Séminaire Lotharingien de Combinatoire, 86B.62 (2022), 12 pp.
Wenjie Fang, Henri Mühle and Jean-Christophe Novelli
Parabolic Tamari Lattices in Linear Type B
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.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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