Semisimplification of the category of tilting modules for GL_n

Brundan, Jonathan; Entova-Aizenbud, Inna; Etingof, Pavel; Ostrik, Victor

doi:10.48550/arxiv.2002.01900

Cited by 2 publications

(3 citation statements)

References 10 publications

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“…In fact, the pair (S, U ) is universal among pairs, cf. for example [AK02,EtO18], and also [H15,BEEO20].…”

Section: Notationmentioning

confidence: 99%

Jacobson-Morozov Lemma for Algebraic Supergroups

Entova-Aizenbud¹,

Serganova²

2020

Preprint

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Given a quasi-reductive algebraic supergroup G, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functor Φ x : Rep(G) → Rep(OSp(1|2)) associated to any given element x ∈ Lie(G)1. For nilpotent elements x, we show that the functor Φ x can be defined using the Deligne filtration associated to x.We use this approach to prove an analogue of the Jacobson-Morozov Lemma for algebraic supergroups. Namely, we give a necessary and sufficient condition on odd nilpotent elements x ∈ Lie(G)1 which define an embedding of supergroups OSp(1|2) → G so that x lies in the image of the corresponding Lie algebra homomorphism.

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“…In fact, the pair (S, U ) is universal among pairs, cf. for example [AK02,EtO18], and also [H15,BEEO20].…”

Section: Notationmentioning

confidence: 99%

Jacobson-Morozov Lemma for Algebraic Supergroups

Entova-Aizenbud¹,

Serganova²

2020

Preprint

Self Cite

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“…In this section we describe a linear monoidal category called the polynomial web category for GL n over a field k [CKM14], [BEAEO20]. Our presentation closely follows [BEAEO20].…”

Section: Polynomial Gl N Websmentioning

confidence: 99%

“…In this section we describe a linear monoidal category called the polynomial web category for GL n over a field k [CKM14], [BEAEO20]. Our presentation closely follows [BEAEO20]. Indeed, in the description below, our presentation is easily derived as the quotient of Web [BEAEO20, Definition 4.7] by the monoidal ideal generated by the identities of objects of weight m, where m > n. If k is algebraically closed, the monoidal category described below is equivalent to the monoidal category of polynomial representations of Gl n , which can be equivalently described as the full subcategory of tilting modules for GL n generated by wedge powers of the standard n-dimensional representation (see [BEAEO20, Remark 4.15]).…”

Section: Polynomial Gl N Websmentioning

confidence: 99%

Triangle presentations and tilting modules for $\text{SL}_{2k+1}$

Jones¹

2020

Preprint

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Triangle presentations are combinatorial structures on finite projective geometries which characterize groups acting simply transitively on the vertices of a locally finite building of type Ãn−1 (n ≥ 3). From a type Ãn−1 triangle presentation on a geometry of order q, we construct a fiber functor on the diagrammatic monoidal category Web(SL − n )over any field k with characteristic p ≥ n − 1 such that q ≡ 1 mod p. When k is algebraically closed and n odd, this gives new fiber functors on the category of tilting modules for SL n .

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Semisimplification of the category of tilting modules for GL_n

Abstract: We describe the semisimplification of the monoidal category of tilting modules for the algebraic group GLn in characteristic p > 0. In particular, we compute the dimensions of the indecomposable tilting modules modulo p.

Cited by 2 publications

References 10 publications

Jacobson-Morozov Lemma for Algebraic Supergroups

Jacobson-Morozov Lemma for Algebraic Supergroups

Triangle presentations and tilting modules for $\text{SL}_{2k+1}$

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