Puzzles in $K$-homology of Grassmannians

TL;DR

This paper introduces new puzzle pieces of hexagonal shape that provide Littlewood-Richardson rules for the K-homology of Grassmannians, expanding the combinatorial tools for understanding their K-theoretic structure.

Contribution

It presents two novel hexagonal puzzle pieces that establish Littlewood-Richardson rules for K-homology of Grassmannians, complementing existing puzzle-based approaches.

Findings

Two new hexagonal puzzle pieces for K-homology

Eight versions of K-theoretic Littlewood-Richardson tableaux

Extension of puzzle combinatorics to K-homology

Abstract

Knutson, Tao, and Woodward formulated a Littlewood-Richardson rule for the cohomology ring of Grassmannians in terms of puzzles. Vakil and Wheeler-Zinn-Justin have found additional triangular puzzle pieces that allow one to express structure constants for $K$-theory of Grassmannians. Here we introduce two other puzzle pieces of hexagonal shape, each of which gives a Littlewood-Richardson rule for $K$-homology of Grassmannians. We also explore the corresponding eight versions of $K$-theoretic Littlewood-Richardson tableaux.