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Jacob P. Matherne

Publications and source records attributed to Jacob P. Matherne.

At least 19 recordsLinked to original sources
arXiv2026

Chromatic symmetric functions of claw-free graphs are not Schur positive

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize chromatic polynomials of graphs. In 1995, Stanley introduced these symmetric functions and conjectured that they are Schur positive for claw-free graphs. We give examples of a line graphs, which are thus claw-free, whose chromatic symmetric function have a negative coefficient in its Schur expansion. We also give a counterexample to the 2018 conjecture of Monical that Schur positive chromatic symmetric functions have saturated Newton polytope when expanded in any finite number of variables. Both of these examples were found using ChatGPT-5.6 Sol Pro.

math.CO↗
arXiv2025

Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions

In 2022, Huh-Matherne-Mészáros-St. Dizier showed that normalized Schur polynomials are Lorentzian, thereby yielding their continuous (resp. discrete) log-concavity on the positive orthant (resp. on their support, in type $A$ root directions). A reinterpretation of this result is that the characters of finite-dimensional simple representations of $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian (DL). In the same paper, these authors also showed that shifted characters of Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are DL. In this work we extend these results to a larger family of modules that subsumes both of the above: we show that shifted characters of all parabolic Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. The proof involves certain graphs on $[n+1]$; more strongly, we explain why the character (i.e., generating function) of the Kostant partition function of any loopless multigraph on $[n+1]$ is Lorentzian after shifting and normalizing. We then show that parabolic Vermas form a "maximal" class with log-concave (hence DL) characters. Namely, log-concavity fails in greater generality along three natural directions: (1) it does not hold for every simple Lie type, (2) nor for a larger universal family of highest weight modules, the higher order Verma modules, even in type $A$, and (3) it does not always hold for important generalizations of Schur polynomials: the Jack and Macdonald polynomials. Finally, we extend these results to parabolic (i.e. "first order") and higher order Verma modules over the semisimple Lie algebras $\oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$. We also partially resolve a conjecture of Huh et al on the DL property for integral highest weight simple modules.

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arXiv2025

Inverse Kazhdan-Lusztig polynomials of matroids under deletion

We provide a deletion formula for the inverse Kazhdan--Lusztig polynomial and the inverse $Z$-polynomial of a matroid. Our formulas provide analogues to the deletion formulas of Braden--Vysogorets for Kazhdan--Lusztig and $Z$-polynomials. We discuss several consequences, which include closed formulas and recursions for these invariants on uniform matroids, projective geometries, glued cycles, and arbitrary matroids of corank $2$. As a relevant application of our deletion formula, we show the existence of a matroid of rank $19$ which disproves a conjecture of Xie and Zhang concerning a real-rootedness property of inverse Kazhdan--Lusztig polynomials.

math.CO↗
arXiv2025

Syzygies of polymatroidal ideals

We introduce the cave polynomial of a polymatroid and show that it yields a valuative function on polymatroids. The support of this polynomial after homogenization is again a polymatroid. The cave polynomial gives a $K$-theoretic description of a polymatroid in the augmented $K$-ring of a multisymmetric lift. As applications, we settle two conjectures: one by Bandari, Bayati, and Herzog regarding polymatroidal ideals, and another by Castillo, Cid-Ruiz, Mohammadi, and Montaño regarding the Möbius support of a polymatroid.

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arXiv2025

A continuous associahedron of type A

Taking a representation-theoretic viewpoint, we construct a continuous associahedron motivated by the realization of the generalized associahedron in the physical setting. We show that our associahedron shares important properties with the generalized associahedron of type A. Our continuous associahedron is convex and manifests a cluster theory: the points which correspond to the clusters are on its boundary, and the edges that correspond to mutations are given by intersections of hyperplanes. This requires development of several methods that are continuous analogues of discrete methods. We conclude the paper by showing that there is a sequence of embeddings of type A generalized associahedra into our continuous associahedron.

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arXiv2025

Building sets, Chow rings, and their Hilbert series

We establish formulas for the Hilbert series of the Chow ring of a polymatroid using arbitrary building sets. For braid matroids and minimal building sets, our results produce new formulas for the Poincar\'e polynomial of the moduli space $\overline{\mathcal{M}}_{0,n+1}$ of pointed stable rational curves, and recover several previous results by Keel, Getzler, Manin, and Aluffi--Marcolli--Nascimento. We also use our methods to produce examples of matroids and building sets for which the corresponding Chow ring has Hilbert series with non-log-concave coefficients. This contrasts with the real-rootedness and log-concavity conjectures of Ferroni--Schr\"oter for matroids with maximal building sets, and of Aluffi--Chen--Marcolli for braid matroids with minimal building sets.

math.CO↗
arXiv2025

Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids

We study equivariant Kazhdan--Lusztig (KL) and $Z$-polynomials of matroids. We formulate an equivariant generalization of a result by Braden and Vysogorets that relates the equivariant KL and $Z$-polynomials of a matroid with those of a single-element deletion. We also discuss the failure of equivariant $γ$-positivity for the $Z$-polynomial. As an application of our main result, we obtain a formula for the equivariant KL polynomial of the graphic matroid gotten by gluing two cycles. Furthermore, we compute the equivariant KL polynomials of all matroids of corank~$2$ via valuations. This provides an application of the machinery of Elias, Miyata, Proudfoot, and Vecchi to corank $2$ matroids, and it extends results of Ferroni and Schröter.

math.CO↗
arXiv2025

The Newton polytope and Lorentzian property of chromatic symmetric functions

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley--Stembridge conjecture, we show that the allowable coloring weights for indifference graphs of Dyck paths are the lattice points of a permutahedron $\mathcal{P}_λ$, and we give a formula for the dominant weight $λ$. Furthermore, we conjecture that such chromatic symmetric functions are Lorentzian, a property introduced by Brändén and Huh as a bridge between discrete convex analysis and concavity properties in combinatorics, and we prove this conjecture for abelian Dyck paths. We extend our results on the Newton polytope to incomparability graphs of (3+1)-free posets, and we give a number of conjectures and results stemming from our work, including results on the complexity of computing the coefficients and relations with the $ζ$ map from diagonal harmonics.

math.CO↗
arXiv2024

Log-concavity of polynomials arising from equivariant cohomology

We study the equivariant cohomology classes of torus-equivariant subvarieties of the space of matrices. For a large class of torus actions, we prove that the polynomials representing these classes (up to suitably changing signs) are covolume polynomials in the sense of Aluffi. We study the cohomology rings of complex varieties in terms of Macaulay inverse systems over $\mathbb{Z}$. As applications, we show that under certain conditions, the Macaulay dual generator is a denormalized Lorentzian polynomial in the sense of Brändén and Huh, and we give a characteristic-free extension (over $\mathbb{Z}$) of the result of Khovanskii and Pukhlikov describing the cohomology ring of toric varieties in terms of volume polynomials.

math.AG↗
arXiv2024

Chow functions for partially ordered sets

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 the present paper we develop a theory that parallels the KLS theory. To each kernel in a given poset, we associate a polynomial function that we call the \emph{Chow function}. The Chow function often exhibits remarkable properties, and sometimes encodes the graded dimensions of a cohomology or Chow ring. The framework of Chow functions provides natural polynomial analogs of graded module decompositions that appear in algebraic geometry, but that work for arbitrary posets, even when no graded module decomposition is known to exist. In this general framework, we prove a number of unimodality and positivity results 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--Poincar\'e series of matroid Chow rings by Ferroni and Schr\"oter, and flag enumerations on Bruhat intervals of Coxeter groups by Billera and Brenti.

math.CO↗
arXiv2024

Hilbert-Poincaré series of matroid Chow rings and intersection cohomology

We study the Hilbert series of four objects arising in the Chow-theoretic and Kazhdan-Lusztig framework of matroids. These are, respectively, the Hilbert series of the Chow ring, the augmented Chow ring, the intersection cohomology module, and its stalk at the empty flat. We develop a parallelism between the Kazhdan-Lusztig polynomial of a matroid and the Hilbert series of its Chow ring. This extends to a parallelism between the $Z$-polynomial of a matroid and the Hilbert series of its augmented Chow ring. This suggests to bring ideas from one framework to the other. Our two main motivations are the real-rootedness conjecture for all of these polynomials, and the problem of computing them. We provide several intrinsic definitions of these invariants; also, by leveraging that they are valuations under matroid polytope subdivisions, we deduce a fast way for computing them for a large class of matroids. Uniform matroids are a case of combinatorial interest; we link the resulting polynomials with certain real-rooted families such as the (binomial) Eulerian polynomials, and we settle a conjecture of Hameister, Rao, and Simpson. Furthermore, we prove the real-rootedness of the Hilbert series of the augmented Chow rings of uniform matroids via a result of Haglund and Zhang; and in addition, we prove a version of a conjecture of Gedeon in the Chow setting: uniform matroids maximize coefficient-wisely these polynomials for matroids with fixed rank and size. By relying on the nonnegativity of the Kazhdan-Lusztig polynomials and the semi-small decompositions of Braden, Huh, Matherne, Proudfoot, and Wang, we strengthen the unimodality of the Hilbert series of Chow rings, augmented Chow rings, and intersection cohomologies to $γ$-positivity, a property for palindromic polynomials that lies between unimodality and real-rootedness; this settles a conjecture of Ferroni, Nasr, and Vecchi.

math.CO↗
arXiv2023

Generalized splines on graphs with two labels and polynomial splines on cycles

Generalized splines are an algebraic combinatorial framework that generalizes and unifies various established concepts across different fields, most notably the classical notion of splines and the topological notion of GKM theory. The former consists of piecewise polynomials on a combinatorial geometric object like a polytope, whose polynomial pieces agree to a specified degree of differentiability. The latter is a graph-theoretic construction of torus-equivariant cohomology that Shareshian and Wachs used to reformulate the well-known Stanley$\unicode{x2013}$Stembridge conjecture, a reformulation that was recently proven to hold by Brosnan and Chow and independently Guay-Paquet. This paper focuses on the theory of generalized splines. A generalized spline on a graph $G$ with each edge labeled by an ideal in a ring $R$ consists of a vertex-labeling by elements of $R$ so that the labels on adjacent vertices $u, v$ differ by an element of the ideal associated to the edge $uv$. We study the $R$-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: $1)$ for all graphs when the set of possible edge-labelings consists of at most two finitely-generated ideals, and $2)$ for cycles when the set of possible edge-labelings consists of principal ideals generated by elements of the form $(ax+by)^2$ in the polynomial ring $\mathbb{C}[x,y]$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in the theory of classical (analytic) splines.

math.CO↗
arXiv2022

The Deligne-Simpson problem for connections on $\mathbb{G}_m$ with a maximally ramified singularity

The classical additive Deligne-Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne-Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called toral singularities; they are singularities whose leading term with respect to a lattice chain filtration is regular semisimple. We solve this problem in the important special case of connections on $\mathbb{G}_m$ with a maximally ramified singularity at $0$ and possibly an additional regular singular point at infinity. We also give a complete characterization of all such connections which are rigid, under the additional hypothesis of unipotent monodromy at infinity.

math.AG↗
arXiv2021

Equivariant log concavity and representation stability

We expand upon the notion of equivariant log concavity, and make equivariant log concavity conjectures for Orlik--Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik--Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{sl}_n$, we exploit the theory of representation stability to give computer assisted proofs of these conjectures in low degree.

math.CO↗
arXiv2020

A semi-small decomposition of the Chow ring of a matroid

We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The decomposition is used to give simple proofs of Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations for the Chow ring, recovering the main result of [AHK18]. We also show that a similar semi-small orthogonal decomposition holds for the augmented Chow ring of M.

math.AG↗
arXiv2020

Singular Hodge theory for combinatorial geometries

We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy conjecture and the nonnegativity of the coefficients of Kazhdan-Lusztig polynomials for all matroids.

math.CO↗
arXiv2020

Counting linear extensions of posets with determinants of hook lengths

We introduce a class of posets, which includes both ribbon posets (skew shapes) and $d$-complete posets, such that their number of linear extensions is given by a determinant of a matrix whose entries are products of hook lengths. We also give $q$-analogues of this determinantal formula in terms of the major index and inversion statistics. As applications, we give families of tree posets whose numbers of linear extensions are given by generalizations of Euler numbers, we draw relations to Naruse-Okada's positive formulas for the number of linear extensions of skew $d$-complete posets, and we give polynomiality results analogous to those of descent polynomials by Diaz-López, Harris, Insko, Omar, and Sagan.

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