Abelian Groups and Representations of Finite Partially by David Arnold

By David Arnold

The subject of this e-book is an exposition of connections among representations of finite partly ordered units and abelian teams. Emphasis is positioned all through on type, an outline of the gadgets as much as isomorphism, and computation of illustration kind, a degree of whilst type is possible. David M. Arnold is the Ralph and Jean hurricane Professor of arithmetic at Baylor college. he's the writer of "Finite Rank Torsion loose Abelian teams and earrings" released within the Springer-Verlag Lecture Notes in arithmetic sequence, a co-editor for 2 volumes of convention complaints, and the writer of diverse articles in mathematical study journals.

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Um) be an indecomposable representation of S and write Uo = UmE9 Vo for some Vo . There is a representation direct sum ° U is indecomposable, either Um = and U is isomorphic to (k,O, . ,O), or Vo=O and U=(Um,UI "",Um _I,Um)' Next write Um= Since Um - I E9 Vm so that ° Since U is indecomposable, either Um-I = ° and U is isomorphic to (k, 0, . . , 0, k), or else Vm = and U = (Um-I, UI , .. , Um-2 , Um-l, Um-I). Repeating this process completes the list of indecomposables as given. (b) Let X = (Vol Uo , (VI + Uo)1 U» .

4. The final example of this section demonstrates that generic representations in rep(S, k) , necessarily of infinite dimension, can be used to construct indecomposable finite-dimensional representations in rep(S, k) . 3. 3 and g(x) an irreducible polynomial in k[x]. Define V /g(xYV = (Vo/g(xYVo, (Vi + g(x)eVo)/g(xYVo: i E S) E rep(S , k). Then V /g(xYV is indecomposable with endomorphism ring k[x]/(g(xY). The following questions are motivated by analogous results for finitedimensional algebras over algebraically closed fields [Crawley-Boevey 91] .

2 to show that if A = BI EEl .. EEl B, with each B, an indecomposable object of X, then t = n and there is a permutation a of {I, 2, . . , n} such that Ai is isomorphic to Bo(i)for each i . 7. 3 such that RH(U ) = End V for each V E A(S , k). 6) . This description depends only on the poset and is independent of the field k. 7 and references). A finite poset S has finite representation type over a field k if lnd( S, k) is finite and has infinite representation type if lnd(S, k) is infinite. 6(a)).

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