The orientifold Temperley--Lieb algebra

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.