NDLI: Two Double Poset Polytopes

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Sanyal, Raman Chappell, Thomas Friedl, Tobias
Copyright Year	2017
Abstract	To every poset $P$, Stanley [Discrete Comput. Geom., 1 (1986), pp. 9--23] associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of $P$. This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer [J. Combin. Theory Ser. A, 118 (2011), pp. 1322--1333] introduced double posets, that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labeled posets. Many combinatorial constructions can be naturally phrased in terms of double posets. We introduce the double order polytope and the double chain polytope and we amply demonstrate that they geometrically capture double posets, i.e., the interaction between the two partial orders. We describe the facial structures, Ehrhart polynomials, and volumes of these polytopes in terms of the combinatorics of double posets. We also describe a curious connection to Geissinger's valuation polytopes and we characterize 2-level polytopes among our double poset polytopes. Fulkerson's anti-blocking polytopes from combinatorial optimization subsume stable set polytopes of graphs and chain polytopes of posets. We determine the geometry of Minkowski and Cayley sums of anti-blocking polytopes. In particular, we describe a canonical subdivision of Minkowski sums of anti-blocking polytopes that facilitates the computation of Ehrhart (quasi-) polynomials and volumes. This also yields canonical triangulations of double poset polytopes. Finally, we investigate the affine semigroup rings associated to double poset polytopes. We show that they have quadratic Gröbner bases, which gives an algebraic description of the unimodular flag triangulations described in the first part.
Sponsorship	Deutsche Forschungsgemeinschaft. Berlin Mathematical School. Freie Universität Berlin
Starting Page	2378
Ending Page	2413
Page Count	36
File Format	PDF
ISSN	08954801
DOI	10.1137/16M1091800
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	31
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-10-24
Access Restriction	Subscribed
Subject Keyword	double order polytope Lattice polytopes Algebraic aspects of posets Combinatorics of partially ordered sets Ehrhart polynomials Special polytopes volumes double posets anti-blocking polytopes double chain polytope Birkhoff lattice Gröbner bases
Content Type	Text
Resource Type	Article
Subject	Mathematics

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