SoCalDM 2024

Leigh Foster (University of Oregon)- The Squish Map on Plane Partitions and the Dimer Model

We exhibit a measure-preserving map between the single dimer model on the hexagon lattice and Kenyon's SL_2 double dimer model on a courser hexagon lattice, and its analogous version on plane partitions. We compute the most interesting special case of this map, related to plane partition q-enumeration with 2-periodic weights. As an application, we specialize the weights to be certain roots of unity and obtain novel generating functions (some known, some new, and some conjectural).

Lenny Fukshansky (Claremont McKenna College) - On a new absolute version of Siegel's lemma (cancelled)

The classical Siegel's lemma (1929) asserts the existence of a nontrivial integer solution to an underdetermined integer homogeneous linear system, whose "size" is small as compared to the size of the coefficients of the system. Far-reaching generalizations of this theorem, producing a full basis for the solution space, were obtained over number fields by Bombieri & Vaaler (1983), and over the field of algebraic numbers by Roy & Thunder (1996), where the "size" was measured by a height function. We obtain a new version of Siegel's lemma, bridging the Bombieri & Vaaler and Roy & Thunder results in two ways: (1) our basis lies over a fixed number field as in Bombieri & Vaaler's theorem; (2) our height-bound does not depend on the number field in question as in Roy & Thunder's theorem. Our result does not imply the previously established ones and is not implied by them, and our basis has some additional interesting properties. Our method is quite different from the previous ones, using only linear algebra. Joint work with Max Forst.

Shiliang Gao (University of Illinois at Urbana-Champaign) - Dimensions of the Berenstein-Zelevinsky polytopes

The Kostka numbers are weight space dimensions of irreducible representations of complex semisimple Lie algebras. In type A, they count the number of semistandard Young tableaux of a given shape and content. In general, they can be described as the number of lattice points in Berenstein-Zelevinsky polytopes. We give a root-theoretic formula for dimensions of Berenstein-Zelevinsky polytopes in classical Lie types. Equivalently, the formula computes the degree of the stretched Kostka quasi-polynomial. This is based on joint work with Yibo Gao.

Claire Levaillant (University of Southern California) - A combinatorial characterization of Wilson and super Wieferich primes

In this talk, we provide a combinatorial expansion of the Fermat quotient q_p(2) to the modulus p^2. We derive a combinatorial characterization of Wieferich primes, of super Wieferich primes and of Wilson and super Wieferich primes. We hope that these combinatorial characterizations could one day help with the search for such primes. 

Zhongyang Li  (University of Connecticut) - Asymptotics of bounded lecture hall tableaux (cancelled)

Lecture hall tableaux, introduced by Corteel and Kim, generalize both lecture hall partitions and anti-lecture hall compositions. As their boundaries extend to infinity, limit shapes emerge in the scaling limit, depicting large-scale structures. In this discussion, I explore new constructions and analyses of Schur generating functions for these tableaux, revealing non-Gelfand-Tsetlin particle configurations and non-doubly periodic dimer models. Applications include confirming a conjecture by Corteel, Keating, and Nicoletti about rescaled height function slopes obeying a complex Burgers equation, and proving that fluctuations of unrescaled height functions converge to the Gaussian free field.

Andrew Sack (University of California, Los Angeles)- The combinatorics of poset associahedra

For a poset P, Galashin introduced a simple polytope A(P) called the P-associahedron. We will discuss a realization of poset associahedra and we show that the f-vector of A(P) depends only on the comparability graph of P. Furthermore, we show that when P is a rooted tree, the 1-skeleton of A(P) orients to a lattice, answering a question of Laplante-Anfossi. These lattices naturally generalize both the weak order on permutations and the Tamari lattice.

Joshua Swanson (University of Southern California) - Promotion permutations

A classical bijection for Catalan objects sends two-row rectangular standard Young tableaux to non-crossing perfect matchings. Building on work of Hopkins--Rubey, we introduce an injection from r-row rectangular tableaux to (r-1)-tuples of permutations. The r=2 case recovers the classical Catalan bijection. We discuss some of the remarkable properties of this new bijection, relate it to promotion, evacuation, and webs, and describe some open problems.

Mikhail Tikhonov (University of Virginia) - Asymptotics of Noncolliding q-exchangeable random walks and lozenge tilings

We consider a process of noncolliding q-exchangeable random walks on Z making steps 0 ("straight") and -1 ("down"). A single random walk is called q-exchangeable if under an elementary transposition of the neighboring steps (down, straight) to (straight, down) the probability of the trajectory is multiplied by a parameter q in (0,1). Our process of m noncolliding q-exchangeable random walks is obtained from the independent q-exchangeable walks via the Doob's h-transform for a certain nonnegative eigenfunction h with the eigenvalue less than 1. The system of m walks evolves in the presence of an absorbing wall at 0.

We show that the trajectory of the noncolliding q-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel of q-distributed random lozenge tilings of sawtooth polygons.

In the limit as m to infinity, q=e^(-γ/m) with γ>0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel. Based on joint work with L. Petrov, arXiv:2303.02380.

Shiyun Wang (University of Minnesota) - The Stanley-Stembridge Conjecture for  (2+1+1)-avoiding unit interval orders

A natural unit interval order is a naturally labelled partially ordered set that avoids patterns (3+1) and (2+2). To each natural unit interval order one can associate a symmetric function. The Stanley-Stembridge conjecture states that each such symmetric function is positive in the basis of complete homogenous symmetric functions. This conjecture has deep connections to cohomology rings of Hessenberg varieties, and to Kazhdan-Lusztig theory. We prove the conjecture for the special case when the unit interval order additionally avoids pattern (2+1+1).

Damir Yeliussizov (Kazakh-British Technical University) - Latin cubes, Kronecker coefficients, and tensor invariants

I will talk about some problems on positivity and unimodality of Kronecker coefficients and their connections with high-dimensional versions of the Alon--Tarsi conjecture on Latin squares. Joint work with Alimzhan Amanov.