On the one-dimensional representations of the general linear supergroup

TL;DR

This paper constructs explicit one-dimensional representations of the general linear supergroup, focusing on the Berezinian and its inverse, using tensor spaces to clarify their characters.

Contribution

It provides explicit constructions of the one-dimensional representations associated with the Berezinian and its inverse for the general linear supergroup.

Findings

Explicit tensor space representations of Berezinian and its inverse.

Clarification of characters of these representations.

Foundations for further study of supergroup representations.

Abstract

Because of its multiplicativity, the Berezinian is the character of the one-dimensional representation of the general linear supergroup. We give an explicit construction of this representation on a space of tensors. Similarly, we construct the representation such that its character is the inverse of the Berezinian.