Gauss decomposition for Chevalley groups, revisited

• Main Authors: A. Smolensky, B. Sury, N. Vavilov
• Format: Article
• Language: English
• Published: University of Isfahan 2012-03-01
• Series: International Journal of Group Theory
• Subjects: Chevalley groups; elementary Chevalley groups; triangular factorisations; rings of stable rank 1; parabolic subgroups; Gauss decomposition; commutator width
• Online Access: http://www.theoryofgroups.ir/?_action=showPDF&article=467&_ob=05f124c20f62a8f21e87d93a386d75b7&fileName=full_text.pdf
• ISSN: 2251-7650; 2251-7669

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.