Positive skew-symmetric matrices (Veronica Calvo Cortes, Max Planck Institute for Mathematics in the Sciences)
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.
Tropical linear systems (Matthew Dupraz, Freie Universität Berlin)
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.
Weighted Ehrhart Theory (Sofía Garzón Mora, Freie Universität Berlin)
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.
Towards plethystic sl(2)-crystals (Álvaro Gutiérrez, University of Bristol)
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.
Quiver presentations of Hecke categories of type D (Ben Mills, University of York)
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.
Symplectic branching and crystals (Bárbara Muniz, Jagiellonian University Krakow)
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.
Some algebraic properties of LSS ideals (Eliana Tolosa Villarreal, Università degli studi di Genova)
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.
Chow functions for partially ordered sets (Lorenzo Vecchi, KTH - Royal Institute of Technology)
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.
Computing realisation numbers using tropical intersection theory (Daniel Green Tripp, University of Bristol)
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.
Moduli Spaces of Weighted Stable Curves and their Fundamental Groups (Haggai Liu, Simon Fraser University)
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.
Schubert coefficients of matroids (Jon Pål Hamre, KTH)
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.
Cohen-Macaulayness of the Second Symbolic Power of Edge Ideals (Delio Jaramillo Velez, Chalmers University of Technology)
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.
Schur algebras in type B (Dinushi Munasinghe, University of Athens)
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.
Causal Discovery for Max-Linear Bayesian Networks (Francesco Nowell, TU Berlin)
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.
A new bijective proof of the q-Pfaff-Saalschütz identity describing multiplication in the Cartan subalgebra of U_q(sl_2) (Michal Szwej, University of Bristol)
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).
Unrefinable partitions and numerical semigroups (Lorenzo Campioni, Università degli Studi dell'Aquila)
A partition into distinct parts is refinable if one of its parts p can be replaced by two different positive integers which do not belong to the partition and whose sum is p, otherwise the partition is unrefinable. For example the partition (1,2,3,5,9,10,12) is a refinable partition because we can substitute 10=4+6 or 12=4+8, while if we consider one of these two refinements we obtain an unrefinable partition, i.e., in (1,2,3,4,5,8,9,12) we cannot replace any parts.
It can be established an important connection between unrefinable partitions and numerical semigroups, additive submonoids of the non-negative integers which include 0 and such that the complementary sets have finitely many elements. This relationship allowed us to find other methods to recognise when a partition is unrefinable or not only looking the hooksets of the Young tableau associated to the numerical semigroup.
Generating Smooth 3-Polytopes (Kyle Huang, BTU Cottbus)
Smooth lattice polytopes are an important class of lattice polytopes in combinatorial algebraic geometry, corresponding to projective embeddings of smooth toric varieties via complete linear series.
For fixed integers $d$ and $n$, there are only finitely many smooth lattice $d$-polytopes with $\le n$ lattice points. We describe and implement a novel algorithm to classify smooth $3$-polytopes, extending previous classifications (by Haase, Lorenzo, and Paffenholz as well as Lundman) to smooth $3$-polytopes with $\leq 50$ lattice points. We also present theoretical findings on smooth $3$-polytopes--in fact, our theoretical and computational findings are intertwined and inform one another.
This is joint work with Christian Haase.
Color rules for cyclic wreath products and an application to the product of projective toric varieties (Fabián Levicán, University of Vienna)
We give a weight-preserving, sign-reversing involution method for computing the multiplicities of irreducible representations in a certain class of $G \wr S_n$-modules, where $G$ is a finite, abelian group. In general, it is sufficient to consider the case where $G = Z_k$, the cyclic group of order $k$. We do this by "coloring" cycles according to a "color rule", and show that irreducible characters can be enumerated by certain classes of semistandard Young tableaux in the colors.
Although in our paper we give a number of applications, in this talk we will focus on one with geometric motivations, namely the character of affine semigroup algebras coming from the product of projective toric varieties. In particular, we compute the character of a product of simplices (coming from a product of projective spaces), and derive as a consequence certain refined Euler-Mahonian identities. Finally, we mention links to a "refined" version of equivariant Ehrhart theory.
(Joint with Marino Romero, University of Vienna. We expect a preprint to be on the Arxiv soon.)
Elliptic curves in game theory (Elke Neuhaus, MPI MiS Leipzig)
We explain the concept of dependency equilibria in game theory, which generalize the well-known concept of Nash equilibria and allow for better results via assumptions of dependencies. They can be described nicely by an algebraic variety, the so-called Spohn variety. For 2x2 games, this variety generically takes the form of an elliptic curve that is the intersection of two quadrics in P^3. We examine the reduction of Spohn curves to plane curves, analyzing conditions under which they are reducible. We then prove that the real points are dense on the Spohn curve in all cases, which is relevant since we are of course interested in real probabilities. We also address general ways to compute the j-invariant of elliptic curves that arise from the intersection of two quadrics.
Interactions between quantum affine algebras and cluster algebras (Francesca Paganelli, Sapienza Università di Roma)
Quantum affine algebras are a special type of quantum group that arise from Lie algebras and have a complicated representation theory. In contrast, cluster algebras have a more combinatorial nature, since they are defined from an initial quiver (that is an oriented graph) and the so-called "exchange relations". To study the representations of quantum affine algebras, one can focus on certain subcategories of representations, specifically on their Grothendieck rings. Hernandez and Leclerc proved that some of these rings actually possess a cluster algebra structure, thereby establishing a connection between these two types of algebras. This concept is known as the monoidal categorification of cluster algebras and I will explain how it works. The running example will be the quantum affine sl(2)-algebra. If time permits, I will discuss my current research on shifted quantum affine algebras through the framework of quantum cluster algebras.
Pre-Cartier quasi-bialgebras (Andrea Rivezzi, Charles University in Prague)
It is well-known that braided monoidal categories correspond to categories of representations of quasi-triangular bialgebras, the latter having a fundamental role in the theory of Yang-Baxter equation and of representations of braid groups.
If one looks at the linear terms of the $\hbar$-completed hexagon equations, obtain the so-called infinitesimal braid relations in a braided monoidal category. The algebraic counterpart is the notion of a pre-Cartier quasi-triangular bialgebra (due to Ardizzoni, Bottegoni, Sciandra, and Weber), which is a quasi-triangular bialgebra together with an infinitesimal R-matrix. In this poster we present the quasi-counterpart of this notion, namely pre-Cartier quasi-bialgebras. We finally sketch that, given a Drinfeld associator, one can quantize any infinitesimal R-matrix in the framework of pre-Cartier quasi-bialgebras, and under some commutation relation the construction restricts to a honest bialgebra. This is based on a joint work with C. Esposito, J. Schnitzer, and T. Weber.
Enumerating 1324-avoiders with few inversions (Emil Verkama, KTH Royal Institute of Technology)
The problem of determining the number of \(1324\)-avoiding permutations of length \(n\) has received much attention. We work towards this goal by enumerating \(\mathrm{av}_n^k(1324)\), the number of \(1324\)-avoiding \(n\)-permutations with exactly \(k\) inversions, for all \(k\) and \(n \geq (k+7)/2\). This is achieved with a new structural characterization of such permutations in terms of a new notion of almost-decomposability. In particular, our enumeration verifies half of a conjecture of Claesson, Jelínek and Steingrímsson, according to which \(\mathrm{av}_n^k(1324) \leq \mathrm{av}_{n+1}^k(1324)\) for all \(n\) and \(k\). Proving the full conjecture would improve the best known upper bound for the exponential growth rate of the number of \(1324\)-avoiders from \(13.5\) to approximately \(13.002\).
On the solvability of the Lie algebra HH^1(B) for blocks of finite groups (Jialin Wang, City St George’s, University of London)
We give some criteria for the Lie algebra of first degree Hochschild cohomology of the twisted group algebra, i.e.
HH^1(kα(P ⋊ E)), to be solvable, where P is a finite abelian p-group, E is an abelian p'-subgroup of Aut(P) and α ∈ Z^2(E; k^×) inflated to P ⋊ E via the canonical surjection P ⋊E → E. As a special case, this gives the criterion to the solvability of the Lie algebra HH^1(B) where B is a p-block of a finite group algebra with abelian defect P and inertial quotient E.
Connection Matrices in Macaulay2 (Nicolas Alexander Weiss, Max Planck Institute for Mathematics in the Sciences)
Systems of linear partial differential equations can be encoded as left ideals over the Weyl algebra. When the ideal is holonomic, they can be expressed in terms of first-order matrix differential equations using connection matrices. In some cases, changes of basis (a gauge transform) allow to make the system of differential equations more manageable, as for example with the ϵ-factorized form in physics. To algorithmically compute connection matrices and such gauge transforms, one requires Gröbner basis computations and in particular to execute the normal form algorithm in the rational Weyl algebra. With our new package "ConnectionMatrices" we add this functionality to Macaulay2.
This is joint work with: Paul Görlach, Joris Koefler, Anna-Laura Sattelberger, Mahrud Sayrafi, Hendrik Schroeder, and Francesca Zaffalon.