Thursday, 31 March 2016

Which algebraic groups are generated by (lifts of) reflections?

The Cartan-Dieudonne theorem
states that each element $g in O(V)$, where $V$ is a quadratic space of dimension $n$ over a field of characteristic $neq 2$, can be written as a product of $leq n$ reflections.



Something similar is true for $SL_n(k)$ for $char k neq 2$: each element $g$ can be written as a product of elements of order $4.$ Indeed, it suffices to prove this for $n=2$. Then $s_t:=begin{pmatrix} 0 & t \ -frac 1t & 0 end{pmatrix}$ is of order $4$ . Let $h(t):=s_ts_{-1}=begin{pmatrix} t & 0 \ 0 & frac 1t end{pmatrix}$, then each element in $U$, the group of upper triangular matrices, is a product of two conjugates of elements $h(t),h(t')$ (provided $|k|geq 4$). Similarly for the lower triangular matrices $V$, and then $SL_2$ is generated by these subgroups.



Since a simply connected semisimple $k$-split group is generated by $SL_2$'s, the same argument applies.



What can we say about other algebraic groups? Clearly unipotent groups (in characteristic 0) don't have elements of finite order, so nothing there.



What about (the k-rational points of) anisotropic groups?

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