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CRYSTAL STRUCTURES FOR SYMMETRIC GROTHENDIECK POLYNOMIALS

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Abstract

The symmetric Grothendieck polynomials representing Schubert classes in the Ktheory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type An 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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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    CARA MONICAL

  2. Department of Combinatorics & Optimization, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    OLIVER PECHENIK

  3. School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD, 4072, Australia

    TRAVIS SCRIMSHAW

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  1. CARA MONICAL
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  2. OLIVER PECHENIK
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MONICAL, C., PECHENIK, O. & SCRIMSHAW, T. CRYSTAL STRUCTURES FOR SYMMETRIC GROTHENDIECK POLYNOMIALS. Transformation Groups 26, 1025–1075 (2021). https://doi.org/10.1007/s00031-020-09623-y

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