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- W2488263011 abstract "Let k be an algebraically closed field of characteristic p ≥ 0. We will take the numbering of Dynkin diagrams of irreducible root systems as given in Table 9.1. Exercise 20.1 (Existence of graph automorphisms) (a) Show how to reduce the proof of Theorem 11.12 on the existence of graph automorphisms to the case of simple groups of simply connected type. (b) Verify the details of the proof for type SL n , n ≥ 3. (c) Show that a suitable element of GO 2 n induces a non-trivial graph automorphism of SO 2 n , n ≥ 2. [ Hint: For (c) consider the element given in Example 22.9(2).] Exercise 20.2 Let G be a group with a BN-pair, with W = N /( B ∩ N ) generated by a set of involutions S . For w ∈ W write l (w) for the length of a shortest expression w = s 1 … s r with s i ∈ S . Show the following: (a) If s ∈ S, w ∈ W with l (ws) ≥ l (w) then B ẇ B · B ṡ B ⊆ B ẇ sB . (b) If s ∈ S, w ∈ W with l (ws) ≤ l (w) then B ẇ B · B ṡ B has non-empty intersection with B ẇ B . (c) If l (ws) (w) , then ṡ ∈ B ẇ -1 B ẇ B ." @default.
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- W2488263011 date "2012-06-19" @default.
- W2488263011 modified "2023-10-01" @default.
- W2488263011 title "Exercises for Part I" @default.
- W2488263011 doi "https://doi.org/10.1017/cbo9780511994777.013" @default.
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