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<br /> ‎$B^-=B^-(Phi,R)$ 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 M‎sufficient 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.