Identities on Factorial Grothendieck Polynomials

Gustafson and Milne proved an identity on the Schur function indexed by a partition of the form $(\lambda_1-n+k,\lambda_2-n+k,\ldots,\lambda_k-n+k)$. On the other hand, Feh\'{e}r, N\'{e}methi and Rim\'{a}nyi found an identity on the Schur function indexed by a partition of the form $(m-k,\ldots,m-k, \lambda_1,\ldots,\lambda_k)$. Feh\'{e}r, N\'{e}methi and Rim\'{a}nyi gave a geometric explanation of their identity, and they raised the question of finding a combinatorial proof. In this paper, we establish a Gustafson-Milne type identity as well as a Feh\'{e}r-N\'{e}methi-Rim\'{a}nyi type identity for factorial Grothendieck polynomials. Specializing a factorial Grothendieck polynomial to a Schur function, we obtain a combinatorial proof of the Feh\'{e}r-N\'{e}methi-Rim\'{a}nyi identity.