Class Note for MATH 796 at KU
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Date Created: 02/06/15
Monday 421 Restricted and Induced Representations De nition 1 Let H C G be nite groups and let p G A GLV be a representation of G Then the restriction of p to H is a representation of G denoted Res p Likewise the restriction of X Xp to H is a chartacter of H denoted by Resg Notice that restricting a representation does not change its character OTOH whether or not a representation is irreducible can change upon restriction Example 1 Let C denote the conjugacy class in 6 of permutations of cycleshape A Recall that G 63 has an irrep whose character 1 Xp is given by 1110111 27 1021 07 11103 1 Let H ng Q 63 This is an abelian group isomorphic to ZSZ so the twodimensional representation Resg p is not irreducible Speci cally let w e27 393 The table of irreducible characters of ng is as follows 1G 1 2 3 1 3 2 Xtriv 1 1 1 X1 1 w of X2 1 w2 w Now it is evident that ResgiJ 2 71 71 X1 X2 Note by the way that the conjugacy class C3 C 63 splits into two singleton conjugacy classes in ng a common phenomenon when working with restrictions Next we construct a representation of G from a representation of a subgroup H C G De nition 2 Let H C G be nite groups and let p H A GLW be a representation of H De ne the induced representation lnd p as follows 0 Choose a set of left coset representatives B 121 br for H in G That is every 9 E G can be expressed uniquely as g bjh for some bj E B and h E H 0 Let HGH be the Cvector space with basis B o Let V CGH c W 0 Let g E G act on bi Eu 6 V as follows Find the unique bi E B and h E H such that 912 bjh and put gbi w bj hw This makes more sense if we observe that g bjhb1 so that the equation becomes bjhbgl 12429 w b ca hw o Extend this to a representation of G on V by linearity Proposition 1 lnd p is a representation of G that is independent of the choice of B Moreover for all g E G 1 1 Xindgp9 2 M70 19 keG k lgkeH Proof First we verify that lnd p is a representation Let gg E G and bi E w E V Then there is a unique bj E B and h E H such that 1 912 bjh and in turn there is a unique 1 E B and h E H such that 2 g bj bgh We need to verify that 3 9 39 9 39 bi 2910 99 bi 10 lndeed 9 g b ca w g b ca hm b2 h hwi On the other hand by l and 2 gbi bjhb1 and g bzhbgl so g g bi w bghhb1 bi E w bg Him as desired Now that we know that lndg p is a representation of G on V we nd its character on an arbitrary element 9 6 Ci Regard lnd pg as a block matrix with 7 row and column blocks each of size dimW and corresponding to the subspace of V of vectors of the form bi w for some xed bin The block in position M is o a copy of ph if gbi bjh for some h E H 0 zero otherwise Therefore xlndgwgg tr 9 cmH W W a cmH c w Z Xph ism gbbh EheH Z Xpb1gbi ism bflgbleH 1 m E ZXph1bilgbZh iE7 heH bflgbleH 1 71 7 Z X k 976 lHl keG p k lgkeH Here we have put k bih which runs over all elements of G The character of lnd Qz is independent of the choice of B therefore so is the representation itselfi D Theorem 2 Frobenius Reciprocity Let H C G be nite groups Let X be a chammcter ofH and let 1 be a character of G Then lt1nd X7 gtG 06 R683 gtHA Example 2 Sometimes Frobenius reciprocity suf ces to calculate the isomorphism type of an induced representationi Let 1b X1 and X2 be as in Example 1 We would like to compute lndg XL By Frobenius reciprocity G G ltlndH X1 gtG ltX1 ResH 1 But 1 is irreducible Therefore it must be the case that lndg X1 1b and the corresponding representations are isomorphic The same is true if we replace X1 with Xgi Proof next time
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