Discriminants, Resultants, and Multidimensional Determinants

Gel'fand, I. M.; Kapranov, Mikhail; Zelevinsky, Andrey V.

doi:10.1007/978-0-8176-4771-1

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“…, where π is the natural projection, and we consider the composition In fact, this definition generalizes the hyperdeterminant of boundary format as introduced in Theorem 14.3.3 of [GKZ94].…”

Section: The Invariant D and The Proof Of The Main Resultsmentioning

confidence: 99%

“…is surjective, is equivalent to the fact that the matrix A ∈ Hom(V * ⊗ I * , W ) representing E is nondegenerate according to [GKZ94] (see Definition 2.3 for the precise definition).…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Hom(V * ⊗ I * , W ) contains the subvariety Q given by matrices A for which the sequence (1) is a complex, that is, such that By [GKZ94], Theorem 14.3.1, this is equivalent to the standard definition of degeneracy given in chapter 14.1 of [GKZ94]. It is easy to check that degenerate matrices fill an irreducible subvariety N of Hom(V * ⊗ I * , W ) of codimension k (see [WZ96]).…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…It is easy to check that degenerate matrices fill an irreducible subvariety N of Hom(V * ⊗ I * , W ) of codimension k (see [WZ96]). Hence, only in the case k = 1 is it well-defined as a hyperdeterminant according to [GKZ94]. In the next section we will define an SL(I) × Sp(W )-invariant on Hom(V * ⊗ I * , W ), called D, which generalizes the hyperdeterminant and is suitable for our purposes.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…The fact that Q 0 (resp. P 0 ) is affine is a consequence of the existence of an invariant of multidimensional matrices representing the instanton bundles, which generalizes the hyperdeterminant (see [GKZ94] and [AO99]). …”

Section: Introductionmentioning

confidence: 99%

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Nondegenerate multidimensional matrices and instanton bundles

Costa

Ottaviani

2002

Trans. Amer. Math. Soc.

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Abstract. In this paper we prove that the moduli space of rank 2n symplectic instanton bundles on P 2n+1 , defined from the well-known monad condition, is affine. This result was not known even in the case n = 1, where by Atiyah, Drinfeld, Hitchin, and Manin in 1978 the real instanton bundles correspond to self-dual Yang Mills Sp(1)-connections over the 4-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multidimensional matrices representing the instanton bundles.

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Section: The Invariant D and The Proof Of The Main Resultsmentioning

confidence: 99%

“…is surjective, is equivalent to the fact that the matrix A ∈ Hom(V * ⊗ I * , W ) representing E is nondegenerate according to [GKZ94] (see Definition 2.3 for the precise definition).…”

Section: Notation and Preliminariesmentioning

confidence: 99%