Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q)
Henk D.L. Hollmann1
and Qing Xiang2
1Philips Research Laboratories Prof. Holstlaan 4 5656 AA Eindhoven The Netherlands Prof. Holstlaan 4 5656 AA Eindhoven The Netherlands
2University of Delaware Department of Mathematical Sciences Newark DE 19716 USA Newark DE 19716 USA
2University of Delaware Department of Mathematical Sciences Newark DE 19716 USA Newark DE 19716 USA
DOI: 10.1007/s10801-006-0005-8
Abstract
The group PGL(2, q) has an embedding into PGL(3, q) such that it acts as the group fixing a nonsingular conic in PG(2, q). This action affords a coherent configuration  {\cal R}( q) on the set L {\cal L}( q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions  {\cal R} +( q) and  {\cal R} - ( q) of  {\cal R}( q) to the set L {\cal L} +( q) of secant (hyperbolic) lines and to the set L {\cal L} - ( q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme  {\cal R} - ( q) is pseudocyclic.
We further show that the coherent configurations  {\cal R}( q 2) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme  {\cal R} +( q 2), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes  {\cal R} +( q 2) and  {\cal R} - ( q 2). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.
Pages: 157–193
Keywords: keywords association scheme; coherent configuration; conic; cross-ratio; exterior line; fusion; pseudocyclic association scheme; secant line; strongly regular graph; tangent line
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References
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2. A.E. Brouwer and J.H. van Lint, “Strongly regular graphs and partial geometries,” in Enumeration and Design, D.M. Jackson and S.A. Vanstone (eds.), Academic Press, Toronto, 1984, part 7, pp. 85-122.
3. D. de Caen and E.R. van Dam, “Fissioned triangular schemes via the cross-ratio,” European J. Combin. 22 (2001), 297-301.
4. P.J. Cameron, “Coherent configurations, association schemes and permutation groups,” Groups, Combinatorics & Geometry (Durham, 2001), World Sci. Publishing, River Edge, NJ, 2003, pp. 55-71.
5. G. Ebert, S. Egner, H.D.L. Hollmann, and Q. Xiang, “On a four-class association scheme,” J. Combin. Theory (A) 96 (2001), 180-191.
6. G. Ebert, S. Egner, Henk D.L. Hollmann, and Q. Xiang, “Proof of a conjecture of De Caen and van Dam,” Europ. J. Combin. 23 (2002), 201-206.
7. D.G. Higman, “Coherent configurations. I,” Rend. Sem. Mat. Univ. Padova 44 (1970), 1-25.
8. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, 2nd edition. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1998.
9. H.D.L. Hollmann, Association Schemes, Masters Thesis, Eindhoven University of Technology,
1982. Springer
10. H.D.L. Hollmann, A note on the Wilbrink graphs, preprint.
11. H.D.L. Hollmann and Q. Xiang, “Pseudocyclic association schemes arising from the actions of PGL(2, 2m ) and P L(2, 2m ),” to appear in J. Combin. Theory, Ser. A.
12. R. Mathon, “3-class association schemes,” in Proceedings of the Conference on Algebraic Aspects of Combinatorics (Univ. Toronto, Toronto, Ont., 1975), Congressus Numerantium, No. XIII, Utilitas Math., Winnipeg, Man., 1975, pp. 123-155.
13. T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, Ill., 1967
14. H. Tanaka, “A 4-class subscheme of the association scheme coming from the action of PGL(2, 4 f ),” Europ. J. Combin. 23 (2002), 121-129.
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