# Steen Ryom-Hansen (Universidad de Talca)

## Graded cellular basis and Jucys-Murphy elements for the generalized blob algebra

The generalized blob algebra $$b_n$$ was introduced by Martin and Woodcock. It can be considered as a higher level Temperley-Lieb algebra, although there is no natural diagram calculus associated with $$b_n$$. The representation theory of the Temperley-Lieb algebra can be seen as a toy model for modular representation theory, but this is not at all the case for the representation theory for $$b_n$$, which appears to contain 'the full story'. In the talk we shall explain how to construct a graded cellular basis for $$b_n$$ in any characteristic. A first step is here given by the Brundan-Kleshchev and Rouquier isomorphism between the cyclotomic KLR-algebra and the cyclotomic Hecke algebra. A main obstacle is here that the known cellular structure on the cyclotomic Hecke algebra is related to the dominance order on multipartitions, which is badly behaved on $$b_n$$. In the talk we explain how to solve this problem. Our solution gives rise to a family of Jucys-Murphy elements on $$b_n$$.