By Marc Cabanes, Michel Enguehard

On the crossroads of illustration thought, algebraic geometry and finite team concept, this ebook blends jointly a number of the major matters of contemporary algebra, synthesising the previous 25 years of analysis, with complete proofs of a few of the main notable achievements within the region. Cabanes and Enguehard persist with 3 major subject matters: first, functions of étale cohomology, resulting in the facts of the new Bonnafé-Rouquier theorems. the second one is an easy 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 means of many examples and historical past is supplied via a number of introductory chapters on easy effects and appendices on algebraic geometry and derived different types. the result's a vital advent for graduate scholars and reference for all algebraists.

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**Example text**

A BN-pair (or Tits system) consists of the data of a group G, two subgroups B, N and a subset S of the quotient N /B ∩ N such that, denoting T := B ∩ N and W := N /T : (TS1) T N (W is therefore a quotient group), W is generated by S and ∀s ∈ S, s 2 = 1. (TS2) ∀s ∈ S, ∀w ∈ W , s Bw ⊆ Bw B ∪ Bsw B. (TS3) B ∪ N generates G. (TS4) ∀s ∈ S, s Bs = B. 13. The notation Bw is unambiguous since w is a class mod. T and T ⊆ B. Similarly, if X is a subgroup of B normalized by T , the notation X w makes sense (and is widely used in what follows).

Iii) If Y is an indecomposable direct summand of Y , then soc(Y ), hd(Y ) are simple, and H (soc(Y )) = soc(H (Y )), H (hd(Y )) = hd(H (Y )). (iv) If Y , Y are indecomposable direct summands of Y , then soc(Y ) ∼ = soc(Y ) (and hd(Y ) ∼ =Y . 2(ii)). Considering injective hulls, we get the following. 26. If E is a Frobenius algebra, then every finitely generated Emodule embeds into a free module E l for some integer l. We shall use the following notation. Notation. Y := Assume that E is Frobenius.

3(iii), it is + enough to show that w ⊇ v(δ,I ) . e. α ∈ + {δ}∪I \ I thanks to (i) above. Let us write α = λδ δ + δ ∈I λδ δ with λδ > 0 and λδ ≥ 0 for 2 Finite BN-pairs 27 δ ∈ I . Then w(α) = λδ w(δ) + δ ∈I λδ w(δ ). If we had w(α) ∈ + , since w(δ) ∈ − and w(I ) ∈ , the non-zero coefficients in w(δ) would be for elements of w(I ). So w(δ) ∈ w(I ) , or equivalently δ ∈ I . But δ ∈ \ I , a contradiction. v(δ, I )−1 ) = l(w) + l(v(δ, I )). v(δ, I )(δ) ∈ − . We have seen + that δ ∈ v(δ,I ) , so v(δ, I )(δ) ∈ + since w δ∪I .