Identities on factorial Grothendieck polynomials

Gustafson and Milne proved an identity which can be used to express a Schur function s(mu) (x(1), x(2), ..., x(n)) with mu = mu(1), mu(2) ..., mu(k)) in terms of the Schur function s(lambda)(x(1), x(2), ..., x(n)), where lambda = (lambda(1), lambda(2), ..., lambda(k)) is a partition such that lambda(i) = mu(i) = n - k for 1 <= i <= k. On the other hand, Feher, Nemethi and Rimanyi found an identity which relates s(mu) (x(1), x(2), ..., x(n)) to the Schur function s(lambda) (x(1), x(2), ..., x(l)) where lambda = (lambda(1), lambda(2), ..., lambda(l)) is a partition obtained from mu by removing some of the largest parts of mu. Feher, Nemethi and Rimanyi 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 Feher-Nemethi-Rimanyi type identity for factorial Grothendieck polynomials. Specializing a factorial Grothendieck polynomial to a Schur function, we obtain a combinatorial proof of the Feher-Nemethi-Rimanyi identity. (C) 2019 Elsevier Inc. All rights reserved.