Universität zu Köln

Nguyen, An Hoa (2011). One-skeleton galleries and cell combinatorics in type A. PhD thesis, Universität zu Köln.

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Abstract

We describe the image of a cell in the Bialynicki-Birula decomposition of Gaussent and Littelmann's Bott-Samelson type variety, which is a desingularization of an affine Schubert variety, in type A for two particular cases. In the first case, we take a one-skeleton gallery, which is completely included in the dominant Weyl chamber. We can show, that the closure of the image of the cells associated to galleries, that belong to certain parts of the associated crystal, are in fact MV-cycles. In the second case we take a gallery of type N\omega_1, which is completely included in the dominant Weyl chamber. We generalize the notion of a Young tableau and use the 1:1 correspondance between tableaux and one-skeleton galleries to analyze, how the associated cells behave under the bumping algorithm. Finally, we can show that if and only if we take two words, equivalent under the Knuth relations, then the closure of the images of the cells associated to the galleries coming from these word, are the same. In addition, this closure is an MV-cycle. This allows a geometric interpretation of the combinatoric Knuth relations in the plactic monoid.