Lattice Polytopes Associated to Certain Demazure Modules of sln + 1
Raika Dehy
and Rupert W.T. Yu
DOI: 10.1023/A:1018775528824
Abstract
Let w be an element of the Weyl group of sl n + 1. We prove that for a certain class of elements w (which includes the longest element w 0 of the Weyl group), there exist a lattice polytope 
R l(w) , for each fundamental weight 
i of sl n + 1, such that for any dominant weight 
= 
i = 1 n a i i , the number of lattice points in the Minkowski sum w = i = 1 n a i i w is equal to the dimension of the Demazure module E w ( ). We also define a linear map A w : R l(w) 
P 
Z R where P denotes the weight lattice, such that char E w ( ) = e e - A(x) where the sum runs through the lattice points x of w .
R l(w) , for each fundamental weight i of sl n + 1, such that for any dominant weight = i = 1 n a i i , the number of lattice points in the Minkowski sum w = i = 1 n a i i w is equal to the dimension of the Demazure module E w ( ). We also define a linear map A w : R l(w) P Z R where P denotes the weight lattice, such that char E w ( ) = e e - A(x) where the sum runs through the lattice points x of w .Pages: 149–172
Keywords: lattice polytope; Demazure module; Minkowski sum; character formula
Full Text: PDF
References
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2. M. Brion, “Points entiers dans les polytopes convexes,” Séminaire Bourbaki, Vol. 1993/94, Exp. No. 780, Astérisque 227 (1995), 145-169.
3. R. Dehy, “Résultats combinatoires sur les modules de Demazure,” preprint of the Université Louis Pasteur, Strasbourg, France, 1996.
4. R. Dehy, “Degenerations of Schubert varieties of algebraic groups of rank two,” preprint (1997).
5. M. Demazure, “Une nouvelle formule des caract`eres,” Bull. Sci. Math. 98 (1974), 163-172.
6. E. Ehrhart, “Sur les polyh`edres rationnels homothéthiques `a n dimension,” C.R. Acad. Sci. 254 (1962) 616- 618.
7. W. Fulton, Intersection Theory, Springer Verlag, Berlin-New York, 1984.
8. W. Fulton and J. Harris, Representation Theory, GTM 129 Springer Verlag, 1991.
9. I. Gelfand and A. Zelevinsky, Multiplicities and regular bases for gln, in “Group theoretical methods in Physics,” Proc. of the third seminar, Yurmala, Vol. 2, pp. 22-31, 1995.
10. N. Gonciulea and V. Lakshmibai, “Degenerations of flag and Schubert varieties to toric varieties,” Transformation Groups 1(3) (1996), 215-248.
11. J.-M. Kantor, “Triangulations of integral polytopes and Ehrhart polynomials,” preprint (1996).
12. A.N. Kirillov and A.D. Berenstein, “Groups generated by involutions, Gelfand-Tsetlin patterns, and combinatorics of Young tableaux,” St. Petersburg Math. Journal 7(1) (1996), 77-127.
13. V. Lakshmibai, “Kempf varieties,” Journal of the Indian Mathematical Society 40 (1976), 299-349.
14. V. Lakshmibai and C. Seshadri, “Geometry of G/P - V ,” Journal of Algebra 100 (1986), 462-557.
15. P. Littelmann, “A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras,” Invent. Math. 116 (1994), 329-346.
16. P. Littelmann, “Cones, Crystals and Patterns,” Transformation Groups 3(2) (1998), 145-179.
17. P. Littelmann, “A plactic algebra for semisimple Lie algebras,” Adv. in Math. 124 (1996), 312-331.
18. I.G. Macdonald, “Symmetric functions and Hall polynomials,” Oxford Mathematical Monographs, 1979.
19. B. Teissier, Variétés toriques et polytopes, Séminaire Bourbaki Exposé 565, LNM 901, Springer Verlag.