Given a permutation w ∈ S n , we consider a determinantal ideal I w whose generators are certain minors in the generic n × 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 × n matrix;• the Stanley-Reisner simplicial complex of the initial ideal in terms of known combinatorial diagrams [FK96], [BB93] associated to permutations in S n ; and• a procedure inductive on weak Bruhat order for listing the facets of this complex.We show that the initial ideal is Cohen-Macaulay, by identifying the StanleyReisner 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. Our main theorems provide a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. More precisely, we apply these theorems to:• define 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;*AK was partly supported by the Clay Mathematics Institute, Sloan Foundation, and NSF. EM was supported by the Sloan Foundation and NSF. ALLEN KNUTSON AND EZRA MILLER• rederive from a topological perspective Fulton's Schubert polynomial formula for universal cohomology classes of degeneracy loci of maps between flagged vector bundles;• supply new proofs that Schubert and Grothendieck polynomials represent cohomology and K-theory classes on the flag manifold; and• provide determinantal formulae for the multidegrees of ladder determinantal rings.The proofs of the main theorems introduce the technique of "Bruhat induction", consisting of a collection of geometric, algebraic, and combinatorial tools, based on divided and isobaric divided differences, that allow one to prove statements about determinantal ideals by induction on weak Bruhat order.

New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be significant even after controlling for correlations from earlier significant summaries.

This paper introduces the class of strongly endotactic networks, a subclass of the endotactic networks introduced by Craciun, Nazarov, and Pantea. The main result states that the global attractor conjecture holds for complex-balanced systems that are strongly endotactic: every trajectory with positive initial condition converges to the unique positive equilibrium allowed by conservation laws. This extends a recent result by Anderson for systems where the reaction diagram has only one linkage class (connected component). The results here are proved using differential inclusions, a setting that includes power-law systems. The key ideas include a perspective on reaction kinetics in terms of combinatorial geometry of reaction diagrams, a projection argument that enables analysis of a given system in terms of systems with lower dimension, and an extension of Birch's theorem, a well-known result about intersections of affine subspaces with manifolds parameterized by monomials.

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 2005], 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.

Let (Π, Σ) be a Coxeter system. An ordered list of elements in Σ and an element in Π determine a subword complex, as introduced in [KM03]. 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 appeared in [FK94], are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented.

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 ) H_A(\beta ) arising from a d × n d \times n integer matrix  A A and a parameter β ∈ C d \beta \in \mathbb {C}^d . To do so we introduce an Euler–Koszul functor for hypergeometric families over  C d \mathbb {C}^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 β ∈ C d \beta \in \mathbb {C}^d is rank-jumping for H A ( β ) H_A(\beta ) if and only if β \beta lies in the Zariski closure of the set of C d \mathbb {C}^d -graded degrees  α \alpha where the local cohomology ⨁ i > d H m i ( C [ N A ] ) α \bigoplus _{i > d} H^i_\mathfrak m(\mathbb {C}[\mathbb {N} A])_\alpha of the semigroup ring C [ N A ] \mathbb {C}[\mathbb {N} A] supported at its maximal graded ideal  m \mathfrak m is nonzero. Consequently, H A ( β ) H_A(\beta ) has no rank-jumps over  C d \mathbb {C}^d if and only if C [ N A ] \mathbb {C}[\mathbb {N} A] is Cohen–Macaulay of dimension  d d .

Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated ‫ގ‬ n -graded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution of its Alexander dual and contains all of the maps in the minimal free resolution of M over every ‫ޚ‬ n -graded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcm-lattices. Using injective resolutions, theorems of Eagon, Reiner, and Terai are generalized to all ‫ގ‬ n -graded modules: the projective dimension of M equals the support-regularity of its Alexander dual, and M is Cohen᎐Macaulay if and only if its Alexander dual has a support-linear free resolution. Alexander duality is applied in the context of the n i Ž . ‫ޚ‬ -graded local cohomology functors H y for squarefree monomial ideals I in I the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I s ᒊ is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, i Ž . that Terai's formula for the Hilbert series of H S is equivalent to Hochster's for

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99]. 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 [BF99]; 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.Our method begins by identifying quiver polynomials as multidegrees [Jos84, Ros89] via equivariant Chow groups [EG98]. Then we make use of Zelevinsky's map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds [Zel85]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double Schubert polynomials [LS82] associated to Zelevinsky permutations; this is our first formula. In the process, we provide a simple argument that Zelevinsky maps are schemetheoretic isomorphisms (originally proved in [LM98] using standard monomial theory). Writing double Schubert polynomials in terms of pipe dreams [FK96] then provides another geometric formula for double quiver polynomials, via [KM03a]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for double quiver functions in terms of products of Stanley symmetric functions [Sta84]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92,KM03a] identifies the summands in our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del Fra in the representation theory of quivers [AD80]. Finally, we apply the combinatorial theory of key polynomials to pass from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a, RS98], and from there to a 'double stable' generalization of the Buch-Fulton conjecture.