Specht modules and semisimplicity criteria for Brauer and Birman--Murakami--Wenzl Algebras
Abstract
A construction of bases for cell modules of the Birman--Murakami--Wenzl (or B--M--W) algebra Bn(q,r) by lifting bases for cell modules of Bn-1(q,r) is given. By iterating this procedure, we produce cellular bases for B--M--W algebras on which a large abelian subalgebra, generated by elements which generalise the Jucys--Murphy elements from the representation theory of the Iwahori--Hecke algebra of the symmetric group, acts triangularly. The triangular action of this abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters q and r, for B--M--W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori--Hecke algebra of the symmetric group.