Get Abelian Groups and Representations of Finite Partially PDF

By David Arnold

ISBN-10: 038798982X

ISBN-13: 9780387989822

The subject matter of this booklet is an exposition of connections among representations of finite in part ordered units and abelian teams. Emphasis is put all through on category, an outline of the items as much as isomorphism, and computation of illustration kind, a degree of whilst category is possible. David M. Arnold is the Ralph and Jean hurricane Professor of arithmetic at Baylor collage. he's the writer of "Finite Rank Torsion unfastened Abelian teams and jewelry" released within the Springer-Verlag Lecture Notes in arithmetic sequence, a co-editor for 2 volumes of convention complaints, and the writer of various articles in mathematical learn journals.

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1 Suppose T is a subposet ofa finite poset S. (a) There is a correspondence F+: rep(T, k) --+ rep(S, k) with Hom(V, V) isomorphic to Hom(F+V, F+V)for each V, V in rep(T, k). (b) There is a correspondence F- :rep(T, k) --+ rep(S, k) with Hom(V, V) isomorphic to Hom(F-V, F-V)for each V, V in rep(T , k). (c) If S has finite representation type, then so does T . (d) If T has infinite representation type, then so does S. PROOF. (a) The correspondence is given by F+(V) = W = (Wo, Wi: i E S) E rep(S, k) for V = (Vo, Vi: i E T), with Wo = U» , Wi = Vi if i E T, Wi = :E{V j: JET, j < i} if i E S\T and there is JET with j < i, and Wi = 0 otherwise.

It now suffices to assume that w(S) = 3 and prove that S must contain (2,2,2), (1,3,3), (1,2 ,5), or (N,4) as a subposet. There are several proofs of this result. All are lengthy and complicated and as such are beyond the scope of this book. 6] uses the theory of integral quadratic forms. 1]. This proof uses the notion of a Zavadiskij derivative of a poset , as defined below. An elementary outline of this proof is given in [Hahn 97] . 1, elements of Ind(S, k) include those elements of Ind(T, k) for each proper subposet T of S.

These five posets are called critical posets. Following is a list of representations for each critical poset 5, demonstrating that rep( 5, k) has infinite representation type. The idea is to show that if A is an indecomposable n x n k-matrix, then there is a representation U E rep(S', k) with dim U ~ n and End U = C(A) , the centralizer of A . Since Jordan block matrices provide examples of n x n indecomposable matrices, ranks of indecomposable representations in rep(5, k) are unbounded. This shows that rep(5, k) has infinite representation type.

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Abelian Groups and Representations of Finite Partially Ordered Sets by David Arnold


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