On the faithfulness of the representation of $\mathrm{GL}(n)$ on the space of curvature tensors

Abstract

We prove that the standard representation of $GL\left(n\right)$ on the space of algebraic curvature tensors is almost faithful by showing that the kernel of this representation contains only the identity map and its negative. We additionally show that the standard representation of $GL\left(n\right)$ on the space of algebraic covariant derivative curvature tensors is faithful.