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 A

A4. Polynomial and rational representations of $GL(n,ℂ)$

If $V$ is a $GL(n,ℂ)\text{-module}$ of dimension $d$ then, by choosing a basis of $V,$ we can define a map $$\begin{array}{cccc}{\rho }_V:& GL(n,ℂ)& ⟶& GL(d,ℂ)\\ & g& ⟼& \rho \left(g\right),\end{array}$$ where $\rho \left(g\right)$ is the transformation of $V$ that is induced by the action of $g$ on $V\text{.}$ Let

$g_{ij}$ denote the $(i,j)$ entry of the matrix $g,$ and $\rho {\left(g\right)}_{kl}$ denote the $(k,l)$ entry of the matrix $\rho \left(g\right)\text{.}$

The map $\rho$ depends on the choice of the basis of $V,$ but the following definitions do not.

The module $V$ is a polynomial representation if there are polynomials $p_{kl}\left(x_{ij}\right),$ $1\le k,$ $l\le d,$ such that $$\rho {\left(g\right)}_{kl}=p_{kl}\left(g_{ij}\right),\text{for all}\hspace{0.17em}1\le k,\hspace{0.17em}l\le d\text{.}$$ In other words $\rho {\left(g\right)}_{jk}$ is the same as the polynomial $p_{kl}$ evaluated at the entries $g_{ij}$ of the matrix $g\text{.}$

The module $V$ is a rational representation if there are rational functions (quotients of two polynomials) $p_{kl}\left(x_{ij}\right)/q_{kl}\left(x_{ij}\right),$ $1\le k,$ $l\le d,$ such that $$\rho {\left(g\right)}_{kl}=p_{kl}\left(g_{ij}\right)/q_{kl}\left(g_{ij}\right),\text{for all}\hspace{0.17em}1\le k,\hspace{0.17em}l\le n\text{.}$$ Clearly, every polynomial representation is a rational one.

The theory of rational representations of $GL(n,ℂ)$ can be reduced to the theory of polynomial representations of $GL(n,ℂ)\text{.}$ This is accomplished as follows. The determinant det : $GL(n,ℂ)\to ℂ$ defines a 1-dimensional (polynomial) representation of $GL(n,ℂ)\text{.}$ Any integral power $$\begin{array}{cccc}{\text{det}}^k:& GL(n,ℂ)& ⟶& ℂ\\ & g& ⟼& \text{det}{\left(g\right)}^k\end{array}$$ of the determinant also determines a 1-dimensional representation of $GL(n,ℂ)\text{.}$ All irreducible rational representations $GL(n,ℂ)$ can be constructed in the form $${\text{det}}^k\otimes V^{\lambda },$$ for some $k\in \mathbb{Z}$ and some irreducible polynomial representation $V^{\lambda }$ of

There exist representations of $GL(n,ℂ)$ which are not rational representations, for example There is no known classification of representations of $GL(n,ℂ)$ which are not rational.

References

See [Ste1989] for a study of the combinatorics of the rational representations of $GL(n,ℂ)\text{.}$

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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