Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 098, 26 pages      arXiv:2504.07412      https://doi.org/10.3842/SIGMA.2025.098

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

Kamyar Amini a, Irit Huq-Kuruvilla b, Leonardo C. Mihalcea a, Daniel Orr a and Weihong Xu c
a) Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
b) Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Da-an, Taipei 106319, Taiwan
c) Division of Physics, Mathematics, and Astronomy, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA

Received April 16, 2025, in final form November 10, 2025; Published online November 20, 2025

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.

Key words: quantum K theory; partial flag varieties; Toda lattice.

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