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

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

2007-05-29

arxiv.org/abs/0705.4142v1

Mathematics

Representation Theory

Scientific paper

10.1007/s10801-007-0058-3

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.

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