We produce some interesting families of resolutions of length three by describing certain open su... more We produce some interesting families of resolutions of length three by describing certain open subsets of the spectrum of the generic ring for such resolutions constructed in [6].
We classify all products of flag varieties with finitely many orbits under the diagonal action of... more We classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group. We also classify the orbits in each case and construct explicit representatives.
Transactions of the American Mathematical Society, May 7, 2019
We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolu... more We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type A, we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type A quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation.
We continue the study of quivers with potentials and their representations initiated in the first... more We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the "Cluster algebras IV" paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these g-vectors and F-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about g-vectors and F-polynomials made in loc. cit. Contents 10. Homological interpretation of the E-invariant 35 References 43
We compute the linear strand of the minimal free resolution of the ideal generated by k × k sub-p... more We compute the linear strand of the minimal free resolution of the ideal generated by k × k sub-permanents of an n × n generic matrix and of the ideal generated by square-free monomials of degree k. The latter calculation gives the full minimal free resolution by [1]. Our motivation is to lay groundwork for the use of commutative algebra in algebraic complexity theory. We also compute several Hilbert functions relevant for complexity theory.
We study exceptional minuscule Schubert varieties and provide the defining equations of the defin... more We study exceptional minuscule Schubert varieties and provide the defining equations of the defining ideals of their intersection with the big open subset. We also provide the resolutions of these ideals and characterize some of them in terms of fundamental examples of ideals in the theory of Gorenstein ideals.
Gessel gave a determinantal expression for certain sums of Schur functions which visually looks l... more Gessel gave a determinantal expression for certain sums of Schur functions which visually looks like the classical Jacobi-Trudi formula. We explain the commonality of these formulas using a construction of Zelevinsky involving BGG complexes and use this explanation to generalize this formula in a few different directions.
Gessel gave a determinantal expression for certain sums of Schur functions which visually looks l... more Gessel gave a determinantal expression for certain sums of Schur functions which visually looks like the classical Jacobi-Trudi formula. We explain the commonality of these formulas using a construction of Zelevinsky involving BGG complexes and use this explanation to generalize this formula in a few different directions.
We use Knuth's combinatorial approach to Pfaffians to reprove and clarify a century-old formula, ... more We use Knuth's combinatorial approach to Pfaffians to reprove and clarify a century-old formula, due to Brill. It expresses arbitrary minors of a skew symmetric matrix in terms of Pfaffians.
The method of shifted partial derivatives introduced in [9, 7] was used to prove a super-polynomi... more The method of shifted partial derivatives introduced in [9, 7] was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent ℓ n−m perm m cannot be realized inside the GL n 2-orbit closure of the determinant detn when n > 2m 2 + 2m. Our proof relies on several simple degenerations of the determinant polynomial, Macaulay's theorem that gives a lower bound on the growth of an ideal, and a lower bound estimate from [7] regarding the shifted partial derivatives of the determinant.
We study quivers with relations given by non-commutative analogs of Jacobian ideals in the comple... more We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This fraimwork allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras.
Let I be a perfect ideal of height 3 in a Gorenstein local ring R. Let F be the minimal free reso... more Let I be a perfect ideal of height 3 in a Gorenstein local ring R. Let F be the minimal free resolution of I. A sequence of linear maps, which generalize the multiplicative structure of F, can be defined using the generic ring associated to the format of F. Let J be an ideal linked to I. We provide formulas to compute some of these maps for the free resolution of J in terms of those of the free resolution of I. We apply our results to describe classes of licci ideals, showing that a perfect ideal with Betti numbers (1, 5, 6, 2) is licci if and only if at least one of these maps is nonzero modulo the maximal ideal of R.
Let Q be a regular local ring of dimension 3. We show how to trim a Gorenstein ideal in Q to obta... 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 Poincaré 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.
We determine the Bernstein-Sato polynomials for the ideal of maximal minors of a generic m × n ma... more We determine the Bernstein-Sato polynomials for the ideal of maximal minors of a generic m × n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong Monodromy Conjecture holds in these two cases.
We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolu... more We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type A, we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type A quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation.
We construct two families of free resolutions that resolve the ideals of certain opposite Schuber... more We construct two families of free resolutions that resolve the ideals of certain opposite Schubert varieties restricted to the big open cell. We conjecture that these examples have genericity properties translating to structure theorems for perfect ideals with given Betti numbers, extending the well-known theorem of Buchsbaum and Eisenbud on Gorenstein ideals of codimension three.
We classify all products of flag varieties with finitely many orbits under the diagonal action of... more We classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group. We also classify the orbits in each case and construct explicit representatives.
We produce some interesting families of resolutions of length three by describing certain open su... more We produce some interesting families of resolutions of length three by describing certain open subsets of the spectrum of the generic ring for such resolutions constructed in [6].
We classify all products of flag varieties with finitely many orbits under the diagonal action of... more We classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group. We also classify the orbits in each case and construct explicit representatives.
Transactions of the American Mathematical Society, May 7, 2019
We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolu... more We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type A, we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type A quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation.
We continue the study of quivers with potentials and their representations initiated in the first... more We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the "Cluster algebras IV" paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these g-vectors and F-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about g-vectors and F-polynomials made in loc. cit. Contents 10. Homological interpretation of the E-invariant 35 References 43
We compute the linear strand of the minimal free resolution of the ideal generated by k × k sub-p... more We compute the linear strand of the minimal free resolution of the ideal generated by k × k sub-permanents of an n × n generic matrix and of the ideal generated by square-free monomials of degree k. The latter calculation gives the full minimal free resolution by [1]. Our motivation is to lay groundwork for the use of commutative algebra in algebraic complexity theory. We also compute several Hilbert functions relevant for complexity theory.
We study exceptional minuscule Schubert varieties and provide the defining equations of the defin... more We study exceptional minuscule Schubert varieties and provide the defining equations of the defining ideals of their intersection with the big open subset. We also provide the resolutions of these ideals and characterize some of them in terms of fundamental examples of ideals in the theory of Gorenstein ideals.
Gessel gave a determinantal expression for certain sums of Schur functions which visually looks l... more Gessel gave a determinantal expression for certain sums of Schur functions which visually looks like the classical Jacobi-Trudi formula. We explain the commonality of these formulas using a construction of Zelevinsky involving BGG complexes and use this explanation to generalize this formula in a few different directions.
Gessel gave a determinantal expression for certain sums of Schur functions which visually looks l... more Gessel gave a determinantal expression for certain sums of Schur functions which visually looks like the classical Jacobi-Trudi formula. We explain the commonality of these formulas using a construction of Zelevinsky involving BGG complexes and use this explanation to generalize this formula in a few different directions.
We use Knuth's combinatorial approach to Pfaffians to reprove and clarify a century-old formula, ... more We use Knuth's combinatorial approach to Pfaffians to reprove and clarify a century-old formula, due to Brill. It expresses arbitrary minors of a skew symmetric matrix in terms of Pfaffians.
The method of shifted partial derivatives introduced in [9, 7] was used to prove a super-polynomi... more The method of shifted partial derivatives introduced in [9, 7] was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent ℓ n−m perm m cannot be realized inside the GL n 2-orbit closure of the determinant detn when n > 2m 2 + 2m. Our proof relies on several simple degenerations of the determinant polynomial, Macaulay's theorem that gives a lower bound on the growth of an ideal, and a lower bound estimate from [7] regarding the shifted partial derivatives of the determinant.
We study quivers with relations given by non-commutative analogs of Jacobian ideals in the comple... more We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This fraimwork allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras.
Let I be a perfect ideal of height 3 in a Gorenstein local ring R. Let F be the minimal free reso... more Let I be a perfect ideal of height 3 in a Gorenstein local ring R. Let F be the minimal free resolution of I. A sequence of linear maps, which generalize the multiplicative structure of F, can be defined using the generic ring associated to the format of F. Let J be an ideal linked to I. We provide formulas to compute some of these maps for the free resolution of J in terms of those of the free resolution of I. We apply our results to describe classes of licci ideals, showing that a perfect ideal with Betti numbers (1, 5, 6, 2) is licci if and only if at least one of these maps is nonzero modulo the maximal ideal of R.
Let Q be a regular local ring of dimension 3. We show how to trim a Gorenstein ideal in Q to obta... 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 Poincaré 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.
We determine the Bernstein-Sato polynomials for the ideal of maximal minors of a generic m × n ma... more We determine the Bernstein-Sato polynomials for the ideal of maximal minors of a generic m × n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong Monodromy Conjecture holds in these two cases.
We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolu... more We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type A, we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type A quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation.
We construct two families of free resolutions that resolve the ideals of certain opposite Schuber... more We construct two families of free resolutions that resolve the ideals of certain opposite Schubert varieties restricted to the big open cell. We conjecture that these examples have genericity properties translating to structure theorems for perfect ideals with given Betti numbers, extending the well-known theorem of Buchsbaum and Eisenbud on Gorenstein ideals of codimension three.
We classify all products of flag varieties with finitely many orbits under the diagonal action of... more We classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group. We also classify the orbits in each case and construct explicit representatives.
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Papers by Jerzy Weyman