smaller numbers). Since we take a diagrammatic approach, we first explain the diagrams that follow. Each bullet represents a one dimensional $k$ vector space, a southwest arrow “$\swarrow$” connecting two bullets corresponds to the action of $a$ and maps one bullet to the other in the indicated direction, and similarly the south east arrows “$\searrow$” correspond to the action of $b$. If no arrow emanates from a bullet in given direction, then the corresponding linear action is understood to be zero.

*(Kronecker, Weierstrass, Basev, Gelfand, Ponomarev, Conlon, Heller, Reiner, Benson)* [@Gel-Pon; @conlon; @hell-rei; @basev; @ben-1] Let $k$ be a field of characteristic $2$. Every isomorphism class of an indecomposable $V_4$ representation over $k$ is precisely one of the following.

1.  The projective indecomposable module $kV_4$ of dimension 4. $$\xymatrix@=2em{
         & \bullet \ar[dl]_a \ar[dr]^b & \\
         \bullet \ar[dr]&  & \bullet \ar[dl]\\
         & \bullet &
        }$$

2.  The (non-projective) indecomposable even dimensional modules:

    1.  For each even dimension $2n$ and an indecomposable rational canonical from corresponding to the power of an irreducible monic polynomial $f(x)^l = \sum_{i=0}^n \theta_i x^i$, $(\theta_n = 1)$ there is an indecomposable representation given by $$\xymatrix@=1.5em{
            &  & \overset{g_{n-1}}{\bullet} \ar@{..>}[d] \ar[dr]& & \overset{g_{n-2}}{\bullet}\ar[dl]^a \ar[dr] &  & \overset{g_{n-3}}{\bullet} \ar[dl]  