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Poset block equivalence of integral matrices. (English) Zbl 1028.15006

Given square matrices \(B\) and \(B'\) with a poset-indexed block structure (for which an \(ij\) block is zero unless \(i\preccurlyeq j\)), when are there invertible matrices \(U\) and \(V\) with this required-zero-block structure such that \(UBV= B'\)? The authors give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain \({\mathcal R}\).
As one application, when \({\mathcal R}\) is a field one classifies such matrices up to similarity by matrices respecting the block structure. The paper also gives complete invariants for equivalence under the additional requirement that the diagonal blocks of \(U\) and \(V\) have determinant 1. The invariants involve an associated diagram (the “\(K\)-web”) of \({\mathcal R}\)-module homomorphisms. The study is motivated by applications to symbolic dynamics and \(C^*\)-algebras.

MSC:

15A21 Canonical forms, reductions, classification
06A11 Algebraic aspects of posets
06F25 Ordered rings, algebras, modules
15B36 Matrices of integers
16G20 Representations of quivers and partially ordered sets
37B10 Symbolic dynamics
46L35 Classifications of \(C^*\)-algebras
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