Mixed Schur-Weyl duality between general linear Lie algebras and cyclotomic walled Brauer algebras

TL;DR

This paper establishes a new duality between general linear Lie algebras and cyclotomic walled Brauer algebras at arbitrary levels, enabling classification of highest weight vectors and efficient computation of decomposition matrices.

Contribution

It introduces mixed Schur-Weyl duality for arbitrary levels, extending prior work and providing tools for analyzing decomposition matrices of cyclotomic walled Brauer algebras.

Findings

Established mixed Schur-Weyl duality at arbitrary levels.

Classified highest weight vectors using cellular bases.

Provided an efficient method for computing decomposition matrices.

Abstract

Motivated by Brundan-Kleshchev's work on higher Schur-Weyl duality, we establish mixed Schur-Weyl duality between general linear Lie algebras and cyclotomic walled Brauer algebras in an arbitrary level. Using weakly cellular bases of cyclotomic walled Brauer algebras, we classify highest weight vectors of certain mixed tensor modules of general linear Lie algebras. This leads to an efficient way to compute decomposition matrices of cyclotomic walled Brauer algebras arising from mixed Schur-Weyl duality, which generalizes early results on level two walled Brauer algebras.