1 research outputs found
Two double poset polytopes
To every poset P, Stanley (1986) 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 (2011) introduced
'double posets', that is, (finite) sets equipped with two partial orders, as a
generalization of Stanley's labelled 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 Groebner bases, which gives an
algebraic description of the unimodular flag triangulations described in the
first part.Comment: 36 pages, 5 figures, improved exposition, minor fixes, and changes
suggested by referees; accepted at SIAM J Discrete Mat