Extremal Derived Length in the Unipotent Radicals of Cartan-Type Automorphism Groups

Abstract

In every group $G^{(j)}\subseteq\gamma_{2^j}(G)$. We exhibit a natural family in which this inclusion is an equality at every step. Let $k$ be a perfect field of characteristic $p\ge 5$ and let $U_S$ be the unipotent radical of the automorphism group of the height-one special algebra $S(n;1)^{(1)}$, $n\ge 3$, so that $\mathrm{Aut}(S(n;1)^{(1)})_{\mathrm{red}}^{\circ}\simeq U_S\rtimes\mathrm{GL}_n$. Writing $D=n(p-1)$, we prove $U_S^{(j)}=\gamma_{2^j}(U_S)$ for all $j$, so that $U_S$ has nilpotency class $D-2$ and the largest derived length its class permits, namely $\lceil\log_2(D-1)\rceil$. The same holds for $U_S(\mathbf{F}_q)$, whose Frattini quotient and lower $p$-central series are determined in closed form. The mechanism is a congruence filtration whose graded Lie algebra has exact bracket generation outside finitely many degrees, the exceptions governed by a single Cartier quotient, which occurs both in the abelianization and in the five-term homology sequence. The non-smooth automorphism scheme contains a canonical Frobenius thickening of $U_S$, whose Verschiebung filtration and distribution algebra are computed. At arbitrary divided-power height, Cartier-Verschiebung morphisms on the de Rham complexes induce explicit maps on restricted derivations. In height one we determine both series for the Witt family in rank at least two and for the Hamiltonian family, and identify the contact automorphism group scheme with the coordinate contact stabilizer. For the Witt radical the congruence filtration is itself the lower central series, and $\mathrm{cl}(U_W)=D-1$, $\mathrm{dl}(U_W)=\lceil\log_2 D\rceil$. For the Hamiltonian radical it is cut out by the flux character and the socle line; with $D_H=2r(p-1)$, $r\ge 2$, one has $\mathrm{cl}(U_H)=D_H-3$, $\mathrm{dl}(U_H)=\lceil\log_2(D_H-2)\rceil$. Thus all three attain the largest derived length their classes permit.