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By Marc Cabanes, Michel Enguehard

On the crossroads of illustration conception, algebraic geometry and finite crew conception, this ebook blends jointly some of the major issues of recent algebra, synthesising the previous 25 years of analysis, with complete proofs of a few of the main awesome achievements within the region. Cabanes and Enguehard stick with 3 major topics: first, purposes of étale cohomology, resulting in the evidence of the new Bonnafé-Rouquier theorems. the second one is a simple and simplified account of the Dipper-James theorems bearing on irreducible characters and modular representations. the ultimate subject matter is neighborhood illustration concept. one of many major effects here's the authors' model of Fong-Srinivasan theorems. in the course of the textual content is illustrated by way of many examples and historical past is supplied by means of numerous introductory chapters on simple effects and appendices on algebraic geometry and derived different types. the result's a vital creation for graduate scholars and reference for all algebraists.

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E. there is no partition of into two non-empty orthogonal subsets). Let I be a subset of such that CG (U I ) ∩ Bw I B = ∅. Then I = ∅ or . Proof. Let us take g ∈ CG (U I ) ∩ Bw I B. Then g can be written as g = un I u with u, u ∈ U and n I ∈ N such that n I T = w I . 25). So we may assume u ∈ X I . 36 Part I Representing finite BN-pairs Let us show that w I (δ) = δ for all δ ∈ \ I . Assume w I (δ) = δ. Then w I (δ) ∈ + \ {δ} and therefore (X δ )n I ⊆ Uδ . Take x ∈ X δ , x = 1. We have u ∈ X I ⊆ Uδ U , so u x ∈ xUδ .

Let I ⊆ and let J = −w0 (I ). Show that PI ∩ (PJ )w0 = = w∈W I Bw I w Bw I . Deduce that this group has a BN-pair (Bw I , N I ). This allows L I to be defined without assuming that the BN-pair of G is split. 19(v); in particular, show that B ∩ (PJ )w0 = Bw I . 2 Finite BN-pairs 39 5. Let G be a finite group endowed with a split BN-pair of characteristic p. We call the following hypothesis the commutator formula: (C) If α, β ∈ and α = ±β, then [X α , X β ] ⊆ , where Cα,β is the set of roots γ ∈ such that γ = aα + bβ with a > 0 and b > 0.

X −−x ∩↓ a ) for all x ≤ a (resp. x ≤ a ). (b) If a = (P, V )−−a = (P , V ), define a set L P∩P of subquotients of P ∩ P in bijection with the above intervals. Show that cuspk (L P∩P ) injects into cuspk (L) in two ways. 14. (c) See which simplification of that proof can be obtaind by assuming the existence of a subgroup L such that P = L V , P = L V are semi-direct products. 20 Part I Representing finite BN-pairs 10. Assume L is a ∩↓-stable -regular set of subquotients of G. Assume that, for every relation (P, V )−−(P , V ) in L with |P| = |P |, the map x → xe(V ) is an isomorphism from Ge(V ) to Ge(V ).

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