The Stembridge equality for skew stable Grothendieck polynomials and skew dual stable Grothendieck polynomials

Abstract

The Schur polynomials s λ are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For ρ=(n,n-1,⋯,1) a staircase shape and μ⊆ρ a subpartition, the Stembridge equality states that s ρ/μ =s ρ/μ T . This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials G λ , and the dual stable Grothendieck polynomials g λ , developed by Buch, Lam, and Pylyavskyy, are variants of the Schur polynomials and describe the K-theory of the Grassmannians. Using the Hopf algebra structure of the ring of symmetric functions and a generalized Littlewood–Richardson rule, we prove that G ρ/μ =G ρ/μ T and g ρ/μ =g ρ/μ T , the analogues of the Stembridge equality for the skew stable and skew dual stable Grothendieck polynomials.