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17
Block Theory via Stable and Rickard Equivalences
, 2000
"... This paper owes a lot to M. Collins for his renewed encouragements. 2. Symmetric algebras, functors and equivalences ..."
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Cited by 27 (3 self)
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This paper owes a lot to M. Collins for his renewed encouragements. 2. Symmetric algebras, functors and equivalences
Broué’s abelian defect group conjecture holds for the HaradaNorton sporadic simple group HN
, 2009
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Selfequivalences of stable module categories
, 2000
"... Let P be an abelian pgroup, E a cyclic p ′group acting freely on P and k an algebraically closed field of characteristic p>0. In this work, we prove that every selfequivalence of the stable module category of k[P ⋊ E] comes from a selfequivalence of the derived category of k[P ⋊ E]. Work of ..."
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Cited by 5 (2 self)
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Let P be an abelian pgroup, E a cyclic p ′group acting freely on P and k an algebraically closed field of characteristic p>0. In this work, we prove that every selfequivalence of the stable module category of k[P ⋊ E] comes from a selfequivalence of the derived category of k[P ⋊ E]. Work of Puig and Rickard allows us to deduce that if a block B with defect group P and inertial quotient E is Rickard equivalent to k[P ⋊ E], then they are splendidly Rickard equivalent. That is, Broué’s original conjecture implies Rickard’s refinement of the conjecture in this case. All of this follows from a general result concerning the selfequivalences of the thick subcategory generated by the trivial module.
The McKay conjecture and Brauer’s induction theorem
"... Let G be an arbitrary finite group. The McKay conjecture asserts that G and the normaliser NG(P) of a Sylow psubgroup P in G have the same number of characters of degree not divisible by p (that is, of p′degree). We propose a new refinement of the McKay conjecture, which suggests that one may choo ..."
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Cited by 3 (2 self)
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Let G be an arbitrary finite group. The McKay conjecture asserts that G and the normaliser NG(P) of a Sylow psubgroup P in G have the same number of characters of degree not divisible by p (that is, of p′degree). We propose a new refinement of the McKay conjecture, which suggests that one may choose a correspondence between the characters of p′degree of G and NG(P) to be compatible with induction and restriction in a certain sense. This refinement implies, in particular, a conjecture of Isaacs and Navarro. We also state a corresponding refinement of the Broue ́ abelian defect group conjecture. We verify the proposed conjectures in several special cases. 1
Conjectures of Alperin and Broué for 2blocks with elementary abelian defect groups of order 8
 J. Reine Angew. Math
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Nonprincipal Block of SL(2; q)
"... We shall claim that Broue's abelian defect group conjecture holds for the nonprincipal pblock of SL(2; pn). 1 ..."
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We shall claim that Broue's abelian defect group conjecture holds for the nonprincipal pblock of SL(2; pn). 1