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Quantum K-theory of IG(2, 2n)

Vladimiro Benedetti, Nicolas Perrin, Weihong Xu

Abstract

We prove that the Schubert structure constants of the quantum K-theory rings of symplectic Grassmannians of lines have signs that alternate with codimension and vanish for degrees at least 3. We also give closed formulas that characterize the multiplicative structure of these rings, including the Seidel representation and a Chevalley formula.

Quantum K-theory of IG(2, 2n)

Abstract

We prove that the Schubert structure constants of the quantum K-theory rings of symplectic Grassmannians of lines have signs that alternate with codimension and vanish for degrees at least 3. We also give closed formulas that characterize the multiplicative structure of these rings, including the Seidel representation and a Chevalley formula.
Paper Structure (23 sections, 69 theorems, 95 equations)

This paper contains 23 sections, 69 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and proof strategy
  3. Schubert varieties
  4. Quantum $K$-theory and Curve neighborhoods
  5. A first reduction
  6. Proof outline
  7. Replacing stable maps by evaluations
  8. Some results on general fibers
  9. Rational curves passing through points
  10. Forgetting maps
  11. Results on curve neighborhoods
  12. Large degrees
  13. Degree 2
  14. Degree 1
  15. The case of $\operatorname{ev}_3^1(u,v)$
  16. ...and 8 more sections

Key Result

Theorem 1.2

Conjecture conj:pos-intro is true for $X = \mathop{\mathrm{IG}}\nolimits(2,2n)$.

Theorems & Definitions (133)

  • Conjecture 1.1
  • Theorem 1.2
  • Proposition 1.3: see Corollary \ref{['cor:dleq2']} and Remark \ref{['rmk:interval']}
  • Theorem 1.4: see Theorem \ref{['thm_quantum_chevalley']}
  • Theorem 1.5: see Theorem \ref{['thm:seidel-Ktheo']}
  • Theorem 1.6: see Theorem \ref{['thm:seidel-geom']}
  • Proposition 1.7: see Proposition \ref{['prop:diagram']}
  • Theorem 2.1
  • Corollary 2.2
  • Conjecture 2.3
  • ...and 123 more