Symbolic method
In mathematics, the symbolic method in invariant theory is an algorithm developed by Cayley (1846), Siegfried Heinrich Aronhold (1858), Alfred Clebsch (1861), and Paul Gordan (1887) 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.
The symbolic method uses a compact but rather confusing and mysterious notation for invariants, depending on the introduction of new symbols a, b, c, ... (from which the symbolic method gets its name) with apparently contradictory properties.
These symbols can be explained by the following example from (Gordan 1887, volume 2, pages 1-3). Suppose that
is a binary quadratic form with an invariant given by the discriminant
The symbolic representation of the discriminant is
where a and b are the symbols. The meaning of the expression (ab)2 is as follows. 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−j
1aj
2, bn−j
1bj
2, .... to Aj. This mapping does not preserve products of polynomials.
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