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 $\Delta(\lambda)$ 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 $\Delta(\lambda)$ fails for $M(\lambda)$, making $M(\lambda)$ and $\Delta(\lambda)$ different modules in general. Finally, we prove that $M(\lambda)$ is isomorphic to the dual Specht module for the Ariki-Koike algebra.Comment: Improved version

The Automorphism Group of Certain Higher Degree Forms

We consider symmetric d-linear forms of dimension n over an algebraically closed field k of characteristic 0. The "center" of a form is the analogous of the space of symmetric matrices of a bilinear form. For d>2 the center is a commutative subalgebra of $\mathbf{M}_n(k)$. The automorphism group of the form acts naturally on the center. We give a description of this group via this action.Comment: Title changed. Final version, to appear in appear in Journal of Pure and Applied Algebr

Graded cellular bases for Temperley-Lieb algebras of type A and B

We show that the Temperley-Lieb algebra of type $A$ and the blob algebra (also known as the Temperley-Lieb algebra of type $B$) at roots of unity are $\mathbb Z$-graded algebras.We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.Comment: 36 pages. Final version, to appear in Journal of Algebraic Combinatoric