Empty runner removal theorem for Ariki-Koike algebras

TL;DR

This paper extends the empty runner removal theorem from type A Iwahori-Hecke algebras to Ariki-Koike algebras, relating their $v$-decomposition numbers through abacus display modifications.

Contribution

It generalizes a known theorem to a broader class of algebras, providing new insights into their $v$-decomposition numbers.

Findings

Established a similar relationship for $v$-decomposition numbers in Ariki-Koike algebras.

Extended the empty runner removal theorem to a more general algebraic setting.

Abstract

For the Iwahori-Hecke algebras of type $A$, James and Mathas proved a theorem which relates $v$-decomposition numbers for different values of $e$, by adding empty runners to the James' abacus display. This result is often referred to as the empty runner removal theorem. In this paper, we extend this theorem to the Ariki-Koike algebras, establishing a similar relationship for the $v$-decomposition numbers.