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

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.