Toda-Type Presentations for the Quantum K Theory of Partial Flag Varieties

TL;DR

This work extends Toda-type presentations from the quantum K-theory of complete flag varieties to partial flag varieties ${\rm Fl}(\mathbf r,n)$ by pushing forward the known ${\rm Fl}(n)$ relations along Kato’s pushforward, yielding a determinantal description ${\rm QK}_T({\rm Fl}(\mathbf r,n)) \cong R[[Q]]/J_Q$ with $R$ built from exterior powers of tautological bundles. It also links the Toda and Whitney presentations, showing that eliminating Whitney generators recovers the Toda relations, and provides a Whitney-based, quantum-parameter-free construction of polynomial representatives for all quantum K-Schubert classes using left divided difference operators. A key practical outcome is a polynomial, quantum-parameter-free description of Schubert classes that persists under pushforward to partial flags, together with an explicit formula for the Schubert point and a concrete Gr$(2,4)$ example. The appendix offers an alternative Toda-proof for ${\rm Fl}(n)$ via the J-function eigenfunction picture, tying together finite-difference Toda Hamiltonians and quantum K-relations. Overall, the results deepen our understanding of quantum K-theoretic presentations on flag varieties and supply explicit, computationally friendly polynomial representatives for Schubert classes in the partial-flag setting.

Abstract

We prove a determinantal, Toda-type, presentation for the equivariant K theory of a partial flag variety ${\rm Fl}(r_1, \dots, r_k;n)$. The proof relies on pushing forward the Toda presentation obtained by Maeno, Naito and Sagaki for the complete flag variety ${\rm Fl}(n)$, via Kato's ${\rm K}_T({\rm pt})$-algebra homomorphism from the quantum K ring of ${\rm Fl}(n)$ to that of ${\rm Fl}(r_1, \dots, r_k;n)$. Starting instead from the Whitney presentation for ${\rm Fl}(n)$, we show that the same pushforward technique gives a recursive formula for polynomial representatives of quantum K Schubert classes in any partial flag variety which do not depend on quantum parameters. In an appendix, we include another proof of the Toda presentation for the equivariant quantum K ring of ${\rm Fl}(n)$, following Anderson, Chen, and Tseng, which is based on the fact that the ${\rm K}$-theoretic $J$-function is an eigenfunction of the finite difference Toda Hamiltonians.

Theorems & Definitions (38)

• Theorem 1.1: Theorem \ref{['thm:main']}
• Theorem 1.2: Theorem \ref{['thm:w0']}
• Proposition 2.1: Kapranov
• Theorem 2.2: Kato
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Lemma 2.6
• Proposition 2.7: Maeno--Naito--Sagaki
• Corollary 3.1