In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley,[1] Siegfried Heinrich Aronhold,[2] Alfred Clebsch,[3] and Paul Gordan[4] in the 19th century for computing invariants of algebraic forms.
It is based on treating the form as if it were a power of a degree one form, which corresponds to embedding a symmetric power of a vector space into the symmetric elements of a tensor product of copies of it.
First of all, (ab) is a shorthand form for the determinant of a matrix whose rows are a1, a2 and b1, b2, so Squaring this we get Next we pretend that so that and we ignore the fact that this does not seem to make sense if f is not a power of a linear form.
Substituting these values gives More generally if is a binary form of higher degree, then one introduces new variables a1, a2, b1, b2, c1, c2, with the properties What this means is that the following two vector spaces are naturally isomorphic: The isomorphism is given by mapping an−j1aj2, bn−j1bj2, .... to Aj.
The extension to a form f in more than two variables x1, x2, x3,... is similar: one introduces symbols a1, a2, a3 and so on with the properties The rather mysterious formalism of the symbolic method corresponds to embedding a symmetric product Sn(V) of a vector space V into a tensor product of n copies of V, as the elements preserved by the action of the symmetric group.