Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

McKay bijections and decomposition numbers

David Cabrera-Berenguer

TL;DR

This paper proves Navarro's prediction for $p$-solvable finite groups by showing there exists a McKay bijection $f: {\rm Irr}_{p'}(G) \to {\rm Irr}_{p'}({\bf N}_{G}(P))$ that preserves $p'$-decomposition numbers against linear Brauer characters, i.e., $d_{\chi \varphi} = d_{f(\chi) \varphi_{{\bf N}_{G}(P)}}$ for all $\chi$ and linear $\varphi$. The authors implement an induction on $|G|$ using Gallagher, Clifford, and Mackey theory to piece together local bijections over invariant linear characters and extend them to a global correspondence, thus establishing Theorem A. As a corollary, they show $d_{\chi 1} \in \{0,1\}$ with the count of such $\chi$ equal to the number of ${\bf N}_{G}(P)$-orbits on $P/P'$, linking decomposition data to orbit counts. These results advance understanding of how decomposition matrices interact with McKay-type bijections in the solvable setting, while also clarifying limitations outside this context.

Abstract

If $G$ is $p$-solvable, we prove that there exists a McKay bijection that respects the decomposition numbers $d_{χ\varphi}$, whenever $\varphi$ is linear.

McKay bijections and decomposition numbers

TL;DR

This paper proves Navarro's prediction for -solvable finite groups by showing there exists a McKay bijection that preserves -decomposition numbers against linear Brauer characters, i.e., for all and linear . The authors implement an induction on using Gallagher, Clifford, and Mackey theory to piece together local bijections over invariant linear characters and extend them to a global correspondence, thus establishing Theorem A. As a corollary, they show with the count of such equal to the number of -orbits on , linking decomposition data to orbit counts. These results advance understanding of how decomposition matrices interact with McKay-type bijections in the solvable setting, while also clarifying limitations outside this context.

Abstract

If is -solvable, we prove that there exists a McKay bijection that respects the decomposition numbers , whenever is linear.
Paper Structure (2 sections, 8 theorems, 29 equations)

This paper contains 2 sections, 8 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Lemma 2.1

Suppose that $G$ is a finite group, $N\unlhd G$, and $H\leq G$ is such that $G=NH$. Let $M=N\cap H$. Then the restriction map ${\rm Char}(G/N)\to{\rm Char}(H/M)$ is a bijection satisfying for $\alpha,\beta\in{\rm Char}(G/N)$. Hence, the restriction defines a bijection $\operatorname{Irr}(G/N)\to\operatorname{Irr}(H/M)$.

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Lemma 2.2: Gallagher correspondence
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 6 more