By Andrew Mathas
This quantity provides an absolutely self-contained creation to the modular illustration thought of the Iwahori-Hecke algebras of the symmetric teams and of the $q$-Schur algebras. The examine of those algebras was once pioneered by way of Dipper and James in a chain of landmark papers. the first objective of the e-book is to categorise the blocks and the easy modules of either algebras. the ultimate bankruptcy encompasses a survey of modern advances and open difficulties. the most effects are proved via exhibiting that the Iwahori-Hecke algebras and $q$-Schur algebras are mobile algebras (in the experience of Graham and Lehrer). this is often proved by way of displaying traditional bases of either algebras that are listed via pairs of ordinary and semistandard tableaux respectively. utilizing the equipment of mobile algebras, that is constructed in bankruptcy 2, this leads to a fresh and stylish category of the irreducible representations of either algebras. The block conception is approached through first proving an analogue of the Jantzen sum formulation for the $q$-Schur algebras. This booklet is the 1st of its sort masking the subject. It bargains a considerably simplified remedy of the unique proofs. The publication is an effective reference resource for specialists. it is going to additionally function an outstanding advent to scholars and starting researchers due to the fact each one bankruptcy comprises routines and there's an appendix containing a brief improvement of the illustration conception of algebras. A moment appendix supplies tables of decomposition numbers.
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