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Comes, Jonathan; Heidersdorf, Thorsten

doi:10.1016/j.jalgebra.2017.01.050

Cited by 24 publications

(29 citation statements)

References 29 publications

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“…Remark 7.1.4. As already observed in [Cm,CH,CE2], each tensor Ob-ideals in C, for C one of our Deligne categories, consists of the objects which are sent to zero by a monoidal functor C → Rep k G, for an affine algebraic supergroup scheme G. Since in the latter categories X ⊗ Y ≃ 0 means either X ≃ 0 or Y ≃ 0, this shows that all tensor Ob-ideals in Deligne categories are 'prime' in the sense of [Ba, Definition 2.1].…”

Section: Tensor Ideals In Deligne Categoriessupporting

confidence: 81%

“…We write L r (λ) for the corresponding simple module. By [CH,Theorem 3.5], we have a bijection Par ∼ → indeC, given by λ → R(λ). We can take R(λ) = (r, e λ ), where r = |λ| ∈ ObC 0 and e λ is a primitive idempotent in B r (δ) = C 0 (r, r) corresponding to L r (λ).…”

Section: Deligne Categories and Brauer Algebrasmentioning

confidence: 99%

“…We can take R(λ) = (r, e λ ), where r = |λ| ∈ ObC 0 and e λ is a primitive idempotent in B r (δ) = C 0 (r, r) corresponding to L r (λ). From [CH,Section 3], for λ ∈ Par and k ∈ N = ObC 0 , we have (6.1) R(λ) k if and only if k − |λ| ∈ 2N.…”

Section: Deligne Categories and Brauer Algebrasmentioning

confidence: 99%

“…Here, Rep k OSp(m|2n) is the category of algebraic finite dimensional representations of the algebraic supergroup OSp(m|2n). We refer to [ES2,CH] for details on that category.…”

Section: Tensor Ideals In Deligne Categoriesmentioning

confidence: 99%

“…By [CH,Lemma 7.5], every projective object in Rep k OSp(m|2n) is in the image of F m|n . We denote the full subcategory of projective objects by P. By [CH,Lemma 7.16], F m,n (X) is projective in Rep k OSp(m|2n) if and only if X ∈ I j−1 . By Theorem 7.1.1, the functor F m,n thus restricts to an equivalence D j ∼ → P. It follows easily from the Yoneda lemma that we have an equivalence…”

Section: Tensor Ideals In Deligne Categoriesmentioning

confidence: 99%

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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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Section: Tensor Ideals In Deligne Categoriessupporting

confidence: 81%

Section: Deligne Categories and Brauer Algebrasmentioning

confidence: 99%

Section: Deligne Categories and Brauer Algebrasmentioning

confidence: 99%

“…Here, Rep k OSp(m|2n) is the category of algebraic finite dimensional representations of the algebraic supergroup OSp(m|2n). We refer to [ES2,CH] for details on that category.…”

Section: Tensor Ideals In Deligne Categoriesmentioning

confidence: 99%

Section: Tensor Ideals In Deligne Categoriesmentioning

confidence: 99%

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Tensor ideals, Deligne categories and invariant theory

Coulembier

2018

Sel. Math. New Ser.

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Generalized negligible morphisms and their tensor ideals

Heidersdorf

Wenzl

2022

Sel. Math. New Ser.

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We introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $${\mathcal {C}}$$ C over a local ring R. If the maximal ideal of R is generated by a single element, we show that any thick ideal of $${\mathcal {C}}$$ C admits an explicitly given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.

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Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

Stroppel

Wilbert

2018

Math. Z.

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We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types B and D. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. As an application we obtain presentations of the cohomology rings of all two-block Springer fibers of types C and D. Moreover, we deduce explicit isomorphisms between the Kazhdan-Lusztig cell modules attached to the induced trivial module and the irreducible Specht modules in types C and D.

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Thick ideals in Deligne's category Re_p(Oδ)

Cited by 24 publications

References 29 publications

Tensor ideals, Deligne categories and invariant theory

Tensor ideals, Deligne categories and invariant theory

Generalized negligible morphisms and their tensor ideals

Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

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