zbMATH — the first resource for mathematics

Cellular algebras. (English) Zbl 0853.20029
The authors define a class of associative algebras (“cellular”) by means of multiplicative properties of a basis, show that they have cell representations whose structure depends on certain invariant bilinear forms, and then obtain a general description of their irreducible representations and block theory as well as criteria for semisimplicity. These concepts are used to discuss the Brauer centraliser algebras, the Ariki-Koike algebras and the Temperley-Lieb and Jones algebras.

20G05 Representation theory for linear algebraic groups
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
20C30 Representations of finite symmetric groups
Full Text: DOI EuDML
[1] [AK] S. Ariki, K. Koike: A Hecke algebra of (?/r?)/Sn and construction of its irreducible representations. Adv. Math.106, 216-243 (1994) · Zbl 0840.20007
[2] [B] R. Brauer: On algebras which are connected with the semisimple continuous groups. Ann. Math.38, 854-887 (1937) · Zbl 0017.39105
[3] [BV] D. Barbasch, D. Vogan: Primitive ideals and orbital integrals in complex classical groups. Math. Ann.259, 153-199 (1982) · Zbl 0489.22010
[4] [Ca] R. Carter: Finite groups of Lie type: conjugacy classes and complex characters. Wiley, Chichester New York, 1985 · Zbl 0567.20023
[5] [CPS] E. Cline, B. Parshall, L. Scott: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math.391, 85-99 (1988) · Zbl 0657.18005
[6] [Cu] C. W. Curtis: Representations of finite groups of Lie type. Bull. A.M.S.1, 721-757 (1979) · Zbl 0416.20002
[7] [D] R. Dipper: Polynomial Representations of finite general linear groups in nondescribing characteristic. Prog. in Math.95, 343-370 (1991) · Zbl 0755.20008
[8] [DJ1] R. Dipper, G. James: Representations of Hecke algebras of general linear groups. Proc. Lond. Math. Soc. (3)52, 20-52 (1986) · Zbl 0587.20007
[9] [DJ2] R. Dipper, G.D. James: Blocks and idempotents of Hecke algebras of general linear groups. Proc. Lond. Math. Soc. (3)54, 57-82 (1987) · Zbl 0615.20009
[10] [DJ3] R. Dipper, G.D. James: Identification of the irreducible modular representations of GL n (q). J. Algebra,104, 266-288 (1986) · Zbl 0622.20032
[11] [DJM] R. Dipper, G.D. James, G.E. Murphy: Hecke algebras of typeB n at roots of unity (Preprint 1994)
[12] [Dr] V.G. Drinfeld: Quantum groups. Proc. Int. Cong. Math. Berkeley pp. 798-820 1986 (1987)
[13] [FG] S. Fishel, I. Grojnowski: Canonical bases for the Brauer centralizer algebra. Math. Res. Letters2, 1-16 (1995)
[14] [G] M. Geck: On the decomposition Numbers of the finite unitary groups in non-defining characteristic. Math. Z.207, 83-89 (1991) · Zbl 0727.20012
[15] [GM] A.M. Garsia, T.J. McLarnen: Relations between Young’s natural and the Kazhdan Lusztig representations ofS n . Adv. Math.69, 32-92 (1988) · Zbl 0657.20014
[16] [Gr] J. Graham: Modular representations of Hecke algebras and related algebras. PhD Thesis, Sydney University 1995
[17] [GW] F.M. Goodman, H. Wenzl: The Temperley-Lieb algebra at roots of unity. Pac. J. Math.161, 307-334 (1993) · Zbl 0823.16004
[18] [HL1] R.B. Howlett, G.I. Lehrer: Induced cuspidal representations and generalised Hecke rings. Invent. Math.58, 37-64 (1980) · Zbl 0435.20023
[19] [HL2] R.B. Howlett, G.I. Lehrer: Representations of generic algebras and finite groups of Lie type. Trans. A.M.S.280, 753-777 (1983)
[20] [HW] P. Hanlon, D. Wales: A tower construction for the radical in Brauer’s centraliser algebras. J. Algebra164, 773-830 (1994) · Zbl 0834.16015
[21] [J1] V.F.R. Jones: A polynomial invariant for knots via von Neumann algebras. Bull. A.M.S.12, 103-111 (1985) · Zbl 0564.57006
[22] [J2] V.F.R. Jones: Hecke algebra representations of braid groups and link polynomials. Ann. Math.126, 335-388 (1987) · Zbl 0631.57005
[23] [J3] V.F.R. Jones: A quotient of the affine Hecke algebra in the Brauer algebra. L’Enseignement Math.40, 313-344 (1994) · Zbl 0852.20035
[24] [J4] V.F.R. Jones: Subfactors and Knots. C.B.M.S.80, A.M.S., Providence RI 1991
[25] [Ji] M. Jimbo: A q-analogue ofU(gl(n+1)), Hecke algebra and the Yang-Baxeter equation. Lett. Math. Phys.11, 247-252 (1986) · Zbl 0602.17005
[26] [KL1] D. Kazhdan, G. Lusztig: Representations of Coxeter groups and Hecke algebras. Invent. math.53, 165-184 (1979) · Zbl 0499.20035
[27] [KL2] D. Kazhdan, G. Lusztig: Schubert varieties and Poincaré duality. Proc. Sym. Pure Math. A.M.S.36, 185-203 (1980)
[28] [Kn] D. Knuth: The art of computer programming. Addison-Wesley, Reading MA 1975
[29] [L] G.I. Lehrer: A survey of Hecke algebras and the Artin braid groups. Contemp. Math.78, 365-385 (1988)
[30] [Lu1] G. Lusztig: Characters of reductive groups over a finite field. Ann. Math. Studies107, Princeton U.P., NJ, 1984 · Zbl 0556.20033
[31] [Lu2] G. Lusztig: Left cells in Weyl groups, in Springer L.N.M.1024, 99-111, Berlin, Heidelberg, New York, 1983
[32] [Lu3] G. Lusztig: Finite dimensional Hopf algebras arising from quantum groups. J.A.M.S.3, 257-296 (1990) · Zbl 0695.16006
[33] [M] G.E. Murphy: On the representation theory of the symmetric groups and associated Hecke algebras. J. Algebra152, 492-513 (1992) · Zbl 0794.20020
[34] [Sh] J-Y. Shi: The Kazhdan-Lusztig cells in certain affine Weyl groups, Springer L.N.M.1179, Berlin, Heidelberg, New York, 1988
[35] [V] D. Vogan: A generalised ?-invariant for the primitive spectrum of a semisimple Lie algebra. Math. Ann.242, 209-224 (1979) · Zbl 0405.17009
[36] [W] H. Wenzl: On the structure of Brauer’s centralizer algebras. Ann. Math.128, 173-193 (1988) · Zbl 0656.20040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.