The orientifold Temperley--Lieb algebra

TL;DR

This paper introduces gradings on simple modules of 2-boundary Temperley--Lieb and symplectic blob algebras via quotients of orientifold quiver Hecke algebras, revealing their graded cellular structure and proposing a method to compute their graded decomposition matrices.

Contribution

It constructs gradings on these algebras by realizing them as quotients of orientifold quiver Hecke algebras, and establishes their graded cellularity, providing new tools for their representation theory.

Findings

Symplectic blob algebras are graded cellular.

First explicit finite-dimensional graded quotients of orientifold quiver Hecke algebras.

Proposed algorithm for calculating graded decomposition matrices.

Abstract

We construct gradings on the simple modules of 2-boundary Temperley--Lieb algebras and symplectic blob algebras by realising the latter algebras as quotients of Varagnolo--Vasserot's orientifold quiver Hecke algebras. We prove that the symplectic blob algebras are graded cellular and provide a conjectural algorithm for calculating their graded decomposition matrices. In doing so, we give the first explicit family of finite-dimensional graded quotients of the orientifold quiver Hecke algebras, providing a new entry point for the structure of these algebras -- in the spirit of Libedinsky--Plaza's ``blob algebra approach'' to modular representation theory.