Poset block equivalence of integral matrices

Given square matrices B and B' with a poset-indexed block structure (for which an ij block is zero unless i ≤ 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 R. As one application, when 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 R-module homomorphisms. The study is motivated by applications to symbolic dynamics and C*-algebras.

[1]  Wolfgang Krieger,et al.  A class ofC*-algebras and topological Markov chains , 1980 .

[2]  Shmuel Friedland.  Simultaneous similarity of matrices , 1983 .

[3]  J. Rosenberg,et al.  Algebraic K-Theory and Its Applications , 1995 .

[4]  F. Grunewald.  Solution of the Conjugacy Problem in Certain Arithmetic Groups , 1980 .

[5]  V. V. Plahotnik.  REPRESENTATIONS OF PARTIALLY ORDERED SETS OVER COMMUTATIVE RINGS , 1976 .

[6]  D. Simson.  Linear Representations of Partially Ordered Sets and Vector Space Categories , 1993 .

[7]  Mike Boyle,et al.  FLOW EQUIVALENCE OF SHIFTS OF FINITE TYPE VIA POSITIVE FACTORIZATIONS , 2002 .

[8]  L. Nazarova.  Representations of partially ordered sets of infinite type , 1974 .

[9]  Dennis Sullivan,et al.  A topological invariant of flows on 1-dimensional spaces , 1975 .

[10]  Automorphisms of Bowen–Franks groups of shifts of finite type , 2001, Ergodic Theory and Dynamical Systems.

[11]  M. Rørdam.  Classification of Cuntz-Krieger algebras , 1995 .

[12]  Danrun Huang.  Flow equivalence of reducible shifts of finite type , 1994, Ergodic Theory and Dynamical Systems.

[13]  John Franks,et al.  Flow equivalence of subshifts of finite type , 1984, Ergodic Theory and Dynamical Systems.

[14]  Lawrence S. Levy,et al.  Sweeping-similarity of matrices , 1986 .

[15]  A cyclic six-term exact sequence for block matrices over a PID , 2001 .

[16]  Danrun Huang.  The classification of two-component Cuntz-Krieger algebras , 1996 .

[17]  S. Weintraub,et al.  Algebra: An Approach via Module Theory , 1992 .

[18]  J. Cuntz.  A class ofC*-algebras and topological markov chains II: Reducible chains and the ext-functor forC*-algebras , 1981 .

[19]  Danrun Huang.  Flow equivalence of reducible shifts of finite type and Cuntz-Krieger algebras. , 1995 .

[20]  John Franks,et al.  Homology for zero-dimensional nonwandering sets , 1977 .

[21]  Dan Segal,et al.  Some general algorithms. I: Arithmetic groups , 1980 .