GAUSS DECOMPOSITION FOR CHEVALLEY GROUPS, REVISITED

Abstract. In the 1960ʹs Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo Suzuki, and Michael Stein discovered that Chevalley groups G = G(;R) over a semilocal ring admit remarkable Gauss decomposition G = TUU^-U, where T = T(;R) is a split maximal torus, whereas U = U(;R) and U^- = U^-(;R) are unipotent radicals of two opposite Borel subgroups B = B(;R) and B^- = B^-(;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 Tavgenʹs rank reduction theorem implies that for the elementary group E = E(;R) condition sr(R) = 1 is also sufficient for Gauss decomposition. In other words, E = HUU^-U, where H = H(;R) = T \ E. This surprising result shows that stronger conditions on the ground ring, such as being semi-local, asr(R) = 1, sr(R; ) = 1, etc., were only needed to guarantee that for simply connected groups G = E, rather than to verify the Gauss decomposition itself.