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Lattice polytopes associated to certain Demazure modules of \(sl_{n+1}\). (English) Zbl 0966.17004

There is a two-part programme to prove the existence of degenerations of Schubert varieties of \(\mathrm{SL}(n)\) into toric varieties, see for instance [N. Gonciulea, and V. Lakshmibai, Transform. Groups 1, 215–248 (1996; Zbl 0909.14028)]. Here the authors present results concerning the first of the two parts. They prove that for a certain class of elements \(w\) of the Weyl group of \(sl_{n+1}\), there exists a lattice polytope \(\Delta_i^w \subset {\mathbb{R}}^{l(w)}\), such that for every dominant weight \(\lambda = \sum_{i = 1}^na_iw_i\) where the \(w_i\) are fundamental weights the number of lattice points in the Minkowski sum \(\Delta_\lambda^w = \sum_{i=1}^na_i\Delta_i^w\) is equal to the dimension of the Demazure module \(E_w(\lambda)\) (for its definition see [M. Demazure, Bull. Sci. Math., II. Ser. 98, 163–172 (1975; Zbl 0365.17005)]. They also present an explicit formula which gives the character of \(E_w(\lambda)\).
For \(w = w_0\) the longest element of the Weyl group other polytopes satisfying the same condition have been constructed by A. D. Berenstein and A. Zelevinsky [J. Geom. Phys. 5, 453–472 (1988; Zbl 0712.17006)] and P. Littelmann [Transform. Groups 3, 145–179 (1998; Zbl 0908.17010)]. The authors of the present paper believe that their construction can be generalized to every simple algebraic group.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
14M15 Grassmannians, Schubert varieties, flag manifolds
20G05 Representation theory for linear algebraic groups
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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References:

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