Invariant theory studies functions and tensors unchanged by a group action. For a linear representation, the polynomial invariant ring records invariant polynomial functions on the representation space. Tensor invariants and commuting algebra actions connect it with Schur–Weyl duality.
For a group acting linearly on , act on the coordinate ring by . The polynomial invariant ring is the fixed subalgebra . It inherits the degree grading. Even over the complex numbers it need not be a unique factorization domain, as the quadratic cone invariant ring demonstrates.
Let and let the binary dihedral group act on by and . For , the invariantspresent the invariant ring as . First take the cyclic invariants with . The residual involution interchanges and negates , so invariant normal forms are polynomials in plus times such polynomials. For the group has order twelve and the equation is .
The scalar sign action of a cyclic group of order two on has invariant ring . It is not factorial: are pairwise nonassociate irreducibles, since every nonconstant invariant has degree at least two, while gives two distinct factorizations.
If a finite group has no nontrivial homomorphism to , its polynomial invariant ring is a unique factorization domain. Factor an invariant in the ambient polynomial ring and collect its irreducible factors into orbit products. Each orbit product transforms by a linear character, hence is invariant. It is prime in the invariant ring, and the invariant factorization is a product of these primes.
A finite group permutes the associate classes of irreducible polynomial factors of an invariant. The product of one representative from each orbit transforms by a scalar character of the group. If that character is trivial, the product is invariant. Its transitive factor orbit makes it prime in the invariant ring: an invariant polynomial divisible by one orbit factor is divisible by all of them.
For the permutation action on coordinates with , every alternating-group invariant is a symmetric polynomial plus the Vandermonde product times a symmetric polynomial. Thus the invariant ring is , where is the discriminant polynomial. The two summands are independent over the symmetric polynomial ring.
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Invariant theory is a branch of mathematics, particularly in the fields of algebra, geometry, and representation theory, that studies properties of mathematical objects that remain unchanged (or invariant) under transformations from a certain group. The most common transformations considered are linear transformations, but the theory can also apply to more general transformation groups. Historically, invariant theory originated in the 19th century, with significant contributions from mathematicians such as David Hilbert and Hermann Weyl.