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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?

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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).
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The problem with $GL(V)$ simply as a group is that it is to big and has a lot of wild representations. For example, the field $\mathbb{C}$ has a lot of field automorphisms, using them you can twist any normal representation and obtain some strange one. So you need to add some additional structure on $GL(V)$ that you want to preserve.

Study of polynomial representations corresponds to understanding the $GL(V)$ group as an algebraic group. This is the natural choice if you want to have a theory over arbitrary (characteristic zero, algebraically closed) field. There are other versions of classification. For example, you can understand $GL_n$ as a Lie group and study holomorphic representations. The results would not change. There are a lot of results of the type "continuous representations of $U(n)$ extend to algebraic representations of $GL(n)$". So, in general, the Schur-Weil theory extends for any reasonable additional structure.

Unfortunately, I don't know the examples of ``weird'' continuous representations, but intuitevely I am confident that they do exist. For me, the representation $|det|^{\pi}$ is weird enough.

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