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

23 pages. Final version accepted for publication in JPAA