On a classical stability result on invariants of isotypical modules

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The real object of this work is to provide an exposition of a basic fact in the classical invariant theory (after giving a survey of representation theory for reductive algebraic groups). This fact, generally called the first fundamental theorem (due to Capelli and Weyl), provides a concrete and explicit decomposition of the symmetric algebra generated by generic entries of an m × n matrix which is regarded as a representation space simultaneously for GL m and GL n (under the left and right multiplications). Though the result is very classical, its dissemination has been suffering in the want of a treatment from the Lie-theoretic point of view, and the true dimensions of its consequences are generally not realized. We try to dramatize this last statement through a brief introduction where the problem in the title is discussed, viz., how to understand for an arbitrary group (or algebraic group) Γ the symmetric algebra on n copies of a Γ-module when n is considered as a variable; in this, and in most of the paper, we work over a field of characteristic 0, though occasional side remarks have been made for arbitrary characteristic.