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Buildings, spiders, and geometric Satake

Part of: Low-dimensional topology Lie groups Finite geometry and special incidence structures

Published online by Cambridge University Press:  10 July 2013

Bruce Fontaine ,
Joel Kamnitzer  and
Greg Kuperberg
Bruce Fontaine
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada email bfontain@gmail.com
Joel Kamnitzer
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada email jkamnitz@math.toronto.edu
Greg Kuperberg
Affiliation:
Department of Mathematics, University of California, Davis, 1 Shields Avenue, Davis CA 95616, USA email greg@math.ucdavis.edu
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Abstract

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Let $G$ be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine Grassmannian. Via the geometric Satake correspondence, we relate these configuration spaces to the invariant vectors coming from webs. In the case of $G= \mathrm{SL} (3)$, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is $\mathrm{CAT} (0)$, is explained by the fact that affine buildings are $\mathrm{CAT} (0)$.

Keywords

websspidersgeometric Satakebuildings

MSC classification

Secondary: 22E47: Representations of Lie and real algebraic groups 51E24: Buildings and the geometry of diagrams 57M27: Invariants of knots and 3-manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 11 , November 2013 , pp. 1871 - 1912
Copyright
© The Author(s) 2013 

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