Restricted representation
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In mathematics, restriction is a fundamental construction in representation theory of groups. 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).
[edit] Utility of restriction
The idea behind many uses of restriction is that since the subgroup H is smaller than G, it ought to be simpler in some way. Therefore, one hopes that restriction, possibly iterated several times, would reduce a representation of G to a representation of a more manageable group, sometimes, the trivial group. This principle has been successfully applied to representations of finite groups, such as symmetric groups, as well as compact Lie groups. Decomposition of representations under restriction to a subgroup is often called branching, and a large body of literature, especially in physics, is devoted to explicit description of various "branching rules" that specify the irreducible constituents and their multiplicities, and bases in representation spaces compatible with restriction. One famous application is the construction of the Gelfand–Tsetlin basis in an irreducible representation of a compact unitary group by restriction to a chain of smaller unitary subgroups.
[edit] 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 (Brauer theory).
[edit] 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.
