相关论文: Chow rings and augmented Chow rings of uniform mat…
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,…
We provide explicit combinatorial formulas for the Chow polynomial and for the augmented Chow polynomial of uniform matroids, thereby proving a conjecture by Ferroni. These formulas refine existing formulas by Hampe and by Eur, Huh, and…
The Chow ring of a matroid (or more generally, atomic latice) is an invariant whose importance was demonstrated by Adiprasito, Huh and Katz, who used it to resolve the long-standing Heron-Rota-Welsh conjecture. Here, we make a detailed…
In this paper, we prove that the Chow ring and augmented Chow ring of a matroid are equivariantly $\gamma$-positive under the action of any group of automorphisms. Our approach provides an explicit combinatorial interpretation of the…
Motivated by the question of whether Chow polynomials of matroids have only real roots, this article revisits the known relationship between Eulerian polynomials and the Hilbert series of Chow rings of permutohedral varieties. This is done…
We introduce and develop the theory of UMEL-shellable posets. These are posets equipped with an edge-lexicographical labeling satisfying certain uniformity and monotonicity properties. This framework encompasses classical families of…
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…
Chow rings of matroids were instrumental in the resolution of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz and in the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang. The Chow ring of a…
It is well known that the Eulerian polynomial is the Hilbert series of the cohomology of the permutahedral variety. Stanley obtained a formula showing that the cohomology carries a permutation representation of $\mathfrak{S}_n$. We answer a…
It is well known that the Eulerian polynomial is the Hilbert series of the cohomology of the permutohedral variety. We answer a question of Stembridge on finding a geometric explanation of the \emph{permutation representation} this…
We study the Chow classes of arbitrary matroids in the Grassmannian. We develop a new combinatorial approach for computing them, by first focusing on snake matroids and then extending our results via valuativity to any matroid. Our main…
We introduce a presentation of the Chow ring of a matroid by a new set of generators, called "simplicial generators." These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation via…
Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton--Sturmfels, and…
We prove two dual recursive decompositions as a graded $\underline{\mathrm{CH}}(M)\otimes \underline{\mathrm{CH}}(N)$-module of the Chow ring $\underline{\mathrm{CH}}(M\oplus N)$ of the direct sum of matroids. We use this to obtain a…
Given a matroid and a group of its matroid automorphisms, we study the induced group action on the Chow ring of the matroid. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring…
Well-known conjectures of Charney--Davis, Gal, and Nevo--Petersen predict increasingly strong positivity phenomena for the h-vectors of flag simplicial spheres. In this paper, we formulate and prove matroid analogues of these conjectures in…
June Huh and Matthew Stevens conjectured that the Hilbert-Poincar\'e series of the Chow ring of any matroid is a polynomial with only real zeros. We prove this conjecture for the class of uniform matroids. We also prove that the Chow…
Motivated by the notion of the inverse $Z$-polynomial introduced by Ferroni, Matherne, Stevens, and Vecchi, we study the equivariant inverse $Z$-polynomial of a matroid equipped with a finite group. We prove that the coefficients of the…
We give elementary and non-inductive proofs of three fundamental theorems about Chow rings of matroids: the standard monomial basis, Poincare duality, and the dragon-Hall-Rado formula. Our approach, which also works for augmented Chow rings…
We introduce the rank-nullity ring of a matroid $M$, which is a subring of the Chow ring of the permutahedral toric variety. This subring contains the tautological Chern classes of $M$, a fact we deduce from a highly symmetric formula for…