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Title: Steinberg's tensor product theorem and exceptional groups
Author: Abiteboul, Manon
ISNI:       0000 0004 7233 0199
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2015
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Let X be a simple, simply connected algebraic group over an algebraically closed field k of characteristic p. Steinberg's tensor product theorem states that if V is a finite dimensional irreducible kX-module then V = V₁ ^(q1) ⊗ ... ⊗ V_n ^(qn) where each V_i is a restricted kX-module and the q_i are distinct powers of p. This theorem can be reformulated in terms of rational representations instead of modules as follows: any irreducible rational representation Φ : X → SL(V) can be factorised as Φ : X →^ψ X x X x ... x X →^μ SL(V) with Ψ : x → (x^(q_1) , ... , x^(q_j)) where x^(q_i) denotes the image of x under the standard Frobenius q_i map and μ restricts to a completely reducible restricted representation on each factor. Liebeck and Seitz proved a generalisation of this theorem where the target group SL(V) is replaced by an arbitrary simple algebraic group G over k in good characteristic. A consequence of this theorem is that every connected simple G-completely reducible (G-cr) subgroup X of G is contained in a uniquely determined commuting product R₁...R_n in G such that each Rᵢ is a simple restricted subgroup of the same type as X and each projection X →Rᵢ/Z(Rᵢ) is non trivial and involves a different field twist. Here saying that X is G-cr means that whenever X is contained in a parabolic subgroup P of G, it is contained in a Levi subgroup of P. The first new theorem of this thesis is a converse of this result: if G is of exceptional type in good characteristic and X is contained in such a product R₁ ... R_n, then X is G-cr, with two exceptions. If G is of classical type we prove a similar theorem with the extra assumption that the Rᵢ are G-cr. In view of these results, it is of interest to determine the products R₁ ... R_n of more than one restricted subgroup of the same type in exceptional algebraic groups; this is achieved in the rest of this thesis. The above results have a number of consequences. First, every restricted connected simple subgroup of G is G-cr with two exceptions. We also classify all non-restricted G-cr simple subgroups of exceptional groups. We find the centralisers of all products of more than one subgroup of the same type and of all non-restricted G-cr simple subgroups. We also prove that the conjugacy classes of such subgroups are uniquely determined by the restriction of the Lie algebra of G.
Supervisor: Liebeck, Martin Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral