GAUSS DECOMPOSITION FOR CHEVALLEY GROUPS, REVISITED

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SMOLENSKY A.; SURY B.; VAVILOV N.

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Journal: INTERNATIONAL JOURNAL OF GROUP THEORY Year:2012 | Volume:1 | Issue:1 Page(s): 3-16

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Title

GAUSS DECOMPOSITION FOR CHEVALLEY GROUPS, REVISITED

Author(s)

SMOLENSKY A. | SURY B. | VAVILOV N. | Issue Writer Certificate

Keywords

CHEVALLEY GROUPSQ4

ELEMENTARY CHEVALLEY GROUPSQ4

TRIANGULAR FACTORISATIONSQ4

RINGS OF STABLE RANK 1Q4

PARABOLIC SUBGROUPSQ4

GAUSS DECOMPOSITIONQ4

COMMUTATOR WIDTHQ4

Abstract

In the 1960's Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo Suzuki, and Michael Stein discovered that CHEVALLEY GROUPS G = G (F, R) over a semilocal ring admit remarkable GAUSS DECOMPOSITION G = TUU-U, where T = T (F, R) is a split maximal torus, whereas U = U (F, R) and U- = U- (F, R) are unipotent radicals of two opposite Borel subgroups B = B (F, R) and B- = B- (F, 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 (F, R) condition sr (R) =1 is also sufficient for GAUSS DECOMPOSITION. In other words, E = HUU-U, where H = H (F, R) =T Ç E. This surprising result shows that stronger conditions on the ground ring, such as being semi-local, asr (R) = 1, sr (R, L) = 1, etc., were only needed to guarantee that for simply connected groups G = E, rather than to verify the GAUSS DECOMPOSITION itself.

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APA: Copy

SMOLENSKY, A., SURY, B., & VAVILOV, N.. (2012). GAUSS DECOMPOSITION FOR CHEVALLEY GROUPS, REVISITED. INTERNATIONAL JOURNAL OF GROUP THEORY, 1(1), 3-16. SID. https://sid.ir/paper/212626/en

Vancouver: Copy

SMOLENSKY A., SURY B., VAVILOV N.. GAUSS DECOMPOSITION FOR CHEVALLEY GROUPS, REVISITED. INTERNATIONAL JOURNAL OF GROUP THEORY[Internet]. 2012;1(1):3-16. Available from: https://sid.ir/paper/212626/en

IEEE: Copy

A. SMOLENSKY, B. SURY, and N. VAVILOV, “GAUSS DECOMPOSITION FOR CHEVALLEY GROUPS, REVISITED,” INTERNATIONAL JOURNAL OF GROUP THEORY, vol. 1, no. 1, pp. 3–16, 2012, [Online]. Available: https://sid.ir/paper/212626/en

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