Mixed Tensor Products, Capelli Berezinians, and Newton's Formula for $\mathfrak{gl}(m|n)$

TL;DR

This paper extends results on mixed tensor products and Capelli determinants to the superalgebra setting, constructing homomorphisms that relate the centers of superalgebras and deriving a super-analog of Newton's formula for $\,\mathfrak{gl}(m|n)$.

Contribution

It introduces a family of superalgebra homomorphisms linking $U(\,\mathfrak{gl}(m+1|n))$ and $U(\,\mathfrak{gl}(m|n))$, and establishes a super-Newton formula relating Capelli and Gelfand generators.

Findings

Constructed superalgebra homomorphisms $\,\varphi_R$ for $\,\mathfrak{gl}(m|n)$.

Derived a super-Newton formula relating Capelli and Gelfand generators.

Identified the kernel of $\,\varphi_{R_1}$ as the ideal generated by the first Gelfand invariant.

Abstract

In this paper, we extend the results of Grantcharov and Robitaille in 2021 on mixed tensor products and Capelli determinants to the superalgebra setting. Specifically, we construct a family of superalgebra homomorphisms $\varphi_R : U(\mathfrak{gl}(m+1|n)) \rightarrow \mathcal{D}'(m|n) \otimes U(\mathfrak{gl}(m|n))$ for a certain space of differential operators $\mathcal{D}'(m|n)$ indexed by a central element $R$ of $\mathcal{D}'(m|n) \otimes U(\mathfrak{gl}(m|n))$. We then use this homomorphism to determine the image of Gelfand generators of the center of $U(\mathfrak{gl}(m+1|n))$. We achieve this by first relating $\varphi_R$ to the corresponding Harish-Chandra homomorphisms and then proving a super-analog of Newton's formula for $\mathfrak{gl}(m)$ relating Capelli generators and Gelfand generators. We also use the homomorphism $\varphi_R$ to obtain representations of $U(\mathfrak{gl}(m+1|n))$ from those of $U(\mathfrak{gl}(m|n))$, and find conditions under which these inflations are simple. Finally, we show that for a distinguished central element $R_1$ in $\mathcal{D}'(m|n)\otimes U(\mathfrak{gl}(m|n))$, the kernel of $\varphi_{R_1}$ is the ideal of $U(\mathfrak{gl}(m+1|n))$ generated by the first Gelfand invariant $G_1$.