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 $G=TU{U}^{-}U$, 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, $E=HU{U}^{-}U$, where $H=H(\Phi ,R)=T\cap E$. This surprising result shows that stronger conditions on the ground ring, such as being semi-local, $\asr(R)=1$, $\sr(R,Λ)=1$, etc., were only needed to guarantee that for simply connected groups $G=E$, rather than to verify the Gauss decomposition itself.

References in corpus (2)

• Unitriangular factorisations of Chevalley groups
• Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank >1 over noetherian rings

Cited by in corpus (5)

• Structure of Chevalley groups over rings via universal localization
• Commutator width in Chevalley groups
• Decompositions of congruence subgroups of Chevalley groups
• Commutator lengths in general linear group over a skew-field
• Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent