D ec 1 99 7 CELLULAR ALGEBRAS ARISING FROM HECKE ALGEBRAS OF TYPE

We study a finite-dimensional quotient of the Hecke algebra of type H n for general n, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss the cellular structure of the algebra, with certain restrictions on the ground ring. To appear in " Mathematische Zeitschrift " 0. Introduction There has been much recent interest in the Temperley–Lieb algebra and its various generalisations. Graham [4] in his thesis studied a certain quotient, which we will call T L(X), of a Hecke algebra H(X) associated to a Dynkin diagram X. In the case where X is a Dynkin diagram of type A, this quotient was considered by Jones [8], who pointed out that it is nothing other than the Temperley–Lieb algebra, which first appeared in [12]. The Temperley–Lieb algebra has applications in several areas of mathematics, including statistical mechanics and knot theory. A remarkable feature of the algebras T L(X) is that they can be finite dimensional , even when H(X) is infinite dimensional. Graham [4] classified the finite dimensional algebras T L(X) into seven infinite families: (Contrast this to the classification of Hecke algebras associated to irreducible Cox-eter systems, in which there are only finitely many algebras of types E, F and H.) This paper is concerned with the infinite family of type H, in which case the Hecke algebra H(H n) is finite dimensional only for n ≤ 4. The algebra T L(H n) was mentioned briefly by Fan in [1, §7.3], where it was conjectured that T L(H n) is generically semisimple. The dimensions of the generically irreducible modules are also conjectured. In the course of the paper, we will prove these conjectures.