EL-Labelings and Canonical Spanning Trees for Subword Complexes

Abstract

Abstract. We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an EL-labeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar EL-labeling was recently studied by M. Kallipoliti and H. Mühle.

Brick polytopes of spherical subword complexes and generalized associahedra

“…29. Consider the Coxeter element c ex := τ 1 τ 2 τ 3 of Example 6.5, whose corresponding word is c ex w • (c ex ) = τ 1 τ 2 τ 3 τ 1 τ 2 τ 3 τ 1 τ 2 τ 1 .…”

“…• in [29], the authors further study EL-labelings and canonical spanning trees for subword complexes, with applications to generation and order theoretic properties of the subword complex, and discuss their relation to alternative EL labelings of Cambrian lattices in [19], • in [30], the authors show that the vertex barycenter of generalized associahedra coincide with that of their corresponding permutahedra, using the vertex description of generalized associahedra presented here, • in [6], the first author and C. Ceballos provide a combinatorial description of the denominator vectors based on the subword complex approach, • in his recent dissertation [43], N. Williams provides an amazing conjecture together with a huge amount of computational evidence that the present approach to cluster complexes and generalized associahedra as well yields a type-independent and explicit bijection to nonnesting partitions (see also [1] for further background on this connection).…”

Associahedra Via Spines

“…In a recent paper [PS13], V. Pilaud and C. Stump studied natural generalizations of the increasing flip order to arbitrary subword complexes. They described in particular four canonical spanning trees of the flip graph of a subword complex, which led to efficient enumeration algorithms of their facets.…”

Hopf dreams and diagonal harmonics

“…To prove this result, we need to recall a result of [BB93] about the generation of all pipe dreams of a given permutation as a partial order. See also [PS13] for more details about the structure of a related partial order.…”

“…Note that the poset Π ω has a nicer structure if one also considers ladder moves (transpose of chute moves) and more generally all other flips. For example, the poset of all increasing flips on Π(ω) is shown to be shellable in [PS13]. In this section, we restrict to chute moves as it simplifies the proof of our main theorem.…”