On the annihilator ideal in the bt-algebra of tensor space

Abstract

We study the representation theory of the braids and ties algebra, or the bt -algebra, E n ( q ) . Using the cellular basis { m s t } for E n ( q ) obtained in previous joint work with J. Espinoza we introduce two kinds of permutation modules M ( λ ) and M ( Λ ) for E n ( q ) . We show that the tensor product module V ⊗ n for E n ( q ) is a direct sum of M ( λ ) 's. We introduce the dual cellular basis { n s t } for E n ( q ) and study its action on M ( λ ) and M ( Λ ) . We show that the annihilator ideal I in E n ( q ) of V ⊗ n enjoys a nice compatibility property with respect to { n s t } . We finally study the quotient algebra E n ( q ) / I , showing in particular that it is a simultaneous generalization of Härterich's ‘generalized Temperley-Lieb algebra’ and Juyumaya's ‘partition Temperley-Lieb algebra’.