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Shellability in reductive monoids


Author: Mohan S. Putcha
Journal: Trans. Amer. Math. Soc. 354 (2002), 413-426
MSC (2000): Primary 20G99, 20M99, 06A07
DOI: https://doi.org/10.1090/S0002-9947-01-02806-9
Published electronically: August 30, 2001
MathSciNet review: 1859281
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Abstract: The purpose of this paper is to extend to monoids the work of Björner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let $M$ be a reductive monoid with unit group $G$, Borel subgroup $B$ and Weyl group $W$. We study the partially ordered set of $B\times B$-orbits (with respect to Zariski closure inclusion) within a $G\times G$-orbit of $M$. This is the same as studying a $W\times W$-orbit in the Renner monoid $R$. Such an orbit is the retract of a `universal orbit', which is shown to be lexicograhically shellable in the sense of Björner and Wachs.


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  • 1. A. Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), 159-183. MR 81i:06001
  • 2. A. Björner and M. Wachs, Bruhat order of Coxeter groups and shellability, Adv. in Math. 43 (1982), 87-100. MR 83i:20043
  • 3. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Univ. Press, 1993. MR 95h:13020
  • 4. C. Chevalley, Sur les décompositions cellulaires des espaces $G/B$, in Algebraic groups and their generalizations, Proc. Sympos. Pure Math., 56, Amer. Math. Soc. (1991), 1-23. MR 95e:14041
  • 5. A. H. Clifford and G. B. Preston, Algebraic Theory of Semigroups, Vol. 1, AMS Surveys, No. 7, Amer. Math. Soc. (1961). MR 24:A2627
  • 6. G. Danaraj and V. Klee, Shellings of spheres and polytopes, Duke Math. J. 41 (1974), 443-451. MR 49:9852
  • 7. C. DeConcini, Equivariant embeddings of homogeneous spaces, Proc. Internat. Congress of Mathematics (1986), 369-377. MR 89e:14045
  • 8. C. DeConcini and V. Lakshmibai, Arithmetic Cohen-Macaulayness and arithmetic normality of Schubert varieties, Amer. J. Math. 103 (1981), 835-850. MR 83e:14035
  • 9. V.V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), 499-511. MR 86f:20045
  • 10. S. Doty, Polynomial representations, algebraic monoids, and Schur algebras of classical type, J. Pure Appl. Algebra 123 (1998), 165-199. MR 98j:20057
  • 11. W. Fulton An introduction to toric varieties, Ann. Math. Studies 131, Princeton Univ. Press, 1993. MR 94g:14028
  • 12. E.A. Pennell, M. S. Putcha and L. E. Renner, Analogue of the Bruhat-Chevalley order for reductive monoids, J. Algebra 196 (1997), 339-368. MR 98j:20090
  • 13. R. A. Proctor, Classical Bruhat orders and lexicographic shellability, J. Algebra 77 (1982), 104-126. MR 84j:20044
  • 14. M. S. Putcha, A semigroup approach to linear algebraic groups, J. Algebra 80 (1983), 164-185. MR 84j:20045
  • 15. M. S. Putcha, Linear Algebraic Monoids, London Math. Soc. Lecture Note Series 133, Cambridge Univ. Press, 1988. MR 90a:20003
  • 16. M. S. Putcha and L. E. Renner, The system of idempotents and the lattice of $\mathcal I$-classes of reductive algebraic monoids, J. Algebra 116 (1988), 385-399.
  • 17. L. E. Renner, Analogue of the Bruhat decomposition for reductive algebraic monoids, J. Algebra 101 (1986), 303-338. MR 87f:20066
  • 18. L. E. Renner, Analogue of the Bruhat decomposition for algebraic monoids II. The length function and trichotomy, J. Algebra 175 (1995), 695-714. MR 96d:20049
  • 19. L. Solomon, The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field, Geom. Dedicata 36 (1990), 15-49. MR 92e:20035
  • 20. L. Solomon, An introduction to reductive monoids, in Semigroups, Formal Languages and Groups (J. Fountain, ed.), Kluwer (1995), 353-367. MR 99h:20099
  • 21. R. P. Stanley, Combinatorics and Commutative Algebra, 2nd edition, Birkhäuser, 1996. MR 98h:05001
  • 22. E. B. Vinberg, The asymptotic semigroup of a semisimple Lie group, in Semigroups in Algebra, Geometry and Analysis, (K. H. Hofmann, J.D. Lawson and E. B. Vinberg, eds.), de Gruyter, 1995, pp. 293-310. MR 96i:20064

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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/S0002-9947-01-02806-9
Received by editor(s): November 24, 1999
Received by editor(s) in revised form: November 6, 2000
Published electronically: August 30, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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