ltWflv for h E G lt1 E Vquot v E V You need to de ne it this way in order for hi to be a homomorphism 7 try it This is called the dual representation or contragredient representation pf Proposition For every 1 E G 2 Xpdh XpU bl Proof Choose a basis 11 t i i 27 of V consisting of eigenvectors of 1 since we are working over C say hvl Alil In this basis ph diagl ie the diagonal matrix Whose entries are the Al and in the dual basis ph diag1i On the other hand some power of ph is the identity matrix so each 1 must be a root of unity so its inverse is just its complex conjugate D 3 Tensor product Recall that if 21 i i 27 21 i v are bases for V V respectively then V V can be de ned as the vector space With basis vl vl1 i n1 j mi In particular dim V V dim Vdim V Accordingly de ne a representation 17 17 V V by p p hv 7 MW 7 v MW or more concisely h v v hv v v 017 extended bilinearly to all of V V In terms of matrices 17 p h is represented by the block matrix 11113 11113 an 11213 11223 1an M13 anQB am Where ph 111 le and 1701 B In particular 3 Xp ph XphXph 4 Hom Recall that HomGV V Homap p is the vector space of all Gequivariant maps p A pquot Meanwhile Hode W can be made into a Gmodule by 4 h gtgtltvgt hlt gtlth4vgtgt p lthgt ltplth1gtltvgtgt for h E G lt1 E Hode W 1 E V That is 1 sends lt1 to the map 1 lt1 Which acts on V as above You can then verify that this is a genuine group action In general When G acts on a vector space V the subspace of Ginva m39ants is de ned as VGvEVlhvthEGi In our current setup a map lt1 is Gequivariant if and only if h lt1 lt1 for all h E G proof left to the reader That is 5 HomGV W Hode WGi Moreover Hode W E V W as vector spaces so 6 XHompp h W M h The Inner Product Recall that a Class function is a function X G A C that is constant on conjugacy classes of G De ne an inner product on the vector space CHG of class functions by 1 7 ltX gta E Z XMW M l l heG Proposition 1 With this setup 1 dlmc VG E Z Xph ltXtriv XpgtG heG Proof De ne a linear map 7r V A V by 1 7r 7 p h 101 In fact 7rv E VG for all 1 E V and if v E VG then 7rv 1 That is 7r is a projection from V A Va and can be represented by the block matrix I 0 0 l Where the rst and second column blocks resp roW blocks correspond to Va and VGL respectively is now evident that dimc VG tr 7r giving the rst equality The second equality follows because Va is just the direct sum of all copies of the trivial representation occurring as Ginvariant subspaces of V D Example 1 Suppose that p is a permutation representation Then Va is the space of functions that are constant on the orbits Therefore the formula becomes number of orbits i 2 number of xed points of 1 G l l heG Which is Burnside s Lemma Proposition 2 ltXp XpgtG dimc Homgpp Proof ltXp XpgtG E ZWXph heG 1 E Z XHompph by 6 l l heG dimc Homp pG by Proposition 1 dimc Homap p by D
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