- University of Connecticut-storrs, Mathematics, Emeritusadd
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We prove the basic properties of determinantal semi-invariants for presenta- tion spaces over any nite dimensional hereditary algebra over any eld. These include the virtual generic decomposition theorem, stability theorem and the... more
We prove the basic properties of determinantal semi-invariants for presenta- tion spaces over any nite dimensional hereditary algebra over any eld. These include the virtual generic decomposition theorem, stability theorem and the c-vector theorem which says that the c-vectors of a cluster tilting object are, up to sign, the determinantal weights of the determinantal semi-invariants dened on the cluster tilting objects. Applications of these theorems are given in several concurrently written papers.
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We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace in the second wedge product of a vector space. Previously Koszul modules of finite length... more
We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace in the second wedge product of a vector space. Previously Koszul modules of finite length have been used to give a proof of Green's Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the stability of sufficiently positive rank 2 vector bundles on curves is governed by resonance.
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In this paper I give an explicit construction of the generic ring R_gen for finite free resolutions of length 3. The corresponding problem for resolutions of length 2 was solved in 1970'ies by Hochster and Huneke. The key role is... more
In this paper I give an explicit construction of the generic ring R_gen for finite free resolutions of length 3. The corresponding problem for resolutions of length 2 was solved in 1970'ies by Hochster and Huneke. The key role is played by the defect Lie algebra introduced in my old work on the subject. The defect Lie algebra turns out to be a parabolic Lie algebra in a Kac-Moody Lie algebra associated to the graph T_p,q,r corresponding to the format of the resolution. The ring R_gen is Noetherian if and only if the graph T_p.q.r is a Dynkin graph.
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Hochster established the existence of a commutative noetherian ring C̃ and a universal resolution U of the form 0→C̃^e→C̃^f→C̃^g→ 0 such that for any commutative noetherian ring S and any resolution V equal to 0→ S^e→ S^f→ S^g→ 0, there... more
Hochster established the existence of a commutative noetherian ring C̃ and a universal resolution U of the form 0→C̃^e→C̃^f→C̃^g→ 0 such that for any commutative noetherian ring S and any resolution V equal to 0→ S^e→ S^f→ S^g→ 0, there exists a unique ring homomorphism C̃→ S with V=U⊗_C̃ S. In the present paper we assume that f=e+g and we find the minimal resolution of K⊗C̃ by free B-modules, where K is a field of characteristic zero and B is a polynomial ring over K. Our techniques are geometric. We use the Bott algorithm and the Representation Theory of the General Linear Group. As a by-product of our work, we resolve a family of maximal Cohen-Macaulay modules defined over a determinantal ring.
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We give counterexamples to Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients.
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We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a... more
We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for sl_2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve.
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In this paper, we study the isotropic Schur roots of an acyclic quiver Q with n vertices. We study the perpendicular category A(d) of a dimension vector d and give a complete description of it when d is an isotropic Schur δ. This is done... more
In this paper, we study the isotropic Schur roots of an acyclic quiver Q with n vertices. We study the perpendicular category A(d) of a dimension vector d and give a complete description of it when d is an isotropic Schur δ. This is done by using exceptional sequences and by defining a subcategory R(Q,δ) attached to the pair (Q,δ). The latter category is always equivalent to the category of representations of a connected acyclic quiver Q_R of tame type, having a unique isotropic Schur root, say δ_R. The understanding of the simple objects in A(δ) allows us to get a finite set of generators for the ring of semi-invariants SI(Q,δ) of Q of dimension vector δ. The relations among these generators come from the representation theory of the category R(Q,δ) and from a beautiful description of the cone of dimension vectors of A(δ). Indeed, we show that SI(Q,δ) is isomorphic to the ring of semi-invariants SI(Q_R,δ_R) to which we adjoin variables. In particular, using a result of Skowroński a...
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For every quiver (valued) of finite representation type we define a finitely presented group called a picture group. This group is very closely related to the cluster theory of the quiver. For example, positive expressions for the Coxeter... more
For every quiver (valued) of finite representation type we define a finitely presented group called a picture group. This group is very closely related to the cluster theory of the quiver. For example, positive expressions for the Coxeter element in the group are in bijection with maximal green sequences [IT17]. The picture group is derived from the semi-invariant picture for the quiver. We use this picture to construct a finite CW complex which (by [IT16]) is a K(π,1) for this group. The cells are in bijection with cluster tilting objects. For example, in type A_n there are a Catalan number of cells. The main result of this paper is the computation of the cohomology ring of all picture groups of type A_n with any orientation and any coefficient ring.
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We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant's algebraic analog of the P not equal to NP... more
We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant's algebraic analog of the P not equal to NP conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.
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We compute the GL-equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix, as well as of those with support in the ideal of 2n x 2n Pfaffians of a (2n+1)x(2n+1) generic... more
We compute the GL-equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix, as well as of those with support in the ideal of 2n x 2n Pfaffians of a (2n+1)x(2n+1) generic skew-symmetric matrix. As an application, we characterize the Cohen-Macaulay modules of covariants for the action of the special linear group SL(G) on G^m. The main tool we develop is a method for computing certain Ext modules based on the geometric technique for computing syzygies and on Matlis duality.
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Let Q be a regular local ring of dimension 3. We show how to trim a Gorenstein ideal in Q to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality... more
Let Q be a regular local ring of dimension 3. We show how to trim a Gorenstein ideal in Q to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality algebra P padded with a non-zero graded vector space on which P_> 1 acts trivially. We explicitly construct an infinite family of such rings.
This paper is a continuation of arXiv:1201.1102. We investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on the simple Lie algebra of type E_7. The methods for... more
This paper is a continuation of arXiv:1201.1102. We investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on the simple Lie algebra of type E_7. The methods for classifying the orbits for these actions were developed by Vinberg . We give the orbit descriptions, the degeneration partial orders, and indicate normality of the orbit closures. We also investigate the rational singularities, Cohen-Macaulay and Gorenstein properties for the orbit closures. We give the information on the defining ideals of orbit closures.
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In this paper we investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on a simple Lie algebras of type E_6, F_4 and G_2. The methods for classifying the orbits for these... more
In this paper we investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on a simple Lie algebras of type E_6, F_4 and G_2. The methods for classifying the orbits for these actions were developed by Vinberg. We give the orbit descriptions, the degeneration partial orders, and decide the normality of the orbit closures. We also investigate the rational singularities, Cohen-Macaulay and Gorenstein properties for the orbit closures. We give the generators of the defining ideals of orbit closures.
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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... more
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.
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We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual... more
We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semi-invariants and the Tilting Triangulation of the (n-1)-sphere.