Knutson, Tao, and Woodward (2004) formulated a Littlewood-Richardson rule for the cohomology ring of Grassmannians in terms of puzzles. Vakil (2006) and Wheeler and Zinn-Justin (2017) 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 Ktheoretic Littlewood-Richardson tableaux.
Bibliographical noteFunding Information:
We are grateful to Allen Knutson and Joel Lewis for helpful conversations. We thank the anonymous referee for careful reading and useful suggestions. Pylyavskyy is partially supported by NSF grant DMS-1351590 and Sloan Fellowship. Yang is partially supported by NSF RTG grant NSF/DMS-1148634.
- Littlewood-richardson coefficient