Classical Invariants and the General Linear Group

Invariant theory in its classical sense is a theory of algebraic forms, i.e. of polynomials such as $$f(a,x) = \sum {\frac{{r!}}{{{r_1}!...{r_n}!}}{a_{{r_1}...{r_n}}}x_1^{{r_1}}...x_n^{{r_n}},}$$ (1.1) homogeneous of some degree r in a set of n independent variables x = (x1,...,xn). The sum in 1.1 is over all vectors \( \overrightarrow r = ({r_1},...,{r_n}) \) of non-negative integers satisfying r1 +... + rn = r, and the coefficients \( {a_{{r_1}}}...{r_n} = {a_{\overrightarrow r }} \) are, like the variables x, regarded as independent indeterminates over a ground field K (usually the field of real or complex numbers).