Poset block equivalence of integral matrices

Boyle, Mike; Huang, Danrun

doi:10.1090/S0002-9947-03-02947-7

Poset block equivalence of integral matrices
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by Mike Boyle and Danrun Huang

Trans. Amer. Math. Soc. 355 (2003), 3861-3886

DOI: https://doi.org/10.1090/S0002-9947-03-02947-7

Published electronically: June 10, 2003

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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.

References

William A. Adkins and Steven H. Weintraub, Algebra, Graduate Texts in Mathematics, vol. 136, Springer-Verlag, New York, 1992. An approach via module theory. MR 1181420, DOI 10.1007/978-1-4612-0923-2
David M. Arnold, Representations of partially ordered sets and abelian groups, Abelian group theory (Perth, 1987) Contemp. Math., vol. 87, Amer. Math. Soc., Providence, RI, 1989, pp. 91–109. MR 995268, DOI 10.1090/conm/087/995268
Rufus Bowen and John Franks, Homology for zero-dimensional nonwandering sets, Ann. of Math. (2) 106 (1977), no. 1, 73–92. MR 458492, DOI 10.2307/1971159
Mike Boyle, Flow equivalence of shifts of finite type via positive factorizations, Pacific J. Math. 204 (2002), no. 2, 273–317. MR 1907894, DOI 10.2140/pjm.2002.204.273
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
D. K. Faddeev, On the equivalence of systems of integral matrices, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 449–454 (Russian). MR 0194432
John Franks, Flow equivalence of subshifts of finite type, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 53–66. MR 758893, DOI 10.1017/S0143385700002261
Shmuel Friedland, Simultaneous similarity of matrices, Adv. in Math. 50 (1983), no. 3, 189–265. MR 724475, DOI 10.1016/0001-8708(83)90044-0
Fritz J. Grunewald, Solution of the conjugacy problem in certain arithmetic groups, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Studies in Logic and the Foundations of Mathematics, vol. 95, North-Holland, Amsterdam-New York, 1980, pp. 101–139. MR 579942
Fritz Grunewald and Daniel Segal, Some general algorithms. I. Arithmetic groups, Ann. of Math. (2) 112 (1980), no. 3, 531–583. MR 595206, DOI 10.2307/1971091
Danrung Huang, Flow equivalence of reducible shifts of finite type, Ergodic Theory Dynam. Systems 14 (1994), no. 4, 695–720. MR 1304139, DOI 10.1017/S0143385700008129
Danrun Huang, The classification of two-component Cuntz-Krieger algebras, Proc. Amer. Math. Soc. 124 (1996), no. 2, 505–512. MR 1301504, DOI 10.1090/S0002-9939-96-03079-1
Danrun Huang, Flow equivalence of reducible shifts of finite type and Cuntz-Krieger algebras, J. Reine Angew. Math. 462 (1995), 185–217. MR 1329907, DOI 10.1515/crll.1995.462.185
Danrun Huang, Automorphisms of Bowen-Franks groups of shifts of finite type, Ergodic Theory Dynam. Systems 21 (2001), no. 4, 1113–1137. MR 1849604, DOI 10.1017/S0143385701001535
Danrun Huang, A cyclic six-term exact sequence for block matrices over a PID, Linear and Multilinear Algebra 49 (2001), no. 2, 91–114. MR 1885668, DOI 10.1080/03081080108818687
D. Huang, The K-web invariant and flow equivalence of reducible shifts of finite type, in preparation.
Lee Klingler and Lawrence S. Levy, Sweeping-similarity of matrices, Linear Algebra Appl. 75 (1986), 67–104. MR 825400, DOI 10.1016/0024-3795(86)90182-5
L.A. Nazarova and A.V. Roiter, Representations of partially ordered sets, J. Soviet Math. 23 (1975), 585-607.
Morris Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972. MR 0340283
Bill Parry and Dennis Sullivan, A topological invariant of flows on $1$-dimensional spaces, Topology 14 (1975), no. 4, 297–299. MR 405385, DOI 10.1016/0040-9383(75)90012-9
Wilhelm Wirtinger, Über eine Minimalaufgabe im Gebiete der analytischen Funktionen von mehreren Veränderlichen, Monatsh. Math. Phys. 47 (1939), 426–431 (German). MR 56, DOI 10.1007/BF01695512
Mikael Rørdam, Classification of Cuntz-Krieger algebras, $K$-Theory 9 (1995), no. 1, 31–58. MR 1340839, DOI 10.1007/BF00965458
Jonathan Rosenberg, Algebraic $K$-theory and its applications, Graduate Texts in Mathematics, vol. 147, Springer-Verlag, New York, 1994. MR 1282290, DOI 10.1007/978-1-4612-4314-4
Daniel Simson, Linear representations of partially ordered sets and vector space categories, Algebra, Logic and Applications, vol. 4, Gordon and Breach Science Publishers, Montreux, 1992. MR 1241646