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 | References | Similar Articles | Additional Information

Abstract: Given square matrices and with a poset-indexed block structure (for which an block is zero unless ), when are there invertible matrices and with this required-zero-block structure such that ? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain . As one application, when 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 and have determinant . The invariants involve an associated diagram (the ``-web'') of -module homomorphisms. The study is motivated by applications to symbolic dynamics and -algebras.

**[AW]**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****[Ar]**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**, 10.1090/conm/087/995268**[BowF]**Rufus Bowen and John Franks,*Homology for zero-dimensional nonwandering sets*, Ann. of Math. (2)**106**(1977), no. 1, 73–92. MR**0458492****[B]**Mike Boyle,*Flow equivalence of shifts of finite type via positive factorizations*, Pacific J. Math.**204**(2002), no. 2, 273–317. MR**1907894**, 10.2140/pjm.2002.204.273**[C]**J. Cuntz,*A class of 𝐶*-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for 𝐶*-algebras*, Invent. Math.**63**(1981), no. 1, 25–40. MR**608527**, 10.1007/BF01389192**[CK]**Joachim Cuntz and Wolfgang Krieger,*A class of 𝐶*-algebras and topological Markov chains*, Invent. Math.**56**(1980), no. 3, 251–268. MR**561974**, 10.1007/BF01390048**[Fa]**D. K. Faddeev,*On the equivalence of systems of integral matrices*, Izv. Akad. Nauk SSSR Ser. Mat.**30**(1966), 449–454 (Russian). MR**0194432****[F]**John Franks,*Flow equivalence of subshifts of finite type*, Ergodic Theory Dynam. Systems**4**(1984), no. 1, 53–66. MR**758893**, 10.1017/S0143385700002261**[Fri]**Shmuel Friedland,*Simultaneous similarity of matrices*, Adv. in Math.**50**(1983), no. 3, 189–265. MR**724475**, 10.1016/0001-8708(83)90044-0**[G]**Fritz J. Grunewald,*Solution of the conjugacy problem in certain arithmetic groups*, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Stud. Logic Foundations Math., vol. 95, North-Holland, Amsterdam-New York, 1980, pp. 101–139. MR**579942****[GS]**Fritz Grunewald and Daniel Segal,*Some general algorithms. I. Arithmetic groups*, Ann. of Math. (2)**112**(1980), no. 3, 531–583. MR**595206**, 10.2307/1971091**[H1]**Danrung Huang,*Flow equivalence of reducible shifts of finite type*, Ergodic Theory Dynam. Systems**14**(1994), no. 4, 695–720. MR**1304139**, 10.1017/S0143385700008129**[H2]**Danrun Huang,*The classification of two-component Cuntz-Krieger algebras*, Proc. Amer. Math. Soc.**124**(1996), no. 2, 505–512. MR**1301504**, 10.1090/S0002-9939-96-03079-1**[H3]**Danrun Huang,*Flow equivalence of reducible shifts of finite type and Cuntz-Krieger algebras*, J. Reine Angew. Math.**462**(1995), 185–217. MR**1329907**, 10.1515/crll.1995.462.185**[H4]**Danrun Huang,*Automorphisms of Bowen-Franks groups of shifts of finite type*, Ergodic Theory Dynam. Systems**21**(2001), no. 4, 1113–1137. MR**1849604**, 10.1017/S0143385701001535**[H5]**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**, 10.1080/03081080108818687**[H6]**D. Huang,*The K-web invariant and flow equivalence of reducible shifts of finite type*, in preparation.**[KL]**Lee Klingler and Lawrence S. Levy,*Sweeping-similarity of matrices*, Linear Algebra Appl.**75**(1986), 67–104. MR**825400**, 10.1016/0024-3795(86)90182-5**[NR]**L.A. Nazarova and A.V. Roiter,*Representations of partially ordered sets*, J. Soviet Math.**23**(1975), 585-607.**[Ne]**Morris Newman,*Integral matrices*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 45. MR**0340283****[PS]**Bill Parry and Dennis Sullivan,*A topological invariant of flows on 1-dimensional spaces*, Topology**14**(1975), no. 4, 297–299. MR**0405385****[Pl]**V. V. Plahotnik,*Representations of partially ordered sets over commutative rings*, Izv. Akad. Nauk SSSR Ser. Mat.**40**(1976), no. 3, 527–543, 709 (Russian). MR**0447345****[R]**Mikael Rørdam,*Classification of Cuntz-Krieger algebras*, 𝐾-Theory**9**(1995), no. 1, 31–58. MR**1340839**, 10.1007/BF00965458**[Ros]**Jonathan Rosenberg,*Algebraic 𝐾-theory and its applications*, Graduate Texts in Mathematics, vol. 147, Springer-Verlag, New York, 1994. MR**1282290****[S]**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**

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

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