General runner removal and the Mullineux map

TL;DR

This paper introduces a new runner removal theorem for q-decomposition numbers in the context of level 1 Fock space, and links it to the Mullineux map, enabling finite computation of certain decomposition numbers.

Contribution

It generalizes previous runner removal theorems and connects them to the Mullineux map, providing a finite method to compute q-decomposition numbers.

Findings

Established a new runner removal theorem for q-decomposition numbers.

Connected the runner removal theorem to the Mullineux map.

Showed that computing all q-decomposition numbers of a given weight is a finite process.

Abstract

We prove a new `runner removal theorem' for $q$-decomposition numbers of the level 1 Fock space of type $A^{(1)}_{e-1}$, generalising earlier theorems of James--Mathas and the author. By combining this with another theorem relating to the Mullineux map, we show that the problem of finding all $q$-decomposition numbers indexed by partitions of a given weight is a finite computation.