Cohomological Tensor Functors on Representations of the General Linear Supergroup

Heidersdorf, Thorsten; Weissauer, Rainer

doi:10.1090/memo/1320

Cited by 10 publications

(21 citation statements)

References 22 publications

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“…The identity a 1 (n, m) = a 2 (m, n) for integers m, n > 0 becomes obvious after substituting ν → n − ν in a 1 (n, m), noticing that in both sums the summands are actually zero for ν > min(m, n). Since a 2 (n, m) = A(n, m) by ( 2), formula (4) follows from (6). Then, equation (7) holds for all integers x = n > 0 and also for all m > 0 due to the symmetry relation satisfied by p(n, m).…”

Section: Lemma 36 Define the Numbersmentioning

confidence: 98%

“…The isomorphism classes of finite dimensional irreducible representations are again described by dominant weights λ [7,9]. The superdimensions of these representations are zero unless λ is maximal atypical [6,10,13]. The maximal atypical λ are again parameterized by integers…”

Section: Introductionmentioning

confidence: 99%

“…In fact, for atypical irreducible representations [7] these character formulas are not entirely satisfying, because the known formulas turn out to be "rather intricate and difficult to apply" [4]. We therefore looked for a different approach to dimension formulas: In [8], we conjectured that for all irreducible maximal atypical representations of weight λ that are attached to a fixed basic weight λ basic (see [6]), the dimensions of the representations depend on the coefficients of the weight λ in a polynomial way. Notice that, for fixed n, the number of different basic weights λ basic is finite and given by the Catalan number C n .…”

Section: Introductionmentioning

confidence: 99%

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Superbinomial coefficients

Maurischat

Weissauer

2021

J Algebr Comb

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We investigate several families of polynomials that are related to certain Euler type summation operators. Being integer valued at integral points, they satisfy combinatorial properties and nearby symmetries, due to triangle recursion relations involving squares of polynomials. Some of these interpolate the Delannoy numbers. The results are motivated by and strongly related to our study of irreducible Lie supermodules, where dimension polynomials in many cases show similar features.

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Section: Lemma 36 Define the Numbersmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Superbinomial coefficients

Maurischat

Weissauer

2021

J Algebr Comb

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“…Parts of our motivation to study HoT comes from our search of understanding the complicated monoidal structure of T m|n . Recent results in this area include the classification of thick ideals of T m|n in [BKN17], a semisimplicity theorem about the Duflo-Serganova functor [HW14], a structural computation of tensor products up to superdimension 0 [He15] [HW15] and explicit tensor product decompositions for special classes of representations [He17]. One of the interesting aspects of the homotopy category is that it is the natural habitat of the Duflo-Serganova cohomology functor DS : T m|n → T m−1|n−1 and its extension DS : C m|n → C m−1|n−1 to the ind completion.…”

Section: 7mentioning

confidence: 99%

“…In the GL(m|1)-case we determine this semisimple quotient. For more background on T m|n = Rep(GL(m|n)) we refer to [HW14].…”

Section: Thenmentioning

confidence: 99%

Homotopy quotients and comodules of supercommutative Hopf algebras

Heidersdorf,

Weissauer

2019

Preprint

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We study induced model structures on Frobenius categories. In particular we consider the case where C is the category of comodules of a supercommutative Hopf algebra A over a field k. Given a graded Hopf algebra quotient A → B satisfying some finiteness conditions, the Frobenius tensor category D of graded B-comodules with its stable model structure induces a monoidal model structure on C. We consider the corresponding homotopy quotient γ : C → HoC and the induced quotient T → HoT for the tensor category T of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in HoT . We apply these results in the Rep(GL(m|n))-case and study its homotopy category HoT .

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On classical tensor categories attached to the irreducible representations of the general linear supergroups $$GL(n\vert n)$$

Heidersdorf

Weissauer

2023

Sel. Math. New Ser.

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We study the quotient of $$\mathcal {T}_n = Rep(GL(n|n))$$ T n = R e p ( G L ( n | n ) ) by the tensor ideal of negligible morphisms. If we consider the full subcategory $$\mathcal {T}_n^+$$ T n + of $$\mathcal {T}_n$$ T n of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category $$Rep(H_n)$$ R e p ( H n ) where $$H_n$$ H n is a pro-reductive algebraic group. We determine the $$H_n$$ H n and the groups $$H_{\lambda }$$ H λ corresponding to the tannakian subcategory in $$Rep(H_n)$$ R e p ( H n ) generated by an irreducible representation $$L(\lambda )$$ L ( λ ) . This gives structural information about the tensor category Rep(GL(n|n)), including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on 2-torsion in $$\pi _0(H_n)$$ π 0 ( H n ) .

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Cohomological Tensor Functors on Representations of the General Linear Supergroup

Cited by 10 publications

References 22 publications

Superbinomial coefficients

Superbinomial coefficients

Homotopy quotients and comodules of supercommutative Hopf algebras

On classical tensor categories attached to the irreducible representations of the general linear supergroups $$GL(n\vert n)$$

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