Specht modules and semisimplicity criteria for Brauer and Birman--Murakami--Wenzl Algebras

A construction of bases for cell modules of the Birman–Murakami–Wenzl (or B–M–W) algebra $B_n(q,r)$ by lifting bases for cell modules of $B_{n-1}(q,r)$ 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.