
Representations of classical Lie groups

In mathematics, the finite-dimensional representations of the complex classical Lie groups 
formula_1, formula_2, formula_3, formula_4, formula_5,
can be constructed using the general representation theory of semisimple Lie algebras. The groups 
formula_2, formula_4, formula_5 are indeed simple Lie groups, and their finite-dimensional representations coincide with those of their maximal compact subgroups, respectively formula_9, formula_10, formula_11. 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.
General linear group formula_1 and special linear group formula_2.
Weyl's construction.
Let formula_15 be the defining representation of formula_1. Any irreducible finite-dimensional representation of formula_1 is a tensor representation, i.e. a subrepresentation of formula_18 for some integer formula_19. 
The irreducible subrepresentations of formula_18 are the images of formula_21 by Schur functors formula_22 associated to partitions formula_23 of formula_24 into at most formula_25 integers, i.e. to Young diagrams of size formula_26 with formula_27. (If formula_28 then formula_29.) Schur functors are defined using Young symmetrizers of the symmetric group formula_30, which acts naturally on formula_18. We write formula_32.
The dimensions of the irreducible representations are
where formula_34 is the hook length of the cell formula_35 in the Young diagram formula_23. 
Case of formula_2.
Two representations formula_41 of formula_1 are equivalent as representations of formula_2 if and only if there is formula_44 such that formula_45. In particular, the determinant representation formula_46 is trivial in formula_2, i.e. it is equivalent to formula_48. 
Tensor products.
Tensor products of finite-dimensional representations of formula_1 are 
where the natural integers formula_51 are 
Littlewood-Richardson coefficients. For example,
Orthogonal group formula_3 and special orthogonal group formula_4.
"In addition to the Lie group representations described here, the orthogonal and special orthogonal groups have spin representations, which are projective representations of these groups, i.e. representations of their universal covering groups."
Construction of representations.
Since formula_3 is a subgroup of formula_1, any irreducible representation of formula_1 is also a representation of formula_3, which may however not be irreducible. In order for a tensor representation of formula_3 to be irreducible, the tensors must be traceless.
Irreducible representations of formula_3 are parametrized by a subset of the Young diagrams associated to irreducible representations of formula_1: the diagrams such that the sum of the lengths of the first two columns is a most formula_25. The irreducible representation formula_63 that corresponds to such a diagram is a subrepresentation of the corresponding formula_1 representation formula_65. For example, in the case of symmetric tensors, 
Case of formula_4.
The antisymmetric tensor formula_68 is a one-dimensional representation of formula_3, which is trivial for formula_4. Then formula_71 where formula_72 is obtained from formula_23 by acting on the length of the first column as formula_74. 
For example, the irreducible representations of formula_84 correspond to Young diagrams of the types formula_85. The irreducible representations of formula_86 correspond to formula_87, and formula_88. 
On the other hand, the dimensions of the spin representations of formula_86 are even integers.
Dimensions.
The dimensions of irreducible representations of formula_4 are given by a formula that depends on the parity of formula_25: 
There is also an expression as a factorized polynomial in formula_25: 
where formula_96 are respectively row lengths, column lengths and hook lengths. In particular, antisymmetric representations have the same dimensions as their formula_1 counterparts, formula_98, but symmetric representations do not,
Tensor products.
In the stable range formula_100, the tensor product multiplicities that appear in the tensor product decomposition formula_101 are Newell-Littlewood numbers, which do not depend on formula_25. Beyond the stable range, the tensor product multiplicities become formula_25-dependent modifications of the Newell-Littlewood numbers.
Symplectic group formula_5.
Representations.
The finite-dimensional irreducible representations of formula_5 are parametrized by Young diagrams with at most formula_25 rows. The dimension of the corresponding representation is 
There is also an expression as a factorized polynomial in formula_25:
Tensor products.
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.