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If you are willing to replace $SL$ by $GL$ (and none of the comments/answers so far have mentioned without the determinant) then this is a special case of a much more general result.

Apply the theory in the following paper to the quiver with one vertex and $m$ edges.

MR0958897 (90e:16048) Le Bruyn, Lieven; Procesi, Claudio

Semisimple representations of quivers.

Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598.

I believe that the

The special case you are considering was studied before this. The reference has already been given by Agol.

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If you are willing to replace $SL$ by $GL$ (and none of the comments/answers so far have mentioned the determinant) then this is a special case of a much more general result.

Apply the theory in the following paper to the quiver with one vertex and $m$ edges.

MR0958897 (90e:16048) Le Bruyn, Lieven; Procesi, Claudio

Semisimple representations of quivers.

Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598.

I believe that the special case you are considering was studied before this.