In mathematics, the finite-dimensional representations of the complex classical Lie groups
, can be constructed using the general representation theory of semisimple Lie algebras.
are indeed simple Lie groups, and their finite-dimensional representations coincide[1] with those of their maximal compact subgroups, respectively
In the classification of simple Lie algebras, the corresponding algebras are However, since the complex classical Lie groups are linear groups, their representations are tensor representations.
Each irreducible representation is labelled by a Young diagram, which encodes its structure and properties.
be the defining representation of the general linear group
integers, i.e. to Young diagrams of size
are mixed tensor representations, i.e. subrepresentations of
In the end, the set of irreducible representations of
This shows that irreducible representations of
can be labeled by pairs of Young tableaux .
are equivalent as representations of the special linear group
can be indexed by Young tableaux, and are all tensor representations (not mixed).
The unitary group is the maximal compact subgroup of
, which means that complex linear, continuous irreducible representations of the latter are in one-to-one correspondence with complex linear, algebraic irreps of the former, via the inclusion
[5] Tensor products of finite-dimensional representations of
Below are a few examples of such tensor products: In the case of tensor representations, 3-j symbols and 6-j symbols are known.
In order for a tensor representation of
are parametrized by a subset of the Young diagrams associated to irreducible representations of
: the diagrams such that the sum of the lengths of the first two columns is at most
correspond to Young diagrams of the types
On the other hand, the dimensions of the spin representations of
, but symmetric representations do not, In the stable range
[9] Beyond the stable range, the tensor product multiplicities become
The decomposition of a tensor representation is given in terms of Littlewood-Richardson coefficients
The rule is valid in the stable range
The generalization to mixed tensor representations is Similar branching rules can be written for the symplectic group.
[12] The finite-dimensional irreducible representations of the symplectic group
The dimension of the corresponding representation is[8] There is also an expression as a factorized polynomial in
:[4] Just like in the case of the orthogonal group, tensor product multiplicities are given by Newell-Littlewood numbers in the stable range, and modifications thereof beyond the stable range.