The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer, G. I.; Zhang, R. B.

doi:10.1007/s00220-016-2731-7

Cited by 38 publications

(73 citation statements)

References 89 publications

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“…Using however the non-elementary grading results of [7], it is possible to describe also the kernel of our action in general, but it requires to work with a graded version of the Brauer algebra as defined in [12] and will appear in a separate article where this graded Brauer algebra is studied. The extra grading refines the results from [18,19]. We expect that our approach generalizes easily to the quantized (super) case using the quantised walled Brauer algebras.…”

Section: Then There Is a Canonical Isomorphism Of Algebrassupporting

confidence: 72%

Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra

Ehrig¹,

Stroppel²

2016

Math. Z.

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Section: Then There Is a Canonical Isomorphism Of Algebrassupporting

confidence: 72%

Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra

Ehrig¹,

Stroppel²

2016

Math. Z.

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“…The orthogonal case. Fix δ ∈ Z ⊂ k. By [LZ3,Theorem 5.6], for every (m, n) ∈ N × N with δ = m − 2n, we have a full monoidal functor F m,n : RepO δ → Rep k OSp(m|2n), determined by the property that it maps R( ) to the natural representation k m|2n . This is the first fundamental theorem of invariant theory, see also [Se,Theorem 3.4] or [DLZ,Section 3.13].…”

Section: Tensor Ideals In Deligne Categoriesmentioning

confidence: 99%

Tensor ideals, Deligne categories and invariant theory

Coulembier

2018

Sel. Math. New Ser.

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We derive some tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne's universal categories RepO δ , RepGL δ and RepP . These results are then used to obtain new insight into the second fundamental theorem of invariant theory for the algebraic supergroups of types A, B, C, D, P .We also find new short proofs for the classification of tensor ideals in RepSt and in the category of tilting modules for SL2(k) with char(k) > 0 and for Uq(sl2) with q a root of unity. In general, for a simple Lie algebra g of type ADE, we show that the lattice of such tensor ideals for Uq(g) corresponds to the lattice of submodules in a parabolic Verma module for the corresponding affine Kac-Moody algebra.2010 Mathematics Subject Classification. 18D10, 17B45, 17B10, 15A72. Key words and phrases. Monoidal (super)category, tensor ideal, thick tensor ideal, Deligne category, algebraic (super)group, second fundamental theorem of invariant theory, tilting modules, quantum groups. 1 3.2. Proofs. By definition, for a submodule M of P C ½ , we have M(X) ⊂ C(½, X), for all X ∈ ObC.Proposition 3.2.1. (i) For a submodule M of P ½ , we define J M as J M (X, Y ) := ι XY M(X ∨ ⊗ Y ) , for all X, Y ∈ ObC.Then J M is a left-tensor ideal in C.

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“…In recent work [3,9] we have proved a first fundamental theorem (FFT) of invariant theory for the orthosymplectic group super scheme OSp(V ), where V is a super space over C. The proof is geometric and applies to the classical groups as special cases; it does not involve the Capelli identities which appear in traditional proofs (cf. [16,Thm 10.2A], [4, App.…”

Section: This Workmentioning

confidence: 99%

Invariants of the special orthogonal group and an enhanced Brauer category

Lehrer

Zhang

2018

Enseign. Math.

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We first give a short intrinsic, diagrammatic proof of the First Fundamental Theorem of invariant theory (FFT) for the special orthogonal group SO m (C), given the FFT for O m (C). We then define, by means of a presentation with generators and relations, an enhanced Brauer category B(m) by adding a single generator to the usual Brauer category B(m), together with four relations. We prove that our category B(m) is actually (and remarkably) equivalent to the category of representations of SO m generated by the natural representation. The FFT for SO m amounts to the surjectivity of a certain functor F on Hom spaces, while the Second Fundamental Theorem for SO m says simply that F is injective on Hom spaces. This theorem provides a diagrammatic means of computing the dimensions of spaces of homomorphisms between tensor modules for SO m (for any m). These methods will be applied to the case of the orthosymplectic Lie algebras osp(m|2n), where the super-Pfaffian enters, in a future work.

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The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Cited by 38 publications

References 89 publications

Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra

Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra

Tensor ideals, Deligne categories and invariant theory

Invariants of the special orthogonal group and an enhanced Brauer category

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