Lattice polytopes associated to certain Demazure modules of $sl_{n+1}$

Dehy, Raika ; Yu, Rupert W.T.

Journal of Algebraic Combinatorics, Tome 10 (1999) no. 2, pp. 149-172.

Résumé

Summary: 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 $Deltasub$ R $^{ l(w) }$, for each fundamental weight $ohgr _{i}$ of sl $_{ n + 1}$, such that for any dominant weight $lambda = sum_{ i = 1} ^{ n }$ a $_{ i }ohgr_{ i }$, the number of lattice points in the Minkowski sum $Delta _{w} ^$ lambda$ = sum_{ i = 1} ^{ n } a _{ i }Delta_{ i } ^{ w }$ is equal to the dimension of the Demazure module E $_{ w }( lambda)$. We also define a linear map A $^{ w }$ : R $^{ l(w) }rarrPotimes _{Z}$ R where $P$ denotes the weight lattice, such that char E $_{ w }( lambda) = e^{ $lambda$} sum$e ^- A(x) where the sum runs through the lattice points $x$ of $Delta _{ $lambda$}^{ w }$ .

Détail

Keywords: lattice polytope, Demazure module, Minkowski sum, character formula