Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Structure of a Hecke algebra quotient


Author: C. Kenneth Fan
Journal: J. Amer. Math. Soc. 10 (1997), 139-167
MSC (1991): Primary 16G30, 05E99; Secondary 16D70, 20F55
DOI: https://doi.org/10.1090/S0894-0347-97-00222-1
MathSciNet review: 1396894
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $W$ be a Coxeter group with Coxeter graph ${\Gamma } $. Let $\cal H$ be the associated Hecke algebra. We define a certain ideal ${\cal I}$ in $\cal H$ and study the quotient algebra ${\bar {\cal H}} = {\cal H}/{\cal I}$. We show that when ${\Gamma } $ is one of the infinite series of graphs of type $E$, the quotient is semi-simple. We examine the cell structures of these algebras and construct their irreducible representations. We discuss the case where ${\Gamma } $ is of type $B$, $F$, or $H$.


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

  • 1. N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5, et 6, Masson, Paris, 1981. MR 83g:17001
  • 2. C. K. Fan, A Hecke Algebra Quotient and Properties of Commutative Elements of a Weyl Group, Thesis (1995), MIT, supervised by G. Lusztig.
  • 3. C. K. Fan, A Hecke Algebra Quotient and Some Combinatorial Applications, Journal of Algebraic Combinatorics 5 no 3 (1996), 175-189. CMP 96:14
  • 4. C. K. Fan, Schubert Varieties and Short Braidedness, preprint (1996).
  • 5. C. K. Fan and R. M. Green, Monomials and Temperley-Lieb Algebras, preprint (1996).
  • 6. C. K. Fan and J. R. Stembridge, Nilpotent Orbits and Commutative Elements, preprint (1996).
  • 7. J. J. Graham, Modular Representations of Hecke Algebras and Related Algebras, Thesis (1995), Univ. of Sydney.
  • 8. V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Annals of Mathematics, 126 (1987), 335-388. MR 89c:46092
  • 9. , A polynomial invariant for knots via Von Neumann algebras, Bulletin of the Amer. Math. Soc., 12 no 1. (1985), 103-111. MR 86e:57006
  • 10. L. H. Kauffmann and H. Saleur, An Algebraic Approach to the Planar Coloring Problem, Commun. Math. Phys. 152 (1993), 565-590. MR 94f:05056
  • 11. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 81j:20066
  • 12. G. Lusztig, Cells in Affine Weyl Groups, in ``Algebraic Groups and Related Topics,'' Advanced Studies in Pure Math. Vol. 6, Kinokuniya and North-Holland, Amsterdam, 1985. MR 87h:20074
  • 13. J. R. Stembridge, The enumeration of fully commutative elements of Coxeter groups, preprint (1996).
  • 14. J. R. Stembridge, On the Fully Commutative Elements of Coxeter Groups, Journal of Alg. Comb., to appear.
  • 15. J. R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc., to appear. CMP 96:12
  • 16. H. N. V. Temperley and E. H. Lieb, Relations between the percolation and colouring problem..., Proceedings of the Royal Society of London (1971), 251-280. MR 58:16425
  • 17. B. W. Westbury, The representation theory of the Temperley-Lieb Algebras, Math. Z. 219 (1995), 539-565. MR 96h:20029

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 16G30, 05E99, 16D70, 20F55

Retrieve articles in all journals with MSC (1991): 16G30, 05E99, 16D70, 20F55


Additional Information

C. Kenneth Fan
Email: ckfan@math.harvard.edu

DOI: https://doi.org/10.1090/S0894-0347-97-00222-1
Keywords: Iwahori-Hecke algebra, Temperley-Lieb algebra, Coxeter group, cell theory, semi-simple algebra
Received by editor(s): May 14, 1996
Additional Notes: Supported in part by a National Science Foundation postdoctoral fellowship.
Dedicated: Dedicated to my teacher, George Lusztig, on his fiftieth birthday
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society