International Journal of Group Theory (Mar 2012)
Gauss decomposition for Chevalley groups, revisited
Abstract
In the 1960's Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo Suzuki, and Michael Stein discovered that Chevalley groups $G=G(Phi,R)$ over a semilocal ring admit remarkable Gauss decomposition $G=TUU^-U$, where $T=T(Phi,R)$ is a split maximal torus, whereas $U=U(Phi,R)$ and $U^-=U^-(Phi,R)$ are unipotent radicals of two opposite Borel subgroups $B=B(Phi,R)$ and $B^-=B^-(Phi,R)$ containing $T$. It follows from the classical work of Hyman Bass and Michael Stein that for classical groups Gauss decomposition holds under weaker assumptions such as $sr(R)=1$ or $asr(R)=1$. Later the third author noticed that condition $sr(R)=1$ is necessary for Gauss decomposition. Here, we show that a slight variation of Tavgen's rank reduction theorem implies that for the elementary group $E=E(Phi,R)$ condition $sr(R)=1$ is also sufficient for Gauss decomposition. In other words, $E=HUU^-U$, where $H=H(Phi,R)=Tcap E$. This surprising result shows that stronger conditions on the ground ring, such as being semi-local, $asr(R)=1$, $sr(R,Lambda)=1$, etc., were only needed to guarantee that for simply connected groups $G=E$, rather than to verify the Gauss decomposition itself.
