## Combinatorial Representation Theory

Arun Ram

Department of Mathematics and Statistics

University of Melbourne

Parkville, VIC 3010 Australia

aram@unimelb.edu.au

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 $\u2102\text{-manifold}$
and a real Lie group is a differential $\mathbb{R}\text{-manifold}$ 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
$$\rho :G\to GL\left(V\right)$$
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 $GL(n,\u2102)$ is also rational
representation, see [FHa1991].

A *complex semisimple Lie group* is a connected complex Lie group $G$ such that its Lie algebra $\U0001d524$
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..

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