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Tautological bundles of realizations of matroids were introduced in Berget et al. (Invent Math 233:951–1039, 2023) as a unifying geometric model for studying matroids. We compute the cohomologies of exterior and symmetric powers of these vector bundles, and show that they depend only on the matroid of the realization. As an application, we show that the log canonical bundle of a wonderful compactification of a hyperplane arrangement complement, in particular the moduli space of pointed rational curves M ¯ 0 , n , has vanishing higher cohomologies. [ABSTRACT FROM AUTHOR]

We obtain upper bounds for the torsion in the K-groups of the ring of integers of imaginary quadratic number fields, in terms of their discriminants. [ABSTRACT FROM AUTHOR]

We give an explicit combinatorial presentation of the Chow groups of a toric scheme over a DVR. As an application, we compute the Chow groups of several toric schemes over a DVR and of their special fibers. [ABSTRACT FROM AUTHOR]

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in N = 4 super Yang–Mills theory. It generalizes cyclic polytopes and the positive Grassmannian and has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the m = 4 amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for Gr 4 , n . Secondly, we exhibit a tiling of the m = 4 amplituhedron which involves a tile which does not come from the BCFW recurrence—the spurion tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for Gr 4 , n . This paper is a companion to our previous paper “Cluster algebras and tilings for the m = 4 amplituhedron.” [ABSTRACT FROM AUTHOR]

The two best studied toric degenerations of the flag variety are those given by the Gelfand–Tsetlin and FFLV polytopes. Each of them degenerates further into a particular monomial variety which raises the problem of describing the degenerations intermediate between the toric and the monomial ones. Using a theorem of Zhu one may show that every such degeneration is semitoric with irreducible components given by a regular subdivision of the corresponding polytope. This leads one to study the parts that appear in such subdivisions as well as the associated toric varieties. It turns out that these parts lie in a certain new family of poset polytopes which we term relative poset polytopes: each is given by a poset and a weakening of its order relation. In this paper we give an in depth study of (both common and marked) relative poset polytopes and their toric varieties in the generality of an arbitrary poset. We then apply these results to degenerations of flag varieties. We also show that our family of polytopes generalizes the family studied in a series of papers by Fang, Fourier, Litza and Pegel while sharing their key combinatorial properties such as pairwise Ehrhart-equivalence and Minkowski-additivity. [ABSTRACT FROM AUTHOR]

Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are indexed by posets of roots labeled long/short. These labeled posets generalize Young diagrams. We prove that Schubert varieties in potentially different cominuscule flag varieties are isomorphic as varieties if and only if their corresponding labeled posets are isomorphic, generalizing the classification of Grassmannian Schubert varieties using Young diagrams by the last two authors. Our proof is type-independent. [ABSTRACT FROM AUTHOR]

We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory Q K T (G / B) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of Q K T (G / B) . In type A n - 1 , we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory Q K (S L n / B) ; we also obtain very explicit information about the coefficients in the respective Chevalley formula. [ABSTRACT FROM AUTHOR]

Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. This paper solves the long standing open problem of giving a non-iterative formula for straightening a filling. We apply our formula to give a complete generalization of a theorem of Gonciulea and Lakshmibai. [ABSTRACT FROM AUTHOR]

We associate to a projective n-dimensional toric variety X Δ a pair of co-commutative (but generally non-commutative) Hopf algebras H X α , H X T . These arise as Hall algebras of certain categories Coh α (X) , Coh T (X) of coherent sheaves on X Δ viewed as a monoid scheme—i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When X Δ is smooth, the category Coh T (X) has an explicit combinatorial description as sheaves whose restriction to each A n corresponding to a maximal cone σ ∈ Δ is determined by an n-dimensional generalized skew shape. The (non-additive) categories Coh α (X) , Coh T (X) are treated via the formalism of proto-exact/proto-abelian categories developed by Dyckerhoff–Kapranov. The Hall algebras H X α , H X T are graded and connected, and so enveloping algebras H X α ≃ U (n X α) , H X T ≃ U (n X T) , where the Lie algebras n X α , n X T are spanned by the indecomposable coherent sheaves in their respective categories. We explicitly work out several examples, and in some cases are able to relate n X T to known Lie algebras. In particular, when X = P 1 , n X T is isomorphic to a non-standard Borel in gl 2 [ t , t - 1 ] . When X is the second infinitesimal neighborhood of the origin inside A 2 , n X T is isomorphic to a subalgebra of gl 2 [ t ] . We also consider the case X = P 2 , where we give a basis for n X T by describing all indecomposable sheaves in Coh T (X) . [ABSTRACT FROM AUTHOR]

We solve the dual multijoint problem and prove the existence of so-called factorisations for arbitrary fields and multijoints of k j -planes. More generally, we deduce a discrete analogue of a theorem due in essence to Bourgain and Guth. Our result is a universal statement which describes a property of the discrete wedge product without any explicit reference to multijoints and is stated as follows: Suppose that k 1 + ... + k d = n . There is a constant C = C (n) so that for any field F and for any finitely supported function S : F n → R ≥ 0 , there are factorising functions s k j : F n × Gr (k j , F n) → R ≥ 0 such that V 1 ∧ ⋯ ∧ V d S p d ≤ ∏ j = 1 d s k j p , V j , for every p ∈ F n and every tuple of planes V j ∈ Gr (k j , F n) , and ∑ p ∈ π j s (p , e (π j)) = C S d , for every k j -plane π j ⊂ F n , where e (π j) ∈ Gr (k j , F n) , is the translate of π j that contains the origin and ∧ denotes the discrete wedge product. [ABSTRACT FROM AUTHOR]

The goal of this paper is to better understand the quasimap vertex functions of type A Nakajima quiver varieties. To that end, we construct an explicit embedding of any type A quiver variety into a type A quiver variety with all framings at the rightmost vertex of the quiver. Then, we consider quasimap counts, showing that the map induced by this embedding on equivariant K-theory preserves vertex functions. [ABSTRACT FROM AUTHOR]

In this paper, we try to answer some questions raised by Cangelmi (Eur J Comb 33(7):1444–1448, 2012). We reinterpret the Riemann–Hurwitz theorem of orientable algebraic hypermaps by introducing tripartite graph morphisms and obtain Riemann–Roch theorems for orientable hypermaps by defining the divisor of a function f on darts. In addition, we extend Riemann–Roch theorem to non-orientable hypermaps by suitably replacing the orientable genus with the non-orientable genus. Finally, as an application of the Riemann–Hurwitz theorem, we establish the second main theorem from the viewpoint of Nevanlinna theory. [ABSTRACT FROM AUTHOR]

Chow rings of flag varieties have bases of Schubert cycles $\sigma _u $ , indexed by permutations. A major problem of algebraic combinatorics is to give a positive combinatorial formula for the structure constants of this basis. The celebrated Littlewood–Richardson rules solve this problem for special products $\sigma _u \cdot \sigma _v$ , where u and v are p-Grassmannian permutations.

Let $\overline {\mathcal {M}}_{g, m|n}$ denote Hassett’s moduli space of weighted pointed stable curves of genus g for the heavy/light weight data $$\begin{align*}\left(1^{(m)}, 1/n^{(n)}\right),\end{align*}$$

The $\Delta $ -Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall–Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a $\Delta $ -Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field $\mathbb {F}_q$ and partitioning the $\Delta $ -Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geometric meaning to a formula of Haglund, Rhoades and Shimozono for the Hall–Littlewood expansion of the symmetric function in the Delta Conjecture at $t=0$ .

A Schubert variety in the complete flag manifold G L n / B is Levi-spherical if the action of a Borel subgroup in a Levi subgroup of a standard parabolic has an open dense orbit. We give a combinatorial classification of these Schubert varieties. This establishes a conjecture of the latter two authors, and a new formulation in terms of standard Coxeter elements. Our proof uses and contributes to the theory of key polynomials (type A Demazure module characters). [ABSTRACT FROM AUTHOR]

We develop a combinatorial rule to compute the real geometry of type B Schubert curves S (λ ∙) in the orthogonal Grassmannian O G (n , C 2 n + 1) , which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A [J. Algebraic Combin. 45(1), 191–243 (2017)]. First, using the type B Wroński map studied in [Adv. Math. 224(3), 827–862 (2010)], we show that the real locus of the Schubert curve has a natural covering map to R P 1 , with monodromy operator ω defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute ω without rectifying the skew tableau. The first uses the crystal operators introduced in [Algebr. Comb. 3(3), 693–725 (2020)], while the second uses local switches much like jeu de taquin. The switching algorithm further computes the K-theory coefficient of the Schubert curve: its nonadjacent switches precisely enumerate Pechenik and Yong's shifted genomic tableaux. The connection to K-theory also gives rise to a partial understanding of the complex geometry of these curves. [ABSTRACT FROM AUTHOR]

The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing a combinatorial version of the Peterson isomorphism with a result of Lapointe and Morse relating quantum Littlewood-Richardson coefficients for the Grassmannian to k-Schur analogs in the homology of the affine Grassmannian obtained by adding rim hooks. We show that these comparisons on quantum cohomology are equivalent, up to Postnikov's strange duality isomorphism. [ABSTRACT FROM AUTHOR]

We prove that twisted versions of Schubert polynomials defined by S ~ w 0 = x 1 n - 1 x 2 n - 2 ⋯ x n - 1 and S ~ w s i = (s i + ∂ i) S ~ w are monomial positive and give a combinatorial formula for their coefficients. In doing so, we reprove and extend a previous result about positivity of skew divided difference operators and show how it implies the Pieri rule for Schubert polynomials. We also give positive formulas for double versions of the S ~ w as well as their localizations. [ABSTRACT FROM AUTHOR]

Aganagic and Okounkov proved that the elliptic stable envelope provides the pole cancellation matrix for the enumerative invariants of quiver varieties known as vertex functions. This transforms a basis of a system of q-difference equations holomorphic in variables z with poles in variables a to a basis of solutions holomorphic in a with poles in z . The resulting functions are expected to be the vertex functions of the 3d mirror dual variety. In this paper, we prove that the functions obtained by applying the elliptic stable envelope to the vertex functions of the cotangent bundle of the full flag variety are precisely the vertex functions for the same variety under an exchange of the parameters Å ↔ z . As a corollary of this, we deduce the expected 3d mirror relationship for the elliptic stable envelope. [ABSTRACT FROM AUTHOR]

The orbifold Euler characteristic of the moduli space ℳ g ; 1 of genus g smooth curves with one marked point (g ≥ 1) was calculated by Harer and Zagier: χ (ℳ g ; 1) = ζ (1 − 2 g) = − B 2 g / (2 g) , where ζ is the Riemann zeta function and B2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics. [ABSTRACT FROM AUTHOR]

In this paper we prove a formula relating the equivariant Euler characteristic of K-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the index vertex is the power series expansion of a rational function. This result is a consequence of the 3d mirror self-symmetry of the variety considered here. In general, one expects an analogous result to hold for any two varieties related by 3d mirror symmetry. [ABSTRACT FROM AUTHOR]

Let X be a symplectic variety equipped with an action of a torus A . Let ν b ⊂ A be a finite cyclic subgroup. We show that K-theoretic stable envelope of the fixed point set X ν b ⊂ X can be obtained via a limit of the elliptic stable envelopes of X. An example of X given by the Hilbert scheme of points in the complex plane is considered in detail. [ABSTRACT FROM AUTHOR]

A normal variety X is called H-spherical for the action of the complex reductive group H if it contains a dense orbit of some Borel subgroup of H. We resolve a conjecture of Hodges–Yong by showing that their spherical permutations are characterized by permutation pattern avoidance. Together with results of Gao–Hodges–Yong this implies that the sphericality of a Schubert variety X w with respect to the largest possible Levi subgroup is characterized by this same pattern avoidance condition. [ABSTRACT FROM AUTHOR]

We study Brill–Noether existence on a finite graph using methods from polyhedral geometry and lattices. We start by formulating analogues of the Brill–Noether conjectures (both the existence and non-existence parts) for R -divisors, i.e. divisors with real coefficients, on a graph. We then reformulate the Brill–Noether existence conjecture for R -divisors on a graph in geometric terms, that we refer to as the covering radius conjecture and we show a weak version, in support of it. Using this, we show an approximate version of the Brill–Noether existence conjecture for divisors on a graph. As applications, we derive upper bounds on the gonality of a graph and its R -divisor analogue. [ABSTRACT FROM AUTHOR]

We study the harmonic polytope, which arose in Ardila, Denham, and Huh's work on the Lagrangian geometry of matroids. We describe its combinatorial structure, showing that it is a (2 n - 2) -dimensional polytope with (n !) 2 1 + 1 2 + ⋯ + 1 n vertices and 3 n - 3 facets. We also give a formula for its volume: it is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra. [ABSTRACT FROM AUTHOR]

Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$. A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$. Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results. [ABSTRACT FROM AUTHOR]

We prove that projective spaces of Lorentzian and real stable polynomials are homeomorphic to Euclidean balls. This solves a conjecture of June Huh and the author. The proof utilises and refines a connection between the symmetric exclusion process in interacting particle systems and the geometry of polynomials.