TITLE: $SL(n) \times SL(n)$-invariants of $m$-tuples of matrices
QUESTION [6 upvotes]: I work over field of complex numbers. Let $G=SL(n) \times SL(n)$, and $(A,B) \in G$ acts on $m$-tuples of matrices $M_{n \times n}(\mathbb{C})^{\oplus m}$ as follows
$$
(A,B) \cdot (M_1, \ldots, M_m) \mapsto (A^{-1}M_1 B, \ldots, A^{-1}M_m B)
$$
Where can I find a description of the ring of invariants? For me it is very important that I consider $SL$ case (opposed to $GL$ case where this is just representations of a Kronecker quiver with  $n$ arrows and the answer is trivial.)

REPLY [2 votes]: The answer in the generality of quiver representations is given by

H. Derksen, J. Weyman, Semi-invariants of quivers and saturation for Littlewood-Richardson coecients, J. Amer. Math. Soc. 13 (2000), no. 3, 467--479. 
A. Schoeld, M. Van den Bergh, Semi-invariants of quivers for arbitrary dimension vectors, Indag. Math. (N.S.) 12 (2001), no. 1, 125--138.
M Domokos, A Zubkov, Semiinvariants of quivers as determinants.

In your special case, I can tell you the generators. Consider the expansion of
$$\det(\sum_{k=1}^m \Lambda_k \otimes M_k),$$ where $\Lambda_k$ is a $d\times d$-matrix of indeterminants, and $\otimes$ is the Kronecker product. Then all coefficients in the expansion are invariants. They generates all invariants if $d$ is big enough. In fact, for fixed $d$, the coefficients linearly span the space of degree $dn$ invariants.