Geodesic
Smooth and palindromic Schubert varieties in affine Grassmannians
Journal of Algebraic Combinatorics, Tome 31 (2010) no. 2, pp. 169-216.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We completely determine the smooth and palindromic Schubert varieties in affine Grassmannians, in all Lie types. We show that an affine Schubert variety is smooth if and only if it is a closed parabolic orbit. In particular, there are only finitely many smooth affine Schubert varieties in a given Lie type. An affine Schubert variety is palindromic if and only if it is a closed parabolic orbit, a chain, one of an infinite family of "spiral" varieties in type $A$, or a certain 9-dimensional singular variety in type $B _{3}$. In particular, except in type $A$ there are only finitely many palindromic affine Schubert varieties in a fixed Lie type. Moreover, in types $D$ and $E$ an affine Schubert variety is smooth if and only if it is palindromic; in all other types there are singular palindromics. The proofs are for the most part combinatorial. The main tool is a variant of Mozes' numbers game, which we use to analyze the Bruhat order on the coroot lattice. In the proof of the smoothness theorem we also use Chevalley's cup product formula.
Keywords: keywords affine grassmannians, Schubert varieties 
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     author = {Billey, Sara C. and Mitchell, Stephen A.},
     title = {Smooth and palindromic {Schubert} varieties in affine {Grassmannians}},
     journal = {Journal of Algebraic Combinatorics},
     pages = {169--216},
     publisher = {mathdoc},
     volume = {31},
     number = {2},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2010__31_2_a4/}
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Billey, Sara C.; Mitchell, Stephen A. Smooth and palindromic Schubert varieties in affine Grassmannians. Journal of Algebraic Combinatorics, Tome 31 (2010) no. 2, pp. 169-216. http://geodesic.mathdoc.fr/item/JAC_2010__31_2_a4/