The orientifold Temperley--Lieb algebra

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.