[Paper Review] Removahedral congruences versus permutree congruences

Doriann Albertin, Vincent Pilaud+1 more|arXiv (Cornell University)|May 30, 2020

Advanced Combinatorial MathematicsMathematics20 references5 citations

TL;DR

This paper establishes a sharp dichotomy between removahedra and quotientopes in the context of lattice congruences of the weak order: only permutree congruences yield fans realizable as removahedra. It fully characterizes the type cones of permutree fans via combinatorial descriptions of rays, exchangeable ray pairs, and facets, leading to a complete classification of all polytopal realizations of these fans and a characterization of when their type cones are simplicial.

ABSTRACT

The associahedron is classically constructed as a removahedron, i.e. by deleting inequalities in the facet description of the permutahedron. This removahedral construction extends to all permutreehedra (which interpolate between the permutahedron, the associahedron and the cube). Here, we investigate removahedra constructions for all quotientopes (which realize the lattice quotients of the weak order). On the one hand, we observe that the permutree fans are the only quotient fans realized by a removahedron. On the other hand, we show that any permutree fan can be realized by a removahedron constructed from any realization of the braid fan. Our results finally lead to a complete description of the type cone of the permutree fans.

Motivation & Objective

• To resolve the long-standing question of which lattice congruences of the weak order admit polytopal realizations as removahedra.
• To provide a complete combinatorial description of the type cone for every permutree fan, enabling a full classification of all polytopal realizations of these fans.
• To establish a strong dichotomy: removahedra can only realize permutree congruences, not general quotientopes.
• To characterize when the type cone of a permutree fan is simplicial, linking to canonical Minkowski decompositions and kinematic space realizations.
• To unify and generalize previous constructions of associahedra and permutreehedra by showing that any braid fan realization can serve as a base for removahedra constructions of permutree fans.

Proposed method

• Define and analyze the type cone of a fan as the space of all realizations of the fan via wall-crossing inequalities, using the framework of deformation cones.
• Use combinatorial data from the decoration δ of a permutree to describe the rays of the δ-permutree fan via subsets of [n], as formalized in Proposition 32.
• Characterize exchangeable ray pairs in the fan using combinatorial conditions on subsets, as in Proposition 39, to identify adjacent maximal cones.
• Derive the facet description of the type cone of a permutree fan using pairs of subsets corresponding to wall-crossing inequalities, as in Proposition 45.
• Construct a family of polytopes Qδ(u) parameterized by positive vectors u ∈ ℝF>0, showing they realize all possible realizations of the δ-permutree fan.
• Use the parametrization of rays and facets to prove that the type cone of a permutree fan is completely determined by the decoration δ, and derive summation formulas for the number of its facets.

Experimental results

Research questions

• RQ1 Which lattice congruences of the weak order on Sn can be realized as removahedra?
• RQ2 Can every permutree fan be realized as a removahedron constructed from any realization of the braid fan?
• RQ3 What is the complete combinatorial structure of the type cone of a permutree fan?
• RQ4 For which decorations δ is the type cone of the δ-permutree fan simplicial?
• RQ5 How do the wall-crossing inequalities governing the type cone relate to the combinatorics of permutrees?

Key findings

• A lattice congruence of the weak order is realizable as a removahedron if and only if it is a permutree congruence, establishing a sharp dichotomy (Theorem 1).
• Any permutree fan can be realized as a removahedron constructed from any realization of the braid fan, not just the classical permutahedron.
• The rays of the δ-permutree fan are completely described by combinatorial conditions on subsets of [n], as given in Proposition 32.
• The pairs of rays that are exchangeable (i.e., lie in adjacent maximal cones) are characterized in Proposition 39 via specific subset pairs.
• The facets of the type cone of a δ-permutree fan are fully described by pairs of subsets corresponding to wall-crossing inequalities, as in Proposition 45.
• The type cone of a permutree fan is simplicial if and only if certain combinatorial conditions on δ are satisfied, and this leads to canonical Minkowski sum decompositions of all realizations (Corollary 54).