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This paper owes a lot to M. Collins for his renewed encouragements. 2. Symmetric algebras, functors and equivalences

Let P be an abelian p-group, 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 self-equivalence of the stable module category of k[P ⋊ E] comes from a self-equivalence 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 self-equivalences of the thick subcategory generated by the trivial module.

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The aim of this paper is to show that two blocks of symmetric groups with isomorphic defect groups have equivalent derived categories. We deduce in particular that Broué’s abelian defect group conjecture holds for symmetric groups. We prove similar results for general linear groups over finite fields and cyclotomic Hecke algebras.

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