Schur-Weyl duality in positive characteristic

Doty, Stephen

doi:10.1090/conm/478/09316

Cited by 5 publications

(3 citation statements)

References 34 publications

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“…This approach is characteristic-free. For u = + r , this gives the original Schur-Weyl duality, see [9]. For u = + r − s , End(V (u)) is a q-analogue of the walled Brauer algebra.…”

Section: Applicationsmentioning

confidence: 99%

Web bases for the general linear groups

Westbury

2011

J Algebr Comb

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“…This approach is characteristic-free. For u = + r , this gives the original Schur-Weyl duality, see [9]. For u = + r − s , End(V (u)) is a q-analogue of the walled Brauer algebra.…”

Section: Applicationsmentioning

confidence: 99%

Web bases for the general linear groups

Westbury

2011

J Algebr Comb

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“…Our approach is motivated by Schur-Weyl duality (see [11,5,1,4,6]), although its full generality is not used here. All we need is the fact that the action of the Brauer algebra commutes with that of a suitable classical group.…”

Section: Introductionmentioning

confidence: 99%

A characteristic-free decomposition of tensor space as a Brauer algebra module

Doty¹

2011

Preprint

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“…In case R = K is an infinite field, Green, De Concini and Procesi proved that Schur-Weyl duality holds (see [Gre80,DCP76]). Another approach assuming only that Schur-Weyl duality holds for C, which is due to Schur, can be found in [Dot09]. Benson and Doty showed in [BD09] that Schur-Weyl duality holds for finite fields with order strictly larger than d.…”

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confidence: 99%

Schur–Weyl duality over commutative rings

Cruz

2018

Communications in Algebra

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The classical case of Schur-Weyl duality states that the actions of the group algebras of GLn and S d on the d th-tensor power of a free module of finite rank centralize each other. We show that Schur-Weyl duality holds for commutative rings where enough scalars can be chosen whose non-zero differences are invertible. This implies all the known cases of Schur-Weyl duality so far. We also show that Schur-Weyl duality fails for Z and for any finite field when d is sufficiently large. 1 Introduction Schur-Weyl duality is a connection between the general linear group and the symmetric group. More specifically, consider n, d ∈ N and let V = R n be the free module of rank n over a commutative ring with identity R. The symmetric group S d acts on the d th-tensor power, V ⊗ d = V ⊗ R • • • ⊗ R V , of the module V by place permutation, that is, σ(v 1 ⊗ • • • ⊗ v d) = v σ −1 (1) ⊗ • • • ⊗ v σ −1 (d) , σ ∈ S d , v i ∈ V. Definition 1.1. [Gre80] The subalgebra End RS d V ⊗ d of the endomorphism algebra End R V ⊗ d is called the Schur algebra. We will denote it by S R (n, d). On the other hand, the general linear group acts on V by multiplication, and thus on the tensor product V ⊗ d by the diagonal action, that is, g(v 1 ⊗ • • • ⊗ v d) = gv 1 ⊗ • • • ⊗ gv d , g ∈ GL n (R), v i ∈ V. These two actions commute, so, by extending these actions to the group algebras, one gets two natural homomorphisms: ρ : RGL n (R) → S R (n, d), ψ : RS d → End RGLn(R) V ⊗ d. Definition 1.2. We say that Schur-Weyl duality holds if the two algebra homomorphisms ρ and ψ are surjective.

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Schur-Weyl duality in positive characteristic

Cited by 5 publications

References 34 publications

Web bases for the general linear groups

Web bases for the general linear groups

A characteristic-free decomposition of tensor space as a Brauer algebra module

Schur–Weyl duality over commutative rings

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