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Parabolic factorizations of split classical groups. (English. Russian original) Zbl 1283.20057

St. Petersbg. Math. J. 23, No. 4, 637-657 (2012); translation from Algebra Anal. 23, No. 4, 1-30 (2011).
An analog of the Dennis-Vaserstein decomposition is proved for an arbitrary pair of maximal parabolic subgroups in split classical groups, under appropriate stability conditions. Before, such decompositions were only known for pairs of terminal parabolic subgroups.

MSC:

20G35 Linear algebraic groups over adèles and other rings and schemes
19B14 Stability for linear groups
20H25 Other matrix groups over rings
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[1] Hyman Bass, Algebraic \?-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. · Zbl 0174.30302
[2] 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. · Zbl 1120.17002
[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).
[4] 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. Z. I. Borevich and N. A. Vavilov, Arrangement of subgroups containing a group of block diagonal matrices in the general linear group over a ring, Izv. Vyssh. Uchebn. Zaved. Mat. 11 (1982), 12 – 16 (Russian). Z. I. Borevich and N. A. Vavilov, The net determinant over a Bezoutian local ring, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 116 (1982), 5 – 13, 161 (Russian). Integral lattices and finite linear groups. Z. I. Borevich and L. Yu. Kolotilina, The subnormalizer of net subgroups in the general linear group over a ring, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 116 (1982), 14 – 19, 161 (Russian). Integral lattices and finite linear groups.
[5] 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. · Zbl 0499.20033
[6] 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. · Zbl 1215.20049
[7] L. N. Vaseršteĭn, On the stabilization of the general linear group over a ring, Math. USSR-Sb. 8 (1969), 383 – 400.
[8] L. N. Vaseršteĭn, Stabilization of unitary and orthogonal groups over a ring with involution, Mat. Sb. (N.S.) 81 (123) (1970), 328 – 351 (Russian).
[9] 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).
[10] L. N. Vaseršteĭn, Stabilization for the classical groups over rings, Mat. Sb. (N.S.) 93(135) (1974), 268 – 295, 327 (Russian).
[11] 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. · Zbl 1144.20316
[12] -, Overgroups of classical groups, Kand. diss., S.-Peterburg. Gos. Univ., St. Petersburg, 2005, pp. 1-129. (Russian)
[13] 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)
[14] E. B. Plotkin, Surjective stabilization of the \?\(_{1}\)-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. · Zbl 0802.19003
[15] 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)
[16] A. A. Suslin and M. S. Tulenbaev, A theorem on stabilization for Milnor’s \?\(_{2}\)-functor, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 64 (1976), 131 – 152, 162 (Russian). Rings and modules. · Zbl 0356.18014
[17] Eiichi Abe and Kazuo Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. (2) 28 (1976), no. 2, 185 – 198. · Zbl 0336.20033
[18] A. Bak, The stable structure of quadratic modules, Thesis, Columbia Univ., 1969. · Zbl 0192.37202
[19] Anthony Bak, Viktor Petrov, and Guoping Tang, Stability for quadratic \?\(_{1}\), \?-Theory 30 (2003), no. 1, 1 – 11. Special issue in honor of Hyman Bass on his seventieth birthday. Part I. · Zbl 1048.19001
[20] Anthony Bak and Tang Guoping, Stability for Hermitian \?\(_{1}\), J. Pure Appl. Algebra 150 (2000), no. 2, 109 – 121. · Zbl 0961.19002
[21] Anthony Bak and Nikolai Vavilov, Structure of hyperbolic unitary groups. I. Elementary subgroups, Algebra Colloq. 7 (2000), no. 2, 159 – 196. · Zbl 0963.20024
[22] H. Bass, \?-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5 – 60. · Zbl 0248.18025
[23] Hyman Bass, Unitary algebraic \?-theory, Algebraic K-theory, III: Hermitian K-theory and geometric applications (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 57 – 265. Lecture Notes in Math., Vol. 343.
[24] Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin-New York, 1970 (French). Schémas en groupes. III: Structure des schémas en groupes réductifs, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 153, Springer-Verlag, Berlin-New York, 1970 (French). Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin-New York, 1970 (French). Schémas en groupes. III: Structure des schémas en groupes réductifs, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 153, Springer-Verlag, Berlin-New York, 1970 (French). Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin-New York, 1970 (French). Schémas en groupes. III: Structure des schémas en groupes réductifs, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 153, Springer-Verlag, Berlin-New York, 1970 (French). · Zbl 0209.24201
[25] R. Keith Dennis, Stability for \?\(_{2}\), Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972) Springer, Berlin, 1973, pp. 85 – 94. Lecture Notes in Math., Vol. 353.
[26] 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. · Zbl 0649.20048
[27] Igor V. Erovenko, \?\?_{\?}(\?[\?]) is not boundedly generated by elementary matrices: explicit proof, Electron. J. Linear Algebra 11 (2004), 162 – 167. · Zbl 1073.20026
[28] Dennis Estes and Jack Ohm, Stable range in commutative rings, J. Algebra 7 (1967), 343 – 362. · Zbl 0156.27303
[29] G. Habdank, A classification of subgroups of \( \Lambda \)-quadratic groups normalized by relative elementary subgroups, Dissertation Universität Bielefeld, 1987, pp. 1-71.
[30] Günter Habdank, A classification of subgroups of \Lambda -quadratic groups normalized by relative elementary groups, Adv. Math. 110 (1995), no. 2, 191 – 233. · Zbl 0821.19002
[31] Alexander J. Hahn and O. Timothy O’Meara, The classical groups and \?-theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, Springer-Verlag, Berlin, 1989. With a foreword by J. Dieudonné. · Zbl 0683.20033
[32] Roozbeh Hazrat and Nikolai Vavilov, Bak’s work on the \?-theory of rings, J. K-Theory 4 (2009), no. 1, 1 – 65. · Zbl 1183.19001
[33] Wilberd van der Kallen, Injective stability for \?\(_{2}\), Algebraic \?-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 77 – 154. Lecture Notes in Math., Vol. 551. · Zbl 0349.18009
[34] Wilberd van der Kallen, \?\?\(_{3}\)(\?[\?]) does not have bounded word length, Algebraic \?-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 357 – 361. · Zbl 0935.20501
[35] 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. · Zbl 0756.11008
[36] Manfred Kolster, On injective stability for \?\(_{2}\), Algebraic \?-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 128 – 168. · Zbl 0498.18009
[37] 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. · Zbl 0639.16015
[38] Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1 – 62 (French). · Zbl 0261.20025
[39] Viktor Petrov, Overgroups of unitary groups, \?-Theory 29 (2003), no. 3, 147 – 174. · Zbl 1040.20039
[40] E. B. Plotkin, Stability theorems of \?\(_{1}\)-functor for Chevalley groups, Nonassociative algebras and related topics (Hiroshima, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 203 – 217. · Zbl 0801.20019
[41] Eugene Plotkin, On the stability of the \?\(_{1}\)-functor for Chevalley groups of type \?\(_{7}\), J. Algebra 210 (1998), no. 1, 67 – 85. · Zbl 0918.20039
[42] E. Plotkin, M. R. Stein, and N. Vavilov, Stability of \( K\)-functors modeled on Chevalley groups, revisited, 2001 (to appear).
[43] R. W. Sharpe, On the structure of the unitary Steinberg group, Ann. of Math. (2) 96 (1972), 444 – 479. · Zbl 0251.20047
[44] R. W. Sharpe, On the structure of the Steinberg group \?\?(\Lambda ), J. Algebra 68 (1981), no. 2, 453 – 467. · Zbl 0465.18006
[45] J. T. Stafford, Stable structure of noncommutative Noetherian rings, J. Algebra 47 (1977), no. 2, 244 – 267. · Zbl 0391.16009
[46] Jack K. Hale and Pedro Martínez-Amores, Stability in neutral equations, Nonlinear Anal. 1 (1976/77), no. 2, 161 – 173. · Zbl 0359.34070
[47] J. T. Stafford, Absolute stable rank and quadratic forms over noncommutative rings, \?-Theory 4 (1990), no. 2, 121 – 130. · Zbl 0719.16016
[48] Anastasia Stavrova, Normal structure of maximal parabolic subgroups in Chevalley groups over rings, Algebra Colloq. 16 (2009), no. 4, 631 – 648. · Zbl 1187.20060
[49] Michael R. Stein, Surjective stability in dimension 0 for \?\(_{2}\) and related functors, Trans. Amer. Math. Soc. 178 (1973), 165 – 191. · Zbl 0267.18015
[50] Michael R. Stein, Stability theorems for \?\(_{1}\), \?\(_{2}\) and related functors modeled on Chevalley groups, Japan. J. Math. (N.S.) 4 (1978), no. 1, 77 – 108. · Zbl 0403.18010
[51] Alexei Stepanov and Nikolai Vavilov, Decomposition of transvections: a theme with variations, \?-Theory 19 (2000), no. 2, 109 – 153. · Zbl 0944.20031
[52] 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. · Zbl 1288.20063
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