Reducibility modulo 𝑝 of complex representations of finite groups of Lie type: Asymptotical result and small characteristic cases

Tiep, Pham; Zalesskiĭ, A.

doi:10.1090/S0002-9939-02-06459-6

Reducibility modulo $p$ of complex representations of finite groups of Lie type: Asymptotical result and small characteristic cases
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Proc. Amer. Math. Soc. 130 (2002), 3177-3184 Request permission

Abstract:

Let $G$ be a finite group of Lie type in characteristic $p$. This paper addresses the problem of describing the irreducible complex (or $p$-adic) representations of $G$ that remain absolutely irreducible under the Brauer reduction modulo $p$. An efficient approach to solve this problem for $p > 3$ has been elaborated in earlier papers by the authors. In this paper, we use arithmetical properties of character degrees to solve this problem for the groups \[ G \in \{ ^{2} B_{2}(q), ^{2}G_{2}(q),G_{2}(q), ^{2}F_{4}(q),F_{4}(q), ^{3}D_{4}(q)\} \] provided that $p \leq 3$. We also prove an asymptotical result, which solves the problem for all finite groups of Lie type over ${\mathbb F}_{q}$ with $q$ large enough.

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