Representations of classical Lie groups

Lie groups and Lie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
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In mathematics, the finite-dimensional representations of the complex classical Lie groups ${\displaystyle GL(n,\mathbb {C} )}$, ${\displaystyle SL(n,\mathbb {C} )}$, ${\displaystyle O(n,\mathbb {C} )}$, ${\displaystyle SO(n,\mathbb {C} )}$, ${\displaystyle Sp(2n,\mathbb {C} )}$, can be constructed using the general representation theory of semisimple Lie algebras. The groups ${\displaystyle SL(n,\mathbb {C} )}$, ${\displaystyle SO(n,\mathbb {C} )}$, ${\displaystyle Sp(2n,\mathbb {C} )}$ are indeed simple Lie groups, and their finite-dimensional representations coincide^[1] with those of their maximal compact subgroups, respectively ${\displaystyle SU(n)}$, ${\displaystyle SO(n)}$, ${\displaystyle Sp(n)}$. In the classification of simple Lie algebras, the corresponding algebras are

${\displaystyle {\begin{aligned}SL(n,\mathbb {C} )&\to A_{n-1}\\SO(n_{\text{odd}},\mathbb {C} )&\to B_{\frac {n-1}{2}}\\SO(n_{\text{even}},\mathbb {C} )&\to D_{\frac {n}{2}}\\Sp(2n,\mathbb {C} )&\to C_{n}\end{aligned}}}$

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.