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A Bruhat Atlas on the Wonderful Compactification of PSO(2n)/SO(2n-1) and A Kazhdan-Lusztig Atlas on G/P

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Abstract

A stratified variety admits a Bruhat (resp., Kazhdan-Lusztig) atlas if it can be covered by open charts isomorphic to opposite Bruhat cells (resp., Kazhdan-Lusztig varieties) in some Kac-Moody flag manifold via stratified isomorphisms. In the first part of this thesis, we construct an anticanonical stratification on the wonderful compactification of the symmetric space PSO(2n)/SO(2n−1) and show that the open charts are stratified-isomorphic to certain (opposite) Bruhat cells in the type Dn+1 flag manifold. In the second part of this thesis, we show that the partial flag manifold G/P with the projected Richardson stratification has a Kazhdan-Lusztig atlas, with each chart stratified-isomorphic to a Kazhdan-Lusztig variety in the affine flag manifold of the loop group G^ of G. Furthermore, we give an affine analogue of Fulton's matrix Schubert varieties in the appendix and use it as a computational tool to illustrate an example of a Kazhdan-Lusztig atlas on a partial flag variety in type A.

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79 pages

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2020-08

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Knutson, Allen

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Stillman, Michael
Kozen, Dexter

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Mathematics

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Ph. D., Mathematics

Degree Level

Doctor of Philosophy

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Government Document

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dissertation or thesis

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