Identities on factorial Grothendieck polynomials

Abstract

Gustafson and Milne proved an identity which can be used to express a Schur function sμ(x1,x2,…,xn) with μ=(μ1,μ2,…,μk) in terms of the Schur function sλ(x1,x2,…,xn), where λ=(λ1,λ2,…,λk) is a partition such that λi=μi+n−k for 1≤i≤k. On the other hand, Fehér, Némethi and Rimányi found an identity which relates sμ(x1,x2,…,xn) to the Schur function sλ(x1,x2,…,xℓ), where λ=(λ1,λ2,…,λℓ) is a partition obtained from μ by removing some of the largest parts of μ. Fehér, Némethi and Rimányi 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 Fehér-Némethi-Rimányi type identity for factorial Grothendieck polynomials. Specializing a factorial Grothendieck polynomial to a Schur function, we obtain a combinatorial proof of the Fehér-Némethi-Rimányi identity.