Identities on Factorial Grothendieck Polynomials

Abstract

Gustafson and Milne proved an identity on the Schur function indexed by a partition of the form $({\lambda }_1-n+k,{\lambda }_2-n+k,\dots ,{\lambda }_k-n+k)$. On the other hand, Fehér, Némethi and Rimányi found an identity on the Schur function indexed by a partition of the form $(m-k,\dots ,m-k,{\lambda }_1,\dots ,{\lambda }_k)$. 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.

13 pages, 1 figure