arXiv subjects

Steven V Sam

Publications and source records attributed to Steven V Sam.

At least 91 records · Page 5Linked to original sources

arXiv2013

Tensor complexes: Multilinear free resolutions constructed from higher tensors

The most fundamental complexes of free modules over a commutative ring are the Koszul complex, which is constructed from a vector (i.e., a 1-tensor), and the Eagon-Northcott and the Buchsbaum-Rim complexes, which are constructed from a matrix (i.e., a 2-tensor). The subject of this paper is a multilinear analogue of these complexes, which we construct from an arbitrary higher tensor. Our construction provides detailed new examples of minimal free resolutions, as well as a unifying view on a wide variety of complexes including: the Eagon-Northcott, Buchsbaum-Rim and similar complexes, the Eisenbud-Schreyer pure resolutions, and the complexes used by Gelfand-Kapranov-Zelevinsky and Weyman to compute hyperdeterminants. In addition, we provide applications to the study of pure resolutions and Boij-Soederberg theory, including the construction of infinitely many new families of pure resolutions and the first explicit description of the differentials of the Eisenbud-Schreyer pure resolutions.

math.AC↗

arXiv2013

The Universal Kummer Threefold

The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in 7-dimensional projective space. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Göpel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble's quartic hypersurface using current tools from combinatorics, geometry, and commutative algebra. Symbolic and numerical computations for genus 3 moduli spaces appear alongside toric and tropical methods.

math.AG↗

arXiv2013

Combinatorial realizations of crystals via torus actions on quiver varieties

Consider Kashiwara's crystal associated to a highest weight representation of a symmetric Kac-Moody algebra. There is a geometric realization of this object using Nakajima's quiver varieties, but in many particular cases it can also be realized by elementary combinatorial methods. Here we propose a framework for extracting combinatorial realizations from the geometric picture: We construct certain torus actions on the quiver varieties and use Morse theory to index the irreducible components by connected components of the subvariety of torus fixed points. We then discuss the case of affine sl(n). There the fixed point components are just points, and are naturally indexed by multi-partitions. There is some choice in our construction, leading to a family of combinatorial models for each highest weight crystal. Applying this construction to the crystal of the fundamental representation recovers a family of combinatorial realizations recently constructed by Fayers. This gives a more conceptual proof of Fayers' result as well as a generalization to higher level. We also discuss a relationship with Nakajima's monomial crystal.

math.CO↗

arXiv2013

Homology of Littlewood complexes

Let V be a symplectic vector space of dimension 2n. Given a partition λwith at most n parts, there is an associated irreducible representation S_{[λ]}(V) of Sp(V). This representation admits a resolution by a natural complex L^λ, which we call the Littlewood complex, whose terms are restrictions of representations of GL(V). When λhas more than n parts, the representation S_{[λ]}(V) is not defined, but the Littlewood complex L^λstill makes sense. The purpose of this paper is to compute its homology. We find that either L^λis acyclic or that it has a unique non-zero homology group, which forms an irreducible representation of Sp(V). The non-zero homology group, if it exists, can be computed by a rule reminiscent of that occurring in the Borel-Weil-Bott theorem. This result can be interpreted as the computation of the "derived specialization" of irreducible representations of Sp(\infty), and as such categorifies earlier results of Koike-Terada on universal character rings. We prove analogous results for orthogonal and general linear groups. Along the way, we will see two topics from commutative algebra: the minimal free resolutions of determinantal ideals and Koszul homology.

math.RT↗

arXiv2013

Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions

We present a systematic approach to studying the geometric aspects of Vinberg theta-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an "enhanced" version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.

math.AG↗

arXiv2012

Introduction to twisted commutative algebras

This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate rings of determinantal varieties, Segre-Veronese embeddings, and Grassmannians. The article is meant to serve as a gentle introduction to the papers of the two authors on the subject, and also to point out some literature in which these algebras appear. The first part reviews the representation theory of the symmetric groups and general linear groups. The second part introduces a related category and develops its basic properties. The third part develops some basic properties of twisted commutative algebras from the perspective of classical commutative algebra and summarizes some of the results of the authors. We have tried to keep the prerequisites to this article at a minimum. The article is aimed at graduate students interested in commutative algebra, algebraic combinatorics, or representation theory, and the interactions between these subjects.

math.AC↗

arXiv2012

Derived supersymmetries of determinantal varieties

We show that the linear strands of the Tor of determinantal varieties in spaces of symmetric and skew-symmetric matrices are irreducible representations for the periplectic (strange) Lie superalgebra. The structure of these linear strands is explicitly known, so this gives an explicit realization of some representations of the periplectic Lie superalgebra. This complements results of Pragacz and Weyman, who showed an analogous statement for the generic determinantal varieties and the general linear Lie superalgebra. We also give a simpler proof of their result. Via Koszul duality, this is an odd analogue of the fact that the coordinate rings of these determinantal varieties are irreducible representations for a certain classical Lie algebra.

math.RT↗

arXiv2012

Pieri resolutions for classical groups

We generalize the constructions of Eisenbud, Fløystad, and Weyman for equivariant minimal free resolutions over the general linear group, and we construct equivariant resolutions over the orthogonal and symplectic groups. We also conjecture and provide some partial results for the existence of an equivariant analogue of Boij-Söderberg decompositions for Betti tables, which were proven to exist in the non-equivariant setting by Eisenbud and Schreyer. Many examples are given.

math.AC↗

arXiv2012

Koszul homology of codimension 3 Gorenstein ideals

In this note, we calculate the Koszul homology of the codimension 3 Gorenstein ideals. We find filtrations for the Koszul homology in terms of modules with pure free resolutions and completely describe these resolutions. We also consider the Huneke-Ulrich deviation 2 ideals.

math.AC↗

arXiv2012

Matrices with restricted entries and q-analogues of permutations

We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.

math.CO↗

arXiv2011

Shapes of free resolutions over a local ring

We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings. Our key new technique uses asymptotic arguments to study formal Q-Betti sequences.

math.AC↗

arXiv2011

Poset structures in Boij-Söderberg theory

Boij-Söderberg theory is the study of two cones: the cone of cohomology tables of coherent sheaves over projective space and the cone of standard graded minimal free resolutions over a polynomial ring. Each cone has a simplicial fan structure induced by a partial order on its extremal rays. We provide a new interpretation of these partial orders in terms of the existence of nonzero homomorphisms, for both the general and the equivariant constructions. These results provide new insights into the families of sheaves and modules at the heart of Boij-Söderberg theory: supernatural sheaves and Cohen-Macaulay modules with pure resolutions. In addition, our results strongly suggest the naturality of these partial orders, and they provide tools for extending Boij-Söderberg theory to other graded rings and projective varieties.

math.AC↗

arXiv2011

Symmetric quivers, invariant theory, and saturation theorems for the classical groups

Let G denote either a special orthogonal group or a symplectic group defined over the complex numbers. We prove the following saturation result for G: given dominant weights λ^1, ..., λ^r such that the tensor product V_{Nλ^1} \otimes ... \otimes V_{Nλ^r} contains nonzero G-invariants for some N \ge 1, we show that the tensor product V_{2λ^1} \otimes ... \otimes V_{2λ^r} also contains nonzero G-invariants. This extends results of Kapovich-Millson and Belkale-Kumar and complements similar results for the general linear group due to Knutson-Tao and Derksen-Weyman. Our techniques involve the invariant theory of quivers equipped with an involution and the generic representation theory of certain quivers with relations.

math.RT↗

arXiv2011

Equations and syzygies of some Kalman varieties

Given a subspace L of a vector space V, the Kalman variety consists of all matrices of V that have a nonzero eigenvector in L. Ottaviani and Sturmfels described minimal equations in the case that dim L = 2 and conjectured minimal equations for dim L = 3. We prove their conjecture and describe the minimal free resolution in the case that dim L = 2, as well as some related results. The main tool is an exact sequence which involves the coordinate rings of these Kalman varieties and the normalizations of some related varieties. We conjecture that this exact sequence exists for all values of dim L.

math.AC↗

arXiv2011

Schubert complexes and degeneracy loci

Given a generic map between flagged vector bundles on a Cohen-Macaulay variety, we construct maximal Cohen-Macaulay modules with linear resolutions supported on the Schubert-type degeneracy loci. The linear resolution is provided by the Schubert complex, which is the main tool introduced and studied in this paper. These complexes extend the Schubert functors of Kraśkiewicz and Pragacz, and were motivated by the fact that Schur complexes resolve maximal Cohen-Macaulay modules supported on determinantal varieties. The resulting formula in K-theory provides a "linear approximation" of the structure sheaf of the degeneracy locus, which can be used to recover a formula due to Fulton.

math.AG↗

arXiv2010

Generalized Ehrhart polynomials

Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of P(n) to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in P(n) is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related.

math.CO↗

arXiv2010

A finite calculus approach to Ehrhart polynomials

A rational polytope is the convex hull of a finite set of points in $\R^d$ with rational coordinates. Given a rational polytope $P \subseteq \R^d$, Ehrhart proved that, for $t\in\Z_{\ge 0}$, the function $#(tP \cap \Z^d)$ agrees with a quasi-polynomial $L_P(t)$, called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new proof of Ehrhart's theorem. This proof also allows us to quickly prove two other facts about Ehrhart quasi-polynomials: McMullen's theorem about the periodicity of the individual coefficients of the quasi-polynomial and the Ehrhart-Macdonald theorem on reciprocity.

math.CO↗

arXiv2009

Computing inclusions of Schur modules

We describe a software package for constructing minimal free resolutions of GL_n(Q)-equivariant graded modules M over Q[x_1, ..., x_n] such that for all i, the ith syzygy module of M is generated in a single degree. We do so by describing some algorithms for manipulating polynomial representations of the general linear group GL_n(Q) following ideas of Olver and Eisenbud-Floystad-Weyman.

math.AC↗