Geometric realizations of the <i>s</i>‐weak order and its lattice quotients

Abstract For an ‐tuple of nonnegative integers, the ‐weak order is a lattice structure on ‐trees, generalizing the weak order on permutations. We first describe the join irreducible elements, the canonical join representations, and the forcing order of the ‐weak order in terms of combinatorial objects, generalizing the arcs, the noncrossing 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 ‐weak order and all its lattice quotients as polyhedral complexes, generalizing the quotient fans and quotientopes of the weak order.