Abstract: \def\R{{\bbf R}}
The aim of this paper is to introduce a method of invariant decompositions of the tensor space $T^r_s\R^n=\R^n\otimes\R^n \otimes\cdots\otimes\R^n\otimes\R^{n*}\otimes\R^{n*}\otimes\cdots \otimes\R^{n*}$ ($r$ factors $\R^n$, $s$ factors the dual vector space $\R^{n*}$), endowed with the tensor representation of the general linear group $GL_n(\R)$. The method is elementary, and is based on the concept of a natural ($GL_n(\R)$-equivariant) projector in $T^r_s\R^n$. The case $r=0$ corresponds with the Young-Kronecker decompositions of $T^0_s\R^n$ into its primitive components. If $r,s\ne 0$, a new, unified invariant decomposition theory is obtained, including as a special case the decomposition theory of tensor spaces by the trace operation. As an example we find the complete list of natural projectors in $T^1_2\R^n$. We show that there exist families of natural projectors, depending on real parameters, defining new representations of the group $GL_n(\R)$ in certain vector subspaces of $T^1_2\R^n$.
Keywords: tensor space of type $(r,s)$, symmetrization, alternation, trace operation, natural projector, tensor space decomposition
Classification (MSC2000): 15A72, 20C33, 20G05, 53A55
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