Quantization in Representation Theory, Derived Algebraic Geometry, and Gauge Theory

The Dyck path algebra associated to a surface

19 Sept 2024, 17:30

1h

SRS

Speaker

Léa Bittmann (Université de Strasbourg)

Description

The Dyck algebra ${\mathbb{A}}_q$ and the double Dyck path algebra ${\mathbb{A}}_{q,t}$ were introduced by Carlsson and Mellit as part of their proof of the shuffle conjecture and the latter is known to be related to the type A double affine Hecke algebra. In this talk, we will see how to define a skein theoretic version of Dyck path algebra $\mathbb{A}(\mathrm{\Sigma })$ associated to a surface $\mathrm{\Sigma }$. We will focus on the following cases: the disk, the annulus and the torus. These last two give variants of the Dyck and double Dyck path algebra, respectively. By studying these algebras further, we give a basis of $\mathbb{A}(D)$, together with a tableau interpretation, and conjecture one for $\mathbb{A}(A)$. This is based on a joint work in progress with A. Mellit and C. Novarini.