Tensor fundamental theorems of invariant theory

TL;DR

This paper establishes fundamental theorems for $GL(V)$-equivariant polynomial maps from matrix tuples to tensor spaces, extending classical invariant theory and symbolic algebra methods.

Contribution

It introduces first and second fundamental theorems for equivariant polynomial maps involving matrix and tensor spaces, advancing invariant theory.

Findings

Formulated fundamental theorems for $GL(V)$-equivariant maps

Connected invariant theory with symbolic algebra techniques

Extended classical results to tensor space contexts

Abstract

The aim of this paper is to establish a first and second fundamental theorem for $GL(V)$ equivariant polynomial maps from $k$--tuples of matrix variables $End(V)^{ k} $ to tensor spaces $End(V)^{ \otimes n}$ in the spirit of H. Weyl's book {\em The classical groups} \cite{Weyl} and of symbolic algebra.