CanaDAM 2019
SFU Harbour Centre, 29 - 31 mai 2019

Algebraic and geometric methods in combinatorics I
Org: Christophe Hohlweg (Université du Québec à Montréal)

EMILY BARNARD, Northeastern University
Graph Associahedra and the Poset of Maximal Tubings  [PDF]

Given a graph $G$ on n vertices, Postnikov defined a graph associahedron $P_G$ as an example of a generalized permutohedron, a polytope whose normal fan coarsens the braid arrangement. Combinatorially, each face of $P_G$ corresponds to certain collections of compatible subgraphs of $G$ called tubings. Graph associahedra were introduced independently by Carr and Devadoss and by Davis, Januszkiewicz, and Scott. The associahedron and permutohedron are both examples of graph associahedra. In this talk, we consider the poset obtained by orienting the one-skeleton of $P_G$ according to a certain linear functional, and its relationship to the weak order on $S_n$.

Facial weak order in hyperplane arrangements  [PDF]

We describe the facial weak order, a poset structure that extends the poset of regions on a central hyperplane arrangement to the set of all faces of the arrangement which was first introduced on the braid arrangements by Krob, Latapy, Novelli, Phan and Schewer. We provide various characterizations of this poset including a local one, a global one, and a geometric one. We then show that the facial weak order is in fact a lattice for simplicial hyperplane arrangements, generalizing a result by Björner, Edelman and Zieglar showing the poset of regions is a lattice for simplicial arrangements.

Minuscule reverse plane partitions via representations of quivers  [PDF]

We study reverse plane partitions defined on minuscule posets. We show that there is bijection between reverse plane partitions defined on a fixed minuscule poset and isomorphism classes of representations of a Dynkin quiver of the corresponding type, each of whose indecomposable summands is supported at the minuscule vertex. This bijection shows that reverse plane partitions encode the Jordan block sizes of a generic nilpotent endomorphism of the corresponding quiver representation. In addition, this bijection generalizes the Hillman-Grassl correspondence in type $A$.

The Peterson Isomorphism: Moduli of Curves and Alcove Walks  [PDF]

In this talk, I will explain the combinatorial tool of folded alcove walks, in addition to surveying a wide range of applications in combinatorics, representation theory, and algebraic geometry. As a concrete example, I will describe a labeling of the points of the moduli space of genus 0 curves in the complete complex flag variety using the combinatorial machinery of alcove walks. Following Peterson, this geometric labeling partially explains the ``quantum equals affine" phenomenon which relates the quantum cohomology of this flag variety to the homology of the affine Grassmannian. This is joint work with Arun Ram.