Indecomposable Modules over F2(C2 x C2) – Group Theory and Representation Theory

Question

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?

Answer 1

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.