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arXiv · 1209.0739

Schubert calculus of Richardson varieties stable under spherical Levi subgroups

Benjamin J. Wyser
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Abstract

We observe that the expansion in the basis of Schubert cycles for $H^*(G/B)$ of the class of a Richardson variety stable under a spherical Levi subgroup is described by a theorem of Brion. Using this observation, along with a combinatorial model of the poset of certain symmetric subgroup orbit closures, we give positive combinatorial descriptions of certain Schubert structure constants on the full flag variety in type $A$. Namely, we describe $c_{u,v}^w$ when $u$ and $v$ are inverse to Grassmannian permutations with unique descents at $p$ and $q$, respectively. We offer some conjectures for similar rules in types $B$ and $D$, associated to Richardson varieties stable under spherical Levi subgroups of $SO(2n+1,\C)$ and $SO(2n,\C)$, respectively.

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Benjamin J. Wyser. 2012-09-04. Schubert calculus of Richardson varieties stable under spherical Levi subgroups. https://doi.org/10.1007/s10801-013-0427-z

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