Combinatorial Expansions in $K$-Theoretic Bases

  • Jason Bandlow
  • Jennifer Morse
Keywords: Symmetric functions, Grothendieck polynomials, Set-valued tableaux


We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape.  Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space.  In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.
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