By Hsio-Fu Tuan
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Additional info for Group Theory Beijing 1984
3), the associated morphism of restriction functors aP:Q@ R ~ Q@Rf is given on the level of a subgroup H of G by a~(x):= I~I L (-l)nIHoi[Ho,resZ~(PHn(reSZn(x)))JH Ho< .. 4) 32 R. Boltje for X E Q 0 R(H), where the sum runs over all chains of subgroups of H. Note that resZ~ (PHn (resZn (X») is an element of Q 0 Rab(Ho) and that for such an element e = Lipdio aipcp E Q 0 Rab(Ho), where aip are rational numbers, we define [Ho, e]H := LipEHO aip[Ho, cp]H. 3. e. is bH 0 a~ = idiQI0R(H) for all H:::: G?
So we may define the projective p-blocks of H[~] to be the blocks of this unique 91-order D[H]. We denote by Blk(H) the set of all projective p-blocks of H[~]. Any irreducible projective ~-character 1J of H[~] is an irreducible character of 2l[H], and hence belongs to a unique projective p-block B(1J) of H[~]. The natural conjugation action of H on its twisted group algebra D[H] over 91 has the center Z(D[H]) as its set of fixed points. 3]). The defect d(b) of b is then the non-negative integer such that pdCb) is the order of each such D.
But it is not strong enough for an inductive proof. To carry out such a proof we need to calculate, for each p-chain C of G, the number of irreducible J-characters 1/1 of NE(C) with a given defect lying over a fixed irreducible J-character q; of Nc(C). Except in rare cases (which, in fact, are not so rare for simple G), the subgroup N E( C, q;) of E by itself does not determine this number. However, the Clifford extension for q; does determine it. So we reformulate the conjecture in terms of Clifford extensions.