Representation Theory of Finite Reductive Groups (New by Marc Cabanes, Michel Enguehard

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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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 .

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