Combinatorial Algebra Meets  Algebraic Combinatorics 

Fri 14:00–14:50 Alejandro Morales (UQAM)

Title: Enumerating factorizations: from the symmetric group to Hecke algebras

Abstract: Factorizations of permutations within the symmetric group have a rich history, connecting diverse areas such as symmetric functions, graph embeddings on surfaces, representation theory via characters and Jucys–Murphy elements, and the Hurwitz problem in algebraic geometry. In this talk, we will explore some of the striking enumerative results related to factorizations of the long cycle and their connections to these fields. Additionally, we extend these studies to analogous problems in the type A Hecke algebra, uncovering elegant q-analogues that generalize results from the symmetric group by leveraging the Jucys–Murphy elements of the Hecke algebra. This talk is based on joint work with José Bastidas, Sarah Brauner, Mathieu Guay-Paquet, GaYee Park, and Franco Saliola.

Fri 16:00–16:20 Bek Chase (Purdue)

Title: Almost complete intersections and the strong Lefschetz property

Abstract: An Artinian standard graded algebra has the weak Lefschetz property if the map induced by multiplication by a general linear form has maximal rank (i.e., is injective or surjective) in each degree, and has the strong Lefschetz property if the map induced by multiplication by every power of a general linear form has maximal rank in each degree. In this talk, we will investigate the strong Lefschetz property for almost complete intersections, providing a complete classification for a class of codimension three monomial almost complete intersections. Furthermore, we prove the strong Lefschetz property for certain arbitrary monomial almost complete intersections. This is joint work with Filip Jonsson Kling.

Fri 16:30–16:50 Mitsuki Hanada (Berkeley)

Title:  A charge monomial basis of the Garsia–Procesi ring

Abstract: The two well-known monomial bases of the (classical) coinvariant ring, the Artin basis and the descent basis, are indexed by permutations and correspond to the statistics inv and maj respectively. Both bases give combinatorial explanations for the Hilbert series of the coinvariant ring. We construct a basis of the Garsia–Procesi ring, a quotient of the coinvariant ring, using the catabolizability type of standard Young tableaux and the charge statistic. This basis is compatible with its Hilbert series in the same manner. Our construction of the basis provides the first direct connection between the combinatorics of the basis with the combinatorial formula for the modified Hall-Littlewood polynomials, due to Lascoux. We use this description to give an elementary proof of the fact that the graded Frobenius character of Garsia–Procesi ring is given by the modified Hall-Littlewood polynomial.

Sat 09:00–09:50 Sonja Petrović  (Illinois Tech)

Title: Probability and randomness in nonlinear algebra 

Abstract: Many problems in symbolic computation with polynomials have high worst-case complexity. At the same time, in many areas of computational mathematics, significant improvements in efficiency have been obtained by algorithms that involve randomization, rather than deterministic ones. This talk will overview a randomized sampling framework from geometric optimization to applied computational algebra, and demonstrate its usefulness on two problems, including solving large (overdetermined) systems of multivariate polynomial equations. 

Sat 10:30–10:50 Thiago Hollenben   (Dalhousie)

Title:  Coinvariant stresses, inverse systems, and Lefschetz properties

Abstract: Lefschetz properties and inverse systems have played key roles in understanding the h-vector of simplicial spheres. In 1996, Lee established connections between these two algebraic tools and rigidity theory. One of the main points of these connections, is to translate geometric information about a complex, coming from vertex coordinates, to the algebraic notion of a linear system of parameters. In this talk, we begin exploring similar connections in the nonlinear case, by using recent results of Herzog and Moradi (2021) where they prove that a subset of the elementary symmetric functions is always a system of parameters for the Stanley-Reisner ideal of a complex. We then look at connections between our results and the study of Lefschetz properties of monomial almost complete intersections, which started in 2011 by Migliore, Miró-Roig and Nagel. We show that our results can be seen as a generalization of one of their results from the boundary of the simplex, to arbitrary Gorenstein* complexes.

Sat 11:00–11:20 Henry Potts-Rubin (Syracuse)

Title: DG-Sensitivity: matching and pruning

Abstract: Let F be a resolution of a cyclic module over a polynomial ring.  We describe two processes via which, under certain conditions, a differential graded (dg) algebra structure on F (if one exists) induces a dg algebra structure on a related (quotient) resolution.  As a consequence, we show that if the minimal free resolution of Q/I, the quotient of the ambient polynomial ring by the edge ideal I of a graph G, admits the structure of a dg algebra, then so does the minimal free resolution of the quotient by the edge ideal of each induced subgraph of G (over the smaller polynomial ring).  Combined with techniques from discrete Morse theory and homological algebra, this allows us to complete classify the trees and cycles which are ``dg."  This joint work with Hugh Geller and Des Martin.  

Sat 14:00–14:20 Karlee Westrem  (Michigan Tech)

Title:  A new symmetric function identity with an application to symmetric group character values

Abstract:  Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory.  Tewodros Amdeberhan conjectures equalities of characters of the symmetric group ∑ sums over a new set called Ev(λ). When investigating the alternating sum of characters for Ev(λ) written in terms of the inner product of Schur functions and power sum symmetric functions, we found an equality between the alternating sum of power sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of ∑ characters over the set Ev(λ) equals 0.

Sat 14:30–14:50 Sam Armon (USC)

Title: Super major index and Thrall's problem

Abstract: Thrall’s problem asks for the Schur decomposition of the higher Lie modules, which are defined using the free Lie algebra. We generalize Thrall’s problem to the free Lie superalgebra, and prove extensions of some known results to this setting. In particular, we determine the Schur decomposition of the bigraded pieces of the free Lie superaelgebra, generalizing a result of Káskiewicz–Weyman in the classical case. We employ a new version of the major index on super tableaux, which we also use to provide a combinatorial interpretation of a q,t-hook formula of Macdonald.

Sat 16:00–16:20 Félix Gélinas (York)

Title: Source characterization of the hypergraphic posets

Abstract: For a hypergraph H on [n], the hypergraphic poset P is the transitive closure of the oriented 1-skeleton of the hypergraphic polytope ∆(H), which is the Minkowski sum of the standard simplices ∆(H) for each hyperedge H in H. In 2019, C. Benedetti, N. Bergeron, and J. Machacek established a remarkable correspondence between the transitive closure of the oriented 1-skeleton of ∆(H) and the flip graph on acyclic orientations of H. Viewing an orientation of H as a map A from H to [n], we define the sources of the acyclic orientations as the values A(H) for each hyperedge H in H. In a recent paper, N. Bergeron and V. Pilaud provide a characterization of P based on the sources of acyclic orientations for interval hypergraphs. Specifically, two distinct acyclic orientations A and B of H are comparable in P if and only if their sources satisfy A(H) ≤ B(H) for all hyperedges H in H. The goal of this work is to extend this source characterization of P to arbitrary hypergraphs on [n].

Sat 16:30–17:20 Alexandra Seceleanu  (Nebraska)

 Title: General symmetric ideals

Abstract: An ideal in a polynomial ring with several variables is called symmetric if it remains invariant as a set under the action of the symmetric group, which permutes the variables of the polynomials. In mathematics, it is a recurring principle that a "general" or "random" member of a family often exhibits desirable properties. Motivated by this idea, we undertake the task of identifying an appropriate notion of a general symmetric ideal within the family of symmetric ideals with a fixed number of generators, considered up to symmetry. This framework enables us to uncover remarkable homological properties inherent to these ideals. This talk is based on joint work with Megumi Harada and Liana Sega and separately with Liana Sega.

Sun 09:30–09:50 Yu Li   (Toronto)

Title:  Real matroid Schubert varieties and zonotopes

Abstract:  Let A be an essential hyperplane arrangement in a real vector space V. The matroid Schubert variety Y is a certain real multiprojective variety that compactifies V. We prove that Y is homeomorphic to the zonotope generated by the normal vectors of the hyperplanes in A with parallel faces identified. This generalizes previous work by Ilin-Kamnitzer-Li-Przytycki-Rybnikov where A is the Coxeter arrangement. Using our combinatorial model, we compute various topological invariants of Y in terms of the matroid that underlies A. Our main tool is an explicit cell decomposition, which depends only on the oriented matroid structure and can be extended to define a combinatorially interesting topological space for any oriented matroid.  This is joint work with Leo Jiang.

Sun 10:00–10:20 Lisa Nicklasson  (Mälardalen)

Title: The algebraic matroid of a determinantal variety

Abstract:  Consider a generic (m×n)-matrix of rank r over an algebraically closed field, where only a subset of the entries is known. We say that the matrix is finitely completable if there are finitely many ways to fill in the unknown entries so that the matrix has the prescribed rank r. The finitely completable patterns provide the base sets of a matroid. Alternatively, the matroid can be defined algebraically in terms of elimination ideals in the ideal of (r+1)-minors. In this talk we will explore this matroid, and try to get a hold of its base sets using commutative algebra and combinatorics.  This talk is based on a joint work with Manolis Tsakiris. No preprint available yet.

Sun 11:00–11:50 Alex Fink (QMUL)

Title: Speyer's tropical f-vector conjecture and its proof

Abstract:  In 2008, looking to bound the face vectors of tropical linear spaces, Speyer introduced the g-invariant of a matroid. He proved its coefficients nonnegative for matroids representable in characteristic zero and conjectured this in general. After introducing the objects, this talk will give an overview of the ingredients in recent work with Andrew Berget that proves the conjecture.  A main character is the variety of coordinatewise quotients of points in two linear subspaces, and its initial degenerations which encode a new generalization of external activity to a pair of matroids.

Sun 12:00–12:20 François Bergeron (UQAM)

Title: Refined h-polynomial of associahedral complexes

Abstract: TBA