We determine the generating function for the $\mathbb{S}_n$-equivariant Chow polynomials of the braid matroid $B_n$. The Chow polynomial of $B_n$ is the Poincaré polynomial of the wonderful compactification of the complement of the braid arrangement, with respect to the maximal building set. A key input to our result is the identification of this wonderful compactification with a moduli space of multiscale differentials, recently established by Devkota, Robotis, and Zahariuc. In this way, our work also contributes to the literature on topology of moduli spaces of multiscale differentials. To prove our main formula, we study the classes of these moduli spaces in the Grothendieck ring of varieties via the formalism of $\mathbb{S}_n$-spaces developed by Getzler and Pandharipande. We also give a new interpretation of the numerical Chow polynomial of $B_n$ as the Poincaré polynomial of a moduli space of genus-zero relative stable maps to $\mathbb{P}^1$.
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.
Motivated by the intersection theory of moduli spaces of curves, we introduce psi classes in matroid Chow rings and prove a number of properties that naturally generalize properties of psi classes in Chow rings of Losev-Manin spaces. We use these properties of matroid psi classes to give new proofs of (1) a Chow-theoretic interpretation for the coefficients of the reduced characteristic polynomials of matroids, (2) explicit formulas for the volume polynomials of matroids, and (3) Poincare duality for matroid Chow rings.
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 matroid is a commutative, graded, Artinian, Gorenstein algebra with linear and quadratic relations defined by the matroid. Dotsenko conjectured that the Chow ring of any matroid is Koszul. The purpose of this paper is to prove Dotsenko's conjecture. We also show that the augmented Chow ring of a matroid is Koszul. As a corollary, we show that the Chow rings a…
For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the $h$-vector of a matroid complex, improving on an old conjecture of Dawson.
The equivariant Kazhdan-Lusztig polynomial of a braid matroid may be interpreted as the intersection cohomology of a certain partial compactification of the configuration space of n distinct labeled points in the plane, regarded as a graded representation of the symmetric group. We show that, in fixed cohomological degree, this sequence of representations of symmetric groups naturally admits the structure of an FS-module, and that the dual FS^op-module is finitely generated. Using the work of Sam and Snowden, we give an asymptotic formula for the dimensions of these representations and obtain …
A reciprocal linear space is the image of a linear space under coordinate-wise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciproca…
Let $G$ be the product $GL_r(C) \times (C^\times)^n$. We show that the $G$-equivariant Chow class of a $G$ orbit closure in the space of $r$-by-$n$ matrices is determined by a matroid. To do this, we split the natural surjective map from the $G$ equvariant Chow ring of the space of matrices to the torus equivariant Chow ring of the Grassmannian. The splitting takes the class of a Schubert variety to the corresponding factorial Schur polynomial, and also has the property that the class of a subvariety of the Grassmannian is mapped to the class of the closure of those matrices whose row span is …
Centre de Mathématiques Laurent Schwartz, Polytechnique（洛朗·施瓦茨数学中心，巴黎综合理工学院）；Dipartimento di Matematica, Università di Pisa（比萨大学数学系）；Department of Mathematics, University of Michigan（密歇根大学数学系）
We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed by the Giansiracusas in arXiv:1308.0042, so there is a well-defined Hilbert-Chow morphism from a tropical scheme to the underlying tropical cycle. For a subscheme of projective space given by a homogeneous ideal I we show that this tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces I_d for d \geq 0. We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme struct…
We use techniques from Gromov-Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov-Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement's wonderful model is a co…
The literature on maximal torus orbits in the Grassmannian is vast; in this paper we initiate a program to extend this to diagonal subtori. Our main focus is generalizing portions of Kapranov's seminal work on Chow quotient compactifications of these orbit spaces. This leads naturally to discrete polymatroids, generalizing the matroidal framework underlying Kapranov's results. By generalizing the Gelfand-MacPherson isomorphism, these Chow quotients are seen to compactify spaces of arrangements of parameterized linear subspaces, and a generalized Gale duality holds here. A special case is birat…
Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the "characteristic polynomial" of a matroid $M$, polynomial whose coefficients enumerate its subsets of given rank, form a log-concave sequence. K. Adiprasito, J. Huh et E. Katz have proved this conjecture using methods which, although entirely combinatorial, are inspired by algebraic geometry. From the Bergman fan of the matroid $M$, they define a graded "Chow ring" $A(M)$ for which they prove analogs of the Poincaré duality, the Hard Lefschet…
We compute the Chow ring of an arbitrary heavy/light Hassett space $\bar{M}_{0, w}$. These spaces are moduli spaces of weighted pointed stable rational curves, where the associated weight vector $w$ consists of only heavy and light weights. Work of Cavalieri et al. exhibits these spaces as tropical compactifications of hyperplane arrangement complements. The computation of the Chow ring then reduces to intersection theory on the toric variety of the Bergman fan of a graphic matroid. Keel has calculated the Chow ring $A^*(\bar{M}_{0, n})$ of the moduli space $\bar{M}_{0, n}$ of stable nodal $n$…
We compute the $GL_{r+1}$-equivariant Chow class of the $GL_{r+1}$-orbit closure of any point $(x_1, \ldots, x_n) \in (\mathbb{P}^r)^n$ in terms of the rank polytope of the matroid represented by $x_1, \ldots, x_n \in \mathbb{P}^r$. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each $d\times n$ matrix $M$ an $n$-ary operation $[M]_\hbar$ on the small equivariant quantum cohomology ring of $\mathbb{P}^r$, which is the $n$-ary quantum product when $M$ is an invertible matrix. We prove that $M \mapsto [M]_\hbar$ is a …
