Decomposition numbers of quantized walled Brauer algebras

TL;DR

This paper links the decomposition numbers of quantized walled Brauer algebras to those of Hecke and q-Schur algebras, enabling computation via Kazhdan-Lusztig polynomials over complex fields.

Contribution

It establishes explicit relationships between decomposition numbers of quantized walled Brauer algebras and those of related algebraic structures, facilitating their computation.

Findings

Decomposition numbers can be computed using inverse Kazhdan-Lusztig polynomials.

Explicit formulas relate walled Brauer algebras to Hecke and q-Schur algebras.

Results apply over the complex field using known theorems.

Abstract

In this paper, we establish explicit relationship between decomposition numbers of quantized walled Brauer algebras and those for either Hecke algebras associated to certain symmetric groups or (rational) $q$-Schur algebras over a field $\kappa$. This enables us to use Ariki's result \cite{Ar} and Varagnolo-Vasserot's result \cite{VV} to compute such decomposition numbers via inverse Kazhdan-Lusztig polynomials associated with affine Weyl groups of type $A$ if the ground field is $\mathbb C$.