Geometric realizations of the s-weak order and its lattice quotients

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.