On rings of supersymmetric polynomials

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.