2018/06/30 by Mikko Korhonen, MIKKO KORHONEN
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic closure #Algebraic group #Algebraic number #Algebraically closed field #Conjecture #Conjugacy class #Counterexample #Finite Group Theory Research #Unipotent #math.GR #math.RT
paper · pdf · doi:10.1007/s00031-019-09538-3
published as Transform. Groups, 25 (2020) 1209-1222 · to appear in Transform. Groups
openalex created_date 2018/07/10 · openalex publication_date 2019/07/30 · arxiv created 2019/09/03 · arxiv updated 2020/11/02 · openalex updated_date 2026/08/05
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 C5, 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).