Weyl’s Polarization Theorem in Positive Characteristic

Derksen, Harm; Makam, Visu

doi:10.1007/s00031-020-09559-3

Cited by 5 publications

(4 citation statements)

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“…Theorem 25 (Weyl's Polarization Theorem -weak form [5]). Assume that the characteristic of the ground field is zero and let m = dim W .…”

Section: Bound Transference and Weyl's Polarization Theoremmentioning

confidence: 99%

“…We establish the theorem by considering a subspace arrangement of cardinality t = |G| associated to the group G. The regularity bound on the ideal of the subspace arrangement in the exterior algebra from [11] gives us the degree bound on a minimal set of generating invariants. The idea of using polynomial functors to establish results in invariant theory goes back to the times of Weyl and it continues nowadays in our work, the work of Derksen and Makam [5], and the work of Snowden [24].…”

Section: Introductionmentioning

confidence: 99%

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Degree bounds for invariant skew polynomials

Gandini¹

2021

Preprint

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When we consider the action of a finite group on a polynomial ring, a polynomial unchanged by the action is called an invariant polynomial. A famous result of Noether states that in characteristic zero the maximal degree of a minimal invariant polynomial is bounded above by the order of the group. Our work establishes that the same bound holds for invariant skew polynomials in the exterior algebra. Our approach to the problem relies on a theorem of Derksen that connects invariant theory to the study of ideals of subspace arrangements. We adapt his proof over the polynomial ring to the exterior algebra, reducing the question to establishing a bound on the Castelnuovo-Mumford regularity of intersections of linear ideals in the exterior algebra. We prove the required regularity bound using tools from representation theory. In particular, the proof relies on the existence of a functor on the category of polynomial functors that translates resolutions of ideals of subspace arrangements over the polynomial ring to resolutions of ideals of subspace arrangements over the exterior algebra.

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“…Theorem 25 (Weyl's Polarization Theorem -weak form [5]). Assume that the characteristic of the ground field is zero and let m = dim W .…”

Section: Bound Transference and Weyl's Polarization Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Degree bounds for invariant skew polynomials

Gandini¹

2021

Preprint

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“…Given a homogeneous polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal call the restitution. In addition to being essential in invariant theory, e.g., Weyl's polarization theorem [8,36], the notion has applications in other areas of mathematics, including algebraic geometry, Lie algebras and representation theory. See Section 5 for a brief summary.…”

Section: Introductionmentioning

confidence: 99%

Core Content Selection and Textbook Design for High School Mathematics

Zhang¹,

Zhou²,

Wang³

et al. 2021

School Mathematics Textbooks in China

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This paper is devoted to studying the following fractional L 2 -critical nonlinear Schrödinger equation≥ 0 is an external potential. We obtain normalized L 2 -norm solutions of the above equation by solving the associated constraint minimization problem (1.4). It shows that there is a threshold a * > 0 such that (1.4) has minimizers for 0 < a < a * , and minimizers do not exist for any a > a * . For the case of a = a * , it gives a fact that the existence and non-existence of minimizers depend strongly on the value of V (0). Especially for V (0) = 0, we prove that minimizers occur blow-up behavior and the mass of minimizers concentrates at the origin as a ր a * . Applying implicit function theorem, the uniqueness of minimizers is also proved for a > 0 small enough.

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“…Um outro problema oriundo da finitude de certos conjuntos geradores (ou separadores, respectivamente) de uma álgebra de invariantes F[H] G consiste em obter limitações, ou o valor exato quando possível, para o menor inteiro D tal que o conjunto de todos os invariantes homogêneos de F[H] G com grau ≤ D é um conjunto gerador (conjunto separador, respectivamente), o qual é denotado por β(F[H] G ) (β sep (F[H] G ), respectivamente). Este problema foi vastamente estudado e existem diversos resultados nessa linha obtidos para álgebras específicas, como exemplos indicamos os resultados apresentados em [10], [19], [21], [22] e [12].…”

Section: Introducãounclassified

Invariantes separadores de matrizes

Barbosa Cavalcante

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Esta tese é dedicada ao estudo de conjuntos de invariantes separadores minimais da álgebra de invariantes de matrizes nilpotentes 3 × 3, bem como da álgebra de semi-invariantes de matrizes 2×2. Baseados em conjuntos geradores destas álgebras, construímos explicitamente conjuntos separadores minimais para a álgebra de invariantes de matrizes nilpotentes O(n, m) para o caso n = m = 3, como também estabelecemos limitações para o grau máximo de qualquer conjunto separador minimal da álgebra O(n, m), quando n = 3 e d ≥ 3. Além disso, obtivemos uma descrição de um conjunto separador minimal para a álgebra de semi-invariantes de matrizes SI (2,2) (m, p, q), quaisquer que sejam m, p, q ∈ N não todos nulos.Palavras-chave: teoria dos invariantes, invariantes de matrizes, semi-invariantes de matrizes, invariantes separadores.

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Weyl’s Polarization Theorem in Positive Characteristic

Cited by 5 publications

References 44 publications

Degree bounds for invariant skew polynomials

Degree bounds for invariant skew polynomials

Core Content Selection and Textbook Design for High School Mathematics

Invariantes separadores de matrizes

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