Symbolic method

In mathematics, the symbolic method in invariant theory is a highly formal algorithm developed in the 19th century for computing form invariants — invariants of algebraic forms. It is based on repeated applications of the "Omega process" (which involves symbolic partial differentiation -- hence the name) to increase the number of variables of a homogeneous form while decreasing the degree. By clever mathematics, the invariant of the form is then reduced to a vector invariant of many dependent variables, most of which then cancel out.

In the classical 19th century definition, a form invariant is a function of the coefficients of a (usually binary) form. It is invariant if it remains the same under any transformation in the transformation group in question.

The simplest classic examples of form invariants are the trace and discriminant of conic sections. These are functions of the coefficients of the general conic,

:"ax² + bxy + cy² + dx + ey + f = 0".

The transformation group is the group of translations, reflections and rotations.

References

*Koh, "Invariant Theory"
*Weyl, "The Classical Groups"