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

• Verfasst von: Heidersdorf, Thorsten [VerfasserIn]
• Verfasst von: Weissauer, Rainer [VerfasserIn]
• Titel: On classical tensor categories attached to the irreducible representations of the general linear supergroups GL(n|n)
• Verf.angabe: Th. Heidersdorf, R. Weissauer
• E-Jahr: 2023
• Jahr: 20 April 2023
• Umfang: 101 S.
• Fussnoten: Gesehen am 31.07.2023
• Titel Quelle: Enthalten in: Selecta mathematica
• Ort Quelle: Basel [u.a.] : Birkhäuser, 1995
• Jahr Quelle: 2023
• Band/Heft Quelle: 29(2023), 3, Artikel-ID 34, Seite 1-101
• ISSN Quelle: 1420-9020
• DOI: doi:10.1007/s00029-023-00842-1
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: 17B10
• Sach-SW: 17B20
• Sach-SW: 17B55
• Sach-SW: 18D10
• Sach-SW: 20G05
• K10plus-PPN: 1853980463

Abstract

We study the quotient of $$\mathcal {T}_n = Rep(GL(n|n))$$by the tensor ideal of negligible morphisms. If we consider the full subcategory $$\mathcal {T}_n^+$$of $$\mathcal {T}_n$$of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category $$Rep(H_n)$$where $$H_n$$is a pro-reductive algebraic group. We determine the $$H_n$$and the groups $$H_{\lambda }$$corresponding to the tannakian subcategory in $$Rep(H_n)$$generated by an irreducible representation $$L(\lambda )$$. 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 $$\pi _0(H_n)$$.