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Automorphisms of Dowling Lattices and Related Geometries

Published online by Cambridge University Press:  12 September 2008

Joseph E. Bonin
Joseph E. Bonin
Affiliation:
Department of Mathematics, The George Washington University, Washington, DC. 20052
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Abstract

Dowling lattices are a class of geometric lattices, based on groups, which have been shown to share many properties with projective geometries. In this paper we show that the automorphisms of Dowling lattices are analogs of the automorphisms of projective geometries. We also treat similar results for several related geometric lattices.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 1 , March 1995 , pp. 1 - 9
Copyright
Copyright © Cambridge University Press 1995

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References

[1]Artin, E. (1957) Geometric Algebra, Interscience Publishers.Google Scholar
[2]Bennett, M. K., Bogart, K. P. and Bonin, J. (1994) The geometry of Dowling lattices. Advances in Math. 103 131–161.CrossRefGoogle Scholar
[3]Björner, A. and Ziegler, G. (1991) Broken circuit complexes: factorizations and generalizations. J. Combin. Theory, Ser. B 51 96–126.CrossRefGoogle Scholar
[4]Bonin, J. (1993) Automorphism groups of higher-weight Dowling geometries. J. Combin. Theory, Ser. B 58 161–173.CrossRefGoogle Scholar
[5]Bonin, J. and Bogart, K. P. (1991) A geometric characterization of Dowling lattices. J. Combin. Theory, Ser. A 56 195–202.CrossRefGoogle Scholar
[6]Bonin, J. and Kung, J. P. S. (1994) Every Group is the Automorphism Group of a Rank-3 Matroid. Geom. Dedicata 50 243–246.CrossRefGoogle Scholar
[7]Dowling, T. A. (1971) Codes, packings and the critical problem. In: Barlotti, A. (ed.) Atti del Convegno di Geometria Combinatoria e sue Applicazioni, Univ. Perugia, Perugia, Italy209–224.Google Scholar
[8]Dowling, T. A. (1973) A q-analog of the partition lattice. In: Srivastava, J. N. (ed.) A Survey of Combinatorial Theory, North-Holland 101–115.CrossRefGoogle Scholar
[9]Dowling, T. A. (1973) A class of geometric lattices based on finite groups. J. Combin. Theory, Ser. B 14 (1973) 61–86. (Erratum, same journal 15 211.)CrossRefGoogle Scholar
[10]Halsey, M. (1987) Line-closed combinatorial geometries. Discrete Math. 65 245–248.CrossRefGoogle Scholar
[11]Józefiak, T. and Sagan, B. (preprint) Free hyperplane arrangements interpolating between root system arrangements.Google Scholar
[12]Kahn, J. and Kung, J. P. S. (1982) Varieties of combinatorial geometries. Trans. Amer. Math. Soc. 271 485–499.CrossRefGoogle Scholar
[13]Kahn, J. and Kung, J. P. S. (1986) A classification of modularly complemented geometric lattices. European J. Combin. 7 243–248.CrossRefGoogle Scholar
[14]Kung, J. P. S. (1990) Combinatorial geometries representable over GF(3) and GF(q). I. The number of points. Discrete Comput. Geom. 5 84–95CrossRefGoogle Scholar
[15]Kung, J. P. S. and Oxley, J. (1988) Combinatorial geometries representable over GF(3) and GF(q). II. Dowling geometries. Graphs and Combin. 4 323–332.CrossRefGoogle Scholar
[16]Whittle, G. (1989) Dowling group geometries and the critical problem. J. Combin. Theory, Ser. B 47 80–92.CrossRefGoogle Scholar
[17]Zaslavsky, T. (1981) The geometry of root systems and signed graphs. Amer. Math. Monthly 88 88–105.CrossRefGoogle Scholar
[18]Zaslavsky, T. (manuscript) Geometric lattices of structured partitions II: Lattices of group-valued partitions based on graphs and sets.Google Scholar
[19]Zaslavsky, T. (1989) Biased graphs. I. Bias, balance, and gains. J. Combin. Theory, Ser. B 47 32–52.CrossRefGoogle Scholar
[20]Zaslavsky, T. (1991) Biased graphs. II. The three matroids. J. Combin. Theory, Ser. B 51 46–72.CrossRefGoogle Scholar
[21]Zaslavsky, T. (to appear) Biased graphs. III. Chromatic and dichromatic invariants. J. Combin. Theory, Ser. B.Google Scholar
[22]Zaslavsky, T. (to appear) Biased graphs. IV. Geometrical realizations. J. Combin. Theory, Ser. B.Google Scholar