We prove that Hori-Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.

Etant donnés $A_1,\cdots ,A_s$ ($s\ge 3$) des $S{L}_2\left(ℂ\right)$-modules non triviaux de dimensions respectives $n_1+1\ge \cdots \ge n_s+1$ (avec $n_1=n_2+\cdots +n_s$) et $\phi \in ℒ(A_2\otimes \cdots \otimes A_s,A_1^{*})$ un $S{L}_2\left(ℂ\right)$-homomorphisme, nous montrons que l’hyperdéterminant de $\phi$ est nul sauf si les modules $A_i$ sont irréductibles et si l’homomorphisme est la multiplication des polynômes homogènes à deux variables.

We prove that the exceptional complex Lie group ${\mathrm{F}}_4$ has a transitive action on the hyperplane section of the complex Cayley plane ${𝕆\mathbb{P}}^2$. Although the result itself is not new, our proof is elementary and constructive. We use an explicit realization of the vector and spin actions of $\mathrm{Spin}(9,ℂ)\le {\mathrm{F}}_4$. Moreover, we identify the stabilizer of the ${\mathrm{F}}_4$-action as a parabolic subgroup ${\mathrm{P}}_4$ (with Levi factor ${\mathrm{B}}_3{\mathrm{T}}_1$) of the complex Lie group ${\mathrm{F}}_4$. In the real case we obtain an analogous realization of ${{\mathrm{F}}_4}^{(-20)}/\mathbb{P}$.