Abstract
Let G be a simple linear algebraic group over a field F. In this
work we prove
several results about G and it's representations. In particular we
determine
the stabilizer of a polynomial f on an irreducible representation V
of G for
several interesting pairs (V; f). We also prove that in most cases
if f is a
polynomial whose stabilizer has identity component G then there is
a correspondence
between similarity classes of twisted forms of f and twisted
forms
of G. In a different direction we determine the group of normalized
degree 3
cohomological invariants for most G which are neither simply
connected nor
adjoint.
Table of Contents
1 Introduction 1 1.1 Algebraic Group Schemes . . . . . . . . . . .
. . . . . . . . . 1 1.2 Morphisms . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 3 1.3 Tori . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 4 1.4 The Lie Algebra of an
Algebraic Group . . . . . . . . . . . . . 4 1.5 Split Semisimple
Algebraic Groups . . . . . . . . . . . . . . . 5 1.6
Representations . . . . . . . . . . . . . . . . . . . . . . . . . .
6 1.7 Tits Algebras . . . . . . . . . . . . . . . . . . . . . . . .
. . . 7 1.8 Galois Cohomology . . . . . . . . . . . . . . . . . . .
. . . . . 7 1.9 Twisted Forms . . . . . . . . . . . . . . . . . . .
. . . . . . . . 8 1.10 Galois Cohomology of Algebraic Groups . . .
. . . . . . . . . 8 2 Linear Preservers and Representations with a
1-dimensional Ring of Invariants 9 2.1 Irreducible representations
and the closed orbit . . . . . . . . 11 2.2 The normalizer of G in
GL(V ) . . . . . . . . . . . . . . . . . . 16 2.3 Linear
transformations preserving minimal elements . . . . . . 19 2.4 The
stabilizer in PGL(V ) . . . . . . . . . . . . . . . . . . . . . 25
2.5 Interlude: non-split groups . . . . . . . . . . . . . . . . . .
. . 26 2.6 Representations with a one-dimensional ring of
invariants . . . 27 2.7 Transformations that preserve minimal
elements and f . . . . 32 2.8 Lines 15 of Table A . . . . . . . . .
. . . . . . . . . . . . . . 34 2.9 Lines 611 of Table A . . . . . .
. . . . . . . . . . . . . . . . . 39 2.10 Some representations
omitted from Table A . . . . . . . . . . 43 2.11 An alternative
formulation of the linear preserver problem . . 45 3 Classifying
forms of simple groups via their invariant poly- nomials 46 3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 46 3.2 Constructing invariants with large stabilizer groups . . .
. . . 48 3.3 Two cohomology sequences . . . . . . . . . . . . . . .
. . . . . 53 3.4 The Fibers . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 55 3.5 The Image . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 63 3.6 Groups of type E8, F4, and G2 . .
. . . . . . . . . . . . . . . . 66 3.7 Conclusion . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 66 4 Degree 3 Cohomological
Invariants of Split Quasi-Simple Groups that are Neither Simply
Connected nor Adjoint 68 4.1 Introduction . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 68 4.2 Decomposable Invariants . . .
. . . . . . . . . . . . . . . . . . 70 4.3 Indecomposable
Invariants . . . . . . . . . . . . . . . . . . . . 71 4.4 SLn =m .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.1
Q(G) for SLn =m . . . . . . . . . . . . . . . . . . . . . 74 4.4.2
Dec(G) for SLn =pr . . . . . . . . . . . . . . . . . . . 74 4.4.3 A
Fibration . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4.4
Examples . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.5
HSpin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81 4.5.1 Q(G) for HSpin . . . . . . . . . . . . . . . . . . . . . .
81 4.5.2 Dec(G) for HSpin4n . . . . . . . . . . . . . . . . . . . .
82 4.6 Restriction of Invariants to Subgroups . . . . . . . . . . .
. . 83 4.6.1 Restrictions in terms of Q(G)= Dec(G) . . . . . . . .
. 83
About this Dissertation
Rights statement
- Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.
| School |
|
|---|
| Department |
|
|---|
| Degree |
|
|---|
| Submission |
|
|---|
| Language |
|
|---|
| Research Field |
|
|---|
| 关键词 |
|
|---|
| Committee Chair / Thesis Advisor |
|
|---|
| Committee Members |
|
|---|