Trivial source character tables of Frobenius groups of type $(C_p \times C_p) \rtimes H$

TL;DR

This work addresses the problem of determining trivial source character tables $ ext{Triv}_p(G)$ for finite Frobenius groups $G$ with an abelian complement and an elementary abelian kernel of order $p^2$, focusing on two fusion-extremal scenarios. The authors develop a framework combining ordinary and Brauer character theory, decomposition matrices, and Green correspondence to express $ ext{Triv}_p(G)$ as explicit block matrices, with complete treatments for the maximal fusion case (one conjugacy class of order $p$ subgroups) and the minimal fusion case (no fusion among the order-$p$ subgroups). The maximal fusion case yields a $3 imes3$ block structure with blocks determined by $X(H)$, $X(N_H(Q_2))$, and $ ext{Dec}_p$ data, while the minimal fusion case uses $G$-normalizers and inflation/restriction to describe all blocks; the paper also provides concrete instances (e.g., $ ext{AGL}_1(p^2)$ and low-order examples) and tabulated results in an appendix. Overall, the results enrich the database of trivial source character tables and advance understanding of trivial source modules in small Frobenius groups, with implications for modular representation theory and related equivalences.

Abstract

Let $p$ be a prime number. We compute the trivial source character tables of finite Frobenius groups $G$ with an abelian Frobenius complement $H$ and an elementary abelian Frobenius kernel of order $p^2$. More precisely, we deal with all infinite families of such groups which occur in the two extremal cases for the fusion of $p$-subgroups: the case in which there exists exactly one $G$-conjugacy class of non-trivial cyclic $p$-subgroups, and the case in which there exist exactly $p+1$ distinct $G$-conjugacy classes of non-trivial cyclic $p$-subgroups.

Theorems & Definitions (29)

• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5: BBthesis
• Lemma 3.1
• proof
• Theorem 3.2: CurtisReinerMethods1
• Proposition 3.3
• proof