Clifford theory

In mathematics, Clifford theory, introduced by Clifford (1937), 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:

Theorem. Let π: G → GL(n,K) be an irreducible representation with K a field. Then the restriction of π to N breaks up into a direct sum of irreducible representations of N of equal dimensions. These irreducible representations of N lie in one orbit for the action of G by conjugation on the equivalence classes of irreducible representations of N. In particular the number of pairwise nonisomorphic summands is no greater than the index of N in G.

Clifford's theorem yields information about the restriction of a complex irreducible character of a finite group G to a normal subgroup N. If μ is a complex character of N, then for a fixed element g of G, another character, μ^(g), of N may be constructed by setting

for all n in N. The character μ^(g) is irreducible if and only if μ is. Clifford's theorem states that if χ is a complex irreducible character of G, and μ is an irreducible character of N with

where e and t are positive integers, and each g_i is an element of G. The integers e and t both divide the index [G:N]. The integer t is the index of a subgroup of G, containing N, known as the inertial subgroup of μ. This is

and is often denoted by

The elements g_i may be taken to be representatives of all the right cosets of the subgroup I_G(μ) in G.

In fact, the integer e divides the index

though the proof of this fact requires some use of Schur's theory of projective representations.

The proof of Clifford's theorem is best explained in terms of modules (and the module-theoretic version works for irreducible modular representations). Let F be a field, V be an irreducible F[G]-module, V_N be its restriction to N and U be an irreducible F[N]-submodule of V_N. For each g in G, U.g is an irreducible F[N]-submodule of V_N, and ${\displaystyle \sum _{g\in G}U.g}$ is an F[G]-submodule of V, so must be all of V by irreducibility. Now V_N 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 F = 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 U.g affords the character μ^(g) and ${\displaystyle \langle \chi _{N},\mu ^{(g)}\rangle =\langle \chi _{N}^{(g)},\mu ^{(g)}\rangle =\langle \chi _{N},\mu \rangle }$. 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.

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