Pavlo Pylyavskyy

A family of bijections between G -parking functions and spanning trees

For a directed graph G on vertices {0, 1, ..., n}, a G-parking function is an n-tuple (b1,...,bn) of non-negative integers such that, for every non-empty subset U ⊆ {1,...,n}, there exists a vertex j ∈ U for which there are more than bj edges going from j to G - U. We construct a family of bijective maps between the set PG of G-parking functions and the set JG of spanning trees of G rooted at 0, thus providing a combinatorial proof of |PG| = |JG|.

Schur positivity and Schur log-concavity

We prove Okounkovs conjecture, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibons conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. We also give an intriguing log-concavity property of Schur functions.

Cell transfer and monomial positivity

We give combinatorial proofs that certain families of differences of products of Schur functions are monomial-positive. We show in addition that such monomial-positivity is to be expected of a large class of generating functions with combinatorial definitions similar to Schur functions. These generating functions are defined on posets with labelled Hasse diagrams and include for example generating functions of Stanleys (P,ω)-partitions.

Inverse problem in cylindrical electrical networks

In this paper we study the inverse Dirichlet-to-Neumann problem for certain cylindrical electrical networks. We define and study a birational transformation acting on cylindrical electrical networks called the electrical

Webs on surfaces, rings of invariants, and clusters

R

Affine geometric crystals in unipotent loop groups

-matrix. We use this transformation to formulate a general conjectural solution to this inverse problem on the cylinder. This conjecture extends work of Curtis, Ingerman, and Morrow [Linear Algebra Appl., 283 (1998), pp. 115--150] and of de Verdiere, Gitler, and Vertigan [Comment. Math. Helv., 71 (1996), pp. 144--167] for circular planar electrical networks. We show that our conjectural solution holds for certain „purely cylindrical” networks. Here we apply the grove combinatorics introduced by Kenyon and Wilson [Trans. Amer. Math. Soc., 363 (2011), pp. 1325--1364].

Y-meshes and generalized pentagram maps

Significance We investigate cluster algebra structures in classical rings of invariants for the special linear group SL3. The key role is played by the combinatorics of webs on marked bordered surfaces. We construct and study cluster algebra structures in rings of invariants of the special linear group action on collections of 3D vectors, covectors, and matrices. The construction uses Kuperberg’s calculus of webs on marked surfaces with boundary.

Combinatorics of K -theory via a K -theoretic Poirier-Reutenauer bialgebra

We study products of the affine geometric crystal of type A corresponding to symmetric powers of the standard representation. The quotient of this product by the R-matrix action is constructed inside the unipotent loop group. This quotient crystal has a semi-infinite limit, where the crystal structure is described in terms of limit ratios previously appearing in the study of total positivity of loop groups.

BOX-BASKET-BALL SYSTEMS

We introduce a rich family of generalizations of the pentagram map sharing the property that each generates an infinite configuration of points and lines with four points on each line. These systems all have a description as

Matrix-Ball Construction of affine Robinson–Schensted correspondence

Y