TITLE: Non-polynomial representations of $GL_n$
QUESTION [9 upvotes]: Recall that every finite-dimensional rational representation of $GL_n$ is of the form $(\det)^{-k} \varrho$ for some integer $k\geq 0$ and polynomial representation $\varrho$ (and $\det$ is the one-dimensional representation $A\mapsto \det(A)$). The irreducible polynomial representations have been classified and are given by the Schur modules. 
My questions are as follows.
Are there simply-described finite-dimensional non-rational representations of $GL_n$?
Are there a lot of them?
Can they be classified?
Also, why do we care about polynomial representations in the first place?

REPLY [3 votes]: There are some pathological examples given in the Appendix of Stanley - Enumerative Combinatorics, vol 2. I'll share some that are relevant to the question:

Let $A \in GL(n,\mathbb{C})$.

$\varphi(A) = |\det(A)|^{\sqrt{2}}$ is a representation of dimension one that is not a rational representation.
$\varphi(A) = [\sigma(a_{ij})]$ with $\sigma$ a field automorphism of $\mathbb{C}$ that is not the identity or complex conjugation (and
  so it must be discontinuous). This is a $n$-dimensional representation
  that is not rational and not continuous.
$\varphi(A) = \begin{pmatrix} 1 & \log|det A | \\ 0 & 1 \end{pmatrix}$. This is a representation of dimension two that is not
  rational, but is continuous.


I don't know of an answer to the question of classification. But altogether, we learn

Yes, in some sense there are a lot of non-rational finite-dimensional representations that are simple to describe.
Perhaps the above examples hint at some ways to classify at least some families (but there's no indication that we can complete this fully).