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Gauss decomposition for Chevalley groups, revisited
In the 1960's Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo
Suzuki, and Michael Stein discovered that Chevalley groups over a
semilocal ring admit remarkable Gauss decomposition , where
is a split maximal torus, whereas and
are unipotent radicals of two opposite Borel subgroups
and containing . 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 condition
\sr(R)=1 is also sufficient for Gauss decomposition. In other words,
, where . 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 , rather than to verify the Gauss decomposition itself