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

2025/04/10 by Amini, Kamyar, Huq-Kuruvilla, Irit, Mihalcea, Leonardo C. +2 · 1 citation
#05E05 #14M15 #14N35 #37K10 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2504.07412

Abstract

We prove a determinantal, Toda-type, presentation for the equivariant K theory of a partial flag variety \rm Fl(r₁, …, 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(r₁, …, 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.

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