On the structure of modules over walled Brauer algebra via normal form and random walks

TL;DR

This paper investigates the structure of modules over the walled Brauer algebra using a normal form, revealing connections to symmetric group combinatorics and differential posets for module dimension calculations.

Contribution

It introduces a normal form for cyclic cell modules over the walled Brauer algebra and relates their basis monomials to those of the symmetric group, providing new structural insights.

Findings

Number of reduced basis monomials matches that of the symmetric group.

Dimensions of modules in the semisimple case are computed via differential posets.

Number of primitive idempotents equals that of the symmetric group.

Abstract

We analyze cyclic cell modules over walled Brauer algebra in terms of a certain normal form. The latter allows us to decompose the algebra into the generating set and annihilator ideal of a certain cyclic vector. In addition, we show that the numbers of reduced basis monomials of given length coincide with those for the symmetric group. For the semisimple case we utilize the theory of differential posets to calculate the dimensions of modules in terms of the paths in Bratelli diagram. It turns out that the number of primitive idempotents is the same as for the symmetric group.