Restricted representation
In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to understand. Rules for decomposing the restriction of an irreducible representation into irreducible representations of the subgroup are called branching rules, and have important applications in physics. For example, in case of explicit symmetry breaking, the symmetry group of the problem is reduced from the whole group to one of its subgroups. In quantum mechanics, this reduction in symmetry appears as a splitting of degenerate energy levels into multiplets, as in the Stark or Zeeman effect.
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. 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
is a representation of H on the same vector space by the same operators:
Classical branching rules
Classical branching rules describe the restriction of an irreducible complex representation of a classical group G to a classical subgroup H, i.e. the multiplicity with which an irreducible representation of H occurs in . By Frobenius reciprocity for compact groups, this is equivalent to finding the multiplicity of in the unitary representation induced from σ. Branching rules for the classical groups were determined by- between successive unitary groups;
- between successive special orthogonal groups and unitary symplectic groups;
- from the unitary groups to the unitary symplectic groups and special orthogonal groups.
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.
Example. The unitary group U has irreducible representations labelled by signatures
where the fi are integers. In fact if a unitary matrix U has eigenvalues zi, then the character of the corresponding irreducible representation f is given by
The branching rule from U to U states that
Example. The unitary symplectic group or quaternionic unitary group, denoted Sp or U, is the group of all transformations of
HN which commute with right multiplication by the quaternions H and preserve the H-valued hermitian inner product
on HN, where q* denotes the quaternion conjugate to q. Realizing quaternions as 2 x 2 complex matrices, the group Sp is just the group of block matrices in SU with
where αij and βij are complex numbers.
Each matrix U in Sp is conjugate to a block diagonal matrix with entries
where |zi| = 1. Thus the eigenvalues of U are. The irreducible representations of Sp are labelled by signatures
where the fi are integers. The character of the corresponding irreducible representation σf is given by
The branching rule from Sp to Sp states that
Here fN + 1 = 0 and the multiplicity m is given by
where
is the non-increasing rearrangement of the 2N non-negative integers, and 0.
Example. The branching from U to Sp relies on two identities of Littlewood:
where Πf,0 is the irreducible representation of U with signature f1 ≥ ··· ≥ fN ≥ 0 ≥ ··· ≥ 0.
where fi ≥ 0.
The branching rule from U to Sp is given by
where all the signature are non-negative and the coefficient M is the multiplicity of the irreducible representation k of U in the tensor product g h. It is given combinatorially by the Littlewood–Richardson rule, the number of lattice permutations of the skew diagram k/h of weight g.
There is an extension of Littelwood's branching rule to arbitrary signatures due to. The Littlewood–Richardson coefficients M are extended to allow the signature f to have 2N parts but restricting g to have even column-lengths. In this case the formula reads
where MN counts the number of lattice permutations of f/h of weight g are counted for which 2j + 1 appears no lower than row N + j of f for 1 ≤ j ≤ |g|/2.
Example. The special orthogonal group SO has irreducible ordinary and spin representations labelled by signatures
- for N = 2n;
- for N = 2n+1.
for N = 2n and by
for N = 2n+1.
The branching rules from SO to SO state that
for N = 2n + 1 and
for N = 2n, where the differences fi − gi must be integers.
Gelfand–Tsetlin basis
Since the branching rules from U to U or SO to SO have multiplicity one, the irreducible summands corresponding to smaller 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 or SO 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.
For the remaining classical group Sp, the branching is no longer multiplicity free, so that if V and W are irreducible representation of Sp and Sp the space of intertwiners HomSp can have dimension greater than one. It turns out that the Yangian Y, a Hopf algebra introduced by Ludwig Faddeev and collaborators, acts irreducibly on this multiplicity space, a fact which enabled to extend the construction of Gelfand–Tsetlin bases to Sp.
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:Theorem. Let : G GL be an irreducible representation with K a field. Then
the restriction of 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".
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 compositionWe 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.