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

By V. B Dlab

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S i . . t l * r E . ) ! e * t 1( E r . s, i . . s l * . , r . sn Q). rr source. s. s|*f It) = o and rt Let X€ _ I a)X=CC'xeP,whereP€ L ( is injective. M , 0 ) . Then L (M,Q)isprojective. r) s ; + 1 E r ) r=* r " n - f 1 . : = . ; ; - . ; - " . ; ; - = = ( s k ( d t usn; . . e. e. C- X = O, or I = C* C Ext' Proof: a) ( c b) If If k €f = a sink, t* (dirn X ) . F* as bimodule. enough to prove a). 4). 3), and. 6) is is sequence with k = 1. 3) and F1 vrould. be injective this is impossible, >1 by general assurnptlon.

Witir g , I/r. ,. x -firtratlon. has an 9 anotherX-filtration of I prove By induction 1s 15 n _r ". 1s a nonzero ) an lsomorphism, o =%SIr We nill a also trlvial. j- Then _ x ,r X resp. image of state$ent q (It)+ ! of =Iand cä ... 8 Proposition: L(x) : = L( M,n ) be a conponent, and ( 2 . 1 . 1 1 ) . S u p p o s eK t o b e ) as in L( {I} Then lnfinite. ) then on both sides. if in true length full tt -!. E" + E^lq tz nonisomorphic with is x) is The Coxeter (Xr, respect (X,X), Extl lengths 2 a subcategory' forn of not finite tYPe.

Vr G. So we qfet an 1-1 correspondence under the same o Y -rvb - Li t X. closure Uv of I. of K let of (, and G. Then [. " tu, E, defined ana G and the over K. F Let v € Uy ö v is be K-rational. defined open in itrs over K. It closure, Then by [Humphreys, p. ,,if the closure of ö ,r-i" V,r. this happens, rof course, if dim ö v = dim 7rr. v. This can be seen in the following way. r,ot" enbed G in M = the kxk-matrix n i€r I'{(yifi2, of F. over K. Letn of c in Ivl is over a field K) by a regular EK M and notj_ce that the closure space of (which also defin.

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