Establishes a first and second fundamental theorem of invariant theory for GL(V)-equivariant polynomial maps from k-tuples of matrices End(V)^k into tensor powers End(V)^{otimes n}, extending Weyl's classical framework out of scalar and matrix targets into the tensor algebra. The FFT identifies every such map as an evaluation of the twisted symbolic algebra T<X>^{otimes n} semidirect Q[S_n] (the free algebra with trace, tensored n times over its central trace algebra, with S_n permuting tensor factors). The SFT, the heart of the paper, describes the kernels of those evaluation maps: every relation among tensor-valued equivariant maps is formally deducible, as a T-ideal in the sense of universal algebra closed under a formal partial trace, from the single antisymmetrizer sum over S_{d+1} of sign(sigma)*sigma together with tr(1)=d. An intermediate family of d+2 tensor Cayley-Hamilton identities C_{k,d}(x) is generated recursively from that antisymmetrizer by the partial trace, recovering the classical Cayley-Hamilton identity as the case k=d and Newton's identity for tr(x^{d+1}) as k=d+1.
This paper by Claudio Procesi establishes first and second fundamental theorems for GL(V) equivariant polynomial maps from k-tuples of matrix variables to tensor spaces, extending the classical frame…