Convegno
“6TH GRADUATE STUDENT MEETING IN APPLIED ALGEBRA AND COMBINATORICS”

We will study the combinatorics (and some representation theory!) of string polytopes, mostly in type A, in terms of so-called wiring diagrams or pseudoline arrangements. We will also study the associated convex order on positive roots. Finally, we will present some new results and open problems!

The composition of sl(2) representations is the 'plethysm' of the respective representations. Crystals are a powerful combinatorial tool for representation theory, but there is no theory on crystals for plethysms of representations. To start such a theory, it suffices to solve a combinatorial problem: decompose Young's poset of partitions into symmetric chains. We review the literature, and present a strategy to do it. Our strategy recovers recently discovered counting formulas for some plethystic coefficients, and new state-of-the-art recursive formulas for some plethysms of Schur functions.

A matrix is totally positive if all of its minors are positive. This notion of positivity coincides with the type A version of Lusztig's more general total positivity in reductive real-split algebraic groups. Since skew-symmetric matrices always have nonpositive entries, they are not totally positive in the classical sense. The space of skew-symmetric matrices is an affine chart of the orthogonal Grassmannian OGr(n,2n). Thus, we define a skew-symmetric matrix to be totally positive if it lies in the totally positive orthogonal Grassmannian. We provide a positivity criterion for these matrices in terms of a fixed collection of minors, and show that their Pfaffians have a remarkable sign pattern. The totally positive orthogonal Grassmannian is a CW cell complex and is subdivided into Richardson cells. We introduce a method to determine which cell a given point belongs to in terms of its associated matroid.

In this talk I would like to talk about metric graphs and chip firing games on metric graphs. These combinatorial analogues exhibit interesting connections with algebraic curves, for example they satisfy an analogue of the Riemann-Roch theorem. Linear systems on metric graphs have a lot of structure - they are generalized polyhedral complexes and also projective tropical spaces. I would like to talk about these and explain how certain geometric quantities relate to some more algebraic ones.

We will discuss a generalization of Stanley's celebrated theorem that the h*-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h*-polynomial as a real-valued function for a larger family of weights. This is joint work with Esme Bajo, Robert Davis, Jesús A. De Loera, Alexey Garber, Katharina Jochemko and Josephine Yu.

Quiver presentations of Hecke categories of type D: We discuss the algebraic structure of the Hecke category corresponding to the parabolic Coxeter system (D_n, A_{n-1}) via the combinatorics of oriented Temperley-Lieb diagrams. This will enable us to fully determine the Ext-quiver and relations presentation for these algebras.

The decomposition of gl-representations when restricted to sp is a classic problem in representation theory, commonly referred to as symplectic branching. The multiplicities that describe this decomposition have a known combinatorial description in terms of certain Littlewood-Richardson tableaux. In this work, we construct an explicit and elementary bijection between the sp-highest weight vectors in the gl-crystals and these tableaux. Thus, we are able to present an alternative interpretation for the symplectic branching, visualizing it at the crystal level.

Every simple finite graph G has an associated Lovász-Saks-Schrijver ring R_G(d) that is related to the d-dimensional orthogonal representations of G. The study of R_G(d) lies at the intersection between algebraic geometry, commutative algebra and combinatorics. We find a link between algebraic properties, such as normality, factoriality, and strong F-regularity, of R_G(d) and combinatorial invariants of the graph G. In particular we prove that if d ≥ pmd(G)+k(G)+1 then R_G(d) is UFD. Here pmd(G) is the positive matching decomposition number of G and k(G) is its degeneracy number.

Three decades ago, Stanley and Brenti initiated the study of the Kazhdan–Lusztig–Stanley (KLS) functions, putting on common ground several polynomials appearing in algebraic combinatorics, discrete geometry, and representation theory. In this talk we will introduce new polynomial functions called Chow functions associated to any graded bounded poset and study their applications to matroid theory, polytopes and Coxeter groups. The Chow functions often exhibit remarkable properties (positivity, palindromicity, unimodality, gamma-positivity), and sometimes encode the graded dimensions of a cohomology or Chow ring. One of the best features of this general framework is that unimodality statements can be proven for posets without relying on versions of the Hard Lefschetz theorem. Our framework shows that there is an unexpected relation between positivity and real-rootedness conjectures about chains on face lattices of polytopes by Brenti and Welker, Hilbert–Poincaré series of matroid Chow rings by Ferroni and Schröter, and flag enumerations on Bruhat intervals of Coxeter groups by Billera and Brenti. This is joint work with Luis Ferroni and Jacob Matherne.

The Deligne-Mumford compactification, $\overline{M_{0,n}}$, of the moduli space of $n$ distinct ordered points on $\mathbb{P}^1$, has many well understood geometric and topological properties. For example, it is a smooth projective variety over its base field. Many interesting properties are known for the manifold $\overline{M_{0,n}}(\mathbb{R})$ of real points of this variety. In particular, its fundamental group, $\pi_1(\overline{M_{0,n}}(\mathbb{R}))$, is related, via a short exact sequence, to another group known as the cactus group. Henriques and Kamnitzer gave an elegant combinatorial presentation of this cactus group. In 2003, Hassett constructed a weighted variant of $\overline{M_{0,n}}(\mathbb{R})$: For each of the $n$ labels, we assign a weight between 0 and 1; points can coincide if the sum of their weights does not exceed one. We seek combinatorial presentations for the fundamental groups of Hassett spaces with certain restrictions on the weights. In particular, we express the Hassett space as a blow-down of $\overline{M_{0,n}}$ and modify the cactus group to produce an analogous short exact sequence. The relations of this modified cactus group involves extensions to the braid relations in $S_n$. To establish the sufficiency of such relations, we consider a certain cell decomposition of these Hassett spaces, which are indexed by ordered planar trees.

We introduce a set of matroid invariants called Schubert coefficients. To define and understand the Schubert coefficients we use tools from algebraic geometry such as Schubert calculus and toric geometry. The main goal is to prove the non-negativity of the Schubert coefficients of sparse paving matroids, and hopefully convince the audience that these are interesting matroid invariants.

In this poster, we present sufficient conditions for the second symbolic power of the edge ideal associated with a graph to be Cohen-Macaulay. These conditions involve the concept of edge-critical graphs. Furthermore, we establish that when the graph has an independence number equal to two, these conditions provide a complete characterization of the Cohen-Macaulayness of the second symbolic power.

The d-realisation number of a graph G counts, roughly speaking, the number of equivalent d-realisations of a generic d-realisation of G. It is known that this number is finite if and only if G is d-rigid. For a minimally 2-rigid graph, we give a way of computing this number as the tropical intersection number of the Bergman fan of the graphic matroid of G with its “reciprocal”. This description allows us to bound the realisation number with some matroid invariant.

Max-linear Bayesian Networks are a class of Directed acyclic graphical (DAG) models which are of interest to statistics and data science due to their relevance to causality and probabilistic inference, particularly of extreme events. They differ from the more extensively studied Gaussian Bayesian Networks in that the structural equations governing the model are tropical polynomials in the random variables. This difference leads to several novel challenges in the task of causal discovery, i.e. the reconstruction of the true DAG underlying a given empirical distribution. More specifically, the combinatorial criteria for separation in the graph equating to conditional independence in the distribution are such that there is no longer a well-defined notion of Markov equivalence. In this talk, we explain how the PC algorithm for causal discovery in Gaussian Bayesian Networks fails in the max-linear setting, and discuss how it may be modified such as to output a well-defined subgraph of the true DAG which encodes its most significant causal relationships. This is a joint work with Carlos Améndola and Benjamin Hollering.

The Pfaff-Saalschütz identity is a result from the theory of hypergeometric series which generalizes many cubic binomial identities. In this talk we present a new bijective proof of its q-analogue. The identity which follows from the main theorem gives the multiplication rule for the quantum deformation of binomial coefficients, defined by Lusztig inside Cartan subalgebra of U_q(sl_2).

We write a natural type B generalization of Hecke-invariant endomorphisms over the tensor product, constructed by Lai and Luo, as an idempotent truncation of the cyclotomic q-Schur algebra of Dipper, James, and Mathas to leverage its established cellular structure in proving quasi-hereditarity results about the newer algebra.