Clifford theory

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 \rightarrow 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

μ(g)(n) = μ(gng − 1)

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

\langle \chi_N,\mu \rangle \neq 0, then
\chi_N = e\left(\sum_{i=1}^{t} \mu^{(g_i)}\right),

where e and t are positive integers, and each gi 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

 \{ g \in G: \mu^{(g)} = \mu \}

and is often denoted by

IG(μ).

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

In fact, the integer e divides the index

[IG(μ):N],

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, VN be its restriction to N and U be an irreducible F[N]-submodule of VN. For each g in G, U.g is an irreducible F[N]-submodule of VN, and \sum_{g \in G} U.g is an F[G]-submodule of V, so 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  F = \mathbb{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 \mathbb{C}N-submodule U.g affords the character μ(g) and \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.

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 (that is, the subgroup of G consisting of those elements which themselves commute with every element 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 which are close (in a sense which can be made precise) to being simple.

Mackey (1976) 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".

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Clifford — is both a given name and a surname of Old English origin that applies to a number of individuals or places. It simply means ford by a cliff .[1] Clifford was a common surname mainly in the 18th century but lost its prominence over the years.… …   Wikipedia

  • Clifford's theorem — may refer to: Clifford s theorem on special divisors Clifford theory in representation theory Hammersley–Clifford theorem in probability This disambiguation page lists mathematics articles associated with the same title. If an …   Wikipedia

  • Clifford Geertz — Born August 23, 1926(1926 08 23) San Francisco Died October 30, 2006(2006 10 30) (aged 80 …   Wikipedia

  • Clifford analysis — Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but… …   Wikipedia

  • Clifford Taubes — Clifford Taubes, 2010. Born 1954 (age 56–57) Roche …   Wikipedia

  • Clifford Victor Johnson — Nationality English Fields theoretical physics, particle physics, mathematical physics …   Wikipedia

  • Clifford Mayes — (born 1953) is an American professor in the Brigham Young University McKay School of Education. A Jungian scholar, Mayes has produced the first [1] book length studies in English on the pedagogical applications of Jungian and neo Jungian… …   Wikipedia

  • Clifford Nass — Nass at TeachAIDS inaugural gala, 2010 Residence Stanford, California, USA Nationality American …   Wikipedia

  • Clifford Hugh Dowker — (1912–1982) was a topologist known for his work in point set topology and also for his contributions in category theory, sheaf theory and knot theory. Contents 1 Biography 2 Work 3 References 4 …   Wikipedia

  • Clifford Martin Will — (born 1946) is a Canadian born mathematical physicist who is well known for his contributions to the theory of general relativity. Will was born in Hamilton, Canada. In 1968, he earned a B.Sc. from McMaster University. At Caltech, he studied… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”