On q-integrals over order polytopes (extended abstract)

International audience A q-integral over an order polytope coming from a poset is interpreted as a generating function of linear extensions of the poset. As an application, theq-beta integral and aq-analog of Dirichlet’s integral are computed. A combinatorial interpretation of aq-Selberg integral is also obtained.

Incompatible Double Posets and Double Order Polytopes

In 1986 Stanley associated to a poset the order polytope. The close interplay between its combinatorial and geometric properties makes the order polytope an object of tremendous interest. Double posets were introduced in 2011 by Malvenuto and Reutenauer as a generalization of Stanleys labelled posets. A double poset is a finite set equipped with two partial orders. To a double poset Chappell, Friedl and Sanyal (2017) associated the double order polytope. They determined the combinatorial structure for the class of compatible double posets. In this paper we generalize their description to all double posets and we classify the 2-level double order polytopes.

Proof of Chapoton's Conjecture on Newton Polygons of $q$-Ehrhart Polynomials

Recently, Chapoton found a $q$-analog of Ehrhart polynomials, which are polynomials in $x$ whose coefficients are rational functions in $q$. Chapoton conjectured the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial of an order polytope. In this paper, we prove Chapoton's conjecture.

Quadratic Gröbner bases arising from partially ordered sets

The order polytope $\mathcal {O}(P)$ and the chain polytope $\mathcal {C}(P)$ associated to a partially ordered set $P$ are studied. In this paper, we introduce the convex polytope $\Gamma (\mathcal {O}(P), -\mathcal {C}(Q))$ which is the convex hull of $\mathcal {O}(P) \cup (-\mathcal {C}(Q))$, where both $P$ and $Q$ are partially ordered sets with $|P|=|Q|=d$. It will be shown that $\Gamma (\mathcal {O}(P), -\mathcal {C}(Q))$ is a normal and Gorenstein Fano polytope by using the theory of reverse lexicographic squarefree initial ideals of toric ideals.

NONCROSSING SETS AND A GRASSMANN ASSOCIAHEDRON

We study a natural generalization of the noncrossing relation between pairs of elements in$[n]$to$k$-tuples in$[n]$that was first considered by Petersenet al.[J. Algebra324(5) (2010), 951–969]. We give an alternative approach to their result that the flag simplicial complex on$\binom{[n]}{k}$induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product$[k]\times [n-k]$of two chains (also called Gelfand–Tsetlin polytope), and that it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). We then observe that this already implies the existence of a flag simplicial polytope generalizing the dual associahedron, whose Stanley–Reisner ideal is an initial ideal of the Grassmann–Plücker ideal, while previous constructions of such a polytope did not guarantee flagness nor reduced to the dual associahedron for$k=2$. On our way we provide general results about order polytopes and their triangulations. We call the simplicial complex thenoncrossing complex, and the polytope derived from it the dualGrassmann associahedron. We extend results of Petersenet al.[J. Algebra324(5) (2010), 951–969] showing that the noncrossing complex and the Grassmann associahedron naturally reflect the relations between Grassmannians with different parameters, in particular the isomorphism$G_{k,n}\cong G_{n-k,n}$. Moreover, our approach allows us to show that the adjacency graph of the noncrossing complex admits a natural acyclic orientation that allows us to define aGrassmann–Tamari orderon maximal noncrossing families. Finally, we look at the precise relation of the noncrossing complex and the weak separability complex of Leclerc and Zelevinsky [Amer. Math. Soc. Transl.181(2) (1998), 85–108]; see also Scott [J. Algebra290(1) (2005), 204–220] among others. We show that the weak separability complex is not only a subcomplex of the noncrossing complex as noted by Petersenet al.[J. Algebra324(5) (2010), 951–969] but actually its cyclically invariant part.

Unimodular Equivalence of Order and Chain Polytopes

Order polytope and chain polytope are two polytopes that arise naturally from a finite partially ordered set. These polytopes have been deeply studied from viewpoints of both combinatorics and commutative algebra. Even though these polytopes possess remarkable combinatorial and algebraic resemblance, they seem to be rarely unimodularly equivalent. In the present paper, we prove the following simple and elegant result: the order polytope and chain polytope for a poset are unimodularly equivallent if and only if that poset avoid the 5-element "X" shape subposet. We also explore a few equivalent statements of the main result.

Noncrossing sets and a Graßmannian associahedron

International audience We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$-tuples in $[n]$. We show that the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product $[k] \times [n-k]$ of two chains, and it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). This shows the existence of a flag simplicial polytope whose Stanley-Reisner ideal is an initial ideal of the Graßmann-Plücker ideal, while previous constructions of such a polytope did not guaranteed flagness. The simplicial complex and the polytope derived from it naturally reflect the relations between Graßmannians with different parameters, in particular the isomorphism $G_{k,n} \cong G_{n-k,n}$. This simplicial complex is closely related to the weak separability complex introduced by Zelevinsky and Leclerc. Nous étudions une généralisation naturelle de la relation entre les paires d’éléments non-croisés de $[n]$ et les $k$-uplets de $[n]$. Nous montrons que le complexe simplicial de drapeau sur $\binom{[n]}{k}$ induit par cette relation est une triangulation régulière, unimodulaire et de drapeau du polytope d’ordre de l’ensemble partiellement ordonné obtenu par le produit $[k] \times [n-k]$ des deux chaînes, et c’est la jointure d’un simplexe et une sphère (c’est-à-dire qu’elle est une triangulation de Gorenstein). Cela montre l’existence d’un polytope simplicial de drapeau dont l’idéal de Stanley-Reisner est un idéal initial de l’idéal de Graßmann-Plücker, tandis que les constructions précédentes d’un tel polytope ne garantissaient pas la propriété de drapeau. Le complexe simplicial et le polytope qui en découle reflètent naturellement les relations entre les Grassmanniens avec différents paramètres, en particulier l’isomorphisme $G_{k,n} \cong G_{n-k,n}$. Ce complexe simplicial est étroitement lié au complexe de séparabilité faible étudié par Zelevinskyet Leclerc.

Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes

International audience Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras. Stanley (1986) a montré que chaque ensemble fini partiellement ordonné permet de définir deux polyèdres, le polyèdre de l'ordre et le polyèdre des cha\^ınes. Ces polyèdres ont le même polynôme de Ehrhart, bien qu'ils soient tout à fait distincts du point de vue combinatoire. On généralise ce résultat à une famille plus générale de polyèdres, construits à partir d'un ensemble partiellement ordonné ayant des entiers attachés à certains de ses éléments. Par cette construction, on explique en termes combinatoires la relation entre les polyèdres de Gelfand-Tsetlin (1950) et ceux de Feigin-Fourier-Littelmann-Vinberg (2010, 2005), qui apparaissent dans la théorie des représentations des algèbres de Lie linéaires spéciales. On utilise les polyèdres de Gelfand-Tsetlin généralisés par Berenstein et Zelevinsky (1989) afin d'obtenir des analogues (conjecturés) des polytopes de Feigin-Fourier-Littelmann-Vinberg pour les algèbres de Lie symplectiques et orthogonales impaires.