Graded sum formula for $\tilde{A}_1$-Soergel calculus and the nil-blob algebra

TL;DR

This paper explores the representation theory of the Soergel calculus algebra in type A_1, generalizes the nil-blob algebra to a two-parameter setting, and provides explicit diagonalization and sum formulas for related modules.

Contribution

It introduces a two-parameter generalization of the nil-blob algebra and applies it to explicitly diagonalize bilinear forms and derive sum formulas in type A_1.

Findings

Diagonalization of bilinear form matrices using Jones-Wenzl idempotents.

Explicit sum formulas for Jantzen filtrations in the Grothendieck group.

Connection between parameters and simple roots in A_1.

Abstract

We study the representation theory of the Soergel calculus algebra $ A_w := \mbox{End}_{{\mathcal D}_{(W,S)}} (\underline{w}) $ over $\mathbb C$ in type $\tilde{A}_1$. We generalize the recent isomorphism between the nil-blob algebra ${\mathbb{NB}}_n$ and $ A_w $ to deal with the two-parameter blob algebra. Under this generalization, the two parameters correspond to the two simple roots for $\tilde{A}_1$. Using this, together with calculations involving the Jones-Wenzl idempotents for the Temperley-Lieb subalgebra of $ \mathbb{NB}_n$, we obtain a concrete diagonalization of the matrix of the bilinear form on the cell module $\Delta_w(v) $ for $ A_w $. The entries of the diagonalized matrices turn out to be products of roots for $\tilde{A}_1$. We use this to study Jantzen type filtrations of $ \Delta_w(v)$ for $A_w $. We show that at enriched Grothendieck group level the corresponding sum formula has terms $ \Delta_w(s_{\alpha }v)[ l(s_{\alpha }v)- l(v)] $, where $[ \cdot ] $ denotes grading shift.