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Gauss decomposition for Chevalley groups, revisited | ||
| International Journal of Group Theory | ||
| مقاله 2، دوره 1، شماره 1، خرداد 2012، صفحه 3-16 اصل مقاله (342.75 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2012.467 | ||
| نویسندگان | ||
| A. Smolensky؛ B. Sury؛ N. Vavilov* | ||
| چکیده | ||
| 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 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, $asr(R)=1$, $sr(R,\Lambda)=1$, etc., were only needed to guarantee that for simply connected groups $G=E$, rather than to verify the Gauss decomposition itself. | ||
| کلیدواژهها | ||
| Chevalley groups؛ elementary Chevalley groups؛ triangular factorisations؛ rings of stable rank 1؛ parabolic subgroups؛ Gauss decomposition؛ commutator width | ||
| مراجع | ||
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