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Register a new userConjugacy action, induced representations and the Steinberg square for simple groups of Lie type
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Let $G$ be a finite simple group of Lie type, and let $pi_G$ be the permutation representation of $G$ associated with the action of $G$ on itself by conjugation. We prove that every irreducible representation of $G$ is a constituent of $pi_G$, unless $G=PSU_n(q)$ and $n$ is coprime to $2(q+1)$, where precisely one irreducible representation fails. Let St be the Steinberg representation of $G$. We prove that a complex irreducible representation of $G$ is a constituent of the tensor square $Stotimes St$, with the same exceptions as in the previous statement.
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Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on G^n, the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in G^n, generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serres notion of G-complete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We also consider some rationality questions.
In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank $2$ elementary abelian $ell$-subgroups in any finite group of Lie type, for any prime $ell$, which may be of independent interest.
For a finite group $G$, let $mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $mathrm{diam}(G)$ is bounded by ${(log_{2} |G|)}^{O(1)}$ in case $G$ is a non-abelian finite simple group. Let $G$ be a finite simple group of Lie type of Lie rank $n$ over the field $F_{q}$. Babais conjecture has been verified in case $n$ is bounded, but it is wide open in case $n$ is unbounded. Recently, Biswas and Yang proved that $mathrm{diam}(G)$ is bounded by $q^{O( n {(log_{2}n + log_{2}q)}^{3})}$. We show that in fact $mathrm{diam}(G) < q^{O(n {(log_{2}n)}^{2})}$ holds. Note that our bound is significantly smaller than the order of $G$ for $n$ large, even if $q$ is large. As an application, we show that more generally $mathrm{diam}(H) < q^{O( n {(log_{2}n)}^{2})}$ holds for any subgroup $H$ of $mathrm{GL}(V)$, where $V$ is a vector space of dimension $n$ defined over the field $F_q$.
In this paper we measure how efficiently a finite simple group $G$ is generated by its elements of order $p$, where $p$ is a fixed prime. This measure, known as the $p$-width of $G$, is the minimal $kin mathbb{N}$ such that any $gin G$ can be written as a product of at most $k$ elements of order $p$. Using primarily character theoretic methods, we sharply bound the $p$-width of some low rank families of Lie type groups, as well as the simple alternating and sporadic groups.
Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that intersects every conjugacy class of involutions of G.
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