Big cells and LU factorization in reductive monoids
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- by Mohan S. Putcha
- Proc. Amer. Math. Soc. 130 (2002), 3507-3513
- DOI: https://doi.org/10.1090/S0002-9939-02-06515-2
- Published electronically: May 29, 2002
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Abstract:
It is well known that an invertible matrix admits a factorization as a product of a lower triangular matrix $L$ and an upper triangular matrix $U$ if and only if all the principal minors of the matrix are non-zero. The corresponding problem for singular matrices is much more subtle. We study this problem in the general setting of a reductive monoid and obtain a solution in terms of the Bruhat-Chevalley order. In the process we obtain a decomposition of the big cell $\overline B{}^-\overline B$ of a reductive monoid, where $B$ and $B^-$ are opposite Borel subgroups of the unit group.References
- Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
- Edwin A. Pennell, Mohan S. Putcha, and Lex E. Renner, Analogue of the Bruhat-Chevalley order for reductive monoids, J. Algebra 196 (1997), no. 2, 339–368. MR 1475115, DOI 10.1006/jabr.1997.7111
- Mohan S. Putcha, A semigroup approach to linear algebraic groups, J. Algebra 80 (1983), no. 1, 164–185. MR 690712, DOI 10.1016/0021-8693(83)90026-1
- Mohan S. Putcha, Linear algebraic monoids, London Mathematical Society Lecture Note Series, vol. 133, Cambridge University Press, Cambridge, 1988. MR 964690, DOI 10.1017/CBO9780511600661
- M. S. Putcha, Shellability in reductive monoids, Trans. Amer. Math. Soc. 354 (2002), 413–426.
- Lex E. Renner, Analogue of the Bruhat decomposition for algebraic monoids, J. Algebra 101 (1986), no. 2, 303–338. MR 847163, DOI 10.1016/0021-8693(86)90197-3
- Lex E. Renner, Analogue of the Bruhat decomposition for algebraic monoids. II. The length function and the trichotomy, J. Algebra 175 (1995), no. 2, 697–714. MR 1339663, DOI 10.1006/jabr.1995.1208
- Louis Solomon, The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field, Geom. Dedicata 36 (1990), no. 1, 15–49. MR 1065211, DOI 10.1007/BF00181463
- Louis Solomon, An introduction to reductive monoids, Semigroups, formal languages and groups (York, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 466, Kluwer Acad. Publ., Dordrecht, 1995, pp. 295–352. MR 1630625
Bibliographic Information
- Mohan S. Putcha
- Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205
- Email: putcha@math.ncsu.edu
- Received by editor(s): March 19, 2001
- Received by editor(s) in revised form: July 30, 2001
- Published electronically: May 29, 2002
- Communicated by: Stephen D. Smith
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 3507-3513
- MSC (2000): Primary 20G99, 15A23
- DOI: https://doi.org/10.1090/S0002-9939-02-06515-2
- MathSciNet review: 1918826