Symbolic method

These symbols can be explained by the following example from Gordan.[5] Suppose that

${\displaystyle \displaystyle f(x)=A_{0}x_{1}^{2}+2A_{1}x_{1}x_{2}+A_{2}x_{2}^{2}}$

is a binary quadratic form with an invariant given by the discriminant

${\displaystyle \displaystyle \Delta =A_{0}A_{2}-A_{1}^{2}.}$

The symbolic representation of the discriminant is

${\displaystyle \displaystyle 2\Delta =(ab)^{2}}$

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

${\displaystyle \displaystyle (ab)=a_{1}b_{2}-a_{2}b_{1}.}$

Squaring this we get

${\displaystyle \displaystyle (ab)^{2}=a_{1}^{2}b_{2}^{2}-2a_{1}a_{2}b_{1}b_{2}+a_{2}^{2}b_{1}^{2}.}$

Next we pretend that

${\displaystyle \displaystyle f(x)=(a_{1}x_{1}+a_{2}x_{2})^{2}=(b_{1}x_{1}+b_{2}x_{2})^{2}}$

so that

${\displaystyle \displaystyle A_{i}=a_{1}^{2-i}a_{2}^{i}=b_{1}^{2-i}b_{2}^{i}}$

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

${\displaystyle \displaystyle (ab)^{2}=A_{2}A_{0}-2A_{1}A_{1}+A_{0}A_{2}=2\Delta .}$

More generally if

${\displaystyle \displaystyle f(x)=A_{0}x_{1}^{n}+{\binom {n}{1}}A_{1}x_{1}^{n-1}x_{2}+\cdots +A_{n}x_{2}^{n}}$

is a binary form of higher degree, then one introduces new variables a₁, a₂, b₁, b₂, c₁, c₂, with the properties

${\displaystyle f(x)=(a_{1}x_{1}+a_{2}x_{2})^{n}=(b_{1}x_{1}+b_{2}x_{2})^{n}=(c_{1}x_{1}+c_{2}x_{2})^{n}=\cdots .}$

What this means is that the following two vector spaces are naturally isomorphic:

• The vector space of homogeneous polynomials in A₀,...A_n of degree m
• The vector space of polynomials in 2m variables a₁, a₂, b₁, b₂, c₁, c₂, ... that have degree n in each of the m pairs of variables (a₁, a₂), (b₁, b₂), (c₁, c₂), ... and are symmetric under permutations of the m symbols a, b, ....,

The isomorphism is given by mapping a^n−j
₁a^j
₂, b^n−j
₁b^j
₂, .... to A_j. This mapping does not preserve products of polynomials.

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

${\displaystyle f(x)=(a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}+\cdots )^{n}=(b_{1}x_{1}+b_{2}x_{2}+b_{3}x_{3}+\cdots )^{n}=(c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3}+\cdots )^{n}=\cdots .}$