Symmetric subgroup orbit closures on flag varieties: Their equivariant\n geometry, combinatorics, and connections with degeneracy loci

2012/01/20 by Benjamin J. Wyser, Wyser, Benjamin J. · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1201.4397

openalex publication_date 2012/01/20 · openalex created_date 2022/09/01 · openalex updated_date 2026/07/28

Abstract

We give explicit formulas for torus-equivariant fundamental classes of closed\nK-orbits on the flag variety G/B when G is one of the classical groups\nSL(n, C), SO(n, C), or Sp(2n, C), and K is a symmetric subgroup of G.\nWe describe parametrizations of each orbit set and the combinatorics of its\nweak order, allowing us to compute formulas for the equivariant classes of all\nremaining orbit closures using divided difference operators. In each of the\ncases in type A, we realize the K-orbit closures as universal degeneracy loci\nof a certain type, involving a vector bundle V over a scheme X equipped\nwith a flag of subbundles and a further structure determined by K. We\ndescribe how our equivariant formulas can be interpreted as formulas for such\nloci in the Chern classes of the various bundles on X.\n

Citations

• Flags, Schubert polynomials, degeneracy loci, and determinantal formulas

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• Matchings in Corona graph and classical symmetric varieties

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