Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of\n their Coordinate Rings, and Sphericity Consequences

2016/09/29 by Reuven Hodges, Hodges, Reuven, V. Lakshmibai +1
Mathematics · #14B05 #14M27 #20G05 (Primary) #20G20 (Secondary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1609.09538

openalex publication_date 2016/09/29 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let L_w be the Levi part of the stabilizer Q_w in GL_N (for left\nmultiplication) of a Schubert variety X(w) in the Grassmannian G_d,N. For\nthe natural action of L_w on \ℂ[X(w)], the homogeneous coordinate\nring of X(w) (for the Pl "ucker embedding), we give a combinatorial\ndescription of the decomposition of \ℂ[X(w)] into irreducible\nL_w-modules; in fact, our description holds more generally for the action of\nthe Levi part L of any parabolic subgroup Q that is contained in Q_w.\nThis decomposition is then used to show that all smooth Schubert varieties, all\ndeterminantal Schubert varieties, and all Schubert varieties in G_2,N are\nspherical L_w-varieties.\n

Related

• A classification of spherical Schubert varieties in the Grassmannian
• Toroidal Schubert Varieties
• Sphericality and Smoothness of Schubert Varieties
• Schubert varieties and the fusion products
• An effective decomposition theorem for Schubert varieties