Translation functors and decomposition numbers for the periplectic Lie superalgebra $\mathfrak{p}(n)$

Balagović, Martina; Daugherty, Zajj; Entova-Aizenbud, Inna; Halacheva, Iva; Hennig, Johanna; Im, Mee Seong; Letzter, Gail; Norton, Emily; Serganova, Vera; Stroppel, Catharina

doi:10.48550/arxiv.1610.08470

Cited by 6 publications

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“…Equation (3.3) shows the explicit connection between T on PD and Res between the Brauer algebras. This explains the similarities between translation functors for the periplectic Lie superalgebra [BDE+,Corollary 4.4.6] and the restriction functors [CE,Proposition 2.3.1].…”

Section: Tensor Product With the Generatormentioning

confidence: 93%

“…The periplectic Lie superalgebra. For each n ∈ Z >0 , the periplectic Lie superalgebra pe(n) is the subalgebra of the general linear superalgebra gl(n|n), which preserves an odd bilinear form β : V × V → k, see [BDE+,Ch,Co1,KT,Mo,Mu], with V := k n|n . Concretely,…”

Section: Preliminariesmentioning

confidence: 99%

“…The supercategory sF n of integrable modules over pe(n). We consider the category sF n which has as objects all F 2 -graded, finite dimensional, integrable, left pe(n)-modules, see [BDE+,Section 2]. The morphism spaces consist of all pe(n)-linear morphisms of (ungraded) k-vector spaces.…”

Section: Preliminariesmentioning

confidence: 99%

“…Now we prove these two theorems. There is a duality * on sF n , see [BDE+,Section 2]. Furthermore, the right adjoint of −⊗M is −⊗M * , for a module M , see e.g.…”

Section: 3mentioning

confidence: 99%

“…Furthermore, the right adjoint of −⊗M is −⊗M * , for a module M , see e.g. [BDE+,Section 4.4]. This implies that M ⊗ N is projective as soon as either M or N is projective.…”

Section: 3mentioning

confidence: 99%

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The periplectic Brauer algebra III: The Deligne category

Coulembier¹,

Ehrig²

2017

Preprint

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We construct a faithful categorical representation of an infinite Temperley-Lieb algebra on the periplectic analogue of Deligne's category. We use the corresponding combinatorics to classify thick tensor ideals in this periplectic Deligne category. This allows to determine the objects in the kernel of the monoidal functor going to the module category of the periplectic Lie supergroup. We use this to classify indecomposable direct summands in the tensor powers of the natural representation, determine which are projective and determine their simple top.

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Section: Tensor Product With the Generatormentioning

confidence: 93%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…Now we prove these two theorems. There is a duality * on sF n , see [BDE+,Section 2]. Furthermore, the right adjoint of −⊗M is −⊗M * , for a module M , see e.g.…”

Section: 3mentioning

confidence: 99%

“…Furthermore, the right adjoint of −⊗M is −⊗M * , for a module M , see e.g. [BDE+,Section 4.4]. This implies that M ⊗ N is projective as soon as either M or N is projective.…”

Section: 3mentioning

confidence: 99%

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The periplectic Brauer algebra III: The Deligne category

Coulembier¹,

Ehrig²

2017

Preprint

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The affine VW supercategory

Balagović

Daugherty

Entova-Aizenbud

et al. 2020

Sel. Math. New Ser.

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We define the affine VW supercategory s⩔, which arises from studying the action of the periplectic Lie superalgebra p(n) on the tensor product M ⊗ V ⊗a of an arbitrary representation M with several copies of the vector representation V of p(n). It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group; the main obstacle was the lack of a quadratic Casimir element in p(n) ⊗ p(n). When M is the trivial representation, the action factors through the Brauer supercategory sBr . Our main result is an explicit basis theorem for the morphism spaces of s⩔ and, as a consequence, of sBr . The proof utilises the close connection with the representation theory of p(n). As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation.

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Affine periplectic Brauer algebras

Chen

Peng

2018

Journal of Algebra

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We formulate Nazarov-Wenzl type algebrasP − d for the representation theory of the periplectic Lie superalgebras p(n). We establish an Arakawa-Suzuki type functor to provide a connection between p(n)-representations andP − d -representations. We also consider various tensor product representations forP − d . The periplectic Brauer algebra A d developed by Moon is a quotient ofP − d . In particular, actions induced by Jucys-Murphy elements can also be recovered under the tensor product representation ofP − d . Moreover, a Poincare-Birkhoff-Witt type basis forP − d is obtained. A diagram realization ofP − d is also obtained.

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Translation functors and decomposition numbers for the periplectic Lie superalgebra $\mathfrak{p}(n)$

Cited by 6 publications

References 0 publications

The periplectic Brauer algebra III: The Deligne category

The periplectic Brauer algebra III: The Deligne category

The affine VW supercategory

Affine periplectic Brauer algebras

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