Crystal structures for symmetric Grothendieck polynomials. (English) Zbl 1472.05152
Summary: The symmetric Grothendieck polynomials representing Schubert classes in the K theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type \(A_n\) crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand-Tsetlin patterns via the 5-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials.
MSC:
| 05E10 | Combinatorial aspects of representation theory |
| 05E05 | Symmetric functions and generalizations |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14N15 | Classical problems, Schubert calculus |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
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