Gauss decomposition for Chevalley groups, revisited

Document Type : Research Paper



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.


Main Subjects

E. Abe (1969). Chevalley groups over local rings. Tohoku Math. J.. 21 (3), 474-494 E. Abe, K. Suzuki (1976). On normal subgroups of Chevalley groups over commutative rings. Tohoku Math. J.. 28 (1), 185-198 Zheng Bodong, You Hong (2002). Products of commutators of transvections over local rings. Linear Algebra Applications. 357, 45-57 You Hong (1994). Commutators and unipotents in symplectic groups. Acta Math. Sinica, New Ser.. 10, 173-179 N. A. Vavilov, A. V. Smolensky, B. Sury (2011). Unitriangular factorisations of Chevalley groups. Zapiski Nauchn. Semin. POMI. 388, 17-47 N. A. Vavilov, S. S. Sinchuk (2011). Parabolic factorisations of the split classical groups. Algebra and Analysis. 23 (4), 1-30 N. A. Vavilov, S. S. Sinchuk (2010). Dennis---Vaserstein type decomposition. Zapiski Nauchn. Semin. POMI. 375, 48-60 N. Vavilov, E. Plotkin (1996). Chevalley groups over commutative rings. I. Elementary calculations. Acta Applicandae Math.. 45, 73-115 N. A. Vavilov, E. B. Plotkin (1984). Net subgroups of Chevalley groups. II. Gauss decomposition. Journal of Soviet Mathematics. 27 (4), 2874-2885 N. A. Vavilov, E. I. Kovach (2011). SL2-factorisations of Chevalley groups. Zapiski Nauchn. Semin. POMI. N. Vavilov (1991). Structure of Chevalley groups over commutative rings. In: Proc. Conf. Nonassociatve algebras and related topics (Hiroshima -- 1990) , World Sci. Publ., London. , 219-335 N. A. Vavilov (1984). Parabolic subgroups of Chevalley groups over a commutative ring. Journal of Soviet Mathematics. 26 (3), 1848-1860 L. N. Vaserstein, E. Wheland (1990). Commutators and companion matrices over rings of stable rank 1. Linear Algebra Appl.. 142, 263-277 L. N. Vaserstein, E. Wheland (1990). Factorization of invertible matrices over rings of stable rank one. J. Austral. Math. Soc., Ser. A. 48, 455-460 L. N. Vaserstein (1984). Bass's first stable range condition. J. Pure Appl. Algebra. 34 (2--3), 319-330 O. I. Tavgen (1992). Bounded generation of normal and twisted Chevalley groups over the rings of S-integers. Contemp. Math.. 131 (1), 409-421 O. I. Tavgen, (1990). Bounded generation of Chevalley groups over rings of S-integer algebraic numbers. Izv. Acad. Sci. USSR. 54 (1), 97-122 A. A. Suslin, M. S. Tulenbaev (1981). A theorem on stabilization for Milnor's K2-functor. J. Soviet Math.. 17, 1804-1819 A. Stepanov, N. Vavilov (2011). On the length of commutators in Chevalley groups. Israel Math. J.. 185, 253-276 R. Steinberg (1967). Lectures on Chevalley groups. Yale University. M. R. Stein (1978). Stability theorems for K1, K2 and related functors modeled on Chevalley groups. Japan J. Math.. 4 (1), 77-108 M. R. Stein (1973). Surjective stability in dimension 0 for K2 and related functors. Trans. Amer. Math. Soc.. 178, 176-191 A. Sivatski, A. Stepanov (1999). On the word length of commutators in GLn(R). K-theory. 17, 295-302 S. Sinchuk, N. Vavilov, Parabolic factorisations of exceptional Chevalley groups. S. Sinchuk Injective stability of unitary K1 revisited. R. W. Sharpe (1981). On the structure of the Steinberg group St(Lambda). J. Algebra. 68, 453-467 R. W. Sharpe (1972). On the structure of the unitary Steinberg group. Ann. Math.. 96 (3), 444-479 A. S. Rapinchuk, I. A. Rapinchuk Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank >1 over Noetherian rings. K. R. Nagarajan, M. P. Devaasahayam, T. Soundararajan (2002). Products of three triangular matrices over commutative rings. Linear Algebra Applic.. 348, 1-6 M. R. Murty, K.L. Petersen The generalized Artin conjecture and arithmetic orbifolds. D. W. Morris (2007). Bounded generation of SL(n,A) after D. Carter, G. Keller, and E. Paige. New York J. Math.. 13, 383-421 H. Matsumoto (1969). Sur les sous-groupes arithmetiques des groupes semi-simples deployes. Ann. Sci. Ecole Norm. Sup. (4). 2, 1-62 O. Loos (1995). Elementary Groups and stability for Jordan pairs. K-Theory. 9, 77-116 M. Liebeck, L. Pyber (2001). Finite linear groups and bounded generation. Duke Math. J.. 107, 159-171 M. Liebeck, E. A. O'Brien, A. Shalev, Pham Huu Tiep, (2010). The Ore conjecture. J. Europ. Math. Soc.. 12, 939-1008 W. van der Kallen (1982). SL3(C[x]) does not have bounded word length. Springer Lecture Notes Math.. 966, 357-361 W. van der Kallen (1976). Injective stability for K2. Springer Lecture Notes Math.. 551, 77-154 N. Iwahori, H. Matsumoto (1965). On some Bruhat decomposition and structure of the Hecke rings of p-adic Chevalley groups. Publ. Math. Inst. Haut. Etudes Sci.. 25, 5-48 N. Gordeev, J. Saxl (2005). Products of conjugacy classes in Chevalley groups over local rings. St. Petersburg Math. J.. 17 (2), 96-107 D. R. Estes, J. Ohm (1967). Stable range in commutative rings. J. Algebra. 7 (3), 343-362 E. Ellers, N. Gordeev (1998). On the conjectures of J. Thompson and O. Ore. Trans. Amer. Math. Soc.. 350, 3657-3671 R. K. Dennis, L. N. Vaserstein (1989). Commutators in linear groups. K-theory. 2, 761-767 R. K. Dennis, L. N. Vaserstein (1988). On a question of M. Newman on the number of commutators. J. Algebra. 118, 150-161 V. Chernousov, E. Ellers, N. Gordeev (2000). Gauss decomposition with prescribed semisimple part: short proof. J. Algebra. 229, 314-332 Chen Huanyin, Chen Miaosen (2004). On products of three triangular matrices over associative rings. Linear Algebra Applic.. 387, 297-311 R. W. Carter (1972). Simple groups of Lie type. Wiley, London et al.,. D. Carter, G. Keller (1984). Elementary expressions for unimodular matrices. Commun. Algebra. 12, 379-389 D. Carter, G. Keller (1983). Bounded elementary generation of SLn(O). Amer. J. Math.. 195, 673-687 Z. I. Borewicz (1976). Parabolic subgroups of linear groups over a semilocal ring. Vestn. Leningr. Univ.. 13, 16-24 H. Bass (1964). K-theory and stable algebra. Publ. Math. Inst. Hautes Etudes Sci.. 22, 5-60 A. Bak, Guoping Tang (2000). Stability for hermitian K1. J. Pure Appl. Algebra. 150, 107-121 A. Bak, V. Petrov, Guoping Tang (2003). Stability for quadratic K1. K-Theory. 30 (1), 1-11 L. Babai, N. Nikolov, L. Pyber (2008). Product growth and mixing in finite groups. In: 19th Annual ACM--SIAM Symposium on Discrete Algorithms, ACM--SIAM. , 248-257 F. A. Arlinghaus, L. N. Vaserstein, You Hong, (1995). Commutators in pseudo-orthogonal groups. J. Austral. Math. Soc., Ser. A. 59, 353-365