By Andrew Mathas
This quantity offers an absolutely self-contained advent to the modular illustration conception of the Iwahori-Hecke algebras of the symmetric teams and of the $q$-Schur algebras. The learn of those algebras was once pioneered by means of Dipper and James in a sequence of landmark papers. the first target of the booklet is to categorise the blocks and the straightforward modules of either algebras. the ultimate bankruptcy includes a survey of modern advances and open difficulties. the most effects are proved by means of exhibiting that the Iwahori-Hecke algebras and $q$-Schur algebras are mobile algebras (in the feel of Graham and Lehrer). this is often proved by means of showing usual bases of either algebras that are listed through pairs of ordinary and semistandard tableaux respectively. utilizing the equipment of mobile algebras, that is constructed in bankruptcy 2, this ends up in a fresh and chic class of the irreducible representations of either algebras. The block conception is approached by means of first proving an analogue of the Jantzen sum formulation for the $q$-Schur algebras. This publication is the 1st of its type protecting the subject. It bargains a considerably simplified therapy of the unique proofs. The e-book is a high-quality reference resource for specialists. it is going to additionally function a great creation to scholars and starting researchers due to the fact every one bankruptcy includes routines and there's an appendix containing a brief improvement of the illustration thought of algebras. A moment appendix offers tables of decomposition numbers.
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