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Monoid Hecke algebras

Author: Mohan S. Putcha
Journal: Trans. Amer. Math. Soc. 349 (1997), 3517-3534
MSC (1991): Primary 20G40, 20G05, 20M30
MathSciNet review: 1401527
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Abstract: This paper concerns the monoid Hecke algebras $\mathcal {H}$ introduced by Louis Solomon. We determine explicitly the unities of the orbit algebras associated with the two-sided action of the Weyl group $W$. We use this to:

  1. find a description of the irreducible representations of $\mathcal {H}$,
  2. find an explicit isomorphism between $\mathcal {H}$ and the monoid algebra of the Renner monoid $R$,
  3. extend the Kazhdan-Lusztig involution and basis to $\mathcal {H}$, and
  4. prove, for a $W\times W$ orbit of $R$, the existence (conjectured by Renner) of generalized Kazhdan-Lusztig polynomials.

References [Enhancements On Off] (What's this?)

  • 1. R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, 1985. MR 87d:20060
  • 2. A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. 1, Math. Surveys No. 7, Amer. Math. Soc., 1961. MR 24:A2627
  • 3. C. W. Curtis, Representations of Hecke algebras, Astérisque 168 (1988), 13-60. MR 90m:20047
  • 4. C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. 2, Wiley, 1987. MR 88f:20002
  • 5. 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
  • 6. V. V. Deodhar, On some geometric aspects of Bruhat orderings. II: The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), 483-506. MR 89a:20054
  • 7. V. V. Deodhar, A splitting criterion for the Bruhat orderings of Coxeter groups, Comm. Algebra 15 (1987), 1889-1894. MR 88i:20052
  • 8. V. V. Deodhar, Duality in parabolic set up for questions in Kazhdan-Lusztig theory, J. Algebra 142 (1991), 201-209. MR 92j:20049
  • 9. J. M. Douglass, An inversion formula for relative Kazhdan-Lusztig polynomials, Comm. Algebra 18 (1990), 371-387. MR 91c:20064
  • 10. J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990. MR 92h:20002
  • 11. N. Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 215-236. MR 29:2307
  • 12. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 81j:20066
  • 13. G. Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981), 490-498. MR 83a:20053
  • 14. J. Okni\'{n}ski and M. S. Putcha, Complex representations of matrix semigroups, Trans. Amer. Math. Soc. 323 (1991), 563-581. MR 91e:20047
  • 15. E. A. Pennell, M. S. Putcha, and L. E. Renner, Analogue of the Bruhat-Chevalley order for reductive monoids, J. Algebra, to appear.
  • 16. M. S. Putcha, A semigroup approach to linear algebraic groups, J. Algebra 80 (1983), 164-185. MR 84j:20045
  • 17. M. S. Putcha, Linear Algebraic Monoids, London Math. Soc. Lecture Note Series., vol. 133, Cambridge Univ. Press, 1988. MR 90a:20003
  • 18. M. S. Putcha, Monoids on groups with BN-pairs, J. Algebra 120 (1989), 139-169. MR 89k:20091
  • 19. M. S. Putcha, Sandwich matrices, Solomon algebras and Kazhdan-Lusztig polynomials, Trans. Amer. Math. Soc. 340 (1993), 415-428. MR 94a:20112
  • 20. M. S. Putcha, Classification of monoids of Lie type, J. Algebra 163 (1994), 636-662. MR 95b:20089
  • 21. M. S. Putcha and L. E. Renner, The system of idempotents and the lattice of $\mathcal {J}$-classes of reductive algebraic monoids, J. Algebra 116 (1988), 385-399. MR 89k:20098
  • 22. M. S. Putcha and L. E. Renner, The canonical compactification of a finite group of Lie type, Trans. Amer. Math. Soc. 337 (1993), 305-319. MR 93g:20123
  • 23. M. S. Putcha and L. E. Renner, Morphisms and duality of monoids of Lie type, J. Algebra 184 (1996), 1025-1040. CMP 1997:1
  • 24. L. E. Renner, Analogue of the Bruhat decomposition for algebraic monoids, J. Algebra 101 (1986), 303-338. MR 87f:20066
  • 25. L. E. Renner, Analogue of the Bruhat decomposition for algebraic monoids. II: The length function and the trichotomy, J. Algebra 175 (1995), 697-714. MR 96d:20049
  • 26. L. E. Renner, Private communication.
  • 27. 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
  • 28. L. Solomon, Reductive monoids, Notes of a talk given at MSRI Workshop on Representations of Reductive Groups over Finite Fields, Berkeley, November 1990.
  • 29. L. Solomon, Algebraic monoids, Notes of a talk given at the Canadian Math. Soc. Meeting, Victoria, December 1991.
  • 30. J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math., vol. 384, Springer-Verlag, 1974. MR 57:9866

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Additional Information

Mohan S. Putcha
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205

Received by editor(s): December 3, 1993
Additional Notes: Research partially supported by NSF Grant DMS9200077
Article copyright: © Copyright 1997 American Mathematical Society

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