Resolution of Weyl modules: the Rota touch

  • D. A. Buchsbaum


This article is offered as homage to Gian-Carlo Rota. In it, we’ll indicate how the application of Rota’s combinatorial methods to the resolution of Weyl modules changed the way the whole question of finding resolutions was addressed, and what kinds of additional information it led to. To do this, we first have to review briefly the methods that were being used prior to Rota’s involvement. Then we look at how letter-place methods and polarizations provided information that had until his intervention not been forthcoming, such as homotopies and an explicit description of syzygies. Finally, we see how using (generalized) Capelli identities for polarizations leads to a better understanding of the known boundary maps for some resolutions, and provides a direction for finding the general boundary map. We end with a description of the terms of the general resolution of the Weyl module associated to the n-rowed almost skew-shape (see below for definition).


Divided Power Place Polarization Koszul Complex General Resolution Weyl Module 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Akin, K., Buchsbaum, D.A. (1985): Characteristic-free representation theory of the general linear group. Adv. Math. 58, 149–200MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Akin, K., Buchsbaum, D.A. (1988): Characteristic-free representation theory of the general linear group. II. Homological considerations. Adv. Math. 72, 171–210MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Akin, K., Buchsbaum, D.A. (1992): A note on the Poincaré resolution of the coordinate ring of the Grassmannian. J. Algebra 152, 427–433MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Akin, K., Buchsbaum, D.A., Weyman, J. (1982): Schur functors and Schur complexes. Adv. Math. 44, 207–278MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Anick, D., Rota, G.-C. (1991): Higher-order syzygies for the bracket algebra and for the ring of coordinates of the Grassmannian. Proc. Nat. Acad. Sci. U.S.A. 88, 8087–8090MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Buchsbaum, D.A., Rota, G.-C. (1993): Projective resolutions of Weyl modules. Proc. Nat. Acad. Sci. U.S.A. 90, 2448–2450MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Buchsbaum, D.A., Rota, G.-C. (1994): A new construction in homological algebra. Proc. Nat. Acad. Sci. U.S.A. 91, 4115–4119MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Buchsbaum, D.A., Sánchez, R. (1994): On lifting maps between Weyl modules: can bad shapes be resolved by better shapes? Adv. Math. 105, 59–75MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Donkin, S. (1986): Finite resolutions of modules for reductive algebraic groups. J. Algebra 101, 473–488MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Klucznik, M.B. (1998): Exact sequences of Schur complexes. Ph.D. thesis, Brandeis UniversityGoogle Scholar
  11. [11]
    Kulkarni, U.B. (1997): Characteristic-free representation theory of GL n(ℤ): some homological aspects. Ph.D. thesis, Brandeis UniversityGoogle Scholar
  12. [12]
    Lascoux, A. (1977): Polynoms de Schur, fonctions de Schur, et grassmanniennes. Thèse, Université de ParisGoogle Scholar

Copyright information

© Springer-Verlag Italia 2001

Authors and Affiliations

  • D. A. Buchsbaum

There are no affiliations available

Personalised recommendations