[Paper Review] Deformation Cones of nested Braid fans

Federico Castillo, Fu Liu|arXiv (Cornell University)|Oct 5, 2017

Advanced Combinatorial Mathematics13 references3 citations

TL;DR

This paper introduces the nested Braid fan as a refinement of the Braid fan and constructs nested permutohedra as polytopes whose normal fans coarsen this fan. It provides a combinatorial proof of the Submodularity Theorem for generalized permutohedra using deformation cones, establishes a one-to-one correspondence between faces of nested permutohedra and chains in ordered partition posets, and shows the nested Braid fan is the barycentric subdivision of the Braid fan, offering a new combinatorial framework for studying polytopes with edge directions in the form $e_i + e_j - e_k - e_\ell$. The key contribution is a systematic method to compute deformation cones and extend results on generalized permutohedra to the nested setting.

ABSTRACT

Generalized permutohedra are deformations of regular permutohedra, and arise in many different fields of mathematics. One important characterization of generalized permutohedra is the Submodular Theorem, which is related to the deformation cone of the Braid fan. We lay out general techniques for determining deformation cones of a fixed polytope and apply it to the Braid fan to obtain a natural combinatorial proof for the Submodular Theorem. We also consider a refinement of the Braid fan, called the nested Braid fan, and construct usual (respectively, generalized) nested permutohedra which have the nested Braid fan as (respectively, refining) their normal fan. We extend many results on generalized permutohedra to this new family of polytopes, including a one-to-one correspondence between faces of nested permutohedra and chains in ordered partition posets, and a theorem analogous to the Submodular Theorem. Finally, we show that the nested Braid fan is the barycentric subdivision of the Braid fan, which gives another way to construct this new combinatorial object.

Motivation & Objective

• To develop general techniques for computing deformation cones of a fixed polytope, particularly for the Braid fan.
• To provide a natural combinatorial proof of the Submodularity Theorem for generalized permutohedra using deformation cones.
• To introduce and study the nested Braid fan as a refinement of the Braid fan and construct nested permutohedra with this fan as their normal fan.
• To extend results from generalized permutohedra—such as face-face correspondences and submodularity—to the new family of nested permutohedra.
• To show that the nested Braid fan is the barycentric subdivision of the Braid fan, offering a new construction of this combinatorial object.

Proposed method

• Define deformations of a polytope P₀ as facet movements that do not pass through any vertex, using the condition that vertices of P₀ remain vertices in the deformed polytope.
• Use the equivalence between deformations and coarsenings of the normal fan to characterize the deformation cone as the set of vectors b such that the polytope {x ∈ V : Ax ≤ b} is a deformation of P₀.
• Introduce the nested Braid fan Br₂ᵈ as a refinement of the Braid fan by grouping points according to the relative order of coordinates and their first differences.
• Construct usual and generalized nested permutohedra as polytopes whose normal fans coarsen the nested Braid fan, using a parameterization by ordered pairs of permutations and integer vectors.
• Establish a one-to-one correspondence between faces of nested permutohedra and chains in ordered partition posets, generalizing a known result for generalized permutohedra.
• Prove that the nested Braid fan is the barycentric subdivision of the Braid fan by showing it arises from stellar subdivisions in a specific order, and verify this fan is projective.

Experimental results

Research questions

• RQ1 Does there exist a usual or generalized nested permutohedron for which the vertex corresponding to a pair of permutations lies in the normal cone of the corresponding face in the nested Braid fan, analogous to the property in usual permutohedra?
• RQ2 Can the permuto-associahedron be realized as a deformation of the regular nested permutohedron?
• RQ3 Is the fan obtained by further refining the nested Braid fan using second differences a projective fan, and does it admit a combinatorial interpretation?
• RQ4 Which sequences of stellar subdivisions of the simplex's normal fan yield coarsenings of the Braid fan?
• RQ5 What is the structure of the hyperplane arrangement defined by xi + xj = xk + xℓ for all tuples (i,j,k,ℓ), including repeated indices, and how many regions does it have?

Key findings

• The deformation cone of the Braid fan is characterized combinatorially, providing a new proof of the Submodularity Theorem via the correspondence between generalized permutohedra and submodular functions on 2[d+1] with f(∅) = 0.
• The nested Braid fan Br₂ᵈ is the barycentric subdivision of the Braid fan Brd, establishing a new geometric and combinatorial construction of this fan.
• A one-to-one correspondence is established between faces of nested permutohedra and chains in ordered partition posets, generalizing the known face-face correspondence for generalized permutohedra.
• The paper proves that generalized permutohedra are precisely the translations of polymatroids, and provides a direct combinatorial proof of this equivalence.
• The normal fan of the usual nested permutohedron Π²_d(M,N) is the nested Braid fan, and this fan is projective, confirming the existence of a well-behaved family of polytopes with edge directions in the form ei + ej − ek − eℓ.
• The authors show that the deformation cone of the Braid fan corresponds exactly to submodular functions on 2[d+1] with f(∅) = 0, and this correspondence is bijective, thus proving the Submodularity Theorem in a purely combinatorial way.