A Weighted Enumeration of Maximal Chains in the Bruhat Order
John R. Stembridge
DOI: 10.1023/A:1015068609503
Abstract
Given a finite Weyl group W with root system 
, assign the weight 
to each covering pair in the Bruhat order related by the reflection corresponding to . Extending this multiplicatively to chains, we prove that the sum of the weights of all maximal chains in the Bruhat order has an explicit product formula, and prove a similar result for a weighted sum over maximal chains in the Bruhat ordering of any parabolic quotient of W. Several variations and open problems are discussed.
, assign the weight to each covering pair in the Bruhat order related by the reflection corresponding to . Extending this multiplicatively to chains, we prove that the sum of the weights of all maximal chains in the Bruhat order has an explicit product formula, and prove a similar result for a weighted sum over maximal chains in the Bruhat ordering of any parabolic quotient of W. Several variations and open problems are discussed.Pages: 291–301
Keywords: Bruhat order; Weyl group; root system
Full Text: PDF
References
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10. J.R. Stembridge, “Combinatorial models for Weyl characters,” Adv. Math., to appear.
2. J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, Berlin, 1972.
3. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990.
4. V. Lakshmibai and C.S. Seshadri, Standard monomial theory, in Proceedings of the Hyderabad Conference on Algebraic Groups, pp. 279-322, Manoj Prakashan, Madras, 1991.
5. P. Littelmann, “A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras,” Invent. Math. 116 (1994), 329-346.
6. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1995.
7. R.A. Proctor, “Bruhat lattices, plane partition generating functions, and minuscule representations,” Europ. J. Combin. 5 (1984), 331-350.
8. R.A. Proctor, “Minuscule elements of Weyl groups, the numbers game, and d-complete posets,” J. Algebra 213 (1999), 272-303.
9. J.R. Stembridge, “Minuscule elements of Weyl groups,” J. Algebra 235 (2001), 722-743.
10. J.R. Stembridge, “Combinatorial models for Weyl characters,” Adv. Math., to appear.