$\mathrm A_2$-proof of structure theorems for Chevalley groups of type $\mathrm F_4$

Abstract:

A new geometric proof is given for the standard description of subgroups in the Chevalley group $G=G(\mathrm {F}_4,R)$ of type $\mathrm {F}_4$ over a commutative ring $R$ that are normalized by the elementary subgroup $E(\mathrm {F}_4,R)$. There are two major approaches to the proof of such results. Localization proofs (Quillen, Suslin, Bak) are based on a reduction in the dimension. The first proofs of this type for exceptional groups were given by Abe, Suzuki, Taddei and Vaserstein, but they invoked the Chevalley simplicity theorem and reduction modulo the radical. At about the same time, the first author, Stepanov, and Plotkin developed a geometric approach, decomposition of unipotents, based on reduction in the rank of the group. This approach combines the methods introduced in the theory of classical groups by Wilson, Golubchik, and Suslin with ideas of Matsumoto and Stein coming from representation theory and $K$-theory. For classical groups in vector representations, the resulting proofs are quite straightforward, but their generalizations to exceptional groups require an explicit knowledge of the signs of action constants, and of equations satisfied by the orbit of the highest weight vector. They depend on the presence of high rank subgroups of types $\mathrm {A}_l$ or $\mathrm {D}_l$, such as $\mathrm {A}_5\le \mathrm {E}_6$ and $\mathrm {A}_7\le \mathrm {E}_7$. The first author and Gavrilovich introduced a new twist to the method of decomposition of unipotents, which made it possible to give an entirely elementary geometric proof (the proof from the Book) for Chevalley groups of types $\Phi =\mathrm {E}_6,\mathrm {E}_7$. This new proof, like the proofs for classical cases, relies upon the embedding of $\mathrm {A}_2$. Unlike all previous proofs, neither results pertaining to the field case nor an explicit knowledge of structure constants and defining equations is ever used. In the present paper we show that, with some additional effort, we can make this proof work also for the case of $\Phi =\mathrm {F}_4$. Moreover, we establish some new facts about Chevalley groups of type $\mathrm {F}_4$ and their 27-dimensional representation.