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The symmetric Grothendieck polynomials generalize Schur polynomials and areSchur-positive by degree. Combinatorially this is manifested as thegeneralization of semistandard Young tableaux by set-valued tableaux. We definea (weak) symmetric $P$-Grothendieck polynomial which generalizes $P$-Schurpolynomials in the same way. Combinatorially this is manifested as thegeneralization of shifted semistandard Young tableaux by a new type of tableauxwhich we call shifted multiset tableaux.

$P$-Schur positive $P$-Grothendieck Polynomials