La prochaine séance du séminaire de combinatoire du plateau de Saclay aura lieu ce lundi 4 avril à 15h en ligne (informations de connexion ci-dessous) avec retransmission en salle Philippe Flajolet du LIX. Nous aurons le plaisir d’écouter Luis Crespo Ruiz (Universidad de Cantabria) nous parler de « Multitriangulations and tropical Pfaffians ». Le résumé et les informations de connexion sont disponibles ci-dessous.

Le programme du séminaire est disponible sur la page https://galac.lri.fr/fr/pages/combi-seminar.html

Résumé : The k-associahedron Ass_k(n) is the simplicial complex of (k + 1)-crossing-free subgraphs of the complete graph with vertices on a circle. Its facets are called k-triangulations. We explore the connection of Ass_k(n) with the Pfaffian variety $Pf_k(n)\subset K^\binom{n}{2}$ of antisymmetric matrices of rank  ≤ 2k. First, we characterize the Gröbner cone $Grob_k(n)\subset \mathbb{R}^\binom{n}{2}$ producing as initial ideal of I(Pf_k(n)) the Stanley-Reisner ideal of Ass_k(n) (that is, the monomial ideal generated by (k + 1)-crossings). We then look at the tropicalization of Pf_k(n) and show that Ass_k(n) embeds naturally as the intersection of trop(Pf_k(n)) and Grob_k(n), and is contained in the totally positive part trop⁺(Pfk(n)) of it. We show that for k = 1 and for each triangulation T of the n-gon, the projection of this embedding of Ass_k(n) to the n − 3 coordinates corresponding to diagonals in T gives a complete polytopal fan, realizing the associahedron. This fan is linearly isomorphic to the g-vector fan of the cluster algebra of type A, shown to be polytopal by Hohlweg, Pilaud and Stella in 2018.

Informations de connexion :

• Lien : https://ecolepolytechnique.zoom.us/j/84295489822?pwd=QW5Xcjhqbjg5NWJ2L2JQRzhmTkhrUT09
• ID de réunion : 842 9548 9822
• Code secret : 139712