Parabolic factorizations of split classical groups

Vavilov, N.; Sinchuk, S.

doi:10.1090/S1061-0022-2012-01211-2

Parabolic factorizations of split classical groups
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by N. A. Vavilov and S. S. Sinchuk
Translated by: the authors

St. Petersburg Math. J. 23 (2012), 637-657

DOI: https://doi.org/10.1090/S1061-0022-2012-01211-2

Published electronically: April 13, 2012

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Abstract:

An analog of the Dennis–Vaserstein decomposition is proved for an arbitrary pair of maximal parabolic subgroups $P_r$ and $P_s$ in split classical groups, under appropriate stability conditions. Before, such decompositions were only known for pairs of terminal parabolic subgroups.

References

Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629, DOI 10.1007/978-3-540-89394-3
N. A. Vavilov, Subgroups of the general linear group over a ring that contains the group of block triangular matrices. II, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. (1982), 5–9, 118 (Russian, with English summary). MR 672588
N. A. Vavilov, Parabolic subgroups of the Chevalley group over a commutative ring, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 116 (1982), 20–43, 161 (Russian). Integral lattices and finite linear groups. MR 687837
N. A. Vavilov and E. B. Plotkin, Net subgroups of Chevalley groups. II. Gaussian decomposition, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 114 (1982), 62–76, 218–219 (Russian). Modules and algebraic groups. MR 669560
N. A. Vavilov and S. S. Sinchuk, Decompositions of Dennis-Vaserstein type, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 375 (2010), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 19, 48–60, 210 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 171 (2010), no. 3, 331–337. MR 2749274, DOI 10.1007/s10958-010-0138-0
L. N. Vaseršteĭn, On the stabilization of the general linear group over a ring, Math. USSR-Sb. 8 (1969), 383–400. MR 0267009
L. N. Vaseršteĭn, Stabilization of unitary and orthogonal groups over a ring with involution, Mat. Sb. (N.S.) 81 (123) (1970), 328–351 (Russian). MR 0269722
L. N. Vaseršteĭn, The stable range of rings and the dimension of topological spaces, Funkcional. Anal. i Priložen. 5 (1971), no. 2, 17–27 (Russian). MR 0284476
L. N. Vaseršteĭn, Stabilization for the classical groups over rings, Mat. Sb. (N.S.) 93(135) (1974), 268–295, 327 (Russian). MR 0338208
V. A. Petrov, Odd unitary groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 305 (2003), no. Vopr. Teor. Predst. Algebr. i Grupp. 10, 195–225, 241 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 130 (2005), no. 3, 4752–4766. MR 2033642, DOI 10.1007/s10958-005-0372-z
—, Overgroups of classical groups, Kand. diss., S.-Peterburg. Gos. Univ., St. Petersburg, 2005, pp. 1–129. (Russian)
E. B. Plotkin, Net subgroups of Chevalley groups and questions of stabilization of the $K_1$-functor, Kand. diss., Leningrad. Gos. Univ., Leningrad, 1985, pp. 1–118. (Russian)
E. B. Plotkin, Surjective stabilization of the $K_1$-functor for some exceptional Chevalley groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 198 (1991), no. Voprosy Teor. Predstav. Algebr Grupp. 2, 65–88, 112 (Russian); English transl., J. Soviet Math. 64 (1993), no. 1, 751–766. MR 1164860, DOI 10.1007/BF02988480
A. V. Stepanov, Conditions for the stability in the theory of linear groups over rings, Kand. diss., Leningrad Gos. Univ., Leningrad, 1987, pp. 1–112. (Russian)
A. A. Suslin and M. S. Tulenbaev, A theorem on stabilization for Milnor’s $K_{2}$-functor, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 64 (1976), 131–152, 162 (Russian). Rings and modules. MR 0457525
Eiichi Abe and Kazuo Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tohoku Math. J. (2) 28 (1976), no. 2, 185–198. MR 439947, DOI 10.2748/tmj/1178240833
A. Bak, The stable structure of quadratic modules, Thesis, Columbia Univ., 1969.
Anthony Bak, Viktor Petrov, and Guoping Tang, Stability for quadratic $K_1$, $K$-Theory 30 (2003), no. 1, 1–11. Special issue in honor of Hyman Bass on his seventieth birthday. Part I. MR 2061845, DOI 10.1023/B:KTHE.0000015340.00470.a9
Anthony Bak and Tang Guoping, Stability for Hermitian $K_1$, J. Pure Appl. Algebra 150 (2000), no. 2, 109–121. MR 1765866, DOI 10.1016/S0022-4049(99)00035-3
Anthony Bak and Nikolai Vavilov, Structure of hyperbolic unitary groups. I. Elementary subgroups, Algebra Colloq. 7 (2000), no. 2, 159–196. MR 1810843, DOI 10.1007/s100110050017
H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604
Hyman Bass, Unitary algebraic $K$-theory, Algebraic $K$-theory, III: Hermitian $K$-theory and geometric applications (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 343, Springer, Berlin, 1973, pp. 57–265. MR 0371994
Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
R. Keith Dennis, Stability for $K_{2}$, Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972) Lecture Notes in Math., Vol. 353, Springer, Berlin, 1973, pp. 85–94. MR 0347892
R. K. Dennis and L. N. Vaserstein, On a question of M. Newman on the number of commutators, J. Algebra 118 (1988), no. 1, 150–161. MR 961333, DOI 10.1016/0021-8693(88)90055-5
Igor V. Erovenko, $\textrm {SL}_n(F[x])$ is not boundedly generated by elementary matrices: explicit proof, Electron. J. Linear Algebra 11 (2004), 162–167. MR 2111520, DOI 10.13001/1081-3810.1129
Dennis Estes and Jack Ohm, Stable range in commutative rings, J. Algebra 7 (1967), 343–362. MR 217052, DOI 10.1016/0021-8693(67)90075-0
G. Habdank, A classification of subgroups of $\Lambda$-quadratic groups normalized by relative elementary subgroups, Dissertation Universität Bielefeld, 1987, pp. 1–71.
Günter Habdank, A classification of subgroups of $\Lambda$-quadratic groups normalized by relative elementary groups, Adv. Math. 110 (1995), no. 2, 191–233. MR 1317615, DOI 10.1006/aima.1995.1008
Alexander J. Hahn and O. Timothy O’Meara, The classical groups and $K$-theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, Springer-Verlag, Berlin, 1989. With a foreword by J. Dieudonné. MR 1007302, DOI 10.1007/978-3-662-13152-7
Roozbeh Hazrat and Nikolai Vavilov, Bak’s work on the $K$-theory of rings, J. K-Theory 4 (2009), no. 1, 1–65. MR 2538715, DOI 10.1017/is008008012jkt087
Wilberd van der Kallen, Injective stability for $K_{2}$, Algebraic $K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976) Lecture Notes in Math., Vol. 551, Springer, Berlin, 1976, pp. 77–154. MR 0506636
Wilberd van der Kallen, $\textrm {SL}_{3}(\textbf {C}[X])$ does not have bounded word length, Algebraic $K$-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 357–361. MR 689383
Max-Albert Knus, Quadratic and Hermitian forms over rings, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 294, Springer-Verlag, Berlin, 1991. With a foreword by I. Bertuccioni. MR 1096299, DOI 10.1007/978-3-642-75401-2
Manfred Kolster, On injective stability for $K_{2}$, Algebraic $K$-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 128–168. MR 689373
B. A. Magurn, W. van der Kallen, and L. N. Vaserstein, Absolute stable rank and Witt cancellation for noncommutative rings, Invent. Math. 91 (1988), no. 3, 525–542. MR 928496, DOI 10.1007/BF01388785
Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1–62 (French). MR 240214
Viktor Petrov, Overgroups of unitary groups, $K$-Theory 29 (2003), no. 3, 147–174. MR 2028500, DOI 10.1023/B:KTHE.0000006934.95243.91
E. B. Plotkin, Stability theorems of $K_1$-functor for Chevalley groups, Nonassociative algebras and related topics (Hiroshima, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 203–217. MR 1150261
Eugene Plotkin, On the stability of the $K_1$-functor for Chevalley groups of type $E_7$, J. Algebra 210 (1998), no. 1, 67–85. MR 1656415, DOI 10.1006/jabr.1998.7535
E. Plotkin, M. R. Stein, and N. Vavilov, Stability of $K$-functors modeled on Chevalley groups, revisited, 2001 (to appear).
R. W. Sharpe, On the structure of the unitary Steinberg group, Ann. of Math. (2) 96 (1972), 444–479. MR 320076, DOI 10.2307/1970820
R. W. Sharpe, On the structure of the Steinberg group $\textrm {St}(\Lambda )$, J. Algebra 68 (1981), no. 2, 453–467. MR 608545, DOI 10.1016/0021-8693(81)90274-X
J. T. Stafford, Stable structure of noncommutative Noetherian rings, J. Algebra 47 (1977), no. 2, 244–267. MR 447335, DOI 10.1016/0021-8693(77)90224-1
Jack K. Hale and Pedro Martínez-Amores, Stability in neutral equations, Nonlinear Anal. 1 (1976/77), no. 2, 161–173. MR 599038, DOI 10.1016/0362-546X(77)90007-4
J. T. Stafford, Absolute stable rank and quadratic forms over noncommutative rings, $K$-Theory 4 (1990), no. 2, 121–130. MR 1081655, DOI 10.1007/BF00533152
Anastasia Stavrova, Normal structure of maximal parabolic subgroups in Chevalley groups over rings, Algebra Colloq. 16 (2009), no. 4, 631–648. MR 2547091, DOI 10.1142/S1005386709000595
Michael R. Stein, Surjective stability in dimension $0$ for $K_{2}$ and related functors, Trans. Amer. Math. Soc. 178 (1973), 165–191. MR 327925, DOI 10.1090/S0002-9947-1973-0327925-8
Michael R. Stein, Stability theorems for $K_{1}$, $K_{2}$ and related functors modeled on Chevalley groups, Japan. J. Math. (N.S.) 4 (1978), no. 1, 77–108. MR 528869, DOI 10.4099/math1924.4.77
Alexei Stepanov and Nikolai Vavilov, Decomposition of transvections: a theme with variations, $K$-Theory 19 (2000), no. 2, 109–153. MR 1740757, DOI 10.1023/A:1007853629389
N. Vavilov, A. Luzgarev, and A. Stepanov, Calculations in exceptional groups over rings, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 373 (2009), no. Teoriya Predstavleniĭ, Dinamicheskie Sistemy, Kombinatornye Metody. XVII, 48–72, 346 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 168 (2010), no. 3, 334–348. MR 2749253, DOI 10.1007/s10958-010-9984-z