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Geometric realizations of the $s$-weak order and its lattice quotients (2405.02092v2)

Published 3 May 2024 in math.CO

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.

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