Download Representation Theory II. Groups and Orders by V. Dlab PDF

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By V. Dlab

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F;~) implies F* of G~HF. 3: ir vi := i| f(1)v~-1(i)" Fw(i) = Fi, so that )' f(i)v I - (i) is defined. -module M* with under- 28 lying vector space V*. If Fj(g) = (F~k(g)), I ! J ! 10 Ik~ -1 (1) .... 11 For (f(1))... F~i (f(n))). nk~ -I (n) G* is just F*. F* ~ G* = F*. 6: N w = i,' J, k wJ ik aik(f;~)>O and g~k(f;w) wJk = ( r J k ... is the cycle product associated with "Wk-1(r2k)) , then N-U xF*(f; ~) = %s X ~gikkZ; 9 aik(f;w)>0 This formula was stated by Klaiber (Klaiber [I]) without proof. If in addition to such an irreducible representation G~HF.

Recall from part I, p. 26 (i) The pairs of ordinary irreducible representations of $2~S 4 S2~A 4 which are associated with respect to are: 1(410),(1410)l,1(014),(011471,t(2,12t0),(3,110)t, I (012,12), (013,1) l, 1(3117, (13 I1 ) t, t(113), (1 113) 1, t(2127,(12112) }, l(2112),(1212)J. The ordinary irreducible representations of which are selfassociated with respect to S2NS4 S2%A8 are= (2210), (0122), (2,1 I17, (112,1 ). (ii) The pairs of ordinary irreducible representations of $2~S 4 which are associated with respect to $2~$4A 2 are: I(410),(014)1,1(3,110),(013,1)1,1(2210),(0122)I, ~(2,1210),(012,1271, I(1410),(0114)I, I(311),( 1 1371, 1(2,1 117,(112,1)J,1(1311),(1113)t,1(2112),(1212)}.

Of the basis group s,t ~ Z70 and s+t=n. 15 of and irreducible representations S s x S t N A n. 54. S s x St n An Let us s,t ~ 2. 31 A s x A t < S2 x St n An < S x St . 34 [a=a']~[~=~'] Since everyone [a]~[~] in to S s x St n An . 31 that and n o t i c e that s,t ~ 2): SAsXA t = [n]+~Es]++[n]+~[~]-+[~]-~[~]++[~]-~[~]-. 33 that in S s x S t n A n. 36 If n=s+t, where s,t,~2, and a = c ' ~ s,~=~'~ t, then [a]~[~] $ Ss• n = [a]+~[~]+~Ss• + [a]+~[~]-~Ss• is the decomposition of the restriction [a]~[~]$Ss• into its irreducible constituents.

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