GL(p) x GL(q)-orbit closures on the flag variety and Schubert structure constants for (p,q)-pairs

2011/09/02 by Wyser, Benjamin J. · 1 citation
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1109.0336

Abstract

We give positive combinatorial descriptions of Schubert structure constants c_u,v^w for the full flag variety in type A_n-1 when u and v form what we refer to as a "(p,q)-pair" (p+q=n). The key observation is that a certain subset of the GL(p,ℂ) × GL(q,ℂ)-orbit closures on the flag variety (those satisfying an easily stated pattern avoidance condition) are Richardson varieties. The result on structure constants follows when one combines this observation with a theorem of Brion concerning intersection numbers of spherical subgroup orbit closures and Schubert varieties.

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