Last update: 17 September 2013

If $V$ is a $GL(n,\u2102)\text{-module}$ of dimension $d$ then, by choosing a basis of $V,$ we can define a map $$\begin{array}{cccc}{\rho}_{V}:& GL(n,\u2102)& \u27f6& GL(d,\u2102)\\ & g& \u27fc& \rho \left(g\right),\end{array}$$ 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 module $V$ is a *polynomial representation* if there are polynomials
${p}_{kl}\left({x}_{ij}\right),$
$1\le k,$ $l\le d,$ such that
$$\rho {\left(g\right)}_{kl}={p}_{kl}\left({g}_{ij}\right),\phantom{\rule{2em}{0ex}}\text{for all}\hspace{0.17em}1\le k,\hspace{0.17em}l\le d\text{.}$$
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
$$\rho {\left(g\right)}_{kl}={p}_{kl}\left({g}_{ij}\right)/{q}_{kl}\left({g}_{ij}\right),\phantom{\rule{2em}{0ex}}\text{for all}\hspace{0.17em}1\le k,\hspace{0.17em}l\le n\text{.}$$
Clearly, every polynomial representation is a rational one.

The theory of rational representations of $GL(n,\u2102)$
can be reduced to the theory of polynomial representations of $GL(n,\u2102)\text{.}$
This is accomplished as follows. The determinant det : $GL(n,\u2102)\to \u2102$
defines a 1-dimensional (polynomial) representation of $GL(n,\u2102)\text{.}$
Any integral power
$$\begin{array}{cccc}{\text{det}}^{k}:& GL(n,\u2102)& \u27f6& \u2102\\ & g& \u27fc& \text{det}{\left(g\right)}^{k}\end{array}$$
of the determinant also determines a 1-dimensional representation of $GL(n,\u2102)\text{.}$
**All irreducible rational representations** $GL(n,\u2102)$
**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** $GL(n,\u2102)\text{.}$

There exist representations of $GL(n,\u2102)$ which are not rational representations, for example $$g\mapsto \left(\begin{array}{c}1& \text{ln\hspace{0.17em}|det(g)|}\\ 0& 1\end{array}\right)\text{.}$$ There is no known classification of representations of $GL(n,\u2102)$ which are not rational.

**References**

See [Ste1989] for a study of the combinatorics of the rational representations of $GL(n,\u2102)\text{.}$

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..