Complete set of invariants explained

In mathematics, a complete set of invariants for a classification problem is a collection of maps

f _{i} : X \to Y _{i} (where X is the collection of objects being classified, up to some equivalence relation \sim , and the Y _{i} are some sets), such that x \sim x' if and only if f _{i(x)} = f _{i(x')} for all i . In words, such that two objects are equivalent if and only if all invariants are equal.^[1]

Symbolically, a complete set of invariants is a collection of maps such that

\left( \prod f _{i} \right) : (X/\sim) \to \left( \prod Y _{i} \right) is injective.

As invariants are, by definition, equal on equivalent objects, equality of invariants is a necessary condition for equivalence; a complete set of invariants is a set such that equality of these is also sufficient for equivalence. In the context of a group action, this may be stated as: invariants are functions of coinvariants (equivalence classes, orbits), and a complete set of invariants characterizes the coinvariants (is a set of defining equations for the coinvariants).

Examples

• In the classification of two-dimensional closed manifolds, Euler characteristic (or genus) and orientability are a complete set of invariants.
• Jordan normal form of a matrix is a complete invariant for matrices up to conjugation, but eigenvalues (with multiplicities) are not.

Realizability of invariants

A complete set of invariants does not immediately yield a classification theorem: not all combinations of invariants may be realized. Symbolically, one must also determine the image of

\prod f _{i} : X \to \prod Y _{i.}

Notes and References

1. . See in particular p. 97.