Automorphisms on the ring of symmetric functions and stable and dual stable Grothendieck polynomials

TL;DR

This paper explores automorphisms of the symmetric functions ring related to stable and dual stable Grothendieck polynomials, revealing their structure and introducing a parameterized deformation with new identities.

Contribution

It characterizes a specific automorphism as an operator linked to a group-like element and generalizes this to a deformation involving a parameter, expanding understanding of these polynomials.

Findings

Automorphism described as the adjoint operator of multiplication by a group-like element.

Introduction of a parameterized deformation of the automorphism.

Derivation of identities involving stable and dual stable Grothendieck polynomials.

Abstract

The dual stable Grothendieck polynomials $g_\lambda$ and their sums $\sum_{\mu\subset\lambda} g_\mu$ (which represent $K$-homology classes of boundary ideal sheaves and structure sheaves of Schubert varieties in the Grassmannians) have the same product structure constants. In this paper we first explain that the ring automorphism $g_\lambda\mapsto\sum_{\mu\subset\lambda} g_\mu$ on the ring of symmetric functions is described as the operator $F^\perp$, the adjoint of the multiplication $(F\cdot)$, by a "group-like" element $F=\sum_{i} h_i$ where $h_i$ is the complete symmetric function. Next we give a generalization: starting with another "group-like" elements $\sum_{i} t^i h_i$, we obtain a deformation with a parameter $t$ of the ring automorphism above, as well as identities involving stable and dual stable Grothendieck polynomials.