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

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'.