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Singular Loci of Rationally Smooth Orbit Closures in Flag Varieties

  • Author / Creator
    Budd, Valerie Gayle
  • Let G be a semi-simple algebraic group over the complex numbers, B a Borel subgroup, and T a maximal torus contained in B. In the first part of this thesis, we examine the singular loci of rationally smooth T-orbit closures X (of some point x) in the flag variety G/B in types A and D. In type A, we prove that a T-orbit closure X in G/B is smooth if and only if it is rationally smooth. In the type D case, where this statement is known to be false, we investigate how the method used to prove the type A case fails. In particular, for y in X, S the stabilizer of y (assumed to be connected), and Y the S-orbit closure of x, we give a description of the S-weights of the tangent space at y of any irreducible S-stable surface in Y containing y. In the second part of this thesis, we examine the maximal singularities of affine Schubert varieties X(w) in the affine flag variety G/B in affine type A, which are equipped with the action of a particular torus T'. Let E-(X(w),u) be the set consisting of the T'-curves C in X(w) which contain a T'-fixed point u, but whose T'-fixed point set {u, v} satisfies u < v < or = w. We obtain a partial characterization of the set E-(X(w),u), where the T'-fixed point u is a maximal singularity of X(w). Furthermore, we provide a necessary condition for a T'-fixed point of a rationally smooth affine Schubert variety to be a maximal singularity. Finally, we prove that the affine permutation w corresponding to any rationally smooth, but singular, Schubert variety X(w) in affine G/B contains the pattern 3412. Using this result, we provide a proof of a conjecture by Billey-Crites that states that a Schubert variety X(w) in affine G/B is smooth if and only if it is indexed by an affine permutation w that avoids the patterns 3412 and 4231.

  • Subjects / Keywords
  • Graduation date
    Spring 2021
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/r3-a4a2-y720
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.