# Orbites d'un sous-groupe de Borel dans le produit de deux grassmanniennes

Abstract : Let $X$ be the direct product of two grassmannians of $k$- and $l$-planes in a finite-dimensional vector space $V$. We study the orbits of a Borel subgroup $B \subset {\rm GL}(V)$ acting diagonally on $X$, as well as their Zariski closures, in analogy with Schubert cells and Schubert varieties in grassmannians. One easily shows that the number of these orbits is finite. Their combinatorial description was obtained by P. Magyar, J. Weyman, and A. Zelevinsky. We obtain a criterion to check whether an orbit lies in the closure of another one. We also construct a resolution of singularities for the closures of these orbits, which is analogous to the Bott-Samelson desingularizations of Schubert varieties.
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https://tel.archives-ouvertes.fr/tel-00263544
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Submitted on : Wednesday, March 12, 2008 - 1:57:51 PM
Last modification on : Wednesday, November 4, 2020 - 2:00:53 PM
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• HAL Id : tel-00263544, version 1

### Citation

Evgeny Smirnov. Orbites d'un sous-groupe de Borel dans le produit de deux grassmanniennes. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2007. Français. ⟨tel-00263544⟩

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