Let $C\_2$ be the cyclic group of order $2$ and $\mathbb{F}\_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}\_2 (C\_2\times C\_2)$. It is well-known that $A$ has infinite representation type. Is there a classification of the finite dimensional indecomposable $A$-modules (and the Auslander-Reiten quiver of $\text{mod}\,A$) in this case?



What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$?
A complete description of the indecomposable modules for $C\_2\times C\_2$, including the Auslander-Reiten quiver is available in David Benson's book *Modular Representation Theory: New Trends and Methods*, Springer 1984, pp.176-181. This is over $\bar{\mathbb{F}}\_2$; the result over arbitrary fields is given in the following few pages.


