Representations of classical Lie groups

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 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.