Duke Mathematical Journal

A polytope calculus for semisimple groups

Jared E. Anderson

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Abstract

We define a collection of polytopes associated to a semisimple group $\mathsf {G}$. Weight multiplicities and tensor product multiplicities may be computed as the number of such polytopes fitting in a certain region. The polytopes are defined as moment map images of algebraic cycles discovered by I. Mirković and K. Vilonen. These cycles are a canonical basis for the intersection homology of (the closures of the strata of) the loop Grassmannian.

Article information

Source
Duke Math. J. Volume 116, Number 3 (2003), 567-588.

Dates
First available in Project Euclid: 26 May 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1085598302

Digital Object Identifier
doi:10.1215/S0012-7094-03-11636-1

Mathematical Reviews number (MathSciNet)
MR1958098

Zentralblatt MATH identifier
01941441

Subjects
Primary: 20G05: Representation theory
Secondary: 14L99: None of the above, but in this section

Citation

Anderson, Jared E. A polytope calculus for semisimple groups. Duke Math. J. 116 (2003), no. 3, 567--588. doi:10.1215/S0012-7094-03-11636-1. http://projecteuclid.org/euclid.dmj/1085598302.


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References

  • J. Anderson, On Mirković and Vilonen's Intersection Homology Cycles for the Loop Grassmannian, Ph.D. Thesis, Princeton University, Princeton, 2000.
  • J. Anderson and I. Mirković, Crystal Graphs via Polytopes, in preparation.
  • M. Atiyah and A. Pressley, ``Convexity and loop groups'' in Arithmetic and Geometry, II, Progr. Math. 36, Birkhäuser, Boston, 1983, 33--63.
  • A. A. Beilinson, J. Bernstein, and P. Deligne, ``Faisceaux pervers'' in Analyse et Topologie sur les Espaces Singuliers (Luminy, 1981), I, Astérisque 100, Soc. Math. France, Montrouge, 1982, 5--171.
  • A. Beilinson and V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves, preprint, 2000, http://zaphod.uchicago.edu/~benzvi
  • A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), 77--128.
  • A. Braverman and D. Gaitsgory, Crystals via the affine Grassmannian, Duke Math. J. 107 (2001), 561--575.
  • M. Brion, ``Sur l'image de l'application moment'' in Seminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1986), Lecture Notes in Math. 1296, Springer, Berlin, 1987, 177--192.
  • J. B. Carrell, The Bruhat Graph of a Coxeter Group, a Conjecture of Deodhar, and Rational Smoothness of Schubert Varieties, Proc. Sympos. Pure Math. 56, Part 1, Amer. Math. Soc., Providence, 1994, 53--61.
  • P. Deligne and J. Milne, ``Tannakian categories'' in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. 900, Philos. Stud. Ser. Philos. 20, Springer, Berlin, 1982, 101--228.
  • B. Feigin, M. Finkelberg, A. Kuznetsov, and I. Mirković, Semiinfinite flags, II: Local and global intersection cohomology of quasimaps' spaces, preprint.
  • S. Fomin and A. Zelevinsky, Cluster algebras, I: Foundations, J. Amer. Math. Soc. 15 (2002), 497--529. \CMP1 887 642
  • V. Ginzburg, Perverse sheaves on a loop group and Langlands' duality, preprint.
  • M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), 135--162.
  • --. --. --. --., Intersection homology, II, Invent. Math. 72 (1983), 77--129.
  • --. --. --. --., ``On the topology of algebraic torus actions'' in Algebraic Groups (Utrecht, 1986), Lecture Notes in Math. 1271, Springer, Berlin, 1987, 73--90.
  • P. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), 329--346.
  • G. Lusztig, ``Singularities, character formulas, and a $q$-analog of weight multiplicities'' in Analysis and Topology on Singular Spaces (Luminy, 1981), II, III, Astérisque 101 --.102, Soc Math. France, Montrouge, 1983, 208--229.
  • R. MacPherson, Intersection homology and perverse sheaves, unpublished lecture notes, 1991.
  • I. Mirković and K. Vilonen, Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett. 7 (2000), 13--24.
  • A. Pressley and G. Segal, Loop Groups, Oxford Math. Monogr., Oxford Univ. Press, New York, 1986.