Counting lifts of irreducible Brauer characters

TL;DR

This paper addresses counting lifts of irreducible Brauer characters in $p$-solvable groups for an odd prime $p$. It generalizes Cossey–Lewis–Navarro by providing an explicit description of lifts with a vertex pair $(Q,\delta)$ under weaker conditions on $Q$, using nucleus-based factorization and stability to build a bijection between Brauer-induced data and ordinary irreducible lifts. The main result shows a bijection between $I_\varphi(T|Q)$ and $L_\varphi(Q,\delta)$ for suitable $N$ and $T=N\mathbf{N}_G(Q,\delta)$, and proves the sharp bound $|L_\varphi(Q,\delta)|\le |\mathbf{N}_G(Q):\mathbf{N}_G(Q,\delta)|$, with the classical consequence $|L_\varphi|\le |Q:Q'|$ in the normal-vertex case. The work unifies and extends prior vertex results, provides a constructive framework for counting lifts, and notes a $\pi$-analogue, broadening applicability to partial character theory.

Abstract

Let $p$ be an odd prime, and suppose that $G$ is a $p$-solvable group and $\varphi\in {\rm IBr}(G)$ has vertex $Q$. In 2011, Cossey, Lewis and Navarro proved that the number of lifts of $\varphi$ is at most $|Q:Q'|$ whenever $Q$ is normal in $G$. In this paper, we present an explicit description of the set of lifts of $\varphi$ with a given vertex pair $(Q,δ)$ under a weaker condition on $Q$, and thus generalize their result.

Theorems & Definitions (20)

• Conjecture 1.1: Cossey
• Theorem 1.2: Cossey-Lewis-Navarro
• Conjecture 1.3: Cossey-Lewis
• Theorem 1
• Corollary 2
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof