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.[1] 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:
^Weyl 1946, pp. 159–160.
and 24 Related for: Restricted representation information
groups. Often the restrictedrepresentation is simpler to understand. Rules for decomposing the restriction of an irreducible representation into irreducible...
induced representation is a representation of a group, G, which is constructed using a known representation of a subgroup H. Given a representation of H...
function Representation theory of finite groups Modular representation theory Frobenius reciprocity Restrictedrepresentation Induced representation Peter–Weyl...
etc. that would be harmful to children under 15 (indirect and restrictedrepresentation of sexuality, violence, improper language (expletives) and low...
tauberian theorem Representation of a Lie group Unitary representation Irreducible representation Harish-Chandra character Restrictedrepresentation Induced representation...
impractical in general Boltzmann machines, it can be made quite efficient in a restricted Boltzmann machine (RBM) which does not allow intralayer connections between...
boundary representation similar to ISO 10303-514, but restricted to the elementary surfaces only ISO 10303-512 Faceted boundary representation a simplified...
term is often restricted to certain Hopf algebras arising as deformations of groups or their universal enveloping algebras. The representation theory of quantum...
to the same representation. Generative representation learning tasks the model with producing the correct data to either match a restricted input or reconstruct...
'lifting', known in special cases, and so is covariant (whereas a restrictedrepresentation is contravariant). Attempts to specify a direct construction have...
characteristic p is restricted. Let g {\displaystyle {\mathfrak {g}}} be a Lie algebra over a field k of characteristic p>0. The adjoint representation of g {\displaystyle...
ℓ of an ℓ-adic representation. There are numerous conditions on representations given by some property of the representationrestricted to a decomposition...
group homomorphism φ:G → H, then if we have a H-representation, we can obtain a restricted G-representation. So, if we have a rep of Diffx1(M)/Diffxn(M)...
mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the...
portal See Restrictedrepresentation and Induced representation for definitions of the processes to which this theorem applies. See Representation theory...
A mental representation (or cognitive representation), in philosophy of mind, cognitive psychology, neuroscience, and cognitive science, is a hypothetical...
additional parties have minimal or nonexistent influence and almost no representation at the national and state level. Causes for this mainly focus on the...
compact subgroup. A continuous representation (π, V) of G on a complex Hilbert space V is called admissible if π restricted to K is unitary and each irreducible...
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra...
Lorentz-invariant positive definite norm. The (m, n) representation is, however, unitary when restricted to the rotation subgroup SO(3), but these representations...
the imaginary part. The analytic representation is a generalization of the phasor concept: while the phasor is restricted to time-invariant amplitude, phase...
Global Information
