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FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS

Year 2008, Volume: 3 Issue: 3, 1 - 42, 01.06.2008

Abstract

We use the author’s combinatorial theory of full heaps to categorify the action of a large class of Weyl groups on their root systems, and thus to give an elementary and uniform construction of a family of faithful permutation representations of Weyl groups. Examples include the standard representations of affine Weyl groups as permutations of Z and geometrical examples such as the realization of the Weyl group of type E6 as permutations of 27 lines on a cubic surface; in the latter case, we also show how to recover the incidence relations between the lines from the structure of the heap. Another class of examples involves the action of certain Weyl groups on sets of pairs (t, f), where t ∈ Z and f is a function from a suitably chosen set to the two-element set {+, −}. Each of the permutation representations corresponds to a module for a Kac-Moody algebra, and gives an explicit basis for it.

References

  • R. B´edard, Cells for two Coxeter groups, Comm. Algebra, 14 (1986), 1253–
  • A. Bj¨orner and F. Brenti, Affine permutations of type A, Electron. J. Combin. (1996), research paper 18.
  • A. Bj¨orner and F. Brenti, Combinatorics of Coxeter Groups, Springer, New York, 2005.
  • P. Cellini, P. M¨oseneder Frajria and P. Papi, The c W -orbit of ρ, Kostant’s formula for powers of the Euler product and affine Weyl groups as permutations of Z, J. Pure Appl. Algebra, 208 (2007), 1103–1119.
  • H.S.M. Coxeter, The polytope 2 whose twenty-seven vertices correspond to the lines to the general cubic surface, Amer. J. Math. 62 (1940), 457–486.
  • H. Eriksson, Computational and combinatorial aspects of Coxeter groups, Ph.D. thesis, KTH, Stockholm, Sweden, 1994.
  • R.M. Green, Full heaps and representations of affine Kac–Moody algebras, Int. Electron. J. Algebra 2 (2007), 138–188.
  • R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.
  • J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univer- sity Press, Cambridge, 1990.
  • V.G. Kac, Infinite dimensional Lie algebras (third edition), Cambridge Uni- versity Press, Cambridge, UK, 1990.
  • M. Kashiwara, On crystal bases of the q-analogue of universal enveloping al- gebras, Duke Math. J. 63 (1991), 465–516.
  • G. Lusztig, Some examples of square integrable representations of semisimple p-adic groups, Trans. Amer. Math. Soc. 277 (1983), 623–653.
  • L. Manivel, Configurations of lines and models of Lie algebras, J. Algebra 304 (2006), 457–486.
  • N. Perrin, Small resolutions of minuscule Schubert varieties, preprint; math.AG/0601117.
  • J.Y. Shi, The Kazhdan–Lusztig cells in certain affine Weyl groups, Lecture Notes in Mathematics, 1179, Springer, Berlin, 1986.
  • J.Y. Shi, The verification of a conjecture on left cells of certain Coxeter groups, Hiroshima Math. J. 24 (1994), 627–646.
  • R.P. Stanley, Enumerative Combinatorics, Volume 1, Cambridge University Press, Cambridge, UK, 1997.
  • J.R. Stembridge, Minuscule elements of Weyl groups, J. Algebra, 235 (2001), –743.
  • G.X. Viennot, Heaps of pieces, I: basic definitions and combinatorial lemmas, Combinatoire ´Enum´erative, ed. G. Labelle and P. Leroux, Springer-Verlag, Berlin, 1986, pp. 321–350.
  • N.J. Wildberger, A combinatorial construction for simply-laced Lie algebras, Adv. Appl. Math. 30 (2003), 385–396.
  • N.J. Wildberger, A combinatorial construction of G, J. Lie Theory 13 (2003), –165. R.M. Green
  • Department of Mathematics, University of Colorado Campus Box 395 Boulder, CO 80309-0395 USA
  • E-mail: rmg@euclid.colorado.edu
Year 2008, Volume: 3 Issue: 3, 1 - 42, 01.06.2008

Abstract

References

  • R. B´edard, Cells for two Coxeter groups, Comm. Algebra, 14 (1986), 1253–
  • A. Bj¨orner and F. Brenti, Affine permutations of type A, Electron. J. Combin. (1996), research paper 18.
  • A. Bj¨orner and F. Brenti, Combinatorics of Coxeter Groups, Springer, New York, 2005.
  • P. Cellini, P. M¨oseneder Frajria and P. Papi, The c W -orbit of ρ, Kostant’s formula for powers of the Euler product and affine Weyl groups as permutations of Z, J. Pure Appl. Algebra, 208 (2007), 1103–1119.
  • H.S.M. Coxeter, The polytope 2 whose twenty-seven vertices correspond to the lines to the general cubic surface, Amer. J. Math. 62 (1940), 457–486.
  • H. Eriksson, Computational and combinatorial aspects of Coxeter groups, Ph.D. thesis, KTH, Stockholm, Sweden, 1994.
  • R.M. Green, Full heaps and representations of affine Kac–Moody algebras, Int. Electron. J. Algebra 2 (2007), 138–188.
  • R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.
  • J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univer- sity Press, Cambridge, 1990.
  • V.G. Kac, Infinite dimensional Lie algebras (third edition), Cambridge Uni- versity Press, Cambridge, UK, 1990.
  • M. Kashiwara, On crystal bases of the q-analogue of universal enveloping al- gebras, Duke Math. J. 63 (1991), 465–516.
  • G. Lusztig, Some examples of square integrable representations of semisimple p-adic groups, Trans. Amer. Math. Soc. 277 (1983), 623–653.
  • L. Manivel, Configurations of lines and models of Lie algebras, J. Algebra 304 (2006), 457–486.
  • N. Perrin, Small resolutions of minuscule Schubert varieties, preprint; math.AG/0601117.
  • J.Y. Shi, The Kazhdan–Lusztig cells in certain affine Weyl groups, Lecture Notes in Mathematics, 1179, Springer, Berlin, 1986.
  • J.Y. Shi, The verification of a conjecture on left cells of certain Coxeter groups, Hiroshima Math. J. 24 (1994), 627–646.
  • R.P. Stanley, Enumerative Combinatorics, Volume 1, Cambridge University Press, Cambridge, UK, 1997.
  • J.R. Stembridge, Minuscule elements of Weyl groups, J. Algebra, 235 (2001), –743.
  • G.X. Viennot, Heaps of pieces, I: basic definitions and combinatorial lemmas, Combinatoire ´Enum´erative, ed. G. Labelle and P. Leroux, Springer-Verlag, Berlin, 1986, pp. 321–350.
  • N.J. Wildberger, A combinatorial construction for simply-laced Lie algebras, Adv. Appl. Math. 30 (2003), 385–396.
  • N.J. Wildberger, A combinatorial construction of G, J. Lie Theory 13 (2003), –165. R.M. Green
  • Department of Mathematics, University of Colorado Campus Box 395 Boulder, CO 80309-0395 USA
  • E-mail: rmg@euclid.colorado.edu
There are 23 citations in total.

Details

Other ID JA78VR77DH
Journal Section Articles
Authors

R. M. Green This is me

Publication Date June 1, 2008
Published in Issue Year 2008 Volume: 3 Issue: 3

Cite

APA Green, R. M. (2008). FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS. International Electronic Journal of Algebra, 3(3), 1-42.
AMA Green RM. FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS. IEJA. June 2008;3(3):1-42.
Chicago Green, R. M. “FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS”. International Electronic Journal of Algebra 3, no. 3 (June 2008): 1-42.
EndNote Green RM (June 1, 2008) FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS. International Electronic Journal of Algebra 3 3 1–42.
IEEE R. M. Green, “FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS”, IEJA, vol. 3, no. 3, pp. 1–42, 2008.
ISNAD Green, R. M. “FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS”. International Electronic Journal of Algebra 3/3 (June 2008), 1-42.
JAMA Green RM. FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS. IEJA. 2008;3:1–42.
MLA Green, R. M. “FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS”. International Electronic Journal of Algebra, vol. 3, no. 3, 2008, pp. 1-42.
Vancouver Green RM. FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS. IEJA. 2008;3(3):1-42.