A COUNTEREXAMPLE TO A CONJUGACY CONJECTURE OF STEINBERG

Let G be a semisimple algebraic group over an algebraically closed field of characteristic p ≥ 0. At the 1966 International Congress of Mathematicians in Moscow, Robert Steinberg conjectured that two elements a , a ′ ∈ G are conjugate in G if and only if f ( a ) and f ( a ′) are conjugate in GL( V )...

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Bibliographic Details
Published in:Transformation groups 2020-12, Vol.25 (4), p.1209-1222
Main Author: KORHONEN, MIKKO
Format: Article
Language:English
Subjects:
Algebra
Conjugates
Lie Groups
Mathematics
Mathematics and Statistics
Representations
Topological Groups
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Summary:Let G be a semisimple algebraic group over an algebraically closed field of characteristic p ≥ 0. At the 1966 International Congress of Mathematicians in Moscow, Robert Steinberg conjectured that two elements a , a ′ ∈ G are conjugate in G if and only if f ( a ) and f ( a ′) are conjugate in GL( V ) for every rational irreducible representation f : G → GL( V ). Steinberg showed that the conjecture holds if a and a ′ are semisimple, and also proved the conjecture when p = 0. In this paper, we give a counterexample to Steinberg’s conjecture. Specifically, we show that when p = 2 and G is simple of type C 5 , there exist two non-conjugate unipotent elements u , u ′ ∈ G such that f ( u ) and f ( u ′) are conjugate in GL( V ) for every rational irreducible representation f : G → GL( V ).
ISSN:1083-4362
1531-586X
DOI:10.1007/s00031-019-09538-3