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.
Geometric invariant theory constructs algebraic quotients of group actions by a reductive algebraic group using invariant sections and a chosen linearization of a line bundle. An affine quotient is the spectrum of the polynomial invariant ring; a projective quotient uses homogeneous invariants and excludes the points where all positive-degree invariants vanish. Stability separates the orbits which give well-behaved geometric quotient points.
For a complex reductive algebraic group acting linearly on , decompose a nonzero vector into weights for each algebraic one-parameter subgroup. With the displayed sign convention, semistability is equivalent to for every such subgroup, and stability is equivalent to strict positivity for every nontrivial subgroup. One must test conjugates of the subgroups as well as those diagonal in a single fixed coordinate system.
Under , acting by inverse substitution, has weight . If is the smallest active exponent of , the multiplicity at is and the Hilbert-Mumford value is . Conjugating replaces this point by any projective point. Thus strict positivity is exactly the displayed condition; replacing the strict inequality by gives semistability. A binary quartic is stable precisely when it has no repeated projective root.
A point is stable when a positive-degree homogeneous invariant is nonzero there and its orbit is closed in that invariant's nonvanishing affine open set, with finite stabilizer. This is stronger than merely having a closed orbit or a nonzero invariant. For binary forms of degree four, the stable points have four distinct projective roots.
The affine null cone is the set of vectors whose orbit closure contains the origin. For a reductive algebraic group it is also the common zero locus of all positive-degree homogeneous invariants. A nonzero vector in this set represents an unstable projective point. The Hilbert-Mumford criterion for the affine null cone detects membership by a algebraic one-parameter subgroup with only positive active weights.
For conjugation of by complex matrices by the special linear group, the affine null cone consists exactly of the nilpotent matrices. If a conjugate tends to zero along an algebraic one-parameter subgroup, its constant characteristic polynomial must be . Conversely, put a nilpotent matrix in strictly upper triangular form and conjugate it by . Every active matrix entry has positive weight and tends to zero. The Hilbert-Mumford criterion for projective stability therefore identifies the semistable locus of the projective conjugation representation as the complement of the projectivized nilpotent cone.
For the natural linearization of a line bundle of the action on lines in , a point is semistable when some positive-degree homogeneous invariant on is nonzero on a representative. This condition is independent of the representative. The nonsemistable vectors form the affine null cone.
A linearization of a line bundle is a group action on its total space lifting the action on the base, linear on each fibre and obeying the group law. It induces actions on global sections of its tensor powers. For the projective space of lines in a linear representation , the induced action on the dual of the tautological line bundle gives the natural linearization of . Geometric invariant theory uses its invariant sections to define the semistable locus.
Replace a degree- binary form by the formal power and recover its coefficients with the linear rule . For a coefficient invariant of degree , use symbolic letters, each of degree , polarize and then apply this rule to each letter. Products of brackets generate the invariant expressions before the final coefficient substitution.
For forms of degrees , the th transvectant contracts pairs of indices using the invariant alternating form. Explicitly it is a scalar normalization of . These contractions realize the irreducible summands in the symmetric-power tensor-product decomposition.
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.
Realize the standard representation of the symmetric group on . The symmetric power sums , for , freely generate its polynomial invariant ring. To justify the description, average any polynomial lift from to over the symmetric group; the resulting symmetric lift is a polynomial in the elementary symmetric polynomials , and restriction sets . The remaining three generators are algebraically independent. The Newton identities express them equivalently through over the complex numbers. In coordinates , one has and .
For a group representation on , a polynomial invariant is a polynomial satisfying for every . Homogeneous degree- invariants form . Using the symmetric power matters: an alternating invariant tensor contracted with repeated copies of the same commuting vector can vanish.
Let the symmetric group act by and , where and . Rotation invariance requires the exponents of each monomial to differ by a multiple of three. After symmetrizing, factor out a power of ; the remainder is . These power sums are polynomials in and , by the recurrence . Hence the entire polynomial invariant ring is generated by and , which are algebraically independent. Their degrees are two and three, giving one invariant in each of degrees two, three and four, and none in degree one.
The action has invariant monomials exactly when their two exponents differ by a multiple of . Every such monomial is a product of . Reducing by leaves the monomials , and with ; their images are distinct monomials in , proving that this is the only relation. The map is finite and its fibres are the finite-group orbits, so the hypersurface is the geometric quotient.
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Choose finitely many homogeneous positive-degree invariants generating the ideal generated by all positive-degree invariants in the ambient polynomial ring. Such a finite set exists because a polynomial ring is Noetherian. Apply the Reynolds operator on a polynomial invariant ring to an expression for any homogeneous invariant in that ideal. Its coefficients become invariants of smaller degree, so induction proves that the chosen set generates the entire invariant ring as an algebra.
In characteristic zero, averaging over a compact group, or over a Zariski-dense compact form of a complex reductive group, gives a degree-preserving projection onto the polynomial invariant ring. Its module property is for invariant . This property turns finite ideal generation in the ambient polynomial ring into finite algebra generation of the invariant subring.
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.