Minkowski Decomposition of Associahedra and Related Combinatorics

TL;DR

This paper explores Minkowski decompositions of associahedra, providing methods to compute their coefficients efficiently, and offers combinatorial interpretations linked to the geometry of their normal fans.

Contribution

It introduces simplified computation methods for Minkowski coefficients of associahedra and connects these to combinatorial structures and the geometry of their normal fans.

Findings

Efficient computation of tight right-hand sides for redundant inequalities.

Simplification of Minkowski coefficient calculations requiring at most four values.

A combinatorial interpretation of Minkowski coefficients using labeled polygons.

Abstract

Realisations of associahedra with linearly non-isomorphic normal fans can be obtained by alteration of the right-hand sides of the facet-defining inequalities from a classical permutahedron. These polytopes can be expressed as Minkowski sums and differences of dilated faces of a standard simplex as described by Ardila, Benedetti & Doker (2010). The coefficients $y_I$ of such a Minkowski decomposition can be computed by M\"obius inversion if tight right-hand sides $z_I$ are known not just for the facet-defining inequalities of the associahedron but also for all inequalities of the permutahedron that are redundant for the associahedron. We show for certain families of these associahedra: (a) how to compute tight values $z_I$ for the redundant inequalities from the values $z_I$ for the facet-defining inequalities; (b) the computation of the values $y_I$ of Ardila, Benedetti & Doker can be significantly simplified and at most four values $z_{a(I)}$, $z_{b(I)}$, $z_{c(I)}$ and $z_{d(I)}$ are needed to compute $y_I$; (c) the four indices $a(I)$, $b(I)$, $c(I)$ and $d(I)$ are determined by the geometry of the normal fan of the associahedron and are described combinatorially; (d) a combinatorial interpretation of the values $y_I$ using a labeled $n$-gon. This last result is inspired from similar interpretations for vertex coordinates originally described originally by J.-L. Loday and well-known interpretations for the $z_I$-values of facet-defining inequalities.