A cell filtration of mixed tensor space

TL;DR

This paper constructs a cellular basis for the walled Brauer algebra, enabling easier description of cell modules and revealing a filtration of mixed tensor space by these modules, advancing understanding of algebraic structures in representation theory.

Contribution

It introduces a cellular basis for the walled Brauer algebra with properties similar to the Murphy basis, facilitating analysis of cell modules and tensor space filtrations.

Findings

Cell modules of the walled Brauer algebra can be described via the new basis.

Mixed tensor space admits a filtration by cell modules of the algebra's image.

The basis simplifies the restriction of cell modules to subalgebras.

Abstract

We construct a cellular basis of the walled Brauer algebra which has similar properties as the Murphy basis of the group algebra of the symmetric group. In particular, the restriction of a cell module to a certain subalgebra can be easily described via this basis. Furthermore, the mixed tensor space possesses a filtration by cell modules -- although not by cell modules of the walled Brauer algebra itself, but by cell modules of its image in the algebra of endomorphisms of mixed tensor space.