Twisted Gelfand-Ponomarev modules

In this expository paper, given a field $K$ and two automorphisms $\sigma ,\tau \in \mathrm{A}\mathrm{u}\mathrm{t}(K)$, we give a self-contained proof of the classification of finite dimensional $K$-vector spaces equipped with two operators $F$ and $V$, respectively $\sigma$-linear and $\tau$-linear, such that $FV=VF=0$. This classification was originally due to the combined results of Gelfand and Ponomarev (1968), and of Kraft (1975). Following a recent suggestion of Chai (2025), we reworked their classification in light of the notion of Kraft quivers. As an application, we generalize and give an algebraic proof of a theorem by Kottwitz and Rapoport concerning the existence of $F$-crystals.