Littlewood-Richardson coefficients for Grothendieck polynomials from integrability. (English) Zbl 1428.05323
Summary: We study the Littlewood-Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant \(K\)-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang-Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood-Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by A. Knutson and T. Tao [Duke Math. J. 119, No. 2, 221–260 (2003; Zbl 1064.14063)] and R. Vakil [Ann. Math. (2) 164, No. 2, 371–422 (2006; Zbl 1163.05337)].
MSC:
| 05E14 | Combinatorial aspects of algebraic geometry |
| 05E10 | Combinatorial aspects of representation theory |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14N15 | Classical problems, Schubert calculus |
| 57S25 | Groups acting on specific manifolds |
Keywords:
Grassmannian permutationsReferences:
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