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.)

$\begingroup$ What precisely is the (trivial) answer in the $GL$ case? $\endgroup$ – Mikhail Borovoi Apr 19 '14 at 19:44

$\begingroup$ For quivers without oriented cycles any invariant function is constant, so by trivial answer I mean basic field. $\endgroup$ – Sasha Pavlov Apr 21 '14 at 10:50

$\begingroup$ I think you can find the answer to your question in the book "The Classical Groups: Their Invariants and Representations" by Hermann Weyl (I don't have this book on my table). $\endgroup$ – Mikhail Borovoi Apr 22 '14 at 12:32

$\begingroup$ The determinants of your matrices $M_1$, ... $M_m$ are polynomial invariants in the $SL$ case, and a natural guess would be that the algebra of polynomial invariants is generated by these determinants. Is this correct? $\endgroup$ – Mikhail Borovoi Apr 22 '14 at 13:49

$\begingroup$ We can do even more general constriction: lets take coefficients of polynomial $det(z_1 M_1+\ldots+z_m M_m)$, they are all invariant and determinants correspond to coefficients of $z_i^n$. But simple examples for small $m$ and $n$ shows that there are other invariants, not covered by this construction. $\endgroup$ – Sasha Pavlov Apr 26 '14 at 0:49
I think the result in general is unknown. I have tried the case m = n = 3 (arXiv:0906.5525v2, sorry in french). Bruno Blind

$\begingroup$ The present question should be a bit more approachable than what you studied in your article. Here the actions on the left and the right are with two different groups. $\endgroup$ – Abdelmalek Abdesselam Apr 29 '14 at 14:08
The answer in the generality of quiver representations is given by
H. Derksen, J. Weyman, Semiinvariants of quivers and saturation for LittlewoodRichardson coecients, J. Amer. Math. Soc. 13 (2000), no. 3, 467479.
A. Schoeld, M. Van den Bergh, Semiinvariants of quivers for arbitrary dimension vectors, Indag. Math. (N.S.) 12 (2001), no. 1, 125138.
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.
My answer to MO question Invariant polynomials for a product of algebraic groups might help a bit. Here there should be invariants of degree $d$ if and only if $d$ is divisible by $n$. For $d=kn$, a linearly generating set can be given in terms of bipartite graphs with colored edges. The vertex set $V$ with $2k$ elements is partitioned into $I$, $J$, each with $k$ elements. All vertices have degree $n$. Finally the edges are colored with $m$ colors. The corresponding invariant is obtained as in my MO answer above by assigning tensors $\epsilon_{a_1,\ldots,a_n}$ to each vertex of $I$, assigning tensors $\eta_{b_1,\ldots,b_n}$ to the vertices of $J$ and contracting indices as indicated by the graph. However one needs to put in the middle of an $a$$b$ contraction a matrix element $(M_c)_{a,b}$ where $c$ is the color of that edge. The invariants obtained from the multilinear expansion of ${\rm det}(z_1 M_1+\cdots+z_m M_m)$ correspond to the graphs with $k=1$, i.e., just two vertices with a multiple edge repeated $n$ times.
The above is just a consequence of the CayleyClebsch Theorem, a.k.a, the first fundamental theorem of invariant theory for $SL_n$. Now one should be able to reduce this infinite generating set using the GrassmannPlücker relation and the straightening algorithm, but I don't know how complicated this gets in the present situation. A possible entry point in this business: Chapters 8 and 9 from the book "Young Tableaux" by Fulton. You can also search with the keywords "standard monomial theory".