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Polytopes associated to Demazure modules of symmetrizable Kac-Moody algebras of rank two. (English) Zbl 0973.17033

Let \(\omega_1\), \(\omega_2\) be the fundamental weights of a finite or affine Kac-Moody algebra of rank two, and \(\tau\) an element of the Weyl group. The author constructs polytopes \(P_\tau(\omega_1)\), \(P_\tau(\omega_2)\subset\mathbb R^{l(\tau)}\) such that for any integral dominant weight \(\lambda=k_1\omega_1+k_2\omega_2\) there is a canonical bijection between integral points of the Minkowski sum \(k_1P_\tau(\omega_1)+k_2P_\tau(\omega_2)\) and Lakshmibai-Seshadri paths in \(\Pi_\tau(\lambda)\) [cf. P. Littelmann, Adv. Math. 124, 312-331 (1996; Zbl 0892.17009)].
As an application the author gives a combinatorial description of weight multiplicities of a Demazure module \(E_\tau(\lambda)\) by constructing a linear map \(\xi \: \mathbb R^{l(\tau)}\to\mathfrak h^*\) so that \(\text{Char} E_\tau(\lambda) = e^\lambda\sum e^{\xi(x)}\), where the sum is over all integral points \(x\) of the polytope \(k_1P_\tau(\omega_1)+k_2P_\tau(\omega_2)\). The author also shows that there exists a flat deformation of the Schubert variety \(S_\tau\) into the toric variety defined by \(P_\tau(\omega_1)\), \(P_\tau(\omega_2)\).
Reviewer: M.Primc (Zagreb)

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

Citations:

Zbl 0892.17009
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References:

[1] Andersen, H. H., Schubert varieties and Demazure’s character formula, Invent. Math., 79, 611-618 (1985) · Zbl 0591.14036
[2] Berenstein, A. D.; Kirillov, A. N., Groups generated by involution, Gelfand-Tsetlin patterns, and combinatorics of Young tableaux, Algebra i Analiz, 7, 92-152 (1995) · Zbl 0848.20007
[3] Berenstein, A. D.; Zelevinsky, A. V., Tensor product multiplicities and convex polytopes in partition space, J. Geom. Phys., 5, 453-472 (1988) · Zbl 0712.17006
[4] Dehy, R., Combinatorial results on Demazure modules, J. Algebra, 205, 505-524 (1998) · Zbl 0911.17016
[5] Dehy, R., Résultats combinatoires sur les modules de Demazure, C. R. Acad. Sci., 324, 977-980 (1997) · Zbl 0927.17012
[6] Dehy, R.; Yu, R., Polytopes associated to certain Demazure modules of \(sl(n)\), J. Algebraic Combin., 10, 149-172 (1999) · Zbl 0966.17004
[7] Eisenbud, D., Commutative Algebra with a View toward Algebraic Geometry (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0819.13001
[8] Ehrhart, E., Sur les polyhèdres rationnels homothétiques à \(n\) dimension, C. R. Acad. Sci., 254, 616-618 (1962) · Zbl 0100.27601
[9] Fulton, W.; Harris, J., Representation Theory (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0744.22001
[10] Gonciulea, N.; Lakshmibai, V., Degenerations of flag and Schubert varieties to toric varieties, Transformation Groups, 2, 215-249 (1996) · Zbl 0909.14028
[11] Gonciulea, N.; Lakshmibai, V., Gröbner bases and standard monomials, C. R. Acad. Sci., 322, 255-260 (1996) · Zbl 0866.13011
[12] Kempf, G.; Ramanathan, A., Multicones over Schubert varieties, Invent. Math., 87, 353-363 (1987) · Zbl 0615.14028
[13] Kumar, S., Demarzue character formula in arbitrary Kac-Moody setting, Invent. Math., 89, 395-423 (1987) · Zbl 0635.14023
[14] Lakshmibai, V., Standard monomial theory for \(G_2\), J. Algebra, 98, 281-318 (1986) · Zbl 0605.14041
[15] Lakshmibai, V.; Musili, C.; Seshadri, C. S., Geometry of \(G/P\)—IV. Standard monomial theory for classical types, Proc. Ind. Acad. Sci. Sect. A Math. Sci., 88, 279-362 (1979) · Zbl 0447.14013
[16] Lakshmibai, V.; Seshadri, C. S., Geometry of \(G/P—V\), J. Algebra, 100, 462-557 (1986) · Zbl 0618.14026
[17] Littelmann, P., A Littelwood-Richardson rule for Kac-Moody algebras, Invent. Math., 116, 329-346 (1994) · Zbl 0805.17019
[18] Littelmann, P., Cones, crystals, and patterns, Transform. Groups, 3, 145-179 (1998) · Zbl 0908.17010
[19] Littelmann, P., Paths and root operators in representation theory, Ann. Math., 142, 499-525 (1995) · Zbl 0858.17023
[20] Littelmann, P., A plactic algebra for semi-simple Lie algebras, Advances in Math., 124, 312-331 (1997) · Zbl 0892.17009
[21] Mehta, V. B.; Ramanathan, A., Frobenius splitting and cohomology for Schubert varieties, Ann. Math., 122, 27-40 (1984) · Zbl 0601.14043
[22] Mathieu, O., Formules de caractères pour les algèbres de Kac-Moody générales, Astérisque, 159-160 (1988) · Zbl 0683.17010
[23] Sanderson, Y., Dimensions of Demazure modules for rank two affine Lie algebras, Compositio Math, 101, 115-131 (1996) · Zbl 0873.17021
[24] Sturmfels, B., Gröbner Basis and Convex Polytopes. Gröbner Basis and Convex Polytopes, University Lecture Series, 8 (1996), American Mathematical Society: American Mathematical Society Providence · Zbl 0856.13020
[25] Teissier, B., “Variétés Toriques et Polytopes,” Séminaire Bourbaki, exposé 565. “Variétés Toriques et Polytopes,” Séminaire Bourbaki, exposé 565, Lecture Notes in Mathematics, 901 (1981), Springer-Verlag: Springer-Verlag New York, p. 71-84 · Zbl 0494.52010
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