Gauss decomposition for Chevalley groups, revisited
عنوان مقاله: Gauss decomposition for Chevalley groups, revisited
شناسه ملی مقاله: JR_THEGR-1-1_002
منتشر شده در در سال 1391
شناسه ملی مقاله: JR_THEGR-1-1_002
منتشر شده در در سال 1391
مشخصات نویسندگان مقاله:
A. Smolensky
B. Sury
N. Vavilov
خلاصه مقاله:
A. Smolensky
B. Sury
N. Vavilov
In the ۱۹۶۰'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 $sr(R)=۱$ or $asr(R)=۱$. Later the third author noticed that condition $sr(R)=۱$ 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 $sr(R)=۱$ is also Msufficient 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, $asr(R)=۱$, $sr(R,\Lambda)=۱$, etc., were only needed to guarantee that for simply connected groups $G=E$, rather than to verify the Gauss decomposition itself.
کلمات کلیدی: Chevalley groups, elementary Chevalley groups, triangular factorisations, rings of stable rank ۱, parabolic subgroups, Gauss decomposition, commutator width
صفحه اختصاصی مقاله و دریافت فایل کامل: https://civilica.com/doc/1198821/