Symbolic method

In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, 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.

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Symbolic notation

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.

Example: the discriminant of a binary quadratic form

These symbols can be explained by the following example from (Gordan 1887, volume 2, pages 1-3). Suppose that

\displaystyle f(x) = A_0x_1^2%2B2A_1x_1x_2%2BA_2x_2^2

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

\displaystyle \Delta=A_0A_2-A_1^2

The symbolic representation of the discriminant is

\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 (ab)=a_1b_2-a_2b_1

Squaring this we get

\displaystyle (ab)^2=a_1^2b_2^2-2a_1a_2b_1b_2%2Ba_2^2b_1^2

Next we pretend that

\displaystyle f(x)=(a_1x_1%2Ba_2x_2)^2=(b_1x_1%2Bb_2x_2)^2

so that

\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 (ab)^2= A_2A_0-2A_1A_1%2BA_0A_2 = 2\Delta

Higher degrees

More generally if

\displaystyle f(x) = A_0x_1^n%2B\binom{n}{1}A_1x_1^{n-1}x_2%2B\cdots%2BA_nx_2^n

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

f(x)=(a_1x_1%2Ba_2x_2)^n=(b_1x_1%2Bb_2x_2)^n=(c_1x_1%2Bc_2x_2)^n=\cdots

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.

More variables

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

f(x)=(a_1x_1%2Ba_2x_2%2Ba_3x_3%2B\cdots)^n=(b_1x_1%2Bb_2x_2%2Bb_3x_3%2B\cdots)^n=(c_1x_1%2Bc_2x_2%2Bc_3x_3%2B\cdots)^n=\cdots

See also

References