Algebraic and Enumerative
Combinatorics in Okayama

Program

February 19, 2018 (Monday)

February 20, 2018 (Tuesday)

February 22, 2018 (Thursday)

February 23, 2018 (Friday)

Roger Behrend (Cardiff University)

The combinatorics of alternating sign matrices

In these talks, I will discuss various topics related to the combinatorics of alternating sign matrices. These will include alternating sign matrix symmetry classes, alternating sign matrix statistics, descending plane partitions, totally symmetric self-complementary plane partitions, alternating sign triangles, fully packed loop configurations, the alternating sign matrix polytope, and the alternating sign matrix poset.

Mihai Ciucu (Indiana University)

Lozenge tilings with gaps in a 90 degree wedge domain with mixed boundary conditions

We consider a triangular gap of side two in a 90 degree angle on the triangular lattice with mixed boundary conditions: a constrained, zig-zag boundary along one side, and a free lattice line boundary along the other. We study the interaction of the gap with the corner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its three images in the sides of the angle. This provides evidence for a unified way of understanding the interaction of gaps with the boundary under mixed boundary conditions, which we present as a conjecture. Our conjecture is phrased in terms of the steady state heat flow problem in a uniform block of material in which there are a finite number of heat sources and sinks. This new physical analogy is equivalent in the bulk to the electrostatic analogy we developed in previous work, but arises as the correct one for the correlation with the boundary.
The starting point for our analysis is an exact formula we prove for the number of lozenge tilings of certain trapezoidal regions with mixed boundary conditions, which is equivalent to a new, multi-parameter generalization of a classical plane partition enumeration problem (that of enumerating symmetric, self-complementary plane partitions).

Hiroshi Naruse (Yamanashi University)

Equivariant K-theory and hook formula for skew shape on d-complete poset

We use equivariant K-theory of flag variety to represent a hook formula for generating function of reverse plane partitions on skew shape d-complete poset in terms of excited diagrams. This is a joint work with Soichi Okada.

Yasuhide Numata (Shinshu University)

On a framework for Hillman-Grassl algorithms

The classical Hillman-Grassl algorithm induces a bijection from reverse plane partitions to multisets of hooks. We introduce a framework such that the analogues of the algorithm work. We apply the framework to swivels-free $d$-complete posets. This is based on the joint work with K. Nakada and S. Okamura.

Greta Panova (University of Pennsylvania)

Hook formulas for skew shapes -- combinatorics, asymptotics and beyond

We will show several combinatorial and aglebraic proofs of this formula, leading to a bijection between SSYTs or reverse plane partitions of skew shape and certain integer arrays that gives two q-analogues of the formula. These formulas can also be proven via non-intersecting lattice paths interpretations, for example connecting Dyck paths and alternating permutations.
We will also show how excited diagrams give asymptotic results for the number of skew Standard Young Tableaux in various regimes of convergence for both partitions. Multivariate versions of the hook formula with consequences to exact product formulas for certain skew SYTs and lozenge tilings with multivariate weights, which also appear to have interesting behavior in the limit. Joint work with A. Morales and I. Pak

Tom Roby (University of Connecticut)

Paths to understanding birational rowmotion on a product of two chains

Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of a poset P, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) In this context, rowmotion appears to be related to Auslander-Reiten translation on certain quivers, and birational rowmotion to Y-systems of type $A_m \times A_n$ described in Zamolodchikov periodicity. We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths r and s is r+s+2 (first proved by D.~Grinberg and the second author), as well as a proof of the birational analogue of homomesy along files for such posets.

Itaru Terada (University of Tokyo)

On an involution on the set of Littlewood-Richardson tableaux

We interpret a bijection, constructed by O. Azenhas, between the Littlewood--Richardson (LR) tableaux of shape $\lambda/\mu$ and weight $\nu$ on one hand and those of shape $\lambda/\nu$ and weight $\mu$ on the other, as a correspondence between the irreducible components of algebraic varieties $\mathcal{G}_{\mu\nu}^\lambda$ and $\mathcal{G}_{\nu\mu}^\lambda$, which are actually isomorphic, but the irreducible components of $\mathcal{G}_{\mu\nu}^\lambda$ are naturally labelled by the LR tableaux of shape $\lambda/\mu$ and weight $\nu$ while those of $\mathcal{G}_{\nu\mu}^\lambda$ by those of shape $\lambda/\nu$ and weight $\mu$. These varieties are closely related with the Hall polynomials indexed by the same triples of partitions.
To be more explicit, Azenhas' bijection consists of iterating a certain deletion operation starting from a given LR tableau $T$ of shape $\lambda/\mu$ and $\nu$ to produce a sequence of LR tableaux $T=T^{(n)},T^{(n-1)},\dots,T^{(1)},T^{(0)}=\varnothing$ in which each $T^{(r)}$ has outer shape with exactly $r$ rows, and while doing so recording how the inner shape shrinks in the form of another LR tableau with $\mu$ and $\nu$ swiched. Our interpretation is obtained by revealing what is represented by these intermediate LR tableaux, in particular $T^{(n-1)}$ obtained in the first step.

Hiro-Fumi Yamada (Kumamoto University)

On Sato's "On Hirota's Bilinear Equations"

In 1980 Mikio Sato published a hand-written note, which contains some tables on the KdV equations. I have been trying to read out information from these tables for a long time. And recently I finally found what Sato wanted to tell us on the KdV equations. I will report the meaning of Sato's tables, and view the KdV equations as the orthogonality relations of the symmetric group characters.