Let be a finite Weyl group and the corresponding affine Weyl group. A random element of can be obtained as a reduced random walk on the alcoves of . By a theorem of Lam (Ann. Probab. 2015), such a walk almost surely approaches one of many directions. We compute these directions when is , and and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types and in the affirmative and for type in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process called the -TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.
Codimension two Artinian algebras have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three Artinian Gorenstein algebras. Despite much work, the strong Lefschetz property for codimension three Artinian Gorenstein algebra has remained largely mysterious; our results build on and strengthen some of the previous results. We here show that every standard-graded codimension three Artinian Gorenstein algebra A having maximum value of the Hilbert function at most six has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of A, we show that A is almost strong Lefschetz, they are strong Lefschetz except in the extremal pair of degrees.
We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion. The first family we consider consists of stretched hook shapes, where we use the cocharge generating polynomial as CSP-polynomial. The second family contains skew shapes, consisting of disjoint rectangles. Again, the charge generating polynomial together with promotion exhibits the cyclic sieving phenomenon. This generalizes earlier results by B. Rhoades and later B. Fontaine and J. Kamnitzer. Finally, we consider certain skew ribbons, where promotion behaves in a predictable manner. This result is stated in the form of a bicyclic sieving phenomenon. One of the tools we use is a novel method for computing charge of skew semistandard tableaux, in the case when every number in the tableau occurs with the same frequency.
When λ is a partition, the specialized non-symmetric Macdonald polynomial Eλ(x; q; 0) is symmetric and related to a modified Hall–Littlewood polynomial. We show that whenever all parts of the integer partition λ are multiples of n, the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under an n-fold cyclic shift of the columns. The corresponding CSP polynomial is given by Eλ(x; q; 0). In addition, we prove a refined cyclic sieving phenomenon where the content of the fillings is fixed. This refinement is closely related to an earlier result by B. Rhoades. We also introduce a skew version of Eλ(x; q; 0). We show that these are symmetric and Schur positive via a variant of the Robinson–Schenstedt–Knuth correspondence and we also describe crystal raising and lowering operators for the underlying fillings. Moreover, we show that the skew specialized non-symmetric Macdonald polynomials are in some cases vertical-strip LLT polynomials. As a consequence, we get a combinatorial Schur expansion of a new family of LLT polynomials.
We give a variant of Artin algebraization along closed subschemes and closed substacks. Our main application is the existence of étale, smooth or syntomic neighborhoods of closed subschemes and closed substacks. In particular, we prove local structure theorems for stacks and their derived counterparts and the existence of henselizations along linearly fundamental closed substacks. These results establish the existence of Ferrand pushouts, which answers positively a question of Temkin-Tyomkin.
We prove that a sequence h of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property (SLP) if and only if it is an Stanley-Iarrobino-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in P-n such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP.
We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determined by at most three parameters.
This thesis concerns the study of the Lefschetz properties of artinian monomial algebras. An artinian algebra is said to satisfy the strong Lefschetz property if multiplication by all powers of a general linear form has maximal rank in every degree. If it holds for the first power it is said to have the weak Lefschetz property (WLP).
In the first paper, we study the Lefschetz properties of monomial algebras by studying their minimal free resolutions. In particular, we give an afirmative answer to an specific case of a conjecture by Eisenbud, Huneke and Ulrich for algebras having almost linear resolutions. Since many algebras are expected to have the Lefschetz properties, studying algebras failing the Lefschetz properties is of a great interest. In the second paper, we provide sharp lower bounds for the number of generators of monomial ideals failing the WLP extending a result by Mezzetti and Miró-Roig which provides upper bounds for such ideals. In the second paper, we also study the WLP of ideals generated by forms of a certain degree invariant under an action of a cyclic group. We give a complete classication of such ideals satisfying the WLP in terms of the representation of the group generalizing a result by Mezzetti and Miró-Roig.
We study the WLP and SLP of artinian monomial ideals in S = K[x1, . . . , xn]
via studying their minimal free resolutions. We study the Lefschetz properties of such ideals
where the minimal free resolution of S/I is linear for at least n − 2 steps. We give an
affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial
ideals with almost linear resolutions.
We determine the sharp lower bound for the Hilbert function in degree d of a
monomial algebra failing the WLP over a polynomial ring with n variables and generated in
degree d. We consider artinian ideals in the polynomial ring with
n variables generated by homogeneous polynomials of degree d invariant under an action of
the cyclic group Z/dZ. We give a complete classification of
such ideals in terms of the WLP depending on the action.
We give a sharp lower bound for the Hilbert function in degree d of artinian quotients k[x1, … , xn] / I failing the Strong Lefschetz property, where I is a monomial ideal generated in degree d≥ 2. We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.
Equivariant Ehrhart theory enumerates the lattice points in a polytope with respect to a group action. Answering a question of Stapledon, we describe the equivariant Ehrhart theory of the permutahedron, and we prove his Effectiveness Conjecture in this special case.
The Joker is an important finite cyclic module over the mod-2 Steenrod algebra A. We show that the Joker, its first two iterated Steenrod doubles, and their linear duals are realizable by spaces of as low a dimension as the instability condition of modules over the Steenrod algebra permits. This continues and concludes prior work by the first author and yields a complete characterization of which versions of Jokers are realizable by spaces or spectra and which are not. The constructions involve sporadic phenomena in homotopy theory (2-compact groups, topological modular forms) and may be of independent interest.
We show that the functor of p-typical co-Witt vectors on commutative algebras over a perfect field k of characteristic p is defined on, and in fact only depends on, a weaker structure than that of a k-algebra. We call this structure a p-polar k-algebra. By extension, the functors of points for any p-adic affine commutative group scheme and for any formal group are defined on, and only depend on, p-polar structures. In terms of abelian Hopf algebras, we show that a cofree cocommutative Hopf algebra can be defined on any p-polar k-algebra P, and it agrees with the cofree commutative Hopf algebra on a commutative k-algebra A if P is the p-polar algebra underlying A; a dual result holds for free commutative Hopf algebras on finite k-coalgebras.
Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E* to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this paper I set up a framework for studying the algebra of such functors, which I call formal plethories, in the case where E* is a Prüfer ring. I show that the "logarithmic" functors of primitives and indecomposables give linear approximations of formal plethories by bimonoids in the 2-monoidal category of bimodules over a ring.
In this paper we study tensor products of affine abelian group schemes over a perfect field k. We first prove that the tensor product G(1)circle times G(2) of two affine abelian group schemes G(1), G(2) over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G(1)circle times G(2). The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. We describe the unipotent part of G(1)circle times G(2) explicitly. using Dieudonne theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.
We investigate the dynamics of a class of smooth maps of the two-torus T2 of the form T(x, y) = (Nx, f(x)(y)), where f(x) : T -> T is a monotone family (in x) of orientation preserving circle diffeomorphisms and N is an element of Z(+) is large. For our class of maps, we show that the dynamics essentially is the same as that of the projective action of non-uniformly hyperbolic SL(2, R)-cocycles. This generalizes a result by L.S. Young [6] to maps T outside the (projective) matrix cocycle case.
We identify the homotopy type of the moduli of maps with a given homotopy type of the base and the homotopy fiber. A new model for the space of weak equivalences and its classifying space is given.
We present a conjecture on the generic Betti numbers of level algebras. We prove the conjecture in the case of Gorenstein Artin algebras of embedding dimension four, and in the case of Artin level algebras whose socle dimension is large. Furthermore, we present computational evidence for the conjecture.
We give geometric constructions of families of graded Gorenstein Artin algebras, some of which span a component of the space Gor(T) parametrizing Gorenstein Artin algebras with a given Hilbert function T. This gives a lot of examples where Gor(T) is reducible. We also show that the Hilbert function of a codimension four Gorenstein Artin algebra can have an arbitrarily long constant part without having the weak Lefschetz property.
We show the existence of graded Gorenstein algebras whose Betti numbers are not unimodal, contradicting a conjecture by R. Stanley. In fact, we prove that the Betti numbers given by the natural lower bound are non-unimodal in sufficiently high embedding dimension - greater than or equal to nine - and we have calculated two examples where the Betti numbers attain this lower bound.
We describe completely the possible Gorenstein Artin quotients of the coordinate ring of a set of points in projective space. In addition, we give a sufficient condition for an ideal of a Cohen–Macaulay graded k‐algebra A to be isomorphic to the canonical module of A. This leads to a description of the canonical module in the case when A is the coordinate ring of points.
In [4] Stanley showed that the Hilbert function of graded Goren-stein Artin algebras need not be unimodal. In this article we prove the exis¬tence of graded Gorenstein Artin algebras whose Hilbert functions have local maxima at any given set of points with the symmetry and the socle degree as the only restrictions.
We give an explicit expression for the Hilbert function of a large class of graded Gorenstein Artin algebras and give a criterion for this function to be unimodal. As a result we obtain an abundance of graded Gorenstein Artin algebras with nonunimodal Hilbert function.
We show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying the strong Lefschetz property. This is the first example of an explicit dual form with these properties. For complete symmetric forms of any degree in any number of variables, we provide an upper bound for the Waring rank by establishing an explicit power sum decomposition. Moreover, we determine the Waring rank, the cactus rank, the resolution and the strong Lefschetz property for any Gorenstein algebra defined by a symmetric cubic form. In particular, we show that the difference between the Waring rank and the cactus rank of a symmetric cubic form can be made arbitrarily large by increasing the number of variables. We provide upper bounds for the Waring rank of generic symmetric forms of degrees four and five.
We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman-Hausknecht. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work.
In this paper we study tensor products of affine abelian group schemes over a perfect field k. We first prove that the tensor product G_1 ⊗ G_2 of two affine abelian group schemes G_1,G_2 over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G_1 ⊗G_2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. We describe the unipotent part of G_1 ⊗ G_2 explicitly, using Dieudonn\'e theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.
We compute the étale cohomology ring H^*(Spec O_K,Z/nZ) where O_K is the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim.
We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that Aut(PSL(2,q^2)) for q an odd prime power, cannot be realized as an unramified Galois group over K, but its maximal solvable quotient can. To prove this result, we determine the ring structure of the \'etale cohomology ring H^*(Spec O_K;Z/2Z) where O_K is the ring of integers of an arbitrary totally imaginary number field K.
We set the foundations for a new approach to Topological Data Analysis (TDA) based on homotopical methods at the chain complex level. We present the category of tame parametrised chain complexes as a comprehensive environment that includes several cases that usually TDA handles separately, such as persistence modules, zigzag modules, and commutative ladders. We extract new invariants in this category using a model structure and various minimal cofibrant approximations. Such approximations and their invariants retain some of the topological, and not just homological, aspects of the objects they approximate.
Given a finite set A ⊂ ℝd, let Covr,k denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.
We associate with every pure flag simplicial complex Δ a standard graded Gorenstein F-RΔ whose homological features are largely dictated by the combinatorics and topology of Δ . As our main result, we prove that the residue field F has a k-step linear RΔ-resolution if and only if Δ satisfies Serre's condition (S k) over F and that RΔ is Koszul if and only if Δ is Cohen-Macaulay over F. Moreover, we show that RΔ has a quadratic Gröbner basis if and only if Δ is shellable. We give two applications: first, we construct quadratic Gorenstein F-s that are Koszul if and only if the characteristic of F is not in any prescribed set of primes. Finally, we prove that whenever RΔ is Koszul the coefficients of its γ-vector alternate in sign, settling in the negative an ic generalization of a conjecture by Charney and Davis.
In this paper we prove a perturbative result for a class of actions on Heisenberg nilmanifolds that have Diophantine properties. Along the way we prove cohomological rigidity and obtain a tame splitting for the cohomology with coefficients in smooth vector fields for such actions.
We give new proofs of two results of Stafford, which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize Serre's Splitting Theorem by imposing a condition to the splitting maps, which has an application to the case of Cartier algebras.
We introduce numerical algebraic geometry methods for computing lower bounds on the reach, local feature size, and weak feature size of the real part of an equidimensional and smooth algebraic variety using the variety's defining polynomials as input. For the weak feature size, we also show that nonquadratic complete intersections generically have finitely many geometric bottlenecks, and we describe how to compute the weak feature size directly rather than a lower bound in this case. In all other cases, we describe additional computations that can be used to determine feature size values rather than lower bounds.
If a pure simplicial complex is partitionable, then its h-vector has a combinatorial interpretation in terms of any partitioning of the complex. Given a non-partitionable complex increment , we construct a complex Gamma superset of increment of the same dimension such that both Gamma and the relative complex (Gamma , increment ) are partitionable. This allows us to rewrite the h-vector of any pure simplicial complex as the difference of two h-vectors of partitionable complexes, giving an analogous interpretation of the h-vector of a non-partitionable complex. By contrast, for a given complex increment it is not always possible to find a complex Gamma such that both Gamma and (Gamma , increment ) are Cohen- Macaulay. We characterize when this is possible, and we show that the construction of such a Gamma in this case is remarkably straightforward. We end with a note on a similar notion for shellability and a connection to Simon's conjecture on extendable shellability for uniform matroids.
In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals ((p-1)/p)(p) whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.
Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We compute the universal valuative invariant of Coxeter matroids. A key ingredient is the family of Coxeter Schubert matroids, which correspond to the Bruhat cells of flag varieties. In the process, we compute the universal valuation of generalized Coxeter permutohedra, a larger family of polyhedra that model Coxeter analogues of combinatorial objects such as matroids, clusters, and posets.
We use a description based on differential forms to systematically explore the space of scalar-tensor theories of gravity. Within this formalism, we propose a basis for the scalar sector at the lowest order in derivatives of the field and in any number of dimensions. This minimal basis is used to construct a finite and closed set of Lagrangians describing general scalar-tensor theories invariant under local Lorentz transformations in a pseudo-Riemannian manifold, which contains ten physically distinct elements in four spacetime dimensions. Subsequently, we compute their corresponding equations of motion and find which combinations are at most second order in derivatives in four as well as an arbitrary number of dimensions. By studying the possible exact forms (total derivatives) and algebraic relations between the basis components, we discover that there are only four Lagrangian combinations producing second-order equations, which can be associated with Horndeski's theory. In this process, we identify a new second-order Lagrangian, named kinetic Gauss-Bonnet, that was not previously considered in the literature. However, we show that its dynamics is already contained in Horndeski's theory. Finally, we provide a full classification of the relations between different second-order theories. This allows us to clarify, for instance, the connection between different covariantizations of Galileons theory. In conclusion, our formulation affords great computational simplicity with a systematic structure. As a first step, we focus on theories with second-order equations of motion. However, this new formalism aims to facilitate advances towards unveiling the most general scalar-tensor theories.
It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut (A 2) of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir (P 2) up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir (P 2) admits Borel subgroups of any rank r ϵ {0, 1, 2} and that all Borel subgroups of rank r ϵ {1, 2} are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus g ≥ 1. Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus g, and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus g. This moduli space is of dimension 2g - 1.
One way to study a hypergraph is to attach to it a tensor. Tensors are a generalization of matrices, and they are an efficient way to encode information in a compact form. In this paper, we study how properties of weighted hypergraphs are reflected on eigenvalues and eigenvectors of their associated tensors. We also show how to efficiently compute eigenvalues with some techniques from numerical algebraic geometry.
This thesis is constituted of two articles, both related to Hilbert functions and h-vectors. In the first paper, we deal with h-vectorsof reduced zero-dimensional schemes in the projective plane, and, in particular, with the problem of finding the possible h-vectors for the union of two sets of points of given h-vectors. In the second paper, we generalize the Green’s Hyperplane Restriction Theorem to the case of modules over the polynomial ring.
The thesis is a collection of four papers dealing with Hilbert functions and Betti numbers.In the first paper, we study the h-vectors of reduced zero-dimensional schemes in . In particular we deal with the problem of findingthe possible h-vectors for the union of two sets of points of given h-vectors. To answer to this problem, we give two kinds of bounds on theh-vectors and we provide an algorithm that calculates many possibleh-vectors.In the second paper, we prove a generalization of Green’s Hyper-plane Restriction Theorem to the case of finitely generated modulesover the polynomial ring, providing an upper bound for the Hilbertfunction of the general linear restriction of a module M in a degree dby the corresponding Hilbert function of a lexicographic module.In the third paper, we study the minimal free resolution of theVeronese modules, , where by giving a formula for the Betti numbers in terms of the reduced homology of the squarefree divisor complex. We prove that is Cohen-Macaulay if and only if k < d, and that its minimal resolutionis linear when k > d(n − 1) − n. We prove combinatorially that the resolution of is pure. We provide a formula for the Hilbert seriesof the Veronese modules. As an application, we calculate the completeBetti diagrams of the Veronese rings .In the fourth paper, we apply the same combinatorial techniques inthe study of the properties of pinched Veronese rings, in particular weshow when this ring is Cohen-Macaulay. We also study the canonicalmodule of the Veronese modules.
In this paper, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of the squarefree divisor complex. In particular, we study the Cohen-Macaulay property of these rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.
In this work, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of squarefree divisor complexes. We characterize when these rings are Cohen-Macaulay and we study the shape of the Betti tables for the pinched Veronese in the two variables. As a byproduct we obtain information on the linearity of such rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.
This paper summarizes our experiences from an exercise in deductive verification of functional properties of automotive embedded Ccode in an industrial setting. We propose a formal requirements model that supports the way C-code requirements are currently written at Scania. We describe our work, for a safety-critical module of an embedded system, on formalizing its functional requirements and verifying its C-code implementation by means of VCC, an established tool for deductive verification. We describe the obstacles we encountered, and discuss the automation of the specification and annotation effort as a prerequisite for integrating this technology into the embedded software design process.