Type $\tilde{C}$ Temperley-Lieb algebra quotients and Catalan combinatorics

Abstract

We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type C, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.

The nil-blob algebra: An incarnation of type $\tilde{A}_1$ Soergel calculus and of the truncated blob algebra

“…On the other hand, the set N M n is in bijection with the set of positive fully commutative elements of the Coxeter group of type B n . In particular, the cardinality of N M n is known to be 2n n , see for example [1]. Hence we deduce that dim…”

“…1 n and s k ∈ S. Then k and k + 1 are in different columns of t and so we conclude that the functions p t and p ts k are equal except that p t (k) = p ts k (k) ± 2, and hence also the paths P t and P ts k are equal except in the interval [k − 1, k + 1] where they are related in the following two possible ways…”