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.