Natural projectors in tensor spaces (Q1610975)

Let \(\mathbb{R}^n\) be an \(n\)-dimensional vector space over the field of real numbers \(\mathbb{R}\). The tensor space of type \((r,s)\) over the space \(\mathbb{R}^n\) is given by \[ T^r_s\mathbb{R}^n=\underbrace{\mathbb{R}^n\otimes \ldots \otimes \mathbb{R}^n}_{r }\otimes \underbrace{\mathbb{R}^{n*}\otimes \ldots \otimes \mathbb{R}^{n*}}_{s } \] (\(r\) factors of \(\mathbb{R}^n\) and \(s\) factors of the dual vector space \(\mathbb{R}^{n*}\)). The space \(\mathbb{R}^n\) is considered with the canonical left action of the general linear group GL\(_n(\mathbb{R})\) and the tensor space \(T^r_s\mathbb{R}^n\) is endowed with the induced tensor action. The aim of the paper is to describe a method allowing to find all GL\(_n(\mathbb{R})\)-invariant vector subspaces of the vector space \(T^r_s\mathbb{R}^n\). The approach under consideration is based on the observation that finding all GL\(_n(\mathbb{R})\)-invariant vector subspaces of the vector space \(T^r_s\mathbb{R}^n\) is equivalent to classifying all GL\(_n(\mathbb{R})\)-equivariant projectors \(P: T^r_s\mathbb{R}^n \to T^r_s\mathbb{R}^n\). The author investigates natural linear operators in a vector space endowed with a left action of GL\(_n(\mathbb{R})\), he introduces natural projectors in tensor spaces and related concepts such as natural projector equations, decomposability, reducibility, and primitivity. The trace decomposition theory is developed. It is shown that the trace decomposition of a tensor is related to a natural projector determined uniquely by certain conditions. Finally, as an application of the developed method all natural projectors in the tensor space \(T^1_2\mathbb{R}^n\) are described explicitly.