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Please excuse me if this question turns out to be incredibly silly for one reason or another.

Are there tensor/hypermatrix analogues of $GL(n,\mathbb{C})$ that are interesting? What I'm mainly thinking of here is the following: characters of finite abelian groups are homomorphisms to $\mathbb{C}^{\times}$ which are $0$-dimensional arrays, but for nonabelian groups one can get more representations by introducing homomorphisms to $GL(n,\mathbb{C})$ which are second order tensors.

Is there an analogue of $GL(n,\mathbb{C})$ for higher order tensors and a corresponding analogue of ``representation theory of finite groups" for such objects as well?

And if not, is there some simple reason why there isn't - such as maybe that such things might all reduce to homomorphisms to $GL(n,\mathbb{C})$ after all?

I understand that there are things like hyperdeterminants of hypermatrices. Are there also things that can act like the trace to give analogues of characters for these things? Basically, very generally, I'm just wondering if one can get more things from finite groups by introducing tensors/hypermatrices that one cannot get from homomorphisms to $GL(n,\mathbb{C})$. (I suppose this can't work if any groups one can get from tensors/hypermatrices are all related to $GL(n,\mathbb{C})$ in an obvious way and if any analogues of ``trace" on these tensors are also related to matrix traces in an obvious way.)

I'm currently aware of http://galton.uchicago.edu/~lekheng/work/sjsu.pdf and http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.3830v1.pdf.

Thanks.

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    $\begingroup$ It is not directly related with yours question about "higher characters", but let me mention: There is series of papers in arXiv by Dolotin Morozov Shakirov et.al. e.g. arxiv.org/abs/hep-th/0609022 where they push-forward the following analogy - gl_n acts on vectors - we can consider automophims of higher tensor - and get "non-linear" gl_n. The point is that for this "non-linear" gl_n one may develop some kind of analogs of linear algebra theorems. $\endgroup$
    – Alexander Chervov
    Commented Nov 15, 2011 at 8:32
  • $\begingroup$ I don't really understand the question. Any group I can write down using tensors (at least in the obvious ways I currently have in mind) embeds into a general linear group, maybe not $\text{GL}(V)$ for $V$ some vector space but $\text{GL}(V \otimes V)$ or $\text{GL}(V \otimes V^{\ast})$ and so forth. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 15, 2011 at 18:46
  • $\begingroup$ A linear representation of group G is, by definition, a homomorphism $G \to GL(n,\Bbb{K})$ for some field $\Bbb{K}$. So all linear representations have this form, without doubt. Are you asking if any other representations, besides linear, are interesting? $\endgroup$
    – Anton Fetisov
    Commented Nov 15, 2011 at 20:25
  • $\begingroup$ Alexander Chervov, thank you very much for the interesting link. Thank you, KConrad, for the edit. Frankly I was indeed just hoping for some kind of "higher characters", not thinking too rigorously about this problem. Anyway, there is already a much better discussion of hypermatrices on Math Overflow here: mathoverflow.net/questions/48045/… $\endgroup$
    – Timothy Foo
    Commented Nov 17, 2011 at 2:35

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