Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T00:41:15.916Z Has data issue: false hasContentIssue false
  • English
  • Français

Buildings, spiders, and geometric Satake

Part of: Lie groups Low-dimensional topology 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 [email protected]
Joel Kamnitzer
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada email [email protected]
Greg Kuperberg
Affiliation:
Department of Mathematics, University of California, Davis, 1 Shields Avenue, Davis CA 95616, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beilinson, A. and Drinfeld, V., Quantization of Hitchin’s integrable system and Hecke eigensheaves, unpublished book manuscript.Google Scholar
Bruhat, F. and Tits, J., Groupes réductifs sur un corps local, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251.Google Scholar
Cautis, S. and Kamnitzer, J., Knot homology via derived categories of coherent sheaves II, ${\mathfrak{s}\mathfrak{l}}_{m} $ case, Invent. Math. 174 (2008), 165–232, arXiv:0710.3216.Google Scholar
Chriss, N. and Ginzburg, V., Representation theory and complex geometry (Birkhäuser, Boston, 1997).Google Scholar
Fock, V. and Goncharov, A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211, arXiv:math/0311149.Google Scholar
Frenkel, I. and Khovanov, M., Canonical bases in tensor products and graphical calculus for ${U}_{q} ({\mathfrak{s}\mathfrak{l}}_{2} )$, Duke Math. J. 87 (1995), 409–480.Google Scholar
Freyd, P. J. and Yetter, D. N., Braided compact closed categories with applications to low-dimensional topology, Adv. Math. 77 (1989), 156–182.CrossRefGoogle Scholar
Fulton, W., Intersection theory, second edition (Springer, Berlin, 1998).CrossRefGoogle Scholar
Gaitsgory, D., Twisted Whittaker model and factorizable sheaves, Selecta Math. (N.S.) 13 (2008), 617–659, arXiv:0705.4571.Google Scholar
Gaussent, S. and Littelmann, P., LS galleries, the path model, and MV cycles, Duke Math. J. 127 (2005), 35–88, arXiv:math/0307122.Google Scholar
Ginzburg, V., Perverse sheaves on a loop group and Langlands duality, Preprint (1995), arXiv:alg-geom/9511007.Google Scholar
Gromov, M., Hyperbolic groups, in Essays in group theory, Mathematical Sciences Research Institute Publications, vol. 8 (Springer, Berlin, 1987), 75–263.Google Scholar
Haines, T. J., Equidimensionality of convolution morphisms and applications to saturation problems, Adv. Math. 207 (2006), 297–327, arXiv:math/0501504.Google Scholar
Humphreys, J. E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9 (Springer, New York, 1972).Google Scholar
Joyal, A. and Street, R., Braided tensor categories, Adv. Math. 102 (1993), 20–78.Google Scholar
Joyce, D., Constructible functions on Artin stacks, J. Lond. Math. Soc. (2) 74 (2006), 583–606, arXiv:math/0403305.CrossRefGoogle Scholar
Kamnitzer, J., Geometric constructions of the irreducible representations of ${\mathrm{GL} }_{n} $, in Geometric representation theory and extended affine Lie algebras (American Mathematical Society, Providence, RI, 2011), 1–18, arXiv:0912.0569.Google Scholar
Kapovich, M., Leeb, B. and Millson, J. J., The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra, Mem. Amer. Math. Soc. 192 (2008), no. 896, arXiv:math/0210256.Google Scholar
Kashiwara, M. and Saito, Y., Geometric construction of crystal bases, Duke Math. J. 89 (1997), 9–36, arXiv:q-alg/9606009.Google Scholar
Kauffman, L. H., Spin networks and knot polynomials, Int. J. Mod. Phys. A 5 (1990), 93–115.CrossRefGoogle Scholar
Khovanov, M., Crossingless matchings and the cohomology of $(n, n)$ Springer varieties, Commun. Contemp. Math. 6 (2004), 561–577, arXiv:math/0202110.CrossRefGoogle Scholar
Khovanov, M., $\mathfrak{s}\mathfrak{l}(3)$ link homology, Algebr. Geom. Topol. 4 (2004), 1045–1081,arXiv:math/0304375.Google Scholar
Khovanov, M. and Kuperberg, G., Web bases for $sl(3)$ are not dual canonical, Pacific J. Math. 188 (1999), 129–153, arXiv:q-alg/9712046.Google Scholar
Kim, D., Graphical calculus on representations of quantum Lie algebras, PhD thesis, University of California, Davis (2003), arXiv:math/0310143.Google Scholar
Kumar, S., Kac-Moody groups, their flag varieties, and representation theory (Birkhäuser, Boston, 2002).CrossRefGoogle Scholar
Kuperberg, G., Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109–151, arXiv:q-alg/9712003.Google Scholar
Littelmann, P., Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), 499–525.Google Scholar
Lusztig, G., Singularities, character formulas, and a $q$-analog of weight multiplicities, in Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque, vol. 101 (Société Mathématique de France, Paris, 1983), 208–229.Google Scholar
Lusztig, G., Constructible functions on varieties attached to quivers, in Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), Progress in Mathematics, vol. 210 (Birkhäuser, Boston, 2003), 177–223.Google Scholar
MacPherson, R., Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432.CrossRefGoogle Scholar
Mirković, I. and Vilonen, K., Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), 95–143, arXiv:math/0401222.CrossRefGoogle Scholar
Mirković, I. and Vybornov, M., Quiver varieties and Beilinson-Drinfeld Grassmannians of type A, Preprint (2007), arXiv:0712.4160.Google Scholar
Morrison, S., A diagrammatic category for the representation theory of ${U}_{q} (s{l}_{n} )$, PhD thesis, University of California, Berkeley (2007), arXiv:0704.1503.Google Scholar
Nakajima, H., Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365–416.Google Scholar
Penrose, R., Applications of negative dimensional tensors, in Combinatorial mathematics and its applications (Academic Press, London, 1971), 221–244.Google Scholar
Reshetikhin, N. Yu. and Turaev, V. G., Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1–26.Google Scholar
Ronan, M., Lectures on buildings, updated and revised edition (University of Chicago Press, Chicago, 2009).Google Scholar
Selinger, P., A survey of graphical languages for monoidal categories, in New structures for physics, Lecture Notes in Physics, vol. 813 (Springer, Berlin, 2011), 289–355, arXiv:0908.3347.Google Scholar
Sikora, A. S. and Westbury, B. W., Confluence theory for graphs, Algebr. Geom. Topol. 7 (2007), 439–478, arXiv:math/0609832.Google Scholar
Stroppel, C., Parabolic category $\mathscr{O} $, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compositio Math. 145 (2009), 954–992, arXiv:math/0608234.CrossRefGoogle Scholar
Turaev, V. G., Quantum invariants of knots and 3-manifolds (De Gruyter, Berlin, 1994).CrossRefGoogle Scholar
Tymoczko, J., A simple bijection between standard $3\times n$ tableaux and irreducible webs for $s{l}_{3} $, J. Algebraic Combin. 35 (2012), 611–632, arXiv:1005.4724.Google Scholar
Westbury, B. W., Enumeration of non-positive planar trivalent graphs, J. Algebraic Combin. 25 (2007), 357–373, arXiv:math/0507112.Google Scholar
Westbury, B. W., Invariant tensors for the spin representation of $\mathrm{so} (7)$, Math. Proc. Cambridge Philos. Soc. 144 (2008), 217–240, arXiv:math/0601209.Google Scholar
Westbury, B. W., Web bases for the general linear groups, J. Algebraic Combin. 35 (2012), 93–107, arXiv:1011.6542.Google Scholar