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

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


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.


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

Mathematics Subject Classification:

22E47 17B10 13D02 



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.


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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

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