Geometric realizations of the $s$-weak order and its lattice quotients
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914180516282368 |
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| author | Philippe, Eva Pilaud, Vincent |
| author_facet | Philippe, Eva Pilaud, Vincent |
| contents | For an $n$-tuple $s$ of non-negative integers, the $s$-weak order is a lattice structure on $s$-trees, generalizing the weak order on permutations. We first describe the join irreducible elements, the canonical join representations, and the forcing order of the $s$-weak order in terms of combinatorial objects, generalizing the arcs, the non-crossing arc diagrams, and the subarc order for the weak order. We then extend the theory of shards and shard polytopes to construct geometric realizations of the $s$-weak order and all its lattice quotients as polyhedral complexes, generalizing the quotient fans and quotientopes of the weak order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02092 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric realizations of the $s$-weak order and its lattice quotients Philippe, Eva Pilaud, Vincent Combinatorics 06B10, 52B11, 52B12 For an $n$-tuple $s$ of non-negative integers, the $s$-weak order is a lattice structure on $s$-trees, generalizing the weak order on permutations. We first describe the join irreducible elements, the canonical join representations, and the forcing order of the $s$-weak order in terms of combinatorial objects, generalizing the arcs, the non-crossing arc diagrams, and the subarc order for the weak order. We then extend the theory of shards and shard polytopes to construct geometric realizations of the $s$-weak order and all its lattice quotients as polyhedral complexes, generalizing the quotient fans and quotientopes of the weak order. |
| title | Geometric realizations of the $s$-weak order and its lattice quotients |
| topic | Combinatorics 06B10, 52B11, 52B12 |
| url | https://arxiv.org/abs/2405.02092 |