Last update: 17 September 2013
If is a of dimension then, by choosing a basis of we can define a map where is the transformation of that is induced by the action of on Let
entry of the matrix and
denote the entry of the matrix
The module is a polynomial representation if there are polynomials such that In other words is the same as the polynomial evaluated at the entries of the matrix
The module is a rational representation if there are rational functions (quotients of two polynomials) such that Clearly, every polynomial representation is a rational one.
The theory of rational representations of can be reduced to the theory of polynomial representations of This is accomplished as follows. The determinant det : defines a 1-dimensional (polynomial) representation of Any integral power of the determinant also determines a 1-dimensional representation of All irreducible rational representations can be constructed in the form for some and some irreducible polynomial representation of
There exist representations of which are not rational representations, for example There is no known classification of representations of which are not rational.
See [Ste1989] for a study of the combinatorics of the rational representations of
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..