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 second 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 $E(\Phi,R)$ condition $\sr(R)=1$ is also sufficient for Gauss decomposition. In other words, $E=HUU^-U$, where $H=H(\Phi,R)=T\cap E$. 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.