Linear Preserver Problems and Cohomological Invariants Public

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

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Mot-clé	Algebraic Groups Cohomological Invariants Linear Preserver
Committee Chair / Thesis Advisor	Garibaldi, Skip, Emory University
Committee Members	Raman, Parimala, Emory University Venapally, Suresh , Emory University

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