title: Pro-reductive groups attached to irreducible representations of the General Linear Supergroup
creator: Heidersdorf, Thorsten
subject: 500
subject: 500 Natural sciences and mathematics
subject: 510
subject: 510 Mathematics
description: We study tensor product decompositions of representations of the General Linear Supergroup Gl(m|n). We show that the quotient of Rep(Gl(m|n),\epsilon)$ by the tensor ideal of negligible representations is the representation category of a pro-reductive supergroup G red. In the Gl(m|1)-case we show G red = Gl(m-1) \times Gl(1) \times Gl(1). In the general case we study the image of the canonical tensor functor Fmn from Deligne's interpolating category Rep (Gl m-n) to Rep(Gl(m|n),\epsilon). We determine the image of indecomposable elements under Fmn. This implies tensor product decompositions between projective modules and between certain irreducible modules, including all irreducible representations in the Gl(m|1)-case. Using techniques from Deligne's category we derive a closed formula for the tensor product of two maximally atypical irreducible Gl(2|2)-representations. We study cohomological tensor functors DS : Rep(Gl(m|m), epsilon) -> Rep(Gl(m-1|m-1)) and describe the image of an irreducible element under DS. At the end we explain how these results can be used to determine the pro-reductive group G L \hookrightarrow Gl(m|m) red corresponding to the subcategory Rep(G L, epsilon) generated by the image of an irreducible element L in Rep(Gl(m|m) red, epsilon).
date: 2013
type: Dissertation
type: info:eu-repo/semantics/doctoralThesis
type: NonPeerReviewed
format: application/pdf
identifier: https://archiv.ub.uni-heidelberg.de/volltextserverhttps://archiv.ub.uni-heidelberg.de/volltextserver/15435/1/phd-main.pdf
identifier: DOI:10.11588/heidok.00015435
identifier: urn:nbn:de:bsz:16-heidok-154356
identifier:   Heidersdorf, Thorsten  (2013) Pro-reductive groups attached to irreducible representations of the General Linear Supergroup.  [Dissertation]     
relation: https://archiv.ub.uni-heidelberg.de/volltextserver/15435/
rights: info:eu-repo/semantics/openAccess
rights: http://archiv.ub.uni-heidelberg.de/volltextserver/help/license_urhg.html
language: eng