Parabolic Tamari Lattices in Linear Type B

TL;DR

This paper introduces and characterizes parabolic Tamari lattices in type B Coxeter groups, providing a combinatorial model and establishing their lattice structure as a new contribution to algebraic combinatorics.

Contribution

It defines parabolic Tamari lattices for type B Coxeter groups, extending the concept from type A and providing a combinatorial and lattice-theoretic framework.

Findings

Introduces a combinatorial model via pattern avoidance.

Establishes an equivalence relation extending to a lattice congruence.

Defines the type-B parabolic Tamari lattice as a new algebraic structure.

Abstract

We study parabolic aligned elements associated with the type-$B$ Coxeter group and the so-called linear Coxeter element. These elements were introduced algebraically in (M\"uhle and Williams, 2019) for parabolic quotients of finite Coxeter groups and were characterized by a certain forcing condition on inversions. We focus on the type-$B$ case and give a combinatorial model for these elements in terms of pattern avoidance. Moreover, we describe an equivalence relation on parabolic quotients of the type-$B$ Coxeter group whose equivalence classes are indexed by the aligned elements. We prove that this equivalence relation extends to a congruence relation for the weak order. The resulting quotient lattice is the type-$B$ analogue of the parabolic Tamari lattice introduced for type $A$ in (M\"uhle and Williams, 2019). These lattices have not appeared in the literature before.