Publisher Summary Since its emergence in the middle of the last century, the invariant theory has oscillated between two clearly distinguishable poles. The first, and the one that was later to survive the temporary death of the field, is geometry. Invariants were identified with the invariants of surfaces. Their study, the aim was which to give information about the solution of systems of polynomial equations, was to lead to the rise of commutative algebra. The second pole of the invariant theory was algorithmic. All invariant theory is ultimately concerned with the problem to generalize to tensors the eigenvalue theory of matrices. This chapter provides a self-contained combinatorial presentation of the vector invariant theory over an arbitrary infinite field. This is done by proving the Straightening Formula, which is one of the fundamental algorithms of multilinear algebra. The straightening formula has two advantages. First, it holds over the ring of integers. Second, it recognizes the crucial role played by the notion of a bitableau in obtaining a characteristic-free proof of the first fundamental theorem.
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