Algebra Some Current Trends: Proceedings of the 5th National by Jörgen Backelin, Jürgen Herzog, Herbert Sanders (auth.),

By Jörgen Backelin, Jürgen Herzog, Herbert Sanders (auth.), Luchezar L. Avramov, Kerope B. Tchakerian (eds.)

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Zd-1 , say, of finite size. (In fact, each g~ has a linear factorization and thus a lineax m a t r i x factorization with matrices of size 1 × 1, and the gi are monornials in disjoint sets of variables. ) Now, for j = 0 . . . d - 1 we let a j be the m a t r i x formed by/33 by applying ~ to each entry. T h e n indeed f ~ OL0 - . , • O L d _ 1 is a linear m a t r i x factorization of finite size. The other statements of the theorem follow immediately from this. [] 32 REFERENCES [ABS] M. F. ATIYAH, R.

A0EK[x] . If f(x) is an irreducible polyn6mial, and f(x)#x+1, x-l, then (up to isomorphism) exactly one indecomposable KD-module M corresponds to ~. Let I c K(a) be the ideal generated by ~(a) = [ f(a)] n nomial f(x) is not symmetric (a polynomial g(x)eK[ x] is called symmet- ric if g(x -]) = x-rag(x) for some integer m), then M induced KD-module. If the poly- If f(~ )is symmetric, then M (K(a)/I) D is an is isomorphic to the ideals A(e]+Ij -= A(e2+I), where A=KD/I, e1=(1+b)/2; e2=(1-b)/2. If f(x)=x+1 (or x-l) and ~ = (x+1) n (or (x-1)n~ then M is isomorphic to one of ideals A(e1+I), A(e2+I), which are non-isomorphic.

3 we know that ai = X1 + ~i ad,r, with aa,n = ~ r d= 2 ¢0(er)Xr, where ¢0(er) is the multiplication by e~-le, on an indecomposable Co(f, ~)-module. In order to describe ad,n we introduce some more notations: Set t :---- [ - ~ ] and G :-- (Z/dZ) t, and let i denote the residue class of an element i ~ Z in Z/dZ. For g = (~1. . . ie) E G and t c {1 . . . t } w e s e t g(t) = ~ = , (i~-1) C C , and g ( t ) = ($'1 . . , ~ ' t - - l , ~ + 1,$t+1 . . . ~t) e G . After having ordered the elements of G lexicographically with respect to the smallest positive remainder we define square matrices %~,92 of size d~ for all gl, g2 E G by setting if gl = g ~ andg2 =g~ otherwise.

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