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In mathematics, the finite-dimensional representations of the complex classical Lie groups G L {\displaystyle GL} , S L {\displaystyle SL} , O {\displaystyle O} , S O {\displaystyle SO} , S p {\displaystyle Sp} ,can be constructed using the general representation theory of semisimple Lie algebras. The groups S L {\displaystyle SL} , S O {\displaystyle SO} , S p {\displaystyle Sp} are indeed simple Lie groups, and their finite-dimensional representations coincide with those of their maximal compact subgroups, respectively S U {\displaystyle SU} , S O {\displaystyle SO} , S p {\displaystyle Sp}. 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.