Inventiones mathematicae

, Volume 123, Issue 1, pp 1-34

First online:

Cellular algebras

  • J. J. GrahamAffiliated withSchool of Mathematics and Statistics, University of Sydney
  • , G. I. LehrerAffiliated withSchool of Mathematics and Statistics, University of Sydney

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A class of associative algebras (“cellular”) is defined by means of multiplicative properties of a basis. They are shown to have cell representations whose structure depends on certain invariant bilinear forms. One thus obtains 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, whose irreducibles are described in full generality, the Ariki-Koike algebras, which include the Hecke algebras of type A and B and (a generalisation of) the Temperley-Lieb and Jones' recently defined “annular” algebras. In particular the latter are shown to be non-semisimple when the defining paramter δ satisfies\(\gamma _{g(n)} (\tfrac{{ - \delta }}{2}) = 1\), where γ n is then-th Tchebychev polynomial andg(n) is a quadratic polynomial.