Tensor product multiplicities, canonical bases and totally positive varieties

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dc.creatorBerenstein, Arkady
dc.creatorZelevinsky, Andrei
dc.date1999-12-02
dc.date.accessioned2026-07-07T05:32:04Z
dc.date.available2026-07-07T05:32:04Z
dc.descriptionWe obtain a family of explicit "polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Here "polyhedral" means that the multiplicity in question is expressed as the number of lattice points in some convex polytope. Our answers use a new combinatorial concept of $\ii$-trails which resemble Littelmann's paths but seem to be more tractable. We also study combinatorial structure of Lusztig's canonical bases or, equivalently of Kashiwara's global bases. Although Lusztig's and Kashiwara's approaches were shown by Lusztig to be equivalent to each other, they lead to different combinatorial parametrizations of the canonical bases. One of our main results is an explicit description of the relationship between these parametrizations. Our approach to the above problems is based on a remarkable observation by G. Lusztig that combinatorics of the canonical basis is closely related to geometry of the totally positive varieties. We formulate this relationship in terms of two mutually inverse transformations: "tropicalization" and "geometric lifting."
dc.descriptionLatex, 42 pages. For viewing the table of contents, LaTeX the file 3 times
dc.identifierhttps://arxiv.org/abs/math/9912012
dc.identifierhttp://arxiv.org/abs/math/9912012
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/79528
dc.subjectRepresentation Theory
dc.subjectAlgebraic Geometry
dc.subjectCombinatorics
dc.subjectQuantum Algebra
dc.titleTensor product multiplicities, canonical bases and totally positive varieties
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