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Geometric realizations of the $s$-weak order and its lattice quotients
- Publication Year :
- 2024
-
Abstract
- 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.<br />Comment: 50 pages, 33 figures. Version 2: minor corrections in particular in a few pictures, added Remark 98
- Subjects :
- Mathematics - Combinatorics
06B10, 52B11, 52B12
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2405.02092
- Document Type :
- Working Paper