Flocks and Partial Flocks of Hyperbolic Quadrics via Root Systems
Laura Bader
, Nicola Durante2
, Maska Law2
, Guglielmo Lunardon4
and Tim Penttila5
2Dipartimento di Matematica e Applicazioni, Universit`a di Napoli “Federico II”, Complesso di Monte S. Angelo-Edificio T, via Cintia, I-80126 Napoli, Italy
DOI: 10.1023/A:1020878313625
Abstract
We construct three infinite families of partial flocks of sizes 12, 24 and 60 of the hyperbolic quadric of PG(3, q), for q congruent to -1 modulo 12, 24, 60 respectively, from the root systems of type D 4, F 4, H 4, respectively. The smallest member of each of these families is an exceptional flock. We then characterise these partial flocks in terms of the rectangle condition of Benz and by not being subflocks of linear flocks or of Thas flocks. We also give an alternative characterisation in terms of admitting a regular group fixing all the lines of one of the reguli of the hyperbolic quadric.
Pages: 21–30
Keywords: flock; maximal exterior set; root system; rectangle condition; partial flock; exterior set; exceptional flock
Full Text: PDF
References
1. L. Bader, “Some new examples of flocks of Q+(3, q),” Geom. Dedicata 27 (1988), 213-218.
2. L. Bader, N. Durante, M. Law, G. Lunardon, and T. Penttila, “Symmetries of BLT-sets,” Designs, Codes and Cryptography, to appear.
3. L. Bader and G. Lunardon, “On the flocks of Q+(3, q),” Geom. Dedicata 29 (1989), 177-183.
4. R.D. Baker and G.L. Ebert, “A nonlinear flock in the Minkowski plane of order 11,” Congr. Numer. 58 (1987), 75-81.
5. W. Benz, “Permutations and plane sections of a ruled quadric,” in Symposia Mathematica INdAM V, 1970, pp. 325-339.
6. A. Bonisoli, “The regular subgroups of the sharply 3-transitive finite permutation groups,” Ann. Disc. Math. 37 (1988), 75-86.
7. A. Bonisoli, “Automorphisms of (B)-geometries,” Res. Lecture Notes Math., Mediterranean (Rende) 1 (1991), 209-219.
8. A. Bonisoli and G. Korchm`aros, “Flocks of hyperbolic quadrics and linear groups containing homologies,” Geom. Dedicata 42(3) (1992), 295-309.
9. L.E. Dickson, Linear Groups: With an Exposition of the Galois Field Theory, Dover Publications, New York, 1958.
10. N. Durante, “Piani inversivi e flock di quadriche,” Master Thesis, University of Naples, 1992.
11. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1990.
12. N.L. Johnson, “Flocks of hyperbolic quadrics and translation planes admitting affine homologies,” J. Geom. 34(1/2) (1989), 50-73.
13. M. Law and T. Penttila, “Flocks, ovals and generalised quadrangles,” Dipartimento di Matematica ed Applicazioni, Napoli, 2000, p. 40.
14. J.J. Seidel, “Discrete non-Euclidean geometry,” in Handbook of Incidence Geometry, F. Buekenhout (Ed.), North-Holland, 1995.
15. J.A. Thas, “Flocks of nonsingular ruled quadrics in PG(3, q),” Atti Accad. Naz. Lincei Rend. 59 (1975), 83-85.
16. J.A. Thas, “Flocks, maximal exterior sets and inversive planes,” in Finite Geometries and Combinatorial Designs, American Mathematical Society, Providence, RI, 1990, pp. 187-218.
2. L. Bader, N. Durante, M. Law, G. Lunardon, and T. Penttila, “Symmetries of BLT-sets,” Designs, Codes and Cryptography, to appear.
3. L. Bader and G. Lunardon, “On the flocks of Q+(3, q),” Geom. Dedicata 29 (1989), 177-183.
4. R.D. Baker and G.L. Ebert, “A nonlinear flock in the Minkowski plane of order 11,” Congr. Numer. 58 (1987), 75-81.
5. W. Benz, “Permutations and plane sections of a ruled quadric,” in Symposia Mathematica INdAM V, 1970, pp. 325-339.
6. A. Bonisoli, “The regular subgroups of the sharply 3-transitive finite permutation groups,” Ann. Disc. Math. 37 (1988), 75-86.
7. A. Bonisoli, “Automorphisms of (B)-geometries,” Res. Lecture Notes Math., Mediterranean (Rende) 1 (1991), 209-219.
8. A. Bonisoli and G. Korchm`aros, “Flocks of hyperbolic quadrics and linear groups containing homologies,” Geom. Dedicata 42(3) (1992), 295-309.
9. L.E. Dickson, Linear Groups: With an Exposition of the Galois Field Theory, Dover Publications, New York, 1958.
10. N. Durante, “Piani inversivi e flock di quadriche,” Master Thesis, University of Naples, 1992.
11. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1990.
12. N.L. Johnson, “Flocks of hyperbolic quadrics and translation planes admitting affine homologies,” J. Geom. 34(1/2) (1989), 50-73.
13. M. Law and T. Penttila, “Flocks, ovals and generalised quadrangles,” Dipartimento di Matematica ed Applicazioni, Napoli, 2000, p. 40.
14. J.J. Seidel, “Discrete non-Euclidean geometry,” in Handbook of Incidence Geometry, F. Buekenhout (Ed.), North-Holland, 1995.
15. J.A. Thas, “Flocks of nonsingular ruled quadrics in PG(3, q),” Atti Accad. Naz. Lincei Rend. 59 (1975), 83-85.
16. J.A. Thas, “Flocks, maximal exterior sets and inversive planes,” in Finite Geometries and Combinatorial Designs, American Mathematical Society, Providence, RI, 1990, pp. 187-218.