T-Surfaces in the Affine Grassmannian

• Author / Creator

  Cheng, Valerie

• In this thesis we examine singularities of surfaces and affine Schubert varieties in the affine Grassmannian $\mathcal{G}/\mathcal{P}$ of type $A^{(1)}$, by considering the action of a particular torus $\widehat{T}$ on $\mathcal{G}/\mathcal{P}$. Let $\Sigma$ be an irreducible $\widehat{T}$-stable surface in $\mathcal{G}/\mathcal{P}$ and let $u$ be an attractive $\widehat{T}$-fixed point with $\widehat{T}$-stable affine neighborhood $\Sigmau$. We give a description of the $\widehat{T}$-weights of the tangent space $Tu(\Sigma)$ of $\Sigma$ at $u$, give some conditions under which $\Sigma$ is nonsingular at $u$, and provide some explicit criteria for $\Sigmau$ to be normal in terms of the weights of $Tu(\Sigma)$. We will also prove a conjecture regarding the singular locus of an affine Schubert variety in $\mathcal{G}/\mathcal{P}$.

• Subjects / Keywords

  • Torus
  • Schubert variety
  • Peterson translate
  • Affine Grassmannian
  • Attractive point
  • T-surface
  • T-orbit closure

• Graduation date

  Fall 2010

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/R3SQ8QS08

• 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.