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arXiv · 1204.4547

Minkowski Decomposition of Associahedra and Related Combinatorics

Carsten Lange
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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 Möbius 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.

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BibTeXRIS

Carsten Lange. 2015-06-25. Minkowski Decomposition of Associahedra and Related Combinatorics. https://doi.org/10.1007/s00454-013-9546-5

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