Quantum K-structure constants and cylindric Grothendieck polynomials for Grassmannians
Abstract
We use the partition functions of a five-vertex exactly solvable lattice model from statistical mechanics to give new combinatorial formulae for the Schubert structure constants of torus-equivariant quantum K-theory rings of Grassmannians. The proofs build on our earlier results where we identified the transfer matrices of these lattice models geometrically as multiplications by the Hirzebruch λy classes of the tautological subbundle and the (dual) quotient bundle. The structure constants are solutions of a system of linear equations with coefficients given by the partition function. This system may be solved explicitly by specializing the Hirzebruch parameter y to an equivariant parameter, leading to explicit formulae using Cramer's rule. Furthermore, we give an alternative combinatorial construction of the partition functions in terms of sums over cylindric set-valued tableaux, which we call cylindric Grothendieck polynomials. The latter generalise McNamara's factorial skew Grothendieck and Yeliussizov's modified Grothendieck polynomials and can be thought of as an equivariant K-theoretic version of Postnikov's toric Schur function. We also provide new polynomial representatives for equivariant K-theory classes of Richardson varieties and their variants.
Related
- Vexillary Grothendieck polynomials via bumpless pipe dreams
- On the support of double Grothendieck polynomials
- Weak Bruhat interval modules of the 0-Hecke algebras for stable Grothendieck polynomials
- Bounded Ratios of Lorentzian Polynomials I: The Ternary Theory and Optimal Bounding Constants
- Closed formula for Kronecker structure constants of a molecular basis of homogeneous symmetric functions
BibTeX
@misc{gorbounov2026quantum,
title = {{Quantum K-structure constants and cylindric Grothendieck polynomials for Grassmannians}},
author = {Vassily Gorbounov and Christian Korff and Leonardo C. Mihalcea},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31607},
eprint = {2610.31607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31607},
}
RIS
TY - UNPB
TI - Quantum K-structure constants and cylindric Grothendieck polynomials for Grassmannians
AU - Vassily Gorbounov
AU - Christian Korff
AU - Leonardo C. Mihalcea
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31607
UR - https://arxiv.org/abs/2610.31607
AB - We use the partition functions of a five-vertex exactly solvable lattice model from statistical mechanics to give new combinatorial formulae for the Schubert structure constants of torus-equivariant quantum K-theory rings of Grassmannians. The proofs build on our earlier results where we identified the transfer matrices of these lattice models geometrically as multiplications by the Hirzebruch λ_y classes of the tautological subbundle and the (dual) quotient bundle. The structure constants are solutions of a system of linear equations with coefficients given by the partition function. This system may be solved explicitly by specializing the Hirzebruch parameter y to an equivariant parameter, leading to explicit formulae using Cramer's rule. Furthermore, we give an alternative combinatorial construction of the partition functions in terms of sums over cylindric set-valued tableaux, which we call cylindric Grothendieck polynomials. The latter generalise McNamara's factorial skew Grothendieck and Yeliussizov's modified Grothendieck polynomials and can be thought of as an equivariant K-theoretic version of Postnikov's toric Schur function. We also provide new polynomial representatives for equivariant K-theory classes of Richardson varieties and their variants.
ER -