arXiv subjects

Ezra Miller

Publications and source records attributed to Ezra Miller.

At least 55 records · Page 3Linked to original sources

arXiv2008

Duality of antidiagonals and pipe dreams

Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A. Knutson and E. Miller, Gröbner geometry of Schubert polynomials, Ann. Math. 161, 1245-1318]. The original proof of the duality was roundabout, relying on the algebra of certain monomial ideals and a recursive characterization of reduced pipe dreams. This paper provides a direct combinatorial proof.

math.CO↗

arXiv2008

Combinatorics of binomial primary decomposition

An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables.

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arXiv2008

Binomial D-modules

We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary Z^d-graded binomial ideal I along with Euler operators defined by the grading and a parameter in C^d. We determine the parameters for which these D-modules (i) are holonomic (equivalently, regular holonomic, when I is standard-graded); (ii) decompose as direct sums indexed by the primary components of I; and (iii) have holonomic rank greater than the generic rank. In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arrangement in C^d. In the special case of Horn hypergeometric D-modules, when I is a lattice basis ideal, we furthermore compute the generic holonomic rank combinatorially and write down a basis of solutions in terms of associated A-hypergeometric functions. This study relies fundamentally on the explicit lattice point description of the primary components of an arbitrary binomial ideal in characteristic zero, which we derive in our companion article arxiv:0803.3846.

math.AG↗

arXiv2007

Multiplier ideals of sums via cellular resolutions

Fix nonzero ideal sheaves a_1,...,a_r on a normal Q-Gorenstein complex variety X. Fix any positive real number c, and consider the multiplier ideal J of the sum a_1+...+a_r with weighting coefficient c. We construct an exact sequence resolving J by sheaves over X that are direct sums of multiplier ideals for products a_1^{v_1}...a_r^{v_r} for various real vectors v such that v_1+...+v_r = c. The resolution is cellular, in the sense that its boundary maps are encoded by the algebraic chain complex of a regular CW-complex. The CW-complex is naturally expressed as a triangulation T of the simplex of nonnegative real vectors summing to c. The acyclicity of our resolution reduces to that of a cellular free resolution, supported on T, of a related monomial ideal. This acyclicity rests on a comparison between the homology of certain homology-manifolds-with-boundary and the homology of the simplicial complexes obtained by deleting collections of boundary faces from them. Our resolution implies the multiplier ideal sum formula J((a_1+...+a_r)^c) = \sum_{|v|=c} J(a_1^{v_1}...a_r^{v_r}), which implicitly follows from Takagi's proof of the two-summand formula (math.AG/0410612). We recover Howald's multiplier ideal formula for monomial ideals (math.AG/0003232) as a special case. Our resolution also yields a new exactness proof for the Skoda complex.

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arXiv2007

A Kleiman-Bertini Theorem for sheaf tensor products

Fix a variety X with a transitive (left) action by an algebraic group G. Let E and F be coherent sheaves on X. We prove that, for elements g in a dense open subset of G, the sheaf Tor_i^X(E, g F) vanishes for all i > 0. When E and F are structure sheaves of smooth subschemes of X in characteristic zero, this follows from the Kleiman-Bertini theorem; our result has no smoothness hypotheses on the supports of E or F, or hypotheses on the characteristic of the ground field.

math.AG↗

arXiv2006

Tableau complexes

Let X,Y be finite sets and T a set of functions from X -> Y which we will call "tableaux". We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such "tableau complexes" have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions. In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are labeled by Buch's set-valued semistandard tableaux. One vertex decomposition of this "Young tableau complex" parallels Lascoux's transition formula for vexillary double Grothendieck polynomials. Consequently, we obtain formulae (both old and new) for these polynomials. In particular, we present a common generalization of the formulae of Wachs and Buch, each of which implies the classical tableau formula for Schur polynomials.

math.CO↗

arXiv2005

Gröbner geometry of vertex decompositions and of flagged tableaux

We relate a classic algebro-geometric degeneration technique, dating at least to [Hodge 1941], to the notion of vertex decompositions of simplicial complexes. The good case is when the degeneration is reduced, and we call this a "geometric vertex decomposition". Our main example in this paper is the family of vexillary matrix Schubert varieties, whose ideals are also known as (one-sided) ladder determinantal ideals. Using a diagonal term order to specify the (Gröbner) degeneration, we show that these have geometric vertex decompositions into simpler varieties of the same type. From this, together with the combinatorics of the pipe dreams of [Fomin--Kirillov 1996], we derive a new formula for the numerators of their multigraded Hilbert series, the double Grothendieck polynomials, in terms of "flagged set-valued tableaux". This unifies work of [Wachs 1985] on flagged tableaux, and [Buch 2002] on set-valued tableaux, giving geometric meaning to both. This work focuses on diagonal term orders, giving results complementary to those of [Knutson--Miller 2004], where it was shown that the generating minors form a Gröbner basis for any antidiagonal term order and any matrix Schubert variety. We show here that under a diagonal term order, the only matrix Schubert varieties for which these minors form Gröbner bases are the vexillary ones, reaching an end toward which the ladder determinantal literature had been building.

math.AG↗

arXiv2004

Reciprocal domains and Cohen-Macaulay $d$-complexes in $R^d$

We extend a reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities. The proof highlights the roles played by Cohen-Macaulayness and canonical modules. The extension raises the issue of whether a Cohen-Macaulay complex of dimension d embedded piecewise-linearly in d-space is necessarily a d-ball. This is observed to be true for d at most 3, but false for d=4.

math.CO↗

arXiv2004

Homological Methods for Hypergeometric Families

We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems H_A(β) arising from a d x n integer matrix A and a parameter β\in \CC^d. To do so we introduce an Euler-Koszul functor for hypergeometric families over \CC^d, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter βis rank-jumping for H_A(β) if and only if βlies in the Zariski closure of the set of \ZZ^d-graded degrees αwhere the local cohomology \bigoplus_{i<d}H^i_\frakm(\CC[\NN A])_αof the semigroup ring \CC[\NN A] supported at its maximal graded ideal \frakm is nonzero. Consequently, H_A(β) has no rank-jumps over \CC^d if and only if \CC[\NN A] is Cohen-Macaulay of dimension d.

math.AG↗

arXiv2004

Combinatorics of rank jumps in simplicial hypergeometric systems

Let A be an integer (d x n) matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d-1. Write \NA for the semigroup generated by the columns of A. It was proved by M. Saito [math.AG/0012257] that the semigroup ring \CC[\NA] over the complex numbers \CC is Cohen-Macaulay if and only if the rank of the GKZ hypergeometric system H_A(beta) equals the normalized volume of conv(A) for all complex parameters beta in \CC^d. Our refinement here shows, in this simplicial case, that H_A(beta) has rank strictly larger than the volume of conv(A) if and only if beta lies in the Zariski closure in \CC^d of all \ZZ^d-graded degrees where the local cohomology H^i_m(\CC[\NA]) at the maximal ideal m is nonzero for some i < d.

math.AC↗

arXiv2003

Alternating formulas for K-theoretic quiver polynomials

The main theorem here is the K-theoretic analogue of the cohomological `stable double component formula' for quiver functions in [Knutson, Miller, and Shimozono, math.AG/0308142]. This K-theoretic version is still in terms of lacing diagrams, but nonminimal diagrams contribute terms of higher degree. The motivating consequence is a conjecture of A. Buch on the sign-alternation of the coefficients appearing in his expansion of quiver K-polynomials in terms of stable Grothendieck polynomials for partitions [Buch, math.AG/0104029].

math.CO↗

arXiv2003

Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings

This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry. Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v in S, which is the exponential map to S from the tangent space at v. We characterize the `cut locus' (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3-polytopes into R^2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.

math.MG↗

arXiv2003

Gröbner geometry of Schubert polynomials

Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric reasons. This results in a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. Given a permutation w in S_n, we consider a determinantal ideal I_w whose generators are certain minors in the generic n x n matrix (filled with independent variables). Using `multidegrees' as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal I_w: - variously graded multidegrees and Hilbert series in terms of ordinary and double Schubert and Grothendieck polynomials; - a Gröbner basis consisting of minors in the generic n x n matrix; - the Stanley-Reisner complex of the initial ideal in terms of known combinatorial diagrams associated to permutations in S_n; and - a procedure inductive on weak Bruhat order for listing the facets of this complex, thereby generating the coefficients of Schubert polynomials by a positive recursion on combinatorial diagrams. We show that the initial ideal is Cohen-Macaulay, by identifying the Stanley-Reisner complex as a special kind of ``subword complex in S_n'', which we define generally for arbitrary Coxeter groups, and prove to be shellable by giving an explicit vertex decomposition. We also prove geometrically a general positivity statement for multidegrees of subschemes.

math.AG↗

arXiv2003

Algorithms for graded injective resolutions and local cohomology over semigroup rings

Let Q be an affine semigroup generating Z^d, and fix a finitely generated Z^d-graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z^d-graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modules H^i_I(M) supported on any monomial (that is, Z^d-graded) ideal I. Since these local cohomology modules are neither finitely generated nor finitely cogenerated, part of this task is defining a finite data structure to encode them.

math.AC↗

arXiv2003

Subword complexes in Coxeter groups

Let (Π,Σ) be a Coxeter system. An ordered list of elements in Σand an element in Πdetermine a {\em subword complex}, as introduced in our paper on Gröbner geometry of Schubert polynomials (math.AG/0110058). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which is due to Fomin and Kirillov, are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented.

math.CO↗

arXiv2003

Four positive formulae for type A quiver polynomials

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1, and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula was conjectured for by Buch and Fulton in terms of factor sequences of Young tableaux; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in ``doubled'' versions, two for `double quiver polynomials', and the other two for their stable versions, the `double quiver functions', where setting half the variables equal to the other half specializes to the ordinary case.

math.AG↗

arXiv2003

Toric degeneration of Schubert varieties and Gel'fand-Cetlin polytopes

This note constructs the flat toric degeneration of the manifold FL_n of flags in C^n from [Gonciulea-Lakshmibai 96] as an explicit GIT quotient of the Gr"obner degeneration in [Knutson-Miller 03]. This implies that Schubert varieties degenerate to reduced unions of toric varieties, associated to faces indexed by rc-graphs (reduced pipe dreams) in the Gel'fand-Cetlin polytope. Our explicit description of the toric degeneration of FL_n provides a simple explanation of how Gel'fand-Cetlin decompositions for irreducible polynomial representations of GL_n arise via geometric quantization.

math.AG↗

arXiv2002

Mitosis recursion for coefficients of Schubert polynomials

Mitosis is a rule introduced by [Knutson-Miller, 2002] for manipulating subsets of the n by n grid. It provides an algorithm that lists the reduced pipe dreams (also known as rc-graphs) [Fomin-Kirillov, Bergeron-Billey] for a permutation w in S_n by downward induction on weak Bruhat order, thereby generating the coefficients of Schubert polynomials [Lascoux-Schutzenberger] inductively. This note provides a short and purely combinatorial proof of these properties of mitosis.

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