Products of similar matrices
HTML articles powered by AMS MathViewer
- by Dave Witte PDF
- Proc. Amer. Math. Soc. 126 (1998), 1005-1015 Request permission
Abstract:
Let $A$ and $B$ be $n \times n$ matrices of determinant $1$ over a field $K$, with $n >2$ or $|K|>3$. We show that if $A$ is not a scalar matrix, then $B$ is a product of matrices similar to $A$. Analogously, we conjecture that if $a$ and $b$ are elements of a semisimple algebraic group $G$ over a field of characteristic zero, and if there is no normal subgroup of $G$ containing $a$ but not $b$, then $b$ is a product of conjugates of $a$. The conjecture is verified for orthogonal groups and symplectic groups, and for all semisimple groups over local fields. Thus, in a connected, semisimple Lie group with finite center, the only conjugation-invariant subsemigroups are the normal subgroups.References
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Christophe Bavard, Longueur stable des commutateurs, Enseign. Math. (2) 37 (1991), no. 1-2, 109–150 (French). MR 1115747
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI 10.2307/1970210
- Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150 (French). MR 207712, DOI 10.1007/BF02684375
- L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR 0171864
- Étienne Ghys, Groupes d’homéomorphismes du cercle et cohomologie bornée, The Lefschetz centennial conference, Part III (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1987, pp. 81–106 (French, with English summary). MR 893858, DOI 10.24033/msmf.358
- A. M. W. Glass, Stephen H. McCleary, and Matatyahu Rubin, Automorphism groups of countable highly homogeneous partially ordered sets, Math. Z. 214 (1993), no. 1, 55–66. MR 1234597, DOI 10.1007/BF02572390
- 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
- Gerhard P. Hochschild, Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics, vol. 75, Springer-Verlag, New York-Berlin, 1981. MR 620024, DOI 10.1007/978-1-4613-8114-3
- Karl H. Hofmann, A short course on the Lie theory of semigroups. I, Sem. Sophus Lie 1 (1991), no. 1, 33–40. MR 1124609
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
- Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
- David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153–215. MR 862716, DOI 10.1007/BF01389157
- Karl-Hermann Neeb, Invariant subsemigroups of Lie groups, Mem. Amer. Math. Soc. 104 (1993), no. 499, viii+193. MR 1152952, DOI 10.1090/memo/0499
- Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
- M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
- William P. Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974), 347–352. MR 356087, DOI 10.1016/0040-9383(74)90025-1
- V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. MR 746308, DOI 10.1007/978-1-4612-1126-6
- Dave Witte, Arithmetic groups of higher $\textbf {Q}$-rank cannot act on $1$-manifolds, Proc. Amer. Math. Soc. 122 (1994), no. 2, 333–340. MR 1198459, DOI 10.1090/S0002-9939-1994-1198459-5
Additional Information
- Dave Witte
- Email: dwitte@math.okstate.edu
- Received by editor(s): August 2, 1996
- Received by editor(s) in revised form: October 1, 1996
- Communicated by: Ronald M. Solomon
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 1005-1015
- MSC (1991): Primary 06F15, 20G15, 20G25; Secondary 20F99, 20H05
- DOI: https://doi.org/10.1090/S0002-9939-98-04368-8
- MathSciNet review: 1451837