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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Poset block equivalence of integral matrices


Authors: Mike Boyle and Danrun Huang
Journal: Trans. Amer. Math. Soc. 355 (2003), 3861-3886
MSC (2000): Primary 15A21; Secondary 06A11, 06F99, 15A36, 16G20, 37B10, 46L35
Published electronically: June 10, 2003
MathSciNet review: 1990568
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Abstract: Given square matrices $B$ and $B'$ with a poset-indexed block structure (for which an $ij$ block is zero unless $i\preceq j$), when are there invertible matrices $U$ and $V$ with this required-zero-block structure such that $UBV = B'$? We 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 we classify such matrices up to similarity by matrices respecting the block structure. We also give 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.


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Additional Information

Mike Boyle
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015
Email: mmb@math.umd.edu

Danrun Huang
Affiliation: Department of Mathematics, St. Cloud State University, St. Cloud, Minnesota 56301-4498
Email: dhuang@stcloudstate.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-02947-7
PII: S 0002-9947(03)02947-7
Keywords: Block, equivalence, poset, integer, matrix, principal ideal domain, cokernel, flow equivalence, representation, similarity
Received by editor(s): September 13, 2000
Published electronically: June 10, 2003
Additional Notes: The first author gratefully acknowledges support of NSF Grant DMS9706852, and sabbatical support from the University of Maryland and the University of Washington. The second author gratefully acknowledges support of Research Grant 211243 from the St.\ Cloud State University
Article copyright: © Copyright 2003 American Mathematical Society