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An overview of horospherical varieties and coloured fans

Sean Monahan

Abstract

We provide an overview of the combinatorial theory of horospherical varieties using coloured fans, a generalization of the combinatorial theory of toric varieties using polyhedral fans.

An overview of horospherical varieties and coloured fans

Abstract

We provide an overview of the combinatorial theory of horospherical varieties using coloured fans, a generalization of the combinatorial theory of toric varieties using polyhedral fans.
Paper Structure (37 sections, 42 theorems, 67 equations)

This paper contains 37 sections, 42 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Summary of horospherical varieties
  3. Horospherical varieties as a subclass of spherical varieties
  4. Resources
  5. Fixed notation
  6. Review of algebraic groups and roots
  7. Borel subgroups
  8. Characters
  9. Roots
  10. Parabolic subgroups
  11. Bruhat decomposition
  12. Horospherical objects
  13. Horospherical subgroups
  14. Horospherical homogeneous spaces
  15. Horospherical varieties
  16. ...and 22 more sections

Key Result

Theorem 1

There is a precise natural bijective correspondence (Here, $\cong$ refers to $G$-equivariant isomorphisms).

Theorems & Definitions (162)

  • Theorem : \ref{['thm:classification of horospherical varieties']}
  • theorem 1.1: Characterizations of spherical
  • proof
  • Proposition : \ref{['prop:valuation cone']}
  • remark 2.1
  • example 2.2: Characters of $B_n\subseteq \operatorname{SL}_n$
  • example 2.3: Roots of $\operatorname{SL}_n$
  • remark 2.4: Roots of reductive $G$
  • example 2.5: Roots of torus
  • remark 2.6
  • ...and 152 more