Minimal polynomials of $p$-elements of finite groups of Lie type with cyclic Sylow $p$-subgroups

TL;DR

The paper establishes that for finite groups of Lie type with cyclic Sylow $p$-subgroups and for irreducible $\,\ell$-modular representations with $p\neq \ell$, the minimal polynomial degree of a $p$-element $g$ satisfies $\deg\phi(g) \in \{ |g|,|g|-1\}$. The authors develop a unified, case-by-case analysis across classical and exceptional groups, employing regular semisimple torus elements, Clifford theory, Deligne–Lusztig theory, and primitive prime divisor arguments, and they explicitly characterize equality cases for many families (including ${\rm GL}^{\varepsilon}_n(q)$, ${\rm SL}^{\varepsilon}_n(q)$, ${\rm Spin}$, ${\rm Sp}$, and exceptional groups). They also treat universal central extensions and provide detailed decomposition data to handle nonliftable Brauer characters, consolidating Hall–Higman-type bounds in this cyclic-Sylow context. The results sharpen our understanding of how $p$-elements act in cross-characteristic representations of Lie-type groups and inform broader questions in modular representation and block theory. The work thus offers a comprehensive framework for minimal polynomials of $p$-elements across a wide spectrum of Lie-type groups, with explicit criteria and open directions for further exceptional and central-extension cases.

Abstract

Extending earlier results of the authors on minimal polynomials of $p$-elements of finite groups of Lie type in cross-characteristic representations, this paper focuses on the case where Sylow $p$-subgroups are cyclic and $p$ is distinct from the representation field characteristic.

Theorems & Definitions (117)

• Theorem 1
• Theorem 2
• Proposition 3
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof