On Blocks with One Modular Character

TL;DR

This paper proves that for certain blocks with a unique modular character, this character can be lifted to an ordinary character, confirming a key conjecture in modular representation theory.

Contribution

It establishes that blocks with a single modular character have this character liftable to an ordinary, p-rational character, confirming the basic set conjecture for these blocks.

Findings

Unique modular character is liftable to an ordinary character

Lifted character is p-rational for odd p

Confirms the basic set conjecture for these blocks

Abstract

Suppose that $B$ is a Brauer $p$-block of a finite group $G$ with a unique modular character $\varphi$. We prove that $\varphi$ is liftable to an ordinary character of $G$ (which moreover is $p$-rational for odd $p$). This confirms the basic set conjecture for these blocks.