Puzzles in $K$-homology of Grassmannians

Pavlo Pylyavskyy and Jed Yang

Vol. 303 (2019), No. 2, 703–727

DOI: 10.2140/pjm.2019.303.703

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 $K$-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.

Keywords

puzzles, hexagon, tiling, K-theory, Grassmannian, Littlewood–Richardson coefficient

Mathematical Subject Classification 2010

Primary: 05E15

Secondary: 14M15, 52C20

Milestones

Received: 3 April 2018

Revised: 3 March 2019

Accepted: 1 May 2019

Published: 4 January 2020