By V. B Dlab

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_'j . )€ (pt) o* such that = (;E) L( M, Q), . Asain we can assume X^= X and Z^ = Z. Now define ß* =(gt): and Sfi. zi) + x for = (rVr). " Fk = F*zk. ali' for i+ k Then the following arrows lq '+' i in f, again -53- t*"* t*** I I t_ t** t_ "'r"*' l | Y ^I . z . 2 Proposition: indeconposable, Then there -Fk---€. \-L and assume dim Z = dim X + dim y. - I-<1. Let y be We may assume pos(I)

1) = d, all modules -K! and in_ a composition = 12 . every in t(M,n) t,. 2) necessary, dimension. ,e will and therefore as'sump_ give arso -60- The non S 2 homogeneous (r,d) paragraph let In this K-realization (M,n). 1 The dimension function representations non-zero defect positive and all defect- Moreover jective modules in Let X € a(X) < O. order definition Let ax < o (f,d) the indecomPosable of non-zero pre- N -

1) in K. e. * alrt'' . Because zero. cannot be identically so o, becauseK is infinite. v. E f. ,= L e! trl ] [ 1 . ,, . . rtl€ , r\ + l,rn(r) ) as ... = Or (I) , t1,1) so let yr2- (dimension as varleties) Y = (Yrtror)€ and indecomposable. and the definition of prejectives End I Of course Assume Then by (2-2-1) there such 1o€l is = ft = Fi Iclentify Y with (iqi)€U and - o ' - o J r IQ . n G, and there C^ g- centralizer of varieties 1f Y ind orbit ) . dim Y = y. j "/c"tg) + is an isomorphism G 'Y, where G' Y denotes the Ur.

### An Introduction to Diagrammatical Methods in Representation Theory by V. B Dlab

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