By Andrew Mathas

This quantity provides an absolutely self-contained advent to the modular illustration concept of the Iwahori-Hecke algebras of the symmetric teams and of the $q$-Schur algebras. The research of those algebras used to be pioneered by way of Dipper and James in a sequence of landmark papers. the first aim of the publication is to categorise the blocks and the straightforward modules of either algebras. the ultimate bankruptcy incorporates a survey of modern advances and open difficulties. the most effects are proved through exhibiting that the Iwahori-Hecke algebras and $q$-Schur algebras are mobile algebras (in the experience of Graham and Lehrer). this can be proved via showing typical bases of either algebras that are listed by way of pairs of normal and semistandard tableaux respectively. utilizing the equipment of mobile algebras, that is built in bankruptcy 2, this leads to a fresh and chic class of the irreducible representations of either algebras. The block idea is approached by way of 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 a great reference resource for specialists. it's going to additionally function a superb advent to scholars and starting researchers given that every one bankruptcy comprises workouts and there's an appendix containing a brief improvement of the illustration concept of algebras. A moment appendix supplies tables of decomposition numbers.

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**Example text**

Since (3 E m; ~ mi, we have O'(P) E m~ ~ mRo and VRo(O'((3)) > O. Let ~ be a nonzero element of the field FRz = Rz / mRz = R' / mo, -I --1 and let (3' E R' be such that (3 = ~(3 . :::; 0'(3' E TI Pi· Then P E (R' \ mo) nm; and a = ~. :::; 0',0'1, ... , elements of the form 0'0', 0' E G). We 1, ... , s. Indeed, O'i 0'0' for some 0' E G. ), we have 0' t/:. Gz (and 0'-1 t/:. Gz), O'-l(m') oF m', and O'-l(m') n R' = mj for some j > 0 (~k). Since 0' E mj ~ O'-l(m/), we have O'i = cpo' E cp(mj) ~ m'.

Since R ~ Rs, we have R ~ H(R) (~ Rs). 1, H(R) is a Henselian valuation ring. Suppose that R is a Henselian valuation ring, R :;::) R, F =:; q(R) is the quotient field of the ring R, F s is the separable closure of the field F, and Rs is a (unique) valuation ring of F s dominating R. Since F ~ F s, we can assume that Fs ~ F s . We set R: =:; Rs n Fs. Then R: is a valuation ring of Fs dominating R. pR s . p(H(R)) ~ R. pFz. We set Fo =:; F n Fs and R~ =:; R n Fo = R~ n Fo. 1 and the relation R ~ R'o ~ R~ holds.

Since Fo is a separable extension of F, we have ao f aj for 0 < i < n. pa = aj). We set m':::; mRo nRFo. pm" = m'. We note that m" f m'. pm" = m' and, R~~. However, R~~ Ro. pa = a since a E Fz = F~z. pa = aj f a. Consequently, m" f m'. Let m" n R~ mj for some j :::; k. 3) = = = mo f mj. Consequently, j > O. pm" ~ mRo' Therefore, aj = 0 and the reduction of the polynomial f yields 1 = (x - a)xn-1. We have a f 0 since a E R~ \ mo, a E Fz. 2. Thus, f E R[x] is a monic polynomial without multiple roots such that 7 has a simple root a over FR.