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Torus orbit closures and 1-strip-less tableaux

Carl Lian

Abstract

We compare two formulas for the class of a generic torus orbit closure on the Grassmannian, due to Klyachko and Berget-Fink. The naturally emerging combinatorial objects are semi-standard fillings we call 1-strip-less tableaux.

Torus orbit closures and 1-strip-less tableaux

Abstract

We compare two formulas for the class of a generic torus orbit closure on the Grassmannian, due to Klyachko and Berget-Fink. The naturally emerging combinatorial objects are semi-standard fillings we call 1-strip-less tableaux.
Paper Structure (9 sections, 13 theorems, 49 equations, 3 figures)

This paper contains 9 sections, 13 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Young tableaux
  5. Schubert calculus
  6. Application of the Littlewood-Richardson rule
  7. 1-strip-less tableaux
  8. Comparison to Klyachko's formula
  9. 1-strip-less tableau and SYT with $r-1$ descents

Key Result

Theorem 1

k Write in terms of the basis of Schubert cycles $\sigma_\mu$. Then, we have where:

Figures (3)

  • Figure 1: The Mondrian tableau associated to the class $\sigma_\lambda\sigma_{\widetilde{\lambda}}$ when $r=4$.
  • Figure 2: Comparison of the Mondrian tableaux $M,M'$ when $r=4$. Here, $r-1=3$ basis elements have been added in the southwest corner.
  • Figure 3: The Mondrian tableaux $M_\circ,M_+$ when $r=4$. After one first step of the geometric Littlewood-Richardson rule is applied, to $M_\circ$, one obtains $M_+$ and $M$ (Figure \ref{['mondrian2']})

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • proof
  • Proposition 6
  • Definition 7
  • Proposition 8
  • proof
  • ...and 15 more