A contratableau model for K-theoretic Littlewood-Richardson rule
Siddheswar Kundu
TL;DR
The paper provides an explicit combinatorial formula for the K-theoretic Littlewood-Richardson coefficients $C^{\nu}_{\lambda,\mu}$ in terms of set-valued contratableaux. It builds on Buch's framework for stable Grothendieck polynomials and Carré's contratableau model by introducing and counting $\mu$-dominant set-valued contratableaux of shape $\lambda$ with weight $\nu-\mu$, establishing a constructive bijection with $\lambda$-dominant set-valued tableaux of shape $\mu$ and weight $\nu-\mu$. The proof leverages GT-patterns and marked GT-patterns to encode the combinatorics, avoiding hive models while connecting tableau and pattern viewpoints. This yields a direct counting interpretation for $C^{\nu}_{\lambda,\mu}$ in the K-theoretic setting, with potential applications to Grassmannian K-theory and related symmetric-function theory.
Abstract
The K-theoretic Littlewood-Richardson rule, established by A. Buch, is a combinatorial method for counting the structure constants involved in the product of two Grothendieck polynomials of Grassmannian type. In this paper, we provide an explicit combinatorial formula in terms of set-valued contratableau for the K-theoretic Littlewood-Richardson rule generalizing contratableau model for the classical Littlewood-Richardson rule given by Carré.