Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups

Authors: Prakash Belkale and Shrawan Kumar
Journal: J. Algebraic Geom. 19 (2010), 199-242
Published electronically: June 16, 2009
MathSciNet review: 2580675
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Abstract | References | Additional Information

Abstract: In this paper we consider the eigenvalue problem, intersection theory of homogeneous spaces (in particular, the Horn problem) and the saturation problem for the symplectic and odd orthogonal groups. The classical embeddings of these groups in the special linear groups play an important role. We deduce properties for these classical groups from the known properties for the special linear groups.

References [Enhancements On Off] (What's this?)

  • [B$ _1$] P. Belkale, Geometric proofs of Horn and saturation conjectures, J. Algebraic Geom. 15 (2006), 133-173. MR 2177198 (2006i:14051)
  • [B$ _2$] P. Belkale, Invariant theory of $ \mathrm{GL}(n)$ and intersection theory of Grassmannians, Inter. Math. Res. Not., Vol. 2004, no. 69, 3709-3721. MR 2099498 (2005h:14117)
  • [BK] P. Belkale and S. Kumar, Eigenvalue problem and a new product in cohomology of flag varieties, Invent. Math. 166 (2006), 185-228. MR 2242637 (2007k:14097)
  • [BeSj] A. Berenstein and R. Sjamaar, Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion, J. Amer. Math. Soc. 13 (2000), 433-466. MR 1750957 (2001a:53121)
  • [BS] N. Bergeron and F. Sottile, A Pieri-type formula for isotropic flag manifolds, Trans. AMS 354 (2002), 2659-2705. MR 1895198 (2003a:14084)
  • [BGG] I. Bernstein, I. Gelfand and S. Gelfand, Schubert cells and cohomology of the spaces $ G/P$, Russian Math. Surveys 28, no. 3 (1973), 1-26. MR 0429933 (55:2941)
  • [Bo] N. Bourbaki, Groupes et Algèbres de Lie, Chapitres 4,5 et 6, Masson, Paris (1981). MR 0240238 (39:1590)
  • [BL] S. Billey and V. Lakshmibai, Singular Loci of Schubert Varieties, Progress in mathematics, Vol. 182, Birkhäuser (2000). MR 1782635 (2001j:14065)
  • [F$ _1$] W. Fulton, Young Tableaux, Cambridge University Press (1997). MR 1464693 (99f:05119)
  • [F$ _2$] W. Fulton, Intersection Theory, Second Edition, Springer (1998). MR 1644323 (99d:14003)
  • [F$ _3$] W. Fulton,
    Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients. Lin. Alg. Appl 319 (2000), 23-36. MR 1799622 (2002a:15024)
  • [F$ _4$] W. Fulton,
    Eigenvalues, invariant factors, highest weights, and Schubert calculus.
    Bull. Amer. Math. Soc. (N.S.) 37 (2000), 209-249. MR 1754641 (2001g:15023)
  • [KM] M. Kapovich and J. Millson, A path model for geodesics in Euclidean buildings and its applications to representation theory, Groups, Geometry and Dynamics 2 (2008), 405-480.
  • [K] A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. 4(1998), 419-445. MR 1654578 (2000b:14054)
  • [KT] A. Knutson and T. Tao, The honeycomb model of $ {\rm GL}\sb n(\mathbb{C})$ tensor products I: Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), 1055-1090. MR 1671451 (2000c:20066)
  • [KuLM] S. Kumar, B. Leeb and J. Millson, The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type, Contemp. Math. 332 (2003), 171-195. MR 2018339 (2004k:53077)
  • [KS] S. Kumar and J. Stembridge, Special isogenies and tensor product multiplicities, Inter. Math. Res. Not., Vol. 2007 (no. 20), 1-13 (2007). MR 2363307 (2008i:20056)
  • [MTV] E. Mukhin, V. Tarasov and A. Varchenko, Schubert calculus and representations of the general linear group, Preprint (2007).
  • [PS] K. Purbhoo and F. Sottile, The recursive nature of cominuscule Schubert calculus, Adv. Math. 217 (2008), 1962-2004. MR 2388083
  • [R] E. Richmond, A partial Horn recursion in the cohomology of flag varieties, Preprint. (To appear in J. Alg. Comb.)
  • [S] A. Schofield, General representations of quivers, Proc. London Math. Soc. 65 (1992), 46-64. MR 1162487 (93d:16014)
  • [Sj] R. Sjamaar, Convexity properties of the moment mapping re-examined, Adv. Math. 138 (1998), 46-91. MR 1645052 (2000a:53148)
  • [So] F. Sottile, General isotropic flags are general (for Grassmannian Schubert calculus), Preprint (2008). (To appear in J. Algebraic Geom.)

Additional Information

Prakash Belkale
Affiliation: Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599–3250

Shrawan Kumar
Affiliation: Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599–3250

Received by editor(s): August 3, 2007
Received by editor(s) in revised form: July 2, 2008
Published electronically: June 16, 2009
Additional Notes: Both the authors were supported by the FRG grant no DMS-0554247 from NSF

American Mathematical Society