Restricted representation

In mathematics, restriction is a fundamental construction in representation theory of groups. Restriction forms a representation of a subgroup from a representation of the whole group. Often the restricted representation is simpler to understand. Rules for decomposing the restriction of an irreducible representation into irreducibe representations of the subgroup are called branching rules, and have important applications in physics. The induced representation is a related operation that forms a representation of the whole group from a representation of a subgroup. The relation between restriction and induction is described by Frobenius reciprocity and the Mackey theorem. Restriction to a normal subgroup behaves particularly well and is often called "Clifford theory" after the theorem of A. H. Clifford. [harvnb|Weyl|1946|p=159-160. "Clifford theory" is also used to describe the results of
W. K. Clifford on special divisors on algebraic curves.] Restriction can be generalized to other group homomorphisms and to other rings.

For any group "G", its subgroup "H", and a linear representation "ρ" of "G", the restriction of "ρ" to "H", denoted

:ρ|_"H",

is a representation of "H" on the same vector space by the same operators:

:ρ|_"H"("h") = ρ("h").

Classical branching rules

Classical branching rules describe the restriction of an irreducible representation (π, "V") of a classical group "G" to a classical subgroup "H", i.e. the multiplicity with which an irreducible representation (σ, "W") of "H" occurs in π. By Frobenius reciprocity for compact groups, this is is equivalent to finding the multiplicity of π in the unitary representation induced from σ. Branching rules for the classical groups were determined by

The results are usually expressed graphically using Young diagrams to encode the signatures used classically to label irreducible representations, familiar from classical invariant theory. Hermann Weyl and Richard Brauer discovered a systematic method for determining the branching rule when the groups "G" and "H" share a common maximal torus: in this case the Weyl group of "H" is a subgroup of that of "G", so that the rule can be deduced from the Weyl character formula. [harvnb|Weyl|1946] [harvnb|Želobenko|1963] A systematic modern interpretation has been given by harvtxt|Howe|1995 in the context of his theory of dual pairs. The special case where σ is the trivial representation of "H" was first used extensively by Hua in his work on the Szegő kernels of bounded symmetric domains in several complex variables, where the Shilov boundary has the form "G"/"H". [harvnb|Helgason|1978] [harvnb|Hua|1963] More generally the Cartan-Helgason theorem gives the decomposition when "G"/"H" is a compact symmetric space, in which case all multiplicities are one; [harvnb|Helgason|1984|p=534-543] a generalization to arbitrary σ has since been obtained by harvtxt|Kostant|2004. Similar geometric considerations have also been used by harvtxt|Knapp|2005 to rederive Littlewood's rules, which involve the celebrated Littlewood-Richardson rules for tensoring irreducible representations of the unitary groups.harvtxt|Littelmann|1995 has found generalizations of these rules to arbitrary compact semisimple Lie groups, using his path model, an approach to representation theory close in spirit to the theory of crystal bases of Lusztig and Kashiwara. His methods yield branching rules for restrictions to subgroups containing a maximal torus. The study of branching rules is important in classical invariant theory and its modern counterpart, algebraic combinatorics. [ harvnb|Goodman|Wallach|1998] [harvnb|Macdonald|1979]

Example. The unitary group "U(N)" has irreducible representations labelled by signatures

: $mathbf{f} ,colon ,f_1ge f_2ge cdots ge f_N$

where the "f"_"i" are integers. In fact if a unitary matrix "U" has eigenvalues "z"_"i", then the character of the corresponding irreducible representation π_f is given by

: $mathrm{Tr} , pi_{mathbf{f(U) = {mathrm{det}, z_j^{f_i +N -i}over prod_{i$

The branching rule from "U(N)" to "U(N – 1)" states that

where all the signature are non-negative and the coefficient "M" (g, h; k) is the multiplicity of the irreducible representation π_k of U("N") in the tensor product π_g $otimes$ π_h. It is given combinatorially by the Littlewood-Richardson rule, the number of lattice permutations of the skew diagram k/h of weight g. [ harvnb|Macdonald|1979]

There is an extension of Littelwood's branching rule to arbitrary signatures due to harvtxt|Sundaram|1990|p=203. The Littlewood-Richardson coefficients "M" (g, h; f) are extended to allow the signature f to have 2"N" parts but restricting g to haveeven column-lengths ("g"_{2"i" – 1} = "g"_2"i"). In this case the formula reads

for "N" = 2"n"+1 and

: pi_{mathbf{g

for "N" = 2"n", where the differences "f"_"i" - "g"_"i" must be integers.

Gelfand-Tsetlin basis

Since the branching rules from U("N") to U("N"–1) or SO("N") to SO("N"–1) have multiplicity one, the irreducible summands corresponding tosmaller and smaller "N" will eventually terminate in one dimensional subspaces. In this way Gelfand and Tsetlin were able to obtain a basis of any irreducible representation of U("N") or SO("N") labelled by a chain of interleaved signatures, called a Gelfand-Tsetlin pattern.Explicit formulas for the action of the Lie algebra on the Gelfand-Tsetlin basis are given in harvtxt|Želobenko|1973.

For the remaining classical group Sp("N"), the branching is no longer multiplicity free, so that if "V" and "W" are irreducible representation of Sp("N"–1) andSp("N") the space of intertwiners Hom_Sp("N"–1)("V","W") can have dimension greater than one. It turns out that the Yangian "Y"( $mathfrak{gl}$ ₂), a Hopf algebra introduced by Ludwig Faddeev and collaborators, acts irreducibly on this multiplicity space, a fact which enabled harvtxt|Molev|2006 to extend the construction of Gelfand-Tsetlin bases to Sp("N"). [G. I. Olshanski had shown that the twisted Yangian "Y"^–( $mathfrak{gl}$ ₂), a sub-Hopf algebra of "Y"( $mathfrak{gl}$ ₂), acts naturally on the space of intertwiners. Its natural irreducible representations correspond to tensor products of the composition of point evaluations with irreducible representations of $mathfrak{gl}$ ₂. These extend to the Yangian "Y"( $mathfrak{gl}$ ₂) and give a representation theoretic explanation of the product form of the branching coefficients.]

Clifford's theorem

In 1937 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: [harvnb|Weyl|1946|p=159-160,311]

Theorem. Let π: "G" $ightarrow$ 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 inequivalent 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 distinct summands is no greater than the index of "N" in "G".

Twenty years later George Mackey 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". [ citation|first=George W.|last=Mackey|authorlink=George Mackey|title=The theory of unitary group representations|series=Chicago Lectures in Mathematics|year=1976|id=ISBN 0-225-50052-7]

Abstract algebraic setting

From the point of view of category theory, restriction is an instance of a forgetful functor. This functor is exact, and its left adjoint functor is called "induction". The relation between restriction and induction in various contexts is called the Frobenius reciprocity. Taken together, the operations of induction and restriction form a powerful set of tools for analyzing representations. This is especially true whenever the representations have the property of complete reducibility, for example, in representation theory of finite groups over a field of characteristic zero.

Generalizations

This rather evident construction may be extended in numerous and significant ways. For instance we may take any group homomorphism φ from "H" to "G", instead of the inclusion map, and define the restricted representation of "H" by the composition

:ρoφ.

We may also apply the idea to other categories in abstract algebra: associative algebras, rings, Lie algebras, Lie superalgebras, Hopf algebras to name some. Representations or modules "restrict" to subobjects, or via homomorphisms.

Notes

References

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