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Singular Loci of Rationally Smooth Orbit Closures in Flag Varieties

 Author / Creator
 Budd, Valerie Gayle

Let G be a semisimple 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 Torbit closures X (of some point x) in the flag variety G/B in types A and D. In type A, we prove that a Torbit 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 Sorbit closure of x, we give a description of the Sweights of the tangent space at y of any irreducible Sstable 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 BilleyCrites 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.

 Graduation date
 Spring 2021

 Type of Item
 Thesis

 Degree
 Doctor of Philosophy

 License
 This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for noncommercial 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.