#### Vol. 303, No. 2, 2019

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Puzzles in $K$-homology of Grassmannians

### Pavlo Pylyavskyy and Jed Yang

Vol. 303 (2019), No. 2, 703–727
##### Abstract

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 $K\phantom{\rule{0.3em}{0ex}}$-theory of Grassmannians. Here we introduce two other puzzle pieces of hexagonal shape, each of which gives a Littlewood–Richardson rule for $K\phantom{\rule{0.3em}{0ex}}$-homology of Grassmannians. We also explore the corresponding eight versions of $K\phantom{\rule{0.3em}{0ex}}$-theoretic Littlewood–Richardson tableaux.

##### Keywords
puzzles, hexagon, tiling, K-theory, Grassmannian, Littlewood–Richardson coefficient
##### Mathematical Subject Classification 2010
Primary: 05E15
Secondary: 14M15, 52C20