Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Automorphisms of the Quantum Cohomology of the Springer Resolution and Applications

Changzheng Li, Changjian Su, Rui Xiong

Abstract

In this paper, we introduce quantum Demazure--Lusztig operators acting by ring automorphisms on the equivariant quantum cohomology of the Springer resolution. Our main application is a presentation of the torus-equivariant quantum cohomology in terms of generators and relations. We provide explicit descriptions for the classical types. We also recover Kim's earlier results for the complete flag varieties by taking the Toda limit.

Automorphisms of the Quantum Cohomology of the Springer Resolution and Applications

Abstract

In this paper, we introduce quantum Demazure--Lusztig operators acting by ring automorphisms on the equivariant quantum cohomology of the Springer resolution. Our main application is a presentation of the torus-equivariant quantum cohomology in terms of generators and relations. We provide explicit descriptions for the classical types. We also recover Kim's earlier results for the complete flag varieties by taking the Toda limit.
Paper Structure (18 sections, 33 theorems, 161 equations)

This paper contains 18 sections, 33 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivariant cohomology of Springer resolution
  4. Calogero--Moser system
  5. Quantum cohomology
  6. Automorhisms of $QH^*_{\mathbb{T}}(T^*\mathcal{B})$
  7. Stable basis
  8. Proof of Theorem \ref{['Swisring']}
  9. Presentation of Quantum Cohomology
  10. Construction of Relations
  11. Proof of Theorem \ref{['qHpre']}
  12. Explicit formulae for the classical Calogero--Moser map
  13. Computation for Classical Types
  14. Type $A_{n-1}$
  15. Type $B_n$ and type $C_n$
  16. ...and 3 more sections

Key Result

Theorem A

Every quantum Demazure--Lusztig operator is a ring automorphism of $QH^*_{\mathbb{T}}(T^*\mathcal{B})$.

Theorems & Definitions (65)

  • Theorem A: Theorem \ref{['Swisring']}
  • Theorem B: Theorem \ref{['qHpre']}
  • Theorem C: Theorem \ref{['Th:typeApre']}
  • Proposition 2.1: MR51508, see also AFbook
  • Proposition 2.2: MR2782198
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1: Quantum Demazure--Lusztig operators
  • Theorem 3.2
  • Example 3.3
  • ...and 55 more