Clifford theory

the result about curves
In mathematics, Clifford theory, introduced by , describes the relation between representations of a group and those of a normal subgroup.

Alfred H. Clifford proved the following result on the restriction of finite dimensional irreducible representations from a group G to a normal subgroup N of finite index:

The proof of Clifford's theorem is best explained in terms of modules (and the module theoretic version works for irreducible modular representations). Let K be a field, V be an irreducible K[G] module, VN be its restriction to N and U be an irreducible K[N] submodule of VN. For each g in G and n in N, the equality holds, since N was a normal subgroup of G. Therefore, g. U is an irreducible K[N] submodule of VN, and is a K[G] submodule of V, hence must be all of V by irreducibility. Now VN is expressed as a sum of irreducible submodules, and this expression may be refined to a direct sum. The proof of the character theoretic statement of the theorem may now be completed in the case K = C. Let χ be the character of G afforded by V and μ be the character of N afforded by U. For each g in G, the C[N] submodule g. U affords the character μ(g) and The respective equalities follow because χ is a class function of G and N is a normal subgroup. The integer e appearing in the statement of the theorem is this common multiplicity.

A corollary of Clifford's theorem, which is often exploited, is that the irreducible character χ appearing in the theorem is induced from an irreducible character of the inertial subgroup IG(μ). If, for example, the irreducible character χ is primitive (that is, χ is not induced from any proper subgroup of G), then G = IG(μ) and χN = eμ. A case where this property of primitive characters is used particularly frequently is when N is Abelian and χ is faithful (that is, its kernel contains just the identity element). In that case, μ is linear, N is represented by scalar matrices in any representation affording character χ and N is thus contained in the center of G. For example, if G is the symmetric group S4, then G has a faithful complex irreducible character χ of degree 3. There is an Abelian normal subgroup N of order 4 (a Klein 4 subgroup) which is not contained in the center of G. Hence χ is induced from a character of a proper subgroup of G containing N. The only possibility is that χ is induced from a linear character of a Sylow 2 subgroup of G.

Clifford's theorem has led to a branch of representation theory in its own right, now known as Clifford theory. This is particularly relevant to the representation theory of finite solvable groups, where normal subgroups usually abound. For more general finite groups, Clifford theory often allows representation theoretic questions to be reduced to questions about groups that are close (in a sense which can be made precise) to being simple. found a more precise version of this result for the restriction of irreducible unitary representations of locally compact groups to closed normal subgroups in what has become known as the "Mackey machine" or "Mackey normal subgroup analysis".