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

TL;DR

This paper explores the representation theory of the $bt$-algebra, introducing permutation modules and analyzing the annihilator ideal, leading to a quotient algebra that generalizes known Temperley-Lieb algebras.

Contribution

It introduces new permutation modules for the $bt$-algebra and studies the annihilator ideal, connecting it to generalized Temperley-Lieb algebras.

Findings

Tensor product module decomposes into permutation modules.

Annihilator ideal has a compatible cellular basis.

Quotient algebra generalizes Temperley-Lieb algebras.

Abstract

We study the representation theory of the braids and ties algebra, or the $bt$-algebra, $ \cal E$. Using the cellular basis $\{m_{{\mathfrak s} {\mathfrak t}} \}$ for $ \cal E$ obtained in previous joint work with J. Espinoza we introduce two kinds of permutation modules $M(\lambda)$ and $ M(\Lambda) $ for $\cal E$. We show that the tensor product module $V^{\otimes n}$ for $\cal E$ is a direct sum of $ M(\lambda)$'s. We introduce the dual cellular basis $\{n_{{\mathfrak s} {\mathfrak t}} \}$ for $ \cal E $ and study its action on $ M(\lambda) $ and $ M(\Lambda ) $. We show that the annihilator ideal $ \cal I $ in $ \cal E $ of $ V^{\otimes n } $ enjoys a nice compatibility property with respect to $\{n_{{\mathfrak s} {\mathfrak t}} \}$. We finally study the quotient algebra $ {\cal E}/{\cal I} $, showing in particular that it is a simultaneous generalization of H\"arterich's 'generalized Temperley-Lieb algebra' and Juyumaya's 'partition Temperley-Lieb algebra'.