In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, Siegfried Heinrich Aronhold, Alfred Clebsch, and Paul Gordan 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.
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. Suppose that
f ( x ) = A 0 x 1 2 + 2 A 1 x 1 x 2 + A 2 x 2 2 {\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
Δ = A 0 A 2 − A 1 2 . {\displaystyle \displaystyle \Delta =A_{0}A_{2}-A_{1}^{2}.}
The symbolic representation of the discriminant is
2 Δ = ( a b ) 2 {\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
( a b ) = a 1 b 2 − a 2 b 1 . {\displaystyle \displaystyle (ab)=a_{1}b_{2}-a_{2}b_{1}.}
Squaring this we get
( a b ) 2 = a 1 2 b 2 2 − 2 a 1 a 2 b 1 b 2 + a 2 2 b 1 2 . {\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
f ( x ) = ( a 1 x 1 + a 2 x 2 ) 2 = ( b 1 x 1 + b 2 x 2 ) 2 {\displaystyle \displaystyle f(x)=(a_{1}x_{1}+a_{2}x_{2})^{2}=(b_{1}x_{1}+b_{2}x_{2})^{2}}
so that
A i = a 1 2 − i a 2 i = b 1 2 − i b 2 i {\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
… excerpt ends here. Continue reading the full article.
