Ideals of generic minors

Caniglia, Leandro; Guccione, Jorge A.; Guccione, Juan J.

doi:10.1080/00927879008824043

Cited by 26 publications

(34 citation statements)

References 1 publication

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“…In the particular case of the determinant, Corollary 13 can be strengthened to say that the number of monomials of degree at most (n − k + ) with an increasing sequence of length (n − k) is not just a lower bound but is exactly equal to dim( ∂ =k (Det n ) ≤ ). This follows from the following powerful result on Gröbner bases of determinantal ideals which has been proved independently by Narasimhan [Nar86], Sturmfels [Stu90] and Caniglia, Guccione and Guccione [CGG90].…”

Section: Proofs Of Binomial Estimatesmentioning

confidence: 98%

Approaching the Chasm at Depth Four

Gupta

Kamath

Kayal

et al. 2014

J. ACM

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Agrawal-Vinay [AV08], Koiran [Koi12] and Tavenas [Tav13] have recently shown that an exp(ω( √ n log n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √ n translates to super-polynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin.We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √ n computing the permanent (or the determinant) must be of size exp(Ω( √ n)).

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Section: Proofs Of Binomial Estimatesmentioning

confidence: 98%

Approaching the Chasm at Depth Four

Gupta

Kamath

Kayal

et al. 2014

J. ACM

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“…They are used in treatments of questions about Cohen-Macaulayness, rational singularities, multiplicity, dimension, a-invariants, and divisor class groups; see [CGG90], [Stu90]…”

Section: Ladder Determinantal Idealsmentioning

confidence: 99%

Gröbner geometry of Schubert polynomials

2005

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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.

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“…The determinantal result was proved independently by Caniglia et al (1990), Narasimhan (1986) and Sturmfels (1990).…”

Section: A Gröbner Basismentioning

confidence: 81%

“…Their properties turn out to be very different from those of the much better understood determinantal ideals. For example, we prove that the generating permanents of P 2 (M ) are not a Gröbner basis of P 2 (M ) in any diagonal order, whereas Narasimhan (1986), Caniglia et al (1990), and Sturmfels (1990) proved independently that the generating (2 × 2)-minors of M are a Gröbner basis of the ideal they generate in any diagonal order. Also, in contrast to determinantal ideals, we prove that if m and n are both at least 3, then P 2 (M ) is not a radical ideal, is not Cohen-Macaulay, and there are minimal primes of distinct heights over it.…”

Section: Introductionmentioning

confidence: 96%

Permanental Ideals

Laubenbacher

Swanson

2000

Journal of Symbolic Computation

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We show that the (2 × 2)-subpermanents of a generic matrix generate an ideal whose height, unmixedness, primary decomposition, the number and structure of the minimal components, resolutions, radical, integral closure and Gröbner bases all depend on the characteristic of the underlying subfield: if the characteristic of the subfield is two, this ideal is the determinantal ideal for which all of these properties are already well known. We show that as long as the characteristic of the subfield is not two, the results are in marked contrast with those for the determinantal ideals.

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Ideals of generic minors

Cited by 26 publications

References 1 publication

Approaching the Chasm at Depth Four

Approaching the Chasm at Depth Four

Gröbner geometry of Schubert polynomials

Permanental Ideals

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