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

By V. B Dlab

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3)). M ,0), -32define and (sl s) is * u . O. , xI s t{k . 1 Lenma: + j€f o' by followed n (r = ,n* ,ä, ( Zi = t Si x')t - , "ru i€f q 1. 6 l(M,s*0). rt is easy to see, ttrat sl right 11g,sno). the diagram Mk "i i, jti ) isagainamorPhismin "i, then define sf S = ß = (ß1) by ß, = c, for and ßn : Zn and sl i+ k+ j, injection for [+I'= q = (si)r tr = (zr'jni)€t1rtr,s*a) = ! , "i "j L ( M, n)into from Let now k be a source. Then define s; I as follows, for ):lr*irj*x. e. by restriction, tig L( M, Q ), i+ k to be the map induced by also becomes a functor and sf, are additive from l(M,O) and sf, is left exact.

Xi ! Hom". (Xr, ttorn,. na F = ,f,. +*j , *i- j6j map tp : xi tr. oi Mj - xj nap 6' : X. _xj ,o, Thj,s notation wlll "j tire qet be used throughout paper. this Let k € | be a sink, X = (Xr, € L( M, n ). Then define ior) + = st I tYi, t ( rq, sn ) as follows: iVg)€ " First, if i+ k +j, i, j€f , then we set y. = X. and jvi = jei. o. ^ I , xk i-n O Mk. Let j€f "j "j by the projection tto. ). j Mk ol be \, j . --r. 3)). M ,0), -32define and (sl s) is * u . O. , xI s t{k . 1 Lenma: + j€f o' by followed n (r = ,n* ,ä, ( Zi = t Si x')t - , "ru i€f q 1.

E. dim G. , thus the closure hence dense in of G Y in U" is U". In Particular t/, and G thj-s is Y Is oPen true for Y in Ux the groups also write in the V To distinguish z. following G = Gy. =f'n'k e f'O and let z'-x n for + , Vz -Vx o. zi the rrow corresponding l) of matrices conFatible with then we the direct summand Fkzk = Fkxk o F*). Fk of Clearly c is epimorphism. j i. e. delete if jei =nipi *ti 51 - a morphism of varieties, Consider is continuous, in particular the preirnage o-1 {G* x; GU z n o, in fact (cx.

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