A special Debarre-Voisin fourfold

Consider the finite simple group $\mathbf{G}:=\mathrm{P}\mathrm{S}\mathrm{L}(2,{\mathbf{F}}_{11})$ of order 660, which has an irreducible representation $V_{10}$ of dimension 10. In this note, we study a special trivector ${\sigma }_0\in {\bigwedge }^3{V}_{10}^{\vee }$ that is $\mathbf{G}$-invariant. Following the construction of Debarre-Voisin, we obtain a smooth hyperk"ahler fourfold $X_6^{{\sigma }_0}\subset \mathrm{G}\mathrm{r}(6,V_{10})$ with many symmetries. We will also look at the associated Peskine variety $X_1^{{\sigma }_0}\subset \mathbf{P}(V_{10})$, which is highly symmetric as well and admits 55 isolated singular points. It will help us to better understand the geometry of the special Debarre-Voisin fourfold $X_6^{{\sigma }_0}$. We also discuss an application of this example to the global geometry of the moduli space of Debarre-Voisin fourfolds.