Let $G$ be a finite group and $p$ be a prime.
Moreover, let $k$ be a finite field of characteristic $p$, such that $p$ divides $|G|$. Furthermore suppose that $k$ is a splitting field for $G$.
Suppose that there is already stored a finite list $L=[N_1,... ,N_s]$ of indecomposable $kG$-modules, such that they are all non-isomorphic to each other.
Let $M$ be a new decomposable $kG$-module.
I know a priori that $M$ decomposes as a direct sum of indecomposable $kG$-modules $M\cong \bigoplus\limits_{i=1}^{n} {M_i}$ such that
$M_n$ is a new $kG$-module I know nothing about (except that it is indecomposable)
all other $M_i$ are isomorphic to a module $N_j$ (for some $j$) occurring in the list $L$ from above.
Example:
$M\cong M_1\oplus M_2 \oplus M_3 \oplus M_4$ and $M_1\cong N_5$, $M_2\cong N_5$, $M_3\cong N_7$.
$M_4$ is the new module I know nothing about.
Also, I had to test with the MAGMA command "IsIsomorphic" which modules of the list $L$ my direct summands $M_1$, $M_2$ and $M_3$ of $M$ are isomorphic to.
Question:
Is there an easier way to chop the modules $N_j$ off the module $M$ (with noting the multiplicities) than to decompose $M$ into all its indec. dir. summands and to test with the MAGMA command "IsIsomorphic" for every $M_i$ and $N_j$?
I'm asking this, because it costs a lot of time and memory, if $M$ is very big.
Thank you very much for the help.
Here's an idea that I've used, in an ad hoc way, in similar situations. I think it helped, although I can't claim to have rigorously tested how much more efficient it is.
Suppose an indecomposable module $N$ is isomorphic to a direct summand of a module $M$. Then if you choose random homomorphisms $\alpha:N\to M$ and $\beta:M\to N$, there is a reasonable probability (at least $(|k|-1)^2/|k|^2$) that $\beta\alpha$ is an isomorphism, in which case $M\cong N\oplus\ker(\beta)$.
So you could try to make $M$ smaller by randomly splitting off summands that are isomorphic to the indecomposables $N_1,\dots,N_s$.