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Quantum $K$-theoretic Whitney relations for type $C$ flag manifolds

Takafumi Kouno

TL;DR

This work delivers a complete Whitney-type presentation for the quantum $K$-ring $QK_T(Fl^{C_n})$ by identifying quantum $K$-theoretic Whitney relations that deform the classical $K$-theory around the tautological filtration of the type $C_n$ flag manifold. It uses semi-infinite flag manifolds and the quantum alcove model to derive explicit product formulas for line bundles, proving three key $QK$-Whitney identities and showing these generate the full set of relations. The authors then construct a Whitney-type presentation as a quotient of a polynomial ring by a carefully designed ideal, with an explicit correspondence between generators and tautological bundles, and lift this classical presentation to the quantum setting via a Nakayama-type lemma. The results provide a practical, computable framework for $QK_T(Fl^{C_n})$, enabling explicit computation of quantum products and opening avenues for analogous type-by-type extensions beyond type $A$.

Abstract

We study relations of $λ_{y}$-classes associated to tautological bundles over the flag manifold of type $C$ in the quantum $K$-ring. These relations are called the quantum $K$-theoretic Whitney relations. The strategy of the proof of the quantum $K$-theoretic Whitney relations is based on the method of semi-infinite flag manifolds and the Borel-type presentation. In addition, we observe that the quantum $K$-theoretic Whitney relations give a complete set of the defining relations of the quantum $K$-ring. This gives a presentation of the quantum $K$-ring of the flag manifold of type $C$, called the Whitney-type presentation, as a quotient of a polynomial ring, different from the Borel-type presentation.

Quantum $K$-theoretic Whitney relations for type $C$ flag manifolds

TL;DR

This work delivers a complete Whitney-type presentation for the quantum -ring by identifying quantum -theoretic Whitney relations that deform the classical -theory around the tautological filtration of the type flag manifold. It uses semi-infinite flag manifolds and the quantum alcove model to derive explicit product formulas for line bundles, proving three key -Whitney identities and showing these generate the full set of relations. The authors then construct a Whitney-type presentation as a quotient of a polynomial ring by a carefully designed ideal, with an explicit correspondence between generators and tautological bundles, and lift this classical presentation to the quantum setting via a Nakayama-type lemma. The results provide a practical, computable framework for , enabling explicit computation of quantum products and opening avenues for analogous type-by-type extensions beyond type .

Abstract

We study relations of -classes associated to tautological bundles over the flag manifold of type in the quantum -ring. These relations are called the quantum -theoretic Whitney relations. The strategy of the proof of the quantum -theoretic Whitney relations is based on the method of semi-infinite flag manifolds and the Borel-type presentation. In addition, we observe that the quantum -theoretic Whitney relations give a complete set of the defining relations of the quantum -ring. This gives a presentation of the quantum -ring of the flag manifold of type , called the Whitney-type presentation, as a quotient of a polynomial ring, different from the Borel-type presentation.
Paper Structure (36 sections, 28 theorems, 135 equations)

This paper contains 36 sections, 28 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation for root systems
  4. Flag manifolds
  5. The classical and the quantum K-ring of the flag manifold
  6. Type C case
  7. K-theoretic Whitney relations
  8. Whitney-type presentation of the quantum K-ring
  9. Quantum K-theoretic Whitney relations
  10. The Whitney-type presentation of the quantum K-ring
  11. Computation of quantum products
  12. The semi-infinite flag manifold and its K-group
  13. The quantum alcove model
  14. The Chevalley formula and line bundle correspondence
  15. The proof of quantum K-theoretic Whitney relations
  16. ...and 21 more sections

Key Result

Theorem A

Theorems & Definitions (50)

  • Theorem A: $=$ Theorem \ref{['thm:QK-Whitney_A']} $+$ Theorem \ref{['thm:QK-Whitney_B']}
  • Theorem B: $=$ Theorem \ref{['thm:Whitney_presentation']}
  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1: Kato
  • Definition 4.2: BFP
  • Definition 4.3: LP
  • Definition 4.4: LL, LNS
  • ...and 40 more