Kazhdan–lusztig Cells and Decomposition Numbers

@inproceedings{Geck1998KazhdanlusztigCA,
  title={Kazhdan–lusztig Cells and Decomposition Numbers},
  author={Meinolf Geck},
  year={1998}
}
We consider a generic Iwahori–Hecke algebra H associated with a finite Weyl group. Any specialization of H gives rise to a corresponding decomposition matrix, and we show that the problem of computing that matrix can be interpreted in terms of Lusztig’s map from H to the asymptotic algebra J . This interpretation allows us to prove that the decomposition matrices always have a lower uni-triangular shape; moreover, we determine these matrices explicitly in the so-called defect 1 case. 
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