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
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. Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
  • 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, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, 10.2307/1971403
  • 9. Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. MR 766964, 10.1090/S0273-0979-1985-15304-2
  • 10. Louis Kauffmann and H. Saleur, An algebraic approach to the planar coloring problem, Comm. Math. Phys. 152 (1993), no. 3, 565–590. MR 1213302
  • 11. David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, 10.1007/BF01390031
  • 12. George Lusztig, Cells in affine Weyl groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 255–287. MR 803338
  • 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 and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1549, 251–280. MR 0498284
  • 17. B. W. Westbury, The representation theory of the Temperley-Lieb algebras, Math. Z. 219 (1995), no. 4, 539–565. MR 1343661, 10.1007/BF02572380

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

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