Geodesic
Simple ${SL}(n)$-modules with normal closures of maximal torus orbits
Journal of Algebraic Combinatorics, Tome 30 (2009) no. 4, pp. 515-538.

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Summary: Let $T$ be the subgroup of diagonal matrices in the group $SL( n)$. The aim of this paper is to find all finite-dimensional simple rational $SL( n)$-modules $V$ with the following property: for each point $v\in V$ the closure [ `$( Tv)$] overlineTv of its $T$-orbit is a normal affine variety. Moreover, for any $SL( n)$-module without this property a $T$-orbit with non-normal closure is constructed. The proof is purely combinatorial: it deals with the set of weights of simple $SL( n)$-modules. The saturation property is checked for each subset in the set of weights.
Keywords: keywords toric variety, normality, saturation 
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     author = {Kuyumzhiyan, Karine},
     title = {Simple ${SL}(n)$-modules with normal closures of maximal torus orbits},
     journal = {Journal of Algebraic Combinatorics},
     pages = {515--538},
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     number = {4},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2009__30_4_a3/}
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Kuyumzhiyan, Karine. Simple ${SL}(n)$-modules with normal closures of maximal torus orbits. Journal of Algebraic Combinatorics, Tome 30 (2009) no. 4, pp. 515-538. http://geodesic.mathdoc.fr/item/JAC_2009__30_4_a3/