Big cells and LU factorization in reductive monoids

Author:
Mohan S. Putcha

Journal:
Proc. Amer. Math. Soc. **130** (2002), 3507-3513

MSC (2000):
Primary 20G99, 15A23

Published electronically:
May 29, 2002

MathSciNet review:
1918826

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that an invertible matrix admits a factorization as a product of a lower triangular matrix and an upper triangular matrix 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 of a reductive monoid, where and are opposite Borel subgroups of the unit group.

**1.**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****2.**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**, 10.1006/jabr.1997.7111**3.**Mohan S. Putcha,*A semigroup approach to linear algebraic groups*, J. Algebra**80**(1983), no. 1, 164–185. MR**690712**, 10.1016/0021-8693(83)90026-1**4.**Mohan S. Putcha,*Linear algebraic monoids*, London Mathematical Society Lecture Note Series, vol. 133, Cambridge University Press, Cambridge, 1988. MR**964690****5.**M. S. Putcha,*Shellability in reductive monoids*, Trans. Amer. Math. Soc.**354**(2002), 413-426.**6.**Lex E. Renner,*Analogue of the Bruhat decomposition for algebraic monoids*, J. Algebra**101**(1986), no. 2, 303–338. MR**847163**, 10.1016/0021-8693(86)90197-3**7.**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**, 10.1006/jabr.1995.1208**8.**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**, 10.1007/BF00181463**9.**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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
20G99,
15A23

Retrieve articles in all journals with MSC (2000): 20G99, 15A23

Additional Information

**Mohan S. Putcha**

Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205

Email:
putcha@math.ncsu.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06515-2

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

Article copyright:
© Copyright 2002
American Mathematical Society