In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory

Invariant theory

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...

 of finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

s states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections. In the case of subgroups of the complex general linear group the theorem was first proved by who gave a case-by-case proof. soon afterwards gave a uniform proof. It has been extended to finite linear groups over an arbitrary field in the non-modular case by Jean-Pierre Serre

Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...

.

Statement of the theorem

Let V be a finite-dimensional vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 over a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 K and let G be a finite subgroup of the general linear group

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...

 GL(V). An element s of GL(V) is called a pseudoreflection if it fixes a codimension one subspace of V and is not the identity transformation I, or equivalently, if the kernel Ker (s − I) has codimension

Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...

 one in V. Assume that the order of G is relatively prime to the characteristic

Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...

 of K (the so-called non-modular case). Then the following three properties are equivalent:

• The group G is generated by pseudoreflections.
• The algebra of invariants K[V]^G is a (free) polynomial algebra.
• The algebra K[V] is a free module
  Free module
  In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
  
   over K[V]^G.

In the case when the field K is the field C of complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, the first condition is usually stated as "G is a complex reflection group

Complex reflection group

In mathematics, a complex reflection group is a group acting on a finite-dimensional complex vector space, that is generated by complex reflections: non-trivial elements that fix a complex hyperplane in space pointwise...

". Shephard and Todd derived a full classification of such groups.

Examples

• Let V be one-dimensional. Then any finite group faithfully acting on V is a subgroup of the multiplicative group of the field K, and hence a cyclic group
  Cyclic group
  In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
  
  . It follows that G consists of roots of unity of order dividing n, where n is its order, so G is generated by pseudoreflections. In this case, K[V] = K[x] is the polynomial ring in one variable and the algebra of invariants of G is the subalgebra generated by xⁿ, hence it is a polynomial algebra.

• Let V = Kⁿ be the standard n-dimensional vector space and G be the symmetric group
  Symmetric group
  In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
  
   S_n acting by permutations of the elements of the standard basis. The symmetric group is generated by transpositions (ij), which act by reflections on V. On the other hand, by the main theorem of symmetric function
  Symmetric function
  In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...
  
  s, the algebra of invariants is the polynomial algebra generated by the elementary symmetric functions e₁, … e_n.

• Let V = K² and G be the cyclic group of order 2 acting by ±I. In this case, G is not generated by pseudoreflections, since the nonidentity element s of G acts without fixed points, so that dim Ker (s − I) = 0. On the other hand, the algebra of invariants is the subalgebra of K[V] = K[x, y] generated by the homogeneous elements x², xy, and y² of degree 2. This subalgebra is not a polynomial algebra because of the relation x²y² = (xy)².

Generalizations

gave an extension of the Chevalley–Shephard–Todd theorem to positive characteristic.

There has been much work on the question of when a reductive algebraic group acting on a vector
space has a polynomial ring of invariants. In the case when the algebraic group is simple and the representation is irreducible all cases when the invariant ring is polynomial have been classified by

In general, the ring of invariants of a finite group acting linearly on a complex vector space is Cohen-Macaulay, so it is a finite rank free module over a polynomial subring.