Download Group Representations : Volume 3 by Gregory Karpilovsky PDF
By Gregory Karpilovsky
This 3rd quantity should be approximately divided into components. the 1st half is dedicated to the research of varied homes of projective characters. designated realization is interested in spin representations and their personality tables and to numerous correpondences for projective characters. between different subject matters, projective Schur index and projective representations of abelian teams are coated. The final subject is investigated via introducing a symplectic geometry on finite abelian teams. the second one half is dedicated to Clifford idea for graded algebras and its software to the corresponding idea for team algebras. the amount ends with an in depth research of the Schur index for usual representations. A prominant position is performed within the dialogue via Brauer teams including cyclotomic algebras and cyclic algebras.
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Example text
Ii) If d = ( G : H ) , then d = d e g x . (iii) degx = (G : H ) if and only if there exists a simple monomial FaGmodule V with dimFV = d e g x . Proof. Since O, is a coboundary, F P H E F H and so F P H has a module U of dimension 1. Then r dirnFUG = ( G : H ) = XdimFWi i=l (21 4 Degrees of irreducible projective characters 27 for some simple FaG-modules W1,... ,W,. Since dimFW; 2 degx for each i E (1,. . , r } , we deduce that d e g x 5 (G : H ) . Moreover, which shows that dJ(G: H ) . (ii) Since d l d e g x , the required assertion follows from (i).
Proof. (i) Since S is cyclic, as is a coboundary. 13, (G : H ) 5 (G : S). 13(i) that d e g x 5 (G : S ) . Moreover, since the proof of dl(G : H ) uses only the assumption that a~ is a coboundary, we also have dJ(G: S). (ii) Let P be a Sylow 2-subgroup of G. 1, the restriction map R e s : H2(G, F*)2 -+ H 2 ( P ,F " ) is injective. Hence 2 divides the order of the cohomology class of a p and so there are no ap-characters of P of degree 1. On the other hand, P has a cyclic subgroup of index 2. 14, 2 is the smallest degree of an irreducible ap-character of P.
2(iii), a is a class-function cocycle. 2(ii). Conversely, let g be a-regular and assume, by way of contradiction, that x;(g) = 0 for all i E (1,. . , r ] . Denote by C the conjugacy class of G containing g. Then C 2 Go and therefore, by (i), f(C) = 0 for any class function f from Go t o F . This is, however a contradiction and so the result is established. H + + . - + a It is now an easy matter to characterize class-function cocycles. 4. Let F be a splitting field for F f f Gand let charF Then a is a class-function cocycle if and only if a is normal.