A characteristic free approach to invariant theory

Concini, Corrado De; Procesi, Claudio

doi:10.1016/s0001-8708(76)80003-5

Cited by 309 publications

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“…It is clear that relations (1 ′ )-(5 ′ ) are consequences of relations (1)- (4). For the converse direction, we define T sr := T ρ T s 1 T −1 ρ ; the remaining cases of relations (1)-(4) may then be obtained from relations (…”

Section: Preliminaries and Statement Of Main Resultsmentioning

confidence: 99%

“…Schur observed that the centralizer algebra of each action equals the image of the other action in End((V ′ ) ⊗r ), in characteristic zero, and Schur and Weyl used this observation to transfer information about the representations of S r to information about the representations of GL(V ′ ). That this Schur-Weyl duality holds in arbitrary characteristic was first observed in [4], although a special case was already used in [2]. In recent years, there have appeared various applications of the Schur-Weyl duality viewpoint to modular representations.…”

Section: Introductionmentioning

confidence: 94%

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Presenting affine q-Schur algebras

Doty

Green

2007

Math. Z.

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Abstract. We obtain a presentation of certain affine q-Schur algebras in terms of generators and relations. The presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine gl n . Our results extend and rely on the corresponding result for the q-Schur algebra of the symmetric group, which were proved by the first author and Giaquinto.

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Section: Preliminaries and Statement Of Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

Presenting affine q-Schur algebras

Doty

Green

2007

Math. Z.

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“…De Concini and Procesi [3] calculated the invariant subrings for several important group scheme actions over an arbitrary commutative ring. In [6], a simple proof (for the action of the general linear group and the symplectic group) utilizing a geometric argument over a field is given.…”

Section: Hashimotomentioning

confidence: 99%

“…By the following theorem due to De Concini and Procesi [3], π # : [2, (6.3)]), it suffices to show that π is dominating. By linear algebra, for each i such that 0 ≤ i ≤ u, the set of linear maps V → W of rank i forms one G -orbit.…”

Section: Hashimotomentioning

confidence: 99%

Base change of Invariant subrings

Hashimoto¹

2007

Nagoya Mathematical Journal

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Abstract. Let R be a Dedekind domain, G an affine flat R-group scheme, and B a flat R-algebra on which G acts. Let A → B G be an R-algebra map. Assume that A is Noetherian. We show that if the induced map K ⊗ A → (K ⊗ B) K⊗G is an isomorphism for any algebraically closed field K which is an R-algebra, then S ⊗ A → (S ⊗ B) S⊗G is an isomorphism for any R-algebra S. §1. IntroductionIn this paper, we prove the following.Theorem 1. Let R be a Dedekind domain, G an affine flat R-group scheme, and M an R-flat G-module. Let A be a Noetherian R-algebra, and V a finitely generated A-module.K⊗G is an isomorphism for any algebraically closed field K which is an R-algebra, then the canonical map ϕ S : S ⊗ V → (S ⊗ M ) S⊗G is an isomorphism for any R-algebra S.As a corollary, we have the following.Corollary 2. Let R be a Dedekind domain, G an affine flat R-group scheme, and B a flat R-algebra on which G-acts. Let A be a Noetherian R-algebra, and ϕ : A → B G an R-algebra map. If the induced map ϕ K : K ⊗ A → (K ⊗ B) K⊗G is an isomorphism for any algebraically closed field K which is an R-algebra, then the canonical map ϕ S : S ⊗ A → (S ⊗ B) S⊗G is an isomorphism for any R-algebra S.So we may work only over algebraically closed field instead of general commutative ring, once we know that the action and the candidate of the generator and the relation of the invariant subring are given over a Dedekind domain (e.g., Z), and the group scheme in problem is flat over the Dedekind domain.

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“…For integers u, v > 0, we write M u,v for the set of u × v matrices with entries in K. The general linear group GL t = GL t (K) acts on the variety V = M m,t × M t,n , with action given by These two theorems are respectively known as the first and second fundamental theorems of invariant theory (for GL t ) (see [4]). They give a full description of the algebra O(V ) GLt .…”

Section: Introductionmentioning

confidence: 99%