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arXiv · 1807.03294

Crystal structures for symmetric Grothendieck polynomials

Cara Monical·Oliver Pechenik·Travis Scrimshaw
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Abstract

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.

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BibTeXRIS

Cara Monical, Oliver Pechenik, Travis Scrimshaw. 2018-11-22. Crystal structures for symmetric Grothendieck polynomials. https://doi.org/10.1007/s00031-020-09623-y

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