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

We show that the Ariki-Terasoma-Yamada tensor module and its permutation submodules $M(λ)$ are modules for the blob algebra when the Ariki-Koike algebra is a Hecke algebra of type $B$. We show that $M(λ)$ and the standard modules $Δ(λ)$ 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 $Δ(λ)$ fails for $M(λ)$, making $M(λ)$ and $Δ(λ)$ different modules in general. Finally, we prove that $M(λ)$ is isomorphic to the dual Specht module for the Ariki-Koike algebra.

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