Pfaffians and Skew-Symmetric Matrices

Heymans, P.

doi:10.1112/plms/s3-19.4.730

Cited by 12 publications

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“…The algebra relies heavily on my preparatory paper ( [6]) on the relations between the minors of a skew-symmetric matrix and the multilinear polynomials in the elements of the matrix known as pfaffians. order k + 1.…”

Section: Introductionmentioning

confidence: 99%

“…In order to write down parametric equations for G k it is convenient to use a notation, for coordinates in the ambient space of Y k , involving a double sequence of subscripts. As in [6], I use them vanishes identically when considered as a polynomial in the x^; it follows that G k does not lie in any other hyperplane, independent of (2.17), so that S p is its ambient space.| (2.17) leads to an obvious transformation of the ^-coordinates to make S p into a face of the simplex of reference. For s > 0, replace one out of y { a 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…From this and equations (2.16) it follows that G k is the quadratic transform of some variety M k whose prime sections are modelled in 8 K by a linear system of varieties based on pfaffians. In the next sections we shall use the formulae of [6] to prove stronger results. M k is in fact modelled by the complete system based on all pfaffians in the elements of P and G k is modelled on M k by the system of all quadratic sections of M k .…”

Section: Introductionmentioning

confidence: 99%

“…Parametric equations for G k : the canonical representation in the general case. In order to transform equations (2.16) to canonical form and deduce the parametric equations of the variety M k of which G k is a quadratic transform we apply the following theorems, proved in [6].…”

Section: Introductionmentioning

confidence: 99%

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The Minimum Model of Spaces on a Quadric

Heymans

1971

Proceedings of the London Mathematical Society

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Minimum Model of Spaces on a Quadric

Heymans

1971

Proceedings of the London Mathematical Society

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“…See [3,11,13]. Note that pf n is invariant under the conjugation action of SO 2n on the space of skewsymmetric matrices.…”

Section: Examplesmentioning

confidence: 99%

Lower bound for ranks of invariant forms

Derksen

Teitler

2015

Journal of Pure and Applied Algebra

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This is an author-produced, peer-reviewed version of this article. LOWER BOUND FOR RANKS OF INVARIANT FORMS HARM DERKSEN AND ZACH TEITLERAbstract. We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad-Schreyer-Shafiei lower bounds for the Waring ranks and cactus ranks of determinants of generic matrices, Pfaffians of generic skew-symmetric matrices, and determinants of generic symmetric matrices.

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CENTRALLY GENERATED PRIMITIVE IDEALS OF U(n$$ \mathfrak{n} $$) IN TYPES B AND D

Ignatyev

2018

Transformation Groups

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Let g be a complex semisimple Lie algebra, b be a Borel subalgebra of g, n be the nilradical of b, and U (n) be the universal enveloping algebra of n. We study primitive ideals of U (n). Almost all primitive ideals are centrally generated, i.e., are generated by their intersections with the center Z(n) of U (n). We present an explicit characterization of the centrally generated primitive ideals of U (n) in terms of the Dixmier map and the Kostant cascade in the case when g is a simple algebra of exceptional type. (For classical simple Lie algebras, a similar characterization was obtained by Ivan Penkov and the first author.) As a corollary, we establish a classification of centrally generated primitive ideals of U (n) for an arbitrary semisimple algebra g.It was proved in [IP, Theorem 3.1] and [Ig1, Theorem 2.4] that, when Φ is of classical type (i.e., Φ = A n−1 , B n , C n or D n ), J ∈ Prim U (n) is centrally generated if and only if J = J(f ) for a certain Kostant form f . In this paper, we prove that this fact is also true when Φ is of exceptional type, i.e., Φ = E 6 , E 7 , E 8 , F 4 or G 2 . Namely, let ∆ β , β ∈ B, be the set of canonical generators of Z(n) (see Section 2). Let J be a primitive ideal of U (n). Since J is the annihilator of a simple n-module, given β ∈ B, there exists the unique c β ∈ C such that ∆ β − c β ∈ J. Our main result, Theorem 5.1, claims that the following conditions are equivalent: i) J is centrally generated;ii) all scalars c β , β ∈ B \ ∆, are nonzero;iii) J = J(f ) for a Kostant form f .If these conditions are satisfied, then we present an explicit way how to reconstruct f by J. As a corollary, we conclude that the same is true for arbitrary (probably, reducible) root system.The paper is organized as follows. In Section 2, we briefly recall the Kostant's characterization of Z(n) and present a (more or less) explicit description of the canonical generators of Z(n) based on A. Panov's work [Pa2]. Using this description, in Section 3 we prove that certain centrally generated ideals are primitive (in fact, it is the key ingredient in the proof of the main result, see Proposition 3.3). Section 4 is devoted to some particular classes of coadjoint orbits. Namely, we prove that certain orbits are disjoint, see Proposition 4.3. Finally, in Section 5, combining our results form two previous sections, we prove the main result, Theorem 5.1. As an immediate corollary, we obtain that the similar result is true for an arbitrary semisimple Lie algebra, see Theorem 5.2. The center of U(n)Theorem 3.2. Suppose Φ is an irreducible root system of classical type, i.e., Φ = A n−1 , B n , C n or D n . The following conditions on a primitive ideal J ⊂ U (n) are equivalent : i) J is centrally generated (or, equivalently, J = J c );ii) the scalars c β , β ∈ B \ ∆, are nonzero;iii) J = J(f ξ ) for a Kostant form f ξ ∈ n * .If these conditions are satisfied, then the map ξ can be reconstructed by J.

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Pfaffians and Skew-Symmetric Matrices

Cited by 12 publications

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The Minimum Model of Spaces on a Quadric

The Minimum Model of Spaces on a Quadric

Lower bound for ranks of invariant forms

CENTRALLY GENERATED PRIMITIVE IDEALS OF U(n$$ \mathfrak{n} $$) IN TYPES B AND D

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