Book contents
- Frontmatter
- Contents
- Preface
- List of tables
- Notation
- PART I LINEAR ALGEBRAIC GROUPS
- PART II SUBGROUP STRUCTURE AND REPRESENTATION THEORY OF SEMISIMPLE ALGEBRAIC GROUPS
- 11 BN-pairs and Bruhat decomposition
- 12 Structure of parabolic subgroups, I
- 13 Subgroups of maximal rank
- 14 Centralizers and conjugacy classes
- 15 Representations of algebraic groups
- 16 Representation theory and maximal subgroups
- 17 Structure of parabolic subgroups, II
- 18 Maximal subgroups of classical type simple algebraic groups
- 19 Maximal subgroups of exceptional type algebraic groups
- 20 Exercises for Part II
- PART III FINITE GROUPS OF LIE TYPE
- Appendix A Root systems
- Appendix B Subsystems
- Appendix C Automorphisms of root systems
- References
- Index
20 - Exercises for Part II
from PART II - SUBGROUP STRUCTURE AND REPRESENTATION THEORY OF SEMISIMPLE ALGEBRAIC GROUPS
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- List of tables
- Notation
- PART I LINEAR ALGEBRAIC GROUPS
- PART II SUBGROUP STRUCTURE AND REPRESENTATION THEORY OF SEMISIMPLE ALGEBRAIC GROUPS
- 11 BN-pairs and Bruhat decomposition
- 12 Structure of parabolic subgroups, I
- 13 Subgroups of maximal rank
- 14 Centralizers and conjugacy classes
- 15 Representations of algebraic groups
- 16 Representation theory and maximal subgroups
- 17 Structure of parabolic subgroups, II
- 18 Maximal subgroups of classical type simple algebraic groups
- 19 Maximal subgroups of exceptional type algebraic groups
- 20 Exercises for Part II
- PART III FINITE GROUPS OF LIE TYPE
- Appendix A Root systems
- Appendix B Subsystems
- Appendix C Automorphisms of root systems
- References
- Index
Summary
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 SLn, n ≥ 3.
(c) Show that a suitable element of GO2n induces a non-trivial graph automorphism of SO2n, 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 = s1 … sr with si ∈ S. Show the following:
(a) If s ∈ S, w ∈ W with ℓ(ws) ≥ ℓ(w) then BẇB · BṡB ⊆ BẇsB.
(b) If s ∈ S, w ∈ W with ℓ(ws) ≤ ℓ(w) then BẇB · BṡB has non-empty intersection with BẇB.
(c) If ℓ(ws) < ℓ(w), then ṡ ∈ Bẇ-1BẇB.
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- Linear Algebraic Groups and Finite Groups of Lie Type , pp. 172 - 178Publisher: Cambridge University PressPrint publication year: 2011