## Combinatorial Representation Theory

Last update: 17 September 2013

## Appendix B

### B9. Complex semisimple Lie groups

We shall not define Lie groups and Lie algebras let us only recall that a complex Lie group is a differential $ℂ\text{-manifold}$ and a real Lie group is a differential $ℝ\text{-manifold}$ and that every Lie group has an associated Lie algebra, see [CMS1995].

If $G$ is a complex Lie group then the word representation is usually used to refer to a holomorphic representation, i.e. the homomorphism $ρ:G→GL(V)$ determined by the module $V$ should be a morphism of (complex) analytic manifolds. Strictly speaking there are representations which are not holomorphic but there is a good theory only for holomorphic representations, so one usually abuses language and assumes that representation means holomorphic representation. The terms holomorphic representation and complex analytic representation are used interchangeably. Similarly, if $G$ is a real Lie group then representation usually means real analytic representation. See [Var1984] p. 102 for further details. Every holomorphic representation of $GL\left(n,ℂ\right)$ is also rational representation, see [FHa1991].

A complex semisimple Lie group is a connected complex Lie group $G$ such that its Lie algebra $𝔤$ is a complex semisimple Lie algebra.

## Notes and references

This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.

Key words and phrases. Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..