Talks and Open Problems
11:40 - Danai Deligeorgaki - Inequalities for f*-vectors
The Ehrhart polynomial ehr_P(n) of a lattice polytope P counts the number of integer points in the n-th integral dilate of P. Ehrhart polynomials of polytopes are often described in terms of the vector of coefficients of ehr_P(n) with respect to different binomial bases, under which they have non-negative coefficients. Such vectors give rise to the h* and f*-vector of P, which coincide with the h and f vectors of a regular unimodular triangulation of P, whenever it exists. In joint work with Matthias Beck, Max Hlavacek and Jerónimo Valencia-Porras, we computed examples of f*-vectors of lattice polytopes, including a family of simplices whose f*-vectors are not unimodal. Even though f*-vectors of lattice polytopes are not necessarily unimodal, there are several interesting inequalities that can hold among their coefficients. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f*-coefficients increases and the last quarter decreases. Even though f*-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h*-vector, there is a polytope with the same h*-vector whose f*-vector is unimodal.
12:00 - Sabino Di Trani - Multipath matroids and digraph colorings
A celebrated result in graph theory links the chromatic polynomial of a graph to the Tutte polynomial of the associated graphic matroid. In this talk, we describe how to associate a matroid to a directed graph G, called the multipath matroid of G, which encodes relevant combinatorial information about edge orientation. We also show that a specialization of the Tutte polynomial of the multipath matroid of G is linked to the number of certain "good" digraph colorings. If I have enough time, I will recall the link between the chromatic polynomial and chromatic cohomology, and describe how the polynomial expressing the number of "good" digraph colorings is linked with multipath cohomology, introduced in a work with Caputi and Collari in 2021.
12:20 - Solal Gaudin - Combinatorics of remixed eulerian numbers
We study the finite dynamics of a particle model over Z. The distribution of their possible end-states can be described by a family of numbers, the remixed eulerian numbers, generalising q-binomial coefficients and q-eulerian numbers. We give new formulas for large subfamilies of these remixed eulerian numbers.
12:40 - Elena Hoster - A family of new statistics on the symmetric group
In this talk, I will first introduce the in-order statistic and show how it generalises left-to-right minima of a permutation. Then, I will talk about its connection to the coarse flag Hilbert-Poincaré series ofMaglione-Voll in the case of the braid arrangement.
14:30 - Martina Costa Cesari - Seshadri stratifications: An application to matrix Schubert varieties
Recently Seshadri stratifications on an embedded projective variety have been introduced by R. Chirivì, X. Fang and P. Littelmann. A Seshadri stratification of an embedded projective variety $X$ is the datum of a suitable collection of subvarieties $X_\tau$ that are smooth in codimension one, and a collection of suitable homogeneous functions $f_\tau$ on $X$ indexed by the same finite set. With such a structure, one can construct a Newton-Okounkov simplicial complex and a flat degeneration of the projective variety into a union of toric varieties. Moreover the theory of Seshadri stratifications provides a geometric setup for a standard monomial theory. In the talk, I will introduce the theory of Seshadri stratification and I will give a Seshadri stratification for matrix Schubert varieties, namely varieties of matrices defined by conditions on the rank of some their submatrices.
14:50 - Alice Dell'Arciprete - Quiver presentations for KLR algebras and Hecke categories
We discuss the algebraic structure of KLR algebras by way of the diagrammatic Hecke categories of maximal parabolics of finite symmetric groups. Combinatorics (in the shape of Dyck tableaux) plays a huge role in understanding the structure of these algebras. Instead of looking only at the sets of Dyck tableaux (which enumerate the p-Kazhdan-Lusztig polynomials) we look at the relationships for passing between these Dyck tableaux. In fact, this “meta-Kazhdan-Lusztig combinatorics” is sufficiently rich as to completely determine the complete Ext-quiver and relations of these algebras.
15:10 - Menghao Qu - A parking function interpretation for nabla of a two-column monomial symmetric function
In this talk, we will establish a recursion for $(-1)^{k} m_{2^{k}1^{I}}$ that shows $C$-positivity. The $C$-operator is a key part of the compositional shuffle theorem, initially conjectured by Haglund, Morse, and Zabrocki in 2012, and later proven by Carlsson and Mellit. As a result, we will derive Schur positivity and a parking function interpretation for $(-1)^{k}\nabla m_{2^{k}1^{|}}$, thereby partially resolving a conjecture proposed by Bergeron et al. in the two-column case. This is joint work with Guoce Xin.
16:00 - Divya Setia - Demazure filtrations of tensor product modules and its connections with algebraic combinatorics
Let g be a finite-dimensional simple Lie algebra over the complex field C and g[t] be the Lie algebra of polynomial mappings from C to g, which is its associated current Lie algebra. The notion of Weyl modules for affine Kac-Moody Lie algebras was introduced by V.Chari and A.Pressley. Subsequently it was demonstrated that for current Lie algebra of type ADE, the local Weyl modules are in fact Demazure modules of level 1 and their Demazure characters coincide with non-symmetric Macdonald polynomials, specialized at t=0. We study the structure of the finite-dimensional representations of the current Lie algebra of type A1, sl2[t], which are obtained by taking tensor products of local Weyl modules with Demazure modules. In this talk, we shall show that these representations admit a Demazure flag and obtain a closed formula for the graded multiplicities of the level 2 Demazure modules in the filtration of the tensor product of two local Weyl modules for sl2[t]. Using Pieri formulas,we have also expressed the product of two specialized Macdonald polynomials in terms of specialized Macdonald polyomials. Furthermore, we show that the tensor product of a local Weyl module with an irreducible sl2[t]-module admits a Demazure filtration and derive graded character of such tensor product modules. This helps us express the product of a specialized Macdonald polynomial with a Schur polynomial in terms of Schur polynomial. Our findings provide evidence for the conjecture that the tensor product of Demazure modules of levels m and n respectively has a filtration by Demazuremodules of level m+n.
16:20 - Marino Romero - Dyck paths, tiered trees, and Macdonald polynomials
We will start by looking at some purely combinatorial structures -- certain labeled Dyck paths and rooted tiered trees. Recent results and conjectures involving operators on the modified Macdonald basis predict a weight-preserving bijection between these two combinatorial sets. Finding such a bijection would give significant progress to our understanding of the combinatorial structures surrounding Delta, Theta, and Nabla operators.
16:40 - Riccardo Biagioli - A problem about inversions and descents
TBA