Pieri Type Rules and GL(2|2) Tensor Products

Heidersdorf, Thorsten; Weissauer, Rainer

doi:10.1007/s10468-020-09954-0

Cited by 4 publications

(6 citation statements)

References 16 publications

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“…We say a negligible module N in T n is potentially projective of degree r if DS n−r (N ) ∈ T r is projective and DS i (N ) is not for i ≤ n − r. Now consider the special representations S i . Then we proved in [HW15] the surprising fact that the projection of S i ⊗S j or S i ⊗(S j ) ∨ on the maximal atypical block is clean. To prove the result we establish the n = 2-case by a brute force calculation.…”

Section: This Follows By Comparingmentioning

confidence: 89%

“…Example 12.5. For GL(2|2) we obtained (up to parity shifts) in [HW15] the formula S i ⊗ S i = Ber i−1 ⊕ M for some module M of superdimension 3. Since sdim(S i ) = 2, det(S i ) = Ber i−1 ⊕ negligible.…”

Section: The Picard Group Of T Nmentioning

confidence: 99%

“…Without this, to work out any such decomposition is rather elaborate. For the case n = 2 see [HW15]. In fact the knowledge of the Jordan-Hölder factors usually gives too little information on the indecomposable objects itself.…”

mentioning

confidence: 99%

“…Our strategy is to determine the groups H n or G n inductively using the functor DS. For n = 2 we need the explicit results of [HW15] to give us the fusion rule between two irreducible representations and we describe the corresponding Tannaka group in lemma 9.2. Starting from the special case n = 2 we can proceed by induction on n. For this we use the embedding H n−1 → H n along with the known branching laws and the classification of small representations due to Andreev, Elashvili and Vinberg [AVE67] which allows to determine inductively the connected derived groups G n = (H 0 n ) der for n ≥ 3; see section 10.…”

mentioning

confidence: 99%

See 3 more Smart Citations

On classical tensor categories attached to the irreducible representations of the General Linear Supergroups $GL(n\vert n)$

Heidersdorf¹,

Weissauer²

2018

Preprint

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We study the quotient of Tn = Rep(GL(n|n)) by the tensor ideal of negligible morphisms. If we consider the full subcategory T + n of Tn of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category Rep(Hn) where Hn is a pro-reductive algebraic group. We determine the connected derived subgroup Gn ⊂ Hn and the groups G λ = (H λ ) 0 der corresponding to the tannakian subcategory in Rep(Hn) generated by an irreducible representation 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 π0(Hn). Contents 1. Introduction 2. The superlinear groups 3. Weight and cup diagrams 4. Cohomological tensor functors. 5. Tannakian arguments 6. The structure of the derived connected groups G n 7. Proof of the structure theorem: Overview 8. Small representations 9. The cases n = 2, 3 and the S i -case 10. Tannakian induction: Proof of the structure theorem 11. A conjectural structure theorem 12. The Picard group of T n 13. The Picard group of T n and the group H n Appendix A. Equivalences and derivatives

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Section: This Follows By Comparingmentioning

confidence: 89%

Section: The Picard Group Of T Nmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On classical tensor categories attached to the irreducible representations of the General Linear Supergroups $GL(n\vert n)$

Heidersdorf¹,

Weissauer²

2018

Preprint

Self Cite

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“…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). Exchanging m and n, we obtain (5). Therefore, both being polynomials of degree 2(m − 1) in x, a 2 (m, x) 2 and E 2 2 p(m, x) coincide.…”

Section: Lemma 36 Define the Numbersmentioning

confidence: 96%

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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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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Pieri Type Rules and GL(2|2) Tensor Products

Cited by 4 publications

References 16 publications

On classical tensor categories attached to the irreducible representations of the General Linear Supergroups $GL(n\vert n)$

On classical tensor categories attached to the irreducible representations of the General Linear Supergroups $GL(n\vert n)$

Superbinomial coefficients

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

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