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We present general techniques to determine the structure of Hecke algebras and similar algebras in the non-semisimple case. We apply these to give a complete description of the structure of the TemperleyLieb algebras at a root of unity. Our description implies in particular that the representation of these algebras on tensor space (C 2 )®" is faithful.Introduction. We shall consider ascending sequences of finite dimensional algebras A\ c Aι c , given by generators and relations, where the relations depend on one or several parameters. Moreover,we also assume that the discriminants of these algebras are non-zero polynomials or rational functions in the parameters. This means the algebras are semisimple except for special values of the parameters. For applications (to the construction of topological invariants, the construction of subfactors, or in statistical mechanical models) these algebras are often needed at the critical values of the parameters; in such cases, interesting semisimple quotients have been constructed in [Jl], [W2,4].In this paper, we initiate a systematic study of the structure of such algebras at the critical values of the parameters, where they are not semisimple.For the examples we have in mind such as Hecke algebras, Brauer algebras, etc., the structure is known in the semisimple case and can be described by the Bratteli diagram, which encodes how an irreducible representation of A n , restricted to A n _\, decomposes into irreducible representations of A n _\. If all the multiplicities in the decompositions are 0 or 1, one can use this to define special path idempotents and matrix units (see e.g. [SV], [Wl,2], [RW]), labelled by paths on the Bratteli diagram. These matrix units are only well defined for generic values of the parameters, for which the algebras are semisimple. Nevertheless, their (usually inductive) defining formulas carry a lot of information about the structure of the algebras which can also be exploited at the critical parameter values. In more detail, our main techniques are as follows:
Abstract. We establish connections between the lattices of non-crossing partitions of type B introduced by V. Reiner, and the framework of the free probability theory of D. Voiculescu.Lattices of non-crossing partitions (of type A, up to now) have played an important role in the combinatorics of free probability, primarily via the noncrossing cumulants of R. Speicher. Here we introduce the concept of noncrossing cumulant of type B; the inspiration for its definition is found by looking at an operation of "restricted convolution of multiplicative functions", studied in parallel for functions on symmetric groups (in type A) and on hyperoctahedral groups (in type B).The non-crossing cumulants of type B live in an appropriate framework of "non-commutative probability space of type B", and are closely related to a type B analogue for the R-transform of Voiculescu (which is the free probabilistic counterpart of the Fourier transform). By starting from a condition of "vanishing of mixed cumulants of type B", we obtain an analogue of type B for the concept of free independence for random variables in a non-commutative probability space.
We establish a framework for cellularity of algebras related to the Jones basic construction. Our framework allows a uniform proof of cellularity of Brauer algebras, ordinary and cyclotomic BMW algebras, walled Brauer algebras, partition algebras, and others. Our cellular bases are labeled by paths on certain branching diagrams rather than by tangles. Moreover, for the class of algebras that we study, we show that the cellular structures are compatible with restriction and induction of modules.2000 Mathematics Subject Classification. 20C08, 16G99, 81R50. 1 I n ∼ = A n −1 ⊗ A n −2 A n −1 as A n −1 bimodules; thus I n is a sort of Jones basic construction for the pair A n −2 ⊆ A n −1 . Since our version of cellularity behaves well under extensions, we can conclude that A n is cellular. Our method is related to ideas introduced by König and Xi in their treatment of cellularity and Morita equivalence [39].Following Cox et. al.[9], our approach employs the interaction between induction and restriction functors relating A n −1 -mod and A n -mod, on the one hand, and localization and globalization functions relating A n -mod and A n −2 -mod, on the other hand. (Write e = e n −1 ∈ A n . The localization functor F :Our framework and that of Cox et. al. dovetail nicely; in fact, our main result (Theorem 3.2) says that if (A n ), (Q n ) are two sequences of algebras satisfying our framework axioms, then (A n ) satisfies a cellular version of the axioms for towers of recollement; see [8] for a discussion of cellularity and towers of recollement.Although our techniques do not seem to be adaptable to proving "strict" cellularity in the sense of [23], by combining our results with previous proofs of "strict" cellularity for our examples, we can show the existence of "strictly" cellular Murphy type bases, i.e. bases indexed by paths on the generic branching diagram for the sequence of algebras (A n ) n ≥0 . We will indicate how this can be done for the cyclotomic BMW algebras; other examples are similar.Several other general frameworks have been proposed for cellularity which also successfully encompass many of our examples; see [39,24,57].In a companion paper [19], we refine the framework of this paper to take into account the role played by Jucys-Murphy elements. At the same time, we modify Andrew Mathas's theory [45] of cellular algebras with Jucys-Murphy elements to take into account coherent sequences of such algebras.Acknowledgement. Part of this work was done while both authors were visiting MSRI in 2008. We are grateful to the organizers of the program in Combinatorial Representation Theory and to the staff at MSRI for a pleasant and stimulating visit. We thank the referees for helpful suggestions which resulted in several improvements.
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