Affine periplectic Brauer algebras

Chen, Chih-Whi; Peng, Yung Ning

doi:10.1016/j.jalgebra.2018.01.005

Cited by 17 publications

(14 citation statements)

References 20 publications

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“…For more details, including a presentation and a description of the centre, see Section 5. One can also define cyclotomic quotients of the algebras s⩔ a by mimicking the constructions in [2] for affine VW algebras, see also [11]. We expect Lemma 8 (stating the vanishing of all loop values) to simplify the necessary admissibility conditions from [2] and more explicitly [19] drastically, but do not pursue this here.…”

Section: 2mentioning

confidence: 99%

“…A basis theorem for the endomorphism algebras of objects in s⩔ was obtained independently in [11] by an algebraic method developed in [32], also using the fake Casimir operator. The Brauer superalgebras recently appeared in the literature under the names odd Brauer algebras, marked Brauer algebras or periplectic Brauer algebras, indicating the slightly different points of view on the subject.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The affine VW supercategory

Balagović

Daugherty

Entova-Aizenbud

et al. 2020

Sel. Math. New Ser.

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“…In order to construct interesting calibrated representations of A n , we study the degenerate affine periplectic Brauer algebra sV V n defined in [1], [5]. This is the algebra generated by A n and commuting formal variables y 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…Our sign conventions follow [1,Def. 39] and differ in some relations from those in [5,Def. 3.1], [24], [7].…”

Section: Introductionmentioning

confidence: 99%

Irreducible calibrated representations of periplectic Brauer algebras and hook representations of the symmetric group

Norton

2020

Journal of Algebra

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We construct an infinite tower of irreducible calibrated representations of periplectic Brauer algebras on which the cup-cap generators act by nonzero matrices. As representations of the symmetric group, these are exterior powers of the standard representation (i.e. hook representations). Our approach uses the recently-defined degenerate affine periplectic Brauer algebra, which plays a role similar to that of the degenerate affine Hecke algebra in representation theory of the symmetric group. We write formulas for the representing matrices in the basis of Jucys-Murphy eigenvectors and we completely describe the spectrum of these representations. The tower formed by these representations provides a new, non-semisimple categorification of Pascal's triangle. Along the way, we also prove some basic results about calibrated representations of the degenerate affine periplectic Brauer algebra.

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“…The affine VW-superalgebras were considered independently in [CP18], where a basis theorem is formulated as well. We should also mention some overlap with [Cou18], which independently introduced the Casimir elements and Jucys-Murphy elements and classified blocks using totally different methods.…”

mentioning

confidence: 99%

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

Balagović

Daugherty

Entova-Aizenbud

et al. 2019

Mathematical Research Letters

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We study the category F n of finite-dimensional integrable representations of the periplectic Lie superalgebra p(n). We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter 0 on the category F n by certain translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for p(n) resembling those for gl(m n). We discover two natural highest weight structures. Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of irreducibles in standard and costandard modules and classify the blocks of F n . We also prove the surprising fact that indecomposable projective modules in this category are multiplicity-free. Contents

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Affine periplectic Brauer algebras

Cited by 17 publications

References 20 publications

The affine VW supercategory

The affine VW supercategory

Irreducible calibrated representations of periplectic Brauer algebras and hook representations of the symmetric group

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

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