Parabolic Tamari Lattices in Linear Type B

TL;DR

The paper develops a type $B$ analogue of parabolic Tamari lattices by studying parabolic aligned elements for the hyperoctahedral group with a linear Coxeter element. It provides a pattern-avoidance description of aligned elements, introduces a projection-based congruence to form a quotient lattice, and defines the type $B$ parabolic Tamari lattice $ extsf{Tam}_{B}(oldsymbol{eta})$ as a natural, new combinatorial object. The results establish that $ extsf{Tam}_{B}(oldsymbol{eta})$ is a congruence-uniform, semidistributive, and trim lattice arising as a quotient of the weak order on parabolic quotients $rak{H}_{oldsymbol{eta}}$, with explicit split/join cases and an explicit projection to the aligned set. These constructions extend type $A$ parabolic Catalan structures to type $B$, offering new combinatorial models, potential connections to broader Coxeter–Catalan phenomena, and avenues for further enumerative and algebraic exploration in the hyperoctahedral setting.

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 (Mühle 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 (Mühle and Williams, 2019). These lattices have not appeared in the literature before.

Figures (8)

• Figure 1: The lattice $\mathsf{Tam}_{B}(3)$.
• Figure 2: The skew shape $\lambda/\mu$, where $\lambda=(10,9,8,7,6)$ and $\mu=(5,4,3,2,1)$, consists of the gray boxes.
• Figure 3: The weak order on $\mathfrak{H}_{(1,2)}$ with the congruence classes with respect to $\Theta$ highlighted.
• Figure 4: The weak order on $\mathfrak{H}_{(0,1,2)}$ with the congruence classes with respect to $\Theta$ highlighted.
• Figure 5: The lattice $\mathsf{Tam}_{B}\bigl((1,2)\bigr)$.

Theorems & Definitions (105)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: reading06cambrian
• Theorem 2.2: gratzer11lattice and freese95free
• Proposition 2.3: pudlak74yeast
• Lemma 2.4: day79characterizations
• Lemma 2.5: freese95free
• Proposition 2.6: thomas06analoguethomas19rowmotion
• Theorem 2.7: thomas19rowmotion
• Remark 3.1