On rings of supersymmetric polynomials

TL;DR

This paper studies various rings of supersymmetric polynomials, providing their generators, relations, and bases, and establishes isomorphisms between different types of these rings.

Contribution

It offers explicit descriptions of three types of supersymmetric polynomial rings, including generators, relations, and bases, and proves isomorphisms among them.

Findings

Descriptions of generators and relations for each ring type

Construction of natural bases using Euler characters

Isomorphisms between polynomial and Laurent supersymmetric rings

Abstract

We consider three types of rings of supersymmetric polynomials: polynomial ones $\Lambda_{m,n}$, partially polynomial $\Lambda_{m,n}^{+y}$ and Laurent supersymmetric rings $\Lambda_{m,n}^{\pm}$. For each type of rings we give their descriptions in terms of generators and relations. As a corollary we get for $n\ge m$ an isomorphism $\Lambda_{m,n}^{+y}=\Lambda_{m,m}^{+y}\otimes\Lambda^{+y}_{0,n-m}$. We also have the same sort of isomorphism for polynomial rings, but in this case the isomorphism does not preserve the grading. For each type of rings we also construct some natural basis consisting of Euler characters.