On Matroids Representable over $GF(3)$ and Other Fields
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Abstract:
The matroids that are representable over $GF(3)$ and some other fields depend on the choice of field. This paper gives matrix characterisations of the classes that arise. These characterisations are analogues of the characterisation of regular matroids as the ones that can be represented over the rationals by a totally-unimodular matrix. Some consequences of the theory are as follows. A matroid is representable over $GF(3)$ and $GF(5)$ if and only if it is representable over $GF(3)$ and the rationals, and this holds if and only if it is representable over $GF(p)$ for all odd primes $p$. A matroid is representable over $GF(3)$ and the complex numbers if and only if it is representable over $GF(3)$ and $GF(7)$. A matroid is representable over $GF(3)$, $GF(4)$ and $GF(5)$ if and only if it is representable over every field except possibly $GF(2)$. If a matroid is representable over $GF(p)$ for all odd primes $p$, then it is representable over the rationals.References
- Brylawski, T. H., and Kelly, D., Matroids and combinatorial geometries. Department of Mathematics, University of North Carolina, Chapel Hill, 1980.
- Brylawski, T. H., and Lucas, D., Uniquely representable combinatorial geometries, in Teorie Combinatorie (Proc. 1973 Internat. Colloq.), pp. 83β104, Accademia Nazionale del Lincei, Rome, 1976.
- A. M. H. Gerards, A short proof of Tutteβs characterization of totally unimodular matrices, Linear Algebra Appl. 114/115 (1989), 207β212. MR 986875, DOI 10.1016/0024-3795(89)90461-8
- Joseph P. S. Kung, Combinatorial geometries representable over $\textrm {GF}(3)$ and $\textrm {GF}(q)$. I. The number of points, Discrete Comput. Geom. 5 (1990), no.Β 1, 83β95. MR 1018017, DOI 10.1007/BF02187781
- Kung, J. P. S. and Nguyen, H. Q., Weak Maps, in Theory of Matroids (ed. N. White), pp. 254β271, Cambridge University Press, Cambridge, 1986.
- Joseph P. S. Kung and James G. Oxley, Combinatorial geometries representable over $\textrm {GF}(3)$ and $\textrm {GF}(q)$. II. Dowling geometries, Graphs Combin. 4 (1988), no.Β 4, 323β332. MR 965387, DOI 10.1007/BF01864171
- Jon Lee, Subspaces with well-scaled frames, Linear Algebra Appl. 114/115 (1989), 21β56. MR 986864, DOI 10.1016/0024-3795(89)90450-3
- Jon Lee, The incidence structure of subspaces with well-scaled frames, J. Combin. Theory Ser. B 50 (1990), no.Β 2, 265β287. MR 1081231, DOI 10.1016/0095-8956(90)90082-B
- Dean Lucas, Weak maps of combinatorial geometries, Trans. Amer. Math. Soc. 206 (1975), 247β279. MR 371693, DOI 10.1090/S0002-9947-1975-0371693-2
- James G. Oxley, A characterization of the ternary matroids with no $M(K_4)$-minor, J. Combin. Theory Ser. B 42 (1987), no.Β 2, 212β249. MR 884255, DOI 10.1016/0095-8956(87)90041-4
- James G. Oxley, A characterization of certain excluded-minor classes of matroids, European J. Combin. 10 (1989), no.Β 3, 275β279. MR 1029175, DOI 10.1016/S0195-6698(89)80063-0
- James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
- Oxley, J. G. and Whittle, G. P., On weak maps of ternary matroids, Europ. J. Combin. (to appear).
- P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), no.Β 3, 305β359. MR 579077, DOI 10.1016/0095-8956(80)90075-1
- W. T. Tutte, A homotopy theorem for matroids. I, II, Trans. Amer. Math. Soc. 88 (1958), 144β174. MR 101526, DOI 10.1090/S0002-9947-1958-0101526-0
- W. T. Tutte, Lectures on matroids, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 1β47. MR 179781, DOI 10.6028/jres.069B.001
- Whittle, G. P., Inequivalent representations of ternary matroids, Discrete Math. 149 (1996), 233β238.
- Whittle, G. P., A characterisation of the matroids representable over $GF(3)$ and the rationals, J. Combin. Theory Ser. B. 65 (1995), 222β261.
- Thomas Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982), no.Β 1, 47β74. MR 676405, DOI 10.1016/0166-218X(82)90033-6
Additional Information
- Geoff Whittle
- Affiliation: Department of Mathematics, Victoria University, PO Box 600 Wellington, New Zealand
- MR Author ID: 182520
- Email: whittle@kauri.vuw.ac.nz
- Received by editor(s): August 20, 1994
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 579-603
- MSC (1991): Primary 05B35
- DOI: https://doi.org/10.1090/S0002-9947-97-01893-X
- MathSciNet review: 1407504