dimension is given by the complex spherical harmonics . For the remaining classical group , the branching is no longer multiplicity free, so that if V and W are irreducible representation of and the space of intertwiners can have dimension greater than one. It turns out that the Yangian , 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 . 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(n,K) be an irreducible representation with K a field. Then the restriction of 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 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 composition 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 available online Representation theory Algebraic combinatorics