The marked Brauer category

Kujawa, Jonathan R.; Tharp, Benjiman C.

doi:10.1112/jlms.12015

Cited by 21 publications

(56 citation statements)

References 34 publications

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“…In this section, we introduce the periplectic Brauer category

scriptA

, following . This is a

double-struckk

‐linear small skeletal category.…”

Section: Preliminariesmentioning

confidence: 99%

“…In fact, when

char (k) = 2

, the two coincide. The category

scriptA

has been introduced as

{Rep}_{0} P

in [, § 4.5], and independently as the ‘marked Brauer category’

B (0, - 1)

in [, § 3].…”

Section: Preliminariesmentioning

confidence: 99%

“…A definition of

A_{n}

in terms of generators and relations is given in [, Definition 2.2], [, § 3.5] and [, § 4.1]. In particular,

A_{n}

is generated by the simple reflections

s_{i} \in {double-struckS}_{n}

, for

i \in {1, \dots, n - 1}

, where

s_{i}

interchanges the dots in positions

i

and

i + 1

from the left, and any diagram with precisely one cup and cap.…”

Section: Preliminariesmentioning

confidence: 99%

“…The labelling of simple modules by

Λ_{A}

is originally due to Kujawa and Tharp (see [, Theorem 4.3.1]), but we provide an alternative proof. Theorem If

n

is odd and

p \notin [2, n]

, the algebra

(A_{n}, ⩽)

is quasi‐hereditary, in the sense of .…”

Section: Introductionmentioning

confidence: 98%

“…The paper is organised as follows. In Section 2, we introduce the periplectic Brauer algebra and category following and fix some conventions for symmetric groups. In Section 3, we use the periplectic Brauer category to introduce an algebra

C_{n}

, which we can easily study using the general theory of .…”

Section: Introductionmentioning

confidence: 99%

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The periplectic Brauer algebra

Coulembier

2018

Proc. London Math. Soc.

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We study the periplectic Brauer algebra introduced by Moon in the study of invariant theory for periplectic Lie superalgebras. We determine when the algebra is quasi‐hereditary, when it admits a quasi‐hereditary 1‐cover and, for fields of characteristic zero, describes the block decomposition. To achieve this, we also develop theories of Jucys–Murphy elements, Bratteli diagrams, Murphy bases, obtain a Humphreys‐BGG reciprocity relation and determine some decomposition multiplicities of cell modules. As an application, we determine the blocks in the category of finite dimensional integrable modules of the periplectic Lie superalgebra.

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“…In this section, we introduce the periplectic Brauer category

scriptA

, following . This is a

double-struckk

‐linear small skeletal category.…”

Section: Preliminariesmentioning

confidence: 99%

“…In fact, when

char (k) = 2

, the two coincide. The category

scriptA

has been introduced as

{Rep}_{0} P

in [, § 4.5], and independently as the ‘marked Brauer category’

B (0, - 1)

in [, § 3].…”

Section: Preliminariesmentioning

confidence: 99%

“…A definition of

A_{n}

in terms of generators and relations is given in [, Definition 2.2], [, § 3.5] and [, § 4.1]. In particular,

A_{n}

is generated by the simple reflections

s_{i} \in {double-struckS}_{n}

, for

i \in {1, \dots, n - 1}

, where

s_{i}

interchanges the dots in positions

i

and

i + 1

from the left, and any diagram with precisely one cup and cap.…”

Section: Preliminariesmentioning

confidence: 99%

“…The labelling of simple modules by

Λ_{A}

is originally due to Kujawa and Tharp (see [, Theorem 4.3.1]), but we provide an alternative proof. Theorem If

n

is odd and

p \notin [2, n]

, the algebra

(A_{n}, ⩽)

is quasi‐hereditary, in the sense of .…”

Section: Introductionmentioning

confidence: 98%

C_{n}

, which we can easily study using the general theory of .…”

Section: Introductionmentioning

confidence: 99%

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The periplectic Brauer algebra

Coulembier

2018

Proc. London Math. Soc.

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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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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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The marked Brauer category

Cited by 21 publications

References 34 publications

The periplectic Brauer algebra

The periplectic Brauer algebra

Tensor ideals, Deligne categories and invariant theory

The affine VW supercategory

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