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Geometric invariant theory and the generalized eigenvalue problem

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Abstract

Let G be a connected reductive subgroup of a complex connected reductive group \(\hat{G}\). Fix maximal tori and Borel subgroups of G and \({\hat{G}}\). Consider the cone \(\mathcal{LR}(G,{\hat{G}})\) generated by the pairs \((\nu,{\hat{\nu}})\) of dominant characters such that \(V_{\nu}^{*}\) is a submodule of \(V_{{\hat{\nu}}}\) (with usual notation). Here we give a minimal set of inequalities describing \(\mathcal{LR}(G,{\hat{G}})\) as a part of the dominant chamber. In other words, we describe the facets of \(\mathcal{LR}(G,{\hat{G}})\) which intersect the interior of the dominant chamber. We also describe smaller faces. Finally, we are interested in some classical redundant inequalities.

Along the way, we obtain results about the faces of the Dolgachev-Hu G-ample cone and variations of this cone.

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References

  1. Belkale, P.: Local systems on \(\Bbb{P}\sp 1-S\) for S a finite set. Compos. Math. 129(1), 67–86 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Belkale, P.: Geometric proof of a conjecture of Fulton. Adv. Math. 216(1), 346–357 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Belkale, P., Kumar, S.: Eigenvalue problem and a new product in cohomology of flag varieties. Invent. Math. 166(1), 185–228 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berenstein, A., Sjamaar, R.: Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion. J. Am. Math. Soc. 13(2), 433–466 (2000) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  5. Białynichi-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. 98, 480–497 (1973)

    Article  Google Scholar 

  6. Brion, M.: On the general faces of the moment polytope. Int. Math. Res. Not. 4, 185–201 (1999)

    Article  Google Scholar 

  7. Buch, A.: Quantum calculator—a software maple package. Available at www.math.rutgers.edu/~asbuch/qcalc

  8. Derksen, H., Weyman, J.: The combinatorics of quiver representations. arXiv:math/0608288

  9. Dolgachev, I.V., Hu, Y.: Variation of geometric invariant theory quotients. Inst. Hautes Études Sci. Publ. Math. 87, 5–56 (1998), with an appendix by Nicolas Ressayre

    Article  MATH  MathSciNet  Google Scholar 

  10. Élashvili, A.G.: Invariant algebras. In: Lie Groups, Their Discrete Subgroups, and Invariant Theory. Adv. Soviet Math., vol. 8, pp. 57–64. Amer. Math. Soc., Providence (1992)

    Google Scholar 

  11. Fulton, W.: Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Am. Math. Soc. (N.S.) 37(3), 209–249 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schéma, troisième partie. Inst. Hautes Études Sci. Publ. Math. 28, 5–255 (1966)

    Article  Google Scholar 

  13. Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math. 67(3), 491–513 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  14. Heckman, G.J.: Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups. Invent. Math. 67(2), 333–356 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hemmecke, R.: 4ti2—a software package for algebraic, geometric and combinatorial problems on linear spaces. Available at www.4ti2.de

  16. Hesselink, W.H.: Desingularizations of varieties of nullforms. Invent. Math. 55(2), 141–163 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  17. Horn, A.: Eigenvalues of sums of Hermitian matrices. Pac. J. Math. 12, 225–241 (1962)

    MATH  MathSciNet  Google Scholar 

  18. Kapovich, M., Kumar, S., Millson, J.J.: The eigencone and saturation for Spin(8). Pure Appl. Math. Q. 5(2), 755–780 (2009)

    MATH  MathSciNet  Google Scholar 

  19. Kempf, G.: Instability in invariant theory. Ann. Math. 108, 2607–2617 (1978)

    Article  MathSciNet  Google Scholar 

  20. Kempf, G., Ness, L.: The length of vectors in representation spaces. In: Algebraic Geometry, Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978. Lecture Notes in Math., vol. 732, pp. 233–243. Springer, Berlin (1979)

    Chapter  Google Scholar 

  21. Kirwan, F.C.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Mathematical Notes, vol. 31. Princeton University Press, Princeton (1984)

    MATH  Google Scholar 

  22. Kleiman, S.L.: The transversality of a general translate. Compos. Math. 28, 287–297 (1974)

    MATH  MathSciNet  Google Scholar 

  23. Klyachko, A.A.: Stable bundles, representation theory and Hermitian operators. Sel. Math. (N.S.) 4(3), 419–445 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. Knop, F., Kraft, H., Vust, T.: The Picard group of a G-variety. In: Algebraische Transformationsgruppen und Invariantentheorie. DMV Sem., vol. 13, pp. 77–87. Birkhäuser, Basel (1989)

    Google Scholar 

  25. Knutson, A., Tao, T.: The honeycomb model of GL n (C) tensor products, I: proof of the saturation conjecture. J. Am. Math. Soc. 12(4), 1055–1090 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Knutson, A., Tao, T., Woodward, C.: The honeycomb model of GL n (ℂ) tensor products, II: puzzles determine facets of the Littlewood-Richardson cone. J. Am. Math. Soc. 17(1), 19–48 (2004) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  27. Konarski, J.: A pathological example of an action of k *. In: Group Actions and Vector Fields, Vancouver, BC, 1981, Lecture Notes in Math., vol. 956, pp. 72–78. Springer, Berlin (1982)

    Chapter  Google Scholar 

  28. Luna, D.: Adhérences d’orbite et invariants. Invent. Math. 29(3), 231–238 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  29. Luna, D., Richardson, R.W.: A generalization of the Chevalley restriction theorem. Duke Math. J. 46(3), 487–496 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  30. Luna, D.: Slices étales. In: Sur les Groupes Algébriques. Bull. Soc. Math. France, Paris, Mémoire, vol. 33, pp. 81–105. Soc. Math. France, Paris (1973)

    Google Scholar 

  31. Manivel, L.: Applications de Gauss et pléthysme. Ann. Inst. Fourier (Grenoble) 47(3), 715–773 (1997)

    MATH  MathSciNet  Google Scholar 

  32. Montagard, P.-L., Ressayre, N.: Sur des faces du cône de Littlewood-Richardson généralisé. Bull. Soc. Math. Fr. 135(3), 343–365 (2007)

    MATH  MathSciNet  Google Scholar 

  33. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3d edn. Springer, New York (1994)

    Google Scholar 

  34. Ness, L.: Mumford’s numerical function and stable projective hypersurfaces. In: Algebraic Geometry, Copenhagen. Lecture Notes in Math. Springer, Berlin (1978)

    Google Scholar 

  35. Ness, L.: A stratification of the null cone via the moment map. Am. J. Math. 106, 1281–1325 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  36. Oda, T.: An introduction to the theory of toric varieties In: Convex Bodies and Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15. Springer, Berlin (1988), translated from the Japanese

    Google Scholar 

  37. Popov, V.L., Vinberg, È.B.: Invariant theory. In: Algebraic Geometry IV. Encyclopedia of Mathematical Sciences, vol. 55, pp. 123–284. Springer, Berlin (1991)

    Google Scholar 

  38. Ramanan, S., Ramanathan, A.: Some remarks on the instability flag. Tohoku Math. J. (2) 36(2), 269–291 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  39. Ressayre, N.: The GIT-equivalence for G-line bundles. Geom. Dedic. 81(1–3), 295–324 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  40. Richardson, R.W.: On orbits of algebraic groups and Lie groups. Bull. Aust. Math. Soc. 25(1), 1–28 (1982)

    Article  MATH  Google Scholar 

  41. Schmitt, A.: A simple proof for the finiteness of GIT-quotients. Proc. Am. Math. Soc. 131(2), 359–362 (2003) (electronic)

    Article  MATH  Google Scholar 

  42. Vakil, R.: Schubert induction. Ann. Math. (2) 164(2), 489–512 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  43. Vergne, M.: Convex polytopes and quantization of symplectic manifolds. Proc. Natl. Acad. Sci. USA 93(25), 14238–14242 (1996); National Academy of Sciences Colloquium on Symmetries Throughout the Sciences (Irvine, CA, 1996)

    Article  MATH  MathSciNet  Google Scholar 

  44. Weyl, H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71(4), 441–479 (1912)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Université Montpellier II, CC 51-Place Eugène Bataillon, 34095, Montpellier Cedex 5, France

    N. Ressayre

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The author was partially supported by the French National Research Agency (ANR-09-JCJC-0102-01).

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Ressayre, N. Geometric invariant theory and the generalized eigenvalue problem. Invent. math. 180, 389–441 (2010). https://doi.org/10.1007/s00222-010-0233-3

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