Harman Patil (Editor)

Complete set of invariants

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

Contents

f i : X → Y i

(where X is the collection of objects being classified, up to some equivalence relation, and the Y i are some sets), such that x ∼ 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.

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

∏ f i : ( X / ∼ ) → ∏ Y i

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 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

    ∏ f i : X → ∏ Y i .

    References

    Complete set of invariants Wikipedia
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