Parabolic Tamari Lattices in Linear Type B
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| author | Fang, Wenjie Mühle, Henri Novelli, Jean-Christophe |
| author_facet | Fang, Wenjie Mühle, Henri Novelli, Jean-Christophe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_13400 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Parabolic Tamari Lattices in Linear Type B Fang, Wenjie Mühle, Henri Novelli, Jean-Christophe Combinatorics 05E15 (Primary), 06A07, 20F55(Secondary) 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. |
| title | Parabolic Tamari Lattices in Linear Type B |
| topic | Combinatorics 05E15 (Primary), 06A07, 20F55(Secondary) |
| url | https://arxiv.org/abs/2112.13400 |