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By R. Goebel, E. Walker

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This amounts to is a direct summand of the tensor product. To n investigate the i n d e c o m p o s a b l e action of G with that of ~n tive ~ n - m O d u l e , G-summands of M ~ S t n , (cf. A p p e n d i x U). we compare the Since St n is a projec- M ~ S t n breaks up into a direct sum of PIM's Ql,n' and the sum of those o c c u r r i n g for a fixed I is a G-summand. large that all weights G-composition of M are below (pn-l)~. factor of M ~ S t n which becomes on r e s t r i c t i o n to u .

Recall is a direct sum of PIM's. I) that each projec- Moreover, of pm (the exact power of p dividing ~ = (p-l)6, only irreducible ef. 2), IF]). the Steinberg module KF-module which dim RI is a Since dim MI < pm St = M(p_l)~ is the could possibly be projective. In fact: PROPOSITION. Proof. requires of F: St is a projective The only proof of this statement known to the author a highly nontrivial There exists ef. Appendix S. Reiner fact about the ordinary an irreducible over Q) of degree pm. 8], of defect [~, Part B, w irreducible 0 implies upon reduction modulo p of dimension must be St.

Lemma 2 of w f is not needed, (Alternatively, of. ) (b) It is well known that W is g e n e r a t e d by the reflections oe which it contains; when p does not divide f, the same is true of the 45 stabilizer has also follows <~,s> of ~ r e g a r d e d been that is less In the Then the p, context Main of the can state the m a i n THEOREM. Let construct QX in M divide f. (a) the . of the let class [~, lemma, X in A, w h i c h this ~ in A, it absolute value X = ( ~ - ~)o in A. of Z in A c l e a r l y of w equals we have of the denoted al.

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Abelian Group Theory by R. Goebel, E. Walker


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