Conference abstracts

Session S11 - Representations of Algebras

July 26, 17:30 ~ 17:50

## The blob algebra in positive characteristic and the $p$-Kazhdan-Lusztig polynomials.

### Universidad de Talca, Chile   -   dplaza@inst-mat.utalca.cl

In 2003, Martin and Woodcock observed that, over a field of characteristic zero, the decomposition numbers for the blob algebra are given by certain evaluations at $1$ of the Kazhdan-Lusztig polynomials associated to the infinite dihedral group $W_\infty$.

In this talk, we prove that in characteristic $p>0$ the decomposition numbers for the blob algebra are given by evaluations at $1$ of the $p$-Kazhdan-Lusztig polynomials associated to $W_\infty$. These polynomials arise as the entries of the change of basis matrix from the basis of the Hecke algebra $\mathcal{H}= \mathcal{H}(W_\infty)$ of $W_\infty$ obtained by decategorifying the corresponding indecomposable Soergel bimodules to the standard basis of $\mathcal{H}$.

In general, to calculate the $p$-Kazhdan-Lusztig polynomials is a very hard task. However, for $W_\infty$, we are able to provide an easy algorithm to compute them.