The Ariki-Terasoma-Yamada tensor space and the blob-algebra

We show that the Ariki-Terasoma-Yamada tensor module and its permutation submodules $M(\lambda )$ are modules for the blob algebra when the Ariki-Koike algebra is a Hecke algebra of type $B$. We show that $M(\lambda )$ and the standard modules $\mathrm{\Delta }(\lambda )$ have the same dimensions, the same localization and similar restriction properties and are equal in the Grothendieck group. Still we find that the universal property for $\mathrm{\Delta }(\lambda )$ fails for $M(\lambda )$, making $M(\lambda )$ and $\mathrm{\Delta }(\lambda )$ different modules in general. Finally, we prove that $M(\lambda )$ is isomorphic to the dual Specht module for the Ariki-Koike algebra.