Advertisement

Diagrams of Hermitian type, highest weight modules, and syzygies of determinantal varieties

  • Thomas J. EnrightEmail author
  • Markus Hunziker
  • W. Andrew Pruett
Chapter
Part of the Progress in Mathematics book series (PM, volume 257)

Abstract

In this mostly expository paper, a natural generalization of Young diagrams for Hermitian symmetric spaces is used to give a concrete and uniform approach to a wide variety of interconnected topics including posets of noncompact roots, canonical reduced expressions, rational smoothness of Schubert varieties, parabolic Kazhdan–Lusztig polynomials, equivalences of categories of highest weight modules, BGG resolutions of unitary highest weight modules, and finally, syzygies and Hilbert series of determinantal varieties.

Keywords

Hermitian symmetric spaces Kazhdan–Lusztig polynomials Category \(\mathcal{O}\) Determinantal varieties Syzygies 

Mathematics Subject Classification:

22E47 17B10 13D02 

Notes

Acknowledgements

This paper grew out of a series of lectures that the second author gave at the University of Georgia in Athens during the VIGRE 2010 Summer School on Geometry and Representation Theory. The second author would like to thank the organizers for their hospitality as well as the participants for their contribution to the wonderful atmosphere of the school. Special thanks go to B. Boe, S. Evens, W. Graham, S. Kumar, D. Nakano, P. Trapa, R. Varley, and R. Zierau. We also thank J. Alexander for his careful proofreading.

References

  1. 1.
    Kaan Akin, David A. Buchsbaum, and Jerzy Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), no. 3, 207–278.Google Scholar
  2. 2.
    Jordan Alexander, Markus Hunziker, and Jeb F. Willenbring, Hilbert series of determinantal varieties and strongly orthogonal roots, in preparation.Google Scholar
  3. 3.
    Erik Backelin, Koszul duality for parabolic and singular category \(\mathcal{O}\), Represent. Theory 3 (1999), 139–152 (electronic).Google Scholar
  4. 4.
    Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527.Google Scholar
  5. 5.
    I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Differential operators on the base affine space and a study of \(\mathfrak{g}\) -modules, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 21–64.Google Scholar
  6. 6.
    Brian D. Boe, Kazhdan–Lusztig polynomials for Hermitian symmetric spaces, Trans. Amer. Math. Soc. 309 (1988), no. 1, 279–294.Google Scholar
  7. 7.
    Brian D. Boe and Markus Hunziker, Kostant modules in blocks of category \(\mathcal{O}_{S}\), Comm. Algebra 37 (2009), no. 1, 323–356.Google Scholar
  8. 8.
    Brian D. Boe and Daniel K. Nakano, Representation type of the blocks of category \(\mathcal{O}_{S}\), Adv. Math. 196 (2005), no. 1, 193–256.Google Scholar
  9. 9.
    N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.Google Scholar
  10. 10.
    Michel Brion and Patrick Polo, Generic singularities of certain Schubert varieties, Math. Z. 231 (1999), no. 2, 301–324.Google Scholar
  11. 11.
    James B. Carrell, The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 53–61.Google Scholar
  12. 12.
    James B. Carrell and Jochen Kuttler, Smooth points of T-stable varieties in G∕B and the Peterson map, Invent. Math. 151 (2003), no. 2, 353–379.Google Scholar
  13. 13.
    Luis G. Casian and David H. Collingwood, The Kazhdan–Lusztig conjecture for generalized Verma modules, Math. Z. 195 (1987), no. 4, 581–600.Google Scholar
  14. 14.
    Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), no. 2, 483–506.Google Scholar
  15. 15.
    Thomas J. Enright, Analogues of Kostant’s \(\mathfrak{u}\) -cohomology formulas for unitary highest weight modules, J. Reine Angew. Math. 392 (1988), 27–36.Google Scholar
  16. 16.
    Thomas J. Enright and Markus Hunziker, Resolutions and Hilbert series of determinantal varieties and unitary highest weight modules, J. Algebra 273 (2004), no. 2, 608–639.Google Scholar
  17. 17.
    I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Resolutions and Hilbert series of the unitary highest weight modules of the exceptional groups, Represent. Theory 8 (2004), 15–51 (electronic).Google Scholar
  18. 18.
    Thomas J. Enright, Markus Hunziker, and Nolan R. Wallach, A Pieri rule for Hermitian symmetric pairs. I, Pacific J. Math. 214 (2004), no. 1, 23–30.Google Scholar
  19. 19.
    Thomas J. Enright and Brad Shelton, Categories of highest weight modules: applications to classical Hermitian symmetric pairs, Mem. Amer. Math. Soc. 67 (1987), no. 367, iv+94.Google Scholar
  20. 20.
    I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Highest weight modules for Hermitian symmetric pairs of exceptional type, Proc. Amer. Math. Soc. 106 (1989), no. 3, 807–819.Google Scholar
  21. 21.
    Thomas J. Enright and Jeb F. Willenbring, Hilbert series, Howe duality, and branching rules, Proc. Natl. Acad. Sci. USA 100 (2003), no. 2, 434–437 (electronic).Google Scholar
  22. 22.
    I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Hilbert series, Howe duality and branching for classical groups, Ann. of Math. (2) 159 (2004), no. 1, 337–375.Google Scholar
  23. 23.
    Roe Goodman and Nolan R. Wallach, Symmetry, Representations, and Invariants, Graduate Texts in Mathematics, Vol. 255, Springer, New York, 2009.Google Scholar
  24. 24.
    Harish-Chandra, On a lemma of F. Bruhat, J. Math. Pures Appl. (9) 35 (1956), 203–210.Google Scholar
  25. 25.
    M. Hochster and John A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020–1058.Google Scholar
  26. 26.
    Jaehyun Hong, Rigidity of smooth Schubert varieties in Hermitian symmetric spaces, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2361–2381 (electronic).Google Scholar
  27. 27.
    Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570.Google Scholar
  28. 28.
    Ronald S. Irving, Singular blocks of the category \(\mathcal{O}\), Math. Z. 204 (1990), no. 2, 209–224.Google Scholar
  29. 29.
    Hans Plesner Jakobsen, Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52 (1983), no. 3, 385–412.Google Scholar
  30. 30.
    Anthony Joseph, Annihilators and associated varieties of unitary highest weight modules, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 1, 1–45.Google Scholar
  31. 31.
    T. Józefiak, P. Pragacz, and J. Weyman, Resolutions of determinantal varieties and tensor complexes associated with symmetric and antisymmetric matrices, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980), Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 109–189.Google Scholar
  32. 32.
    Masaki Kashiwara and Michèle Vergne, On the Segal–Shale–Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1–47.Google Scholar
  33. 33.
    David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184.Google Scholar
  34. 34.
    Bertram Kostant, Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. (2) 74 (1961), 329–387.Google Scholar
  35. 35.
    I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Lie algebra cohomology and generalized Schubert cells, Ann. of Math. (2) 77 (1963), 72–144.Google Scholar
  36. 36.
    Thomas Lam and Lauren Williams, Total positivity for cominuscule Grassmannians, New York J. Math. 14 (2008), 53–99.Google Scholar
  37. 37.
    Alain Lascoux, Syzygies des variétés déterminantales, Adv. in Math. 30 (1978), no. 3, 202–237.Google Scholar
  38. 38.
    Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Kazhdan & Lusztig pour les grassmanniennes, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980), Astérisque, Vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266.Google Scholar
  39. 39.
    J. Lepowsky, A generalization of the Bernstein–Gelfand–Gelfand resolution, J. Algebra 49 (1977), no. 2, 496–511.Google Scholar
  40. 40.
    Kenyon J. Platt, Nonzero infinitesimal blocks of category \(\mathcal{O}_{S}\), Algebr. Represent. Theory 14 (2011), no. 4, 665–689.Google Scholar
  41. 41.
    Robert A. Proctor, Interactions between combinatorics, Lie theory and algebraic geometry via the Bruhat orders, Ph.D. thesis, MIT, 1981.Google Scholar
  42. 42.
    W. Andrew Pruett, Diagrams and reduced decompositions for cominuscule flag varieties and affine Grassmannians, Ph.D. thesis, Baylor University, 2010.Google Scholar
  43. 43.
    Wilfried Schmid, Vanishing theorems for Lie algebra cohomology and the cohomology of discrete subgroups of semisimple Lie groups, Adv. in Math. 41 (1981), no. 1, 78–113.Google Scholar
  44. 44.
    H. Schubert, Beiträge zur abzählenden Geometrie, Math. Ann. 10 (1876), no. 1, 1–116.Google Scholar
  45. 45.
    W. Soergel, \(\mathfrak{n}\) -cohomology of simple highest weight modules on walls and purity, Invent. Math. 98 (1989), no. 3, 565–580.Google Scholar
  46. 46.
    John R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5 (1996), no. 4, 353–385.Google Scholar
  47. 47.
    Hugh Thomas and Alexander Yong, A combinatorial rule for (co)minuscule Schubert calculus, Adv. Math. 222 (2009), no. 2, 596–620.Google Scholar
  48. 48.
    Hermann Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939.Google Scholar
  49. 49.
    Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, Vol. 149, Cambridge University Press, Cambridge, 2003.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Thomas J. Enright
    • 1
    Email author
  • Markus Hunziker
    • 2
  • W. Andrew Pruett
    • 3
  1. 1.Department of MathematicsUniversity of California San DiegoLa JollaUSA
  2. 2.Department of MathematicsBaylor UniversityWacoUSA
  3. 3.The University of Mississippi Medical CenterJacksonUSA

Personalised recommendations