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A6-invariant ovoids of the Klein quadric
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Electronic Notes in Discrete Mathematics 40 (2013) 3–7 www.elsevier.com/locate/endm
A6-invariant ovoids of the Klein quadric V. Abatangelo D. Emma B. Larato 1,2 Dipartimento di Meccanica, Matematica e Management Politecnico di Bari Bari, Italy
Abstract Translation planes associated with A6 -invariant ovoids of the Klein quadric are discussed. Keywords: Translation plane, ovoid, Klein quadric, alternating group of degree six
1
Introduction
h Ovoids of the Klein quadric Q+ 5 of P G(5, p ) correspond to translation planes 2h of order p so that the translation complement modulo scalars of the translation plane is isomorphic to the subgroup of the collineation group of Q+ 5 preserving the corresponding ovoid. The ovoids of Q+ 5 considered here are A6 -invariant and hence correspond to the translation planes of type (A) according to the terminology introduced in [6], that is, they are two-dimensional translation planes over GF (ph ) with ph ≥ 5 and have a subgroup G ∼ = SL(2, 9) in their translation complement. The interest around type (A) translation planes arose 1977 when Ostrom [8] 1
Research supported by the Italian Ministry MIUR, Aspetti geometrici, combinatorici e gruppali delle Geometrie di Galois, PRIN 2008 2 Email: [email protected];[email protected]; [email protected] 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.002
4
V. Abatangelo et al. / Electronic Notes in Discrete Mathematics 40 (2013) 3–7
proved that SL(2, 9) is one of the “largest” two possibilities for a subgroup Γ in the linear translation complement of a translation plane as far as Γ′ = Γ acts irreducibly on V (4, q) and the order of Γ is prime to p. It seems that such a plane does not exist for h > 1, but this has been proven so far only for p = 5, 7. From now on the case h = 1 is discussed. Examples for small primes p were constructed in [1,5,6,7]; see also [Chapter 32][2]. Moorhouse [6] pointed out that slicing the Conway-Kleidman-Wilson binary ovoids of O8+ given in [3] provides A6 -invariant ovoids of the Klein quadric P G(5, p) for every p > 3 prime. Dempwolff and Guthmann [4] combined the slicing method with Number theory to construct more examples. Nevertheless, as Moorhouse himself observed, his slicing method is not exhaustive. The question of deciding which of these A6 -invariant ovoids can be obtained by the slicing method has not been treated so far. In investigating A6 -invariant ovoids of the Klein quadric of P G(5, p) we use the approach introduced in [1] that is completely independent from the slicing method. For smaller primes 5 ≤ p ≤ 37, our goal is a detailed data for every A6 -invariant ovoid O including the decomposition of O into A6 -orbits. Our next step will be to create an analogous database for A5 -invariant ovoids which can be obtained by Moorhouse’s slicing method. Comparing the two databases will allow to answer the above question for 5 ≤ p ≤ 37. So far we have dealt with the case p = 23, and we give a brief account of our results. Since the Moorhouse slicing method applies to p = 23 as in [6, Case II Section 3], and provides an (A)-type translation plane of order 232 , the main question is to identify that plane in our classification. Well, we did the necessary computation, and found that the linear collineation group S6 preserving that Moorhouse ovoid in the Klein quadric has six orbits (of lengths 80, 60, 30, 180, 60, 20). This shows that Moorhouse’s ovoid coincides with our ovoid O1 . It may be noted that the subgroup of S6 which preserves the latin and the greek planes has index 2 and hence is A6 , that is, the translation plane is self-polar [6, Section 4]. Therefore, the translation complement modulo scalars must contain A6 , but not necessarily S6 , which actually happens in this case. Moorhouse’s paper [6] also contains a table showing the number of pairwise non-equivalent A6 -invariant ovoids of the Klein quadric of P G(5, p) for all primes 5 ≤ p ≤ 37. Our computation confirms these data; see Table 2.
V. Abatangelo et al. / Electronic Notes in Discrete Mathematics 40 (2013) 3–7
2
5
Methods and Results
∼ Any G-invariant ovoid of Q+ 5 with G = A6 is the union of caps satisfying the following properties: (i) the caps are G-orbits; (ii) any two caps are consistent, that is their union is still a cap; (iii) the sum of their lengths is equal to q 2 + 1 . The first step consists in determining the caps with property (i). To carry out computations, a suitable equation of Q+ 5 is 6 X h=1
x2h −
6 X
xi xj = 0
i=1 i
so that G acts on the coordinates as A6 in its natural representation. The following table shows the integers that occur as the length of a cap with property (i), as far as 5 ≤ p ≤ 37.
p
|O|
length of A6 -invariant caps
5
26
6, 20
7
50
15, 20, 30
11
122
6, 20, 30, 60, 72, 90
13
170
20, 30, 60
17
290
15, 20, 30, 60, 120, 180
19
362
6, 20, 30, 60, 90, 120, 180
23
530
15, 20, 30, 60, 90, 120, 180, 360
29
842
6, 20, 30, 60, 90, 120, 180, 360
31
962 6, 20, 30, 40, 60, 72, 120, 180, 360
37
1370
20, 30, 60, 90, 120, 180, 360 Table 1
6
V. Abatangelo et al. / Electronic Notes in Discrete Mathematics 40 (2013) 3–7
The second step is to construct the graph Γ whose vertices are the caps with property (i) and adjacency is defined so that any two consistent caps are adjacent. Then the A6 -ovoids are the (maximal) cliques of size p2 + 1. Finally, they are classified up to collineations. The table below shows the number of caps which are G-orbits, the number of A6 -invariant and S6 -invariant ovoids, respectively.
p A6-orbit caps A6 -invariant ovoids S6 -invariant ovoids 5
2
1
0
7
4
2
1
11
14
8
5
13
10
3
2
17
20
4
2
19
39
8
6
23
84
23
6
29
107
7
2
31
177
14
12
37
195
11
6
Table 2
References [1] Biliotti, M., and Korchm´ aros, G., Some finite translation planes arising from A6-invariant ovoids of the Klein quadric, J. Geom. 37 (1990), 29-47. [2] Biliotti M., Jha V., and Johnson N. L., “Handbook of Finite Translation Planes” Pure and Applied Mathematics, 289, Boca Raton: Chapman & Hall/CRC. [3] Conway, J. H.; Kleidman, P. B., and Wilson, R. A., New families of ovoids in O8+ , Geom. Dedicata 26 (1988), 157-170.
V. Abatangelo et al. / Electronic Notes in Discrete Mathematics 40 (2013) 3–7
7
[4] Dempwolff, U., and Guthmann, A., Applications of number theory to ovoids and translation planes, Geom. Dedicata 78 (1999), 201-213. [5] Mason, G., Some translation planes of order 72 which admit SL2 (9), Geom. Dedicata 17 (1985), 297-305. [6] Moorhouse, G. E., Ovoids and translation planes from lattices, Mostly finite geometries (Iowa City, IA, 1996), 123134, Lecture Notes in Pure and Appl. Math., 190, Dekker, New York, 1997. [7] Nakagawa, N., Some translation planes of order 112 which admit SL(2, 9), Hokkaido Math. J. 20 (1991), no. 1, 91-107. [8] Ostrom, T. G., Collineation groups whose order is prime to the characteristic, Math. Z. 156 (1977), 59-71.
Electronic Notes in Discrete Mathematics 40 (2013) 3–7 www.elsevier.com/locate/endm
A6-invariant ovoids of the Klein quadric V. Abatangelo D. Emma B. Larato 1,2 Dipartimento di Meccanica, Matematica e Management Politecnico di Bari Bari, Italy
Abstract Translation planes associated with A6 -invariant ovoids of the Klein quadric are discussed. Keywords: Translation plane, ovoid, Klein quadric, alternating group of degree six
1
Introduction
h Ovoids of the Klein quadric Q+ 5 of P G(5, p ) correspond to translation planes 2h of order p so that the translation complement modulo scalars of the translation plane is isomorphic to the subgroup of the collineation group of Q+ 5 preserving the corresponding ovoid. The ovoids of Q+ 5 considered here are A6 -invariant and hence correspond to the translation planes of type (A) according to the terminology introduced in [6], that is, they are two-dimensional translation planes over GF (ph ) with ph ≥ 5 and have a subgroup G ∼ = SL(2, 9) in their translation complement. The interest around type (A) translation planes arose 1977 when Ostrom [8] 1
Research supported by the Italian Ministry MIUR, Aspetti geometrici, combinatorici e gruppali delle Geometrie di Galois, PRIN 2008 2 Email: [email protected];[email protected]; [email protected] 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.002
4
V. Abatangelo et al. / Electronic Notes in Discrete Mathematics 40 (2013) 3–7
proved that SL(2, 9) is one of the “largest” two possibilities for a subgroup Γ in the linear translation complement of a translation plane as far as Γ′ = Γ acts irreducibly on V (4, q) and the order of Γ is prime to p. It seems that such a plane does not exist for h > 1, but this has been proven so far only for p = 5, 7. From now on the case h = 1 is discussed. Examples for small primes p were constructed in [1,5,6,7]; see also [Chapter 32][2]. Moorhouse [6] pointed out that slicing the Conway-Kleidman-Wilson binary ovoids of O8+ given in [3] provides A6 -invariant ovoids of the Klein quadric P G(5, p) for every p > 3 prime. Dempwolff and Guthmann [4] combined the slicing method with Number theory to construct more examples. Nevertheless, as Moorhouse himself observed, his slicing method is not exhaustive. The question of deciding which of these A6 -invariant ovoids can be obtained by the slicing method has not been treated so far. In investigating A6 -invariant ovoids of the Klein quadric of P G(5, p) we use the approach introduced in [1] that is completely independent from the slicing method. For smaller primes 5 ≤ p ≤ 37, our goal is a detailed data for every A6 -invariant ovoid O including the decomposition of O into A6 -orbits. Our next step will be to create an analogous database for A5 -invariant ovoids which can be obtained by Moorhouse’s slicing method. Comparing the two databases will allow to answer the above question for 5 ≤ p ≤ 37. So far we have dealt with the case p = 23, and we give a brief account of our results. Since the Moorhouse slicing method applies to p = 23 as in [6, Case II Section 3], and provides an (A)-type translation plane of order 232 , the main question is to identify that plane in our classification. Well, we did the necessary computation, and found that the linear collineation group S6 preserving that Moorhouse ovoid in the Klein quadric has six orbits (of lengths 80, 60, 30, 180, 60, 20). This shows that Moorhouse’s ovoid coincides with our ovoid O1 . It may be noted that the subgroup of S6 which preserves the latin and the greek planes has index 2 and hence is A6 , that is, the translation plane is self-polar [6, Section 4]. Therefore, the translation complement modulo scalars must contain A6 , but not necessarily S6 , which actually happens in this case. Moorhouse’s paper [6] also contains a table showing the number of pairwise non-equivalent A6 -invariant ovoids of the Klein quadric of P G(5, p) for all primes 5 ≤ p ≤ 37. Our computation confirms these data; see Table 2.
V. Abatangelo et al. / Electronic Notes in Discrete Mathematics 40 (2013) 3–7
2
5
Methods and Results
∼ Any G-invariant ovoid of Q+ 5 with G = A6 is the union of caps satisfying the following properties: (i) the caps are G-orbits; (ii) any two caps are consistent, that is their union is still a cap; (iii) the sum of their lengths is equal to q 2 + 1 . The first step consists in determining the caps with property (i). To carry out computations, a suitable equation of Q+ 5 is 6 X h=1
x2h −
6 X
xi xj = 0
i=1 i
so that G acts on the coordinates as A6 in its natural representation. The following table shows the integers that occur as the length of a cap with property (i), as far as 5 ≤ p ≤ 37.
p
|O|
length of A6 -invariant caps
5
26
6, 20
7
50
15, 20, 30
11
122
6, 20, 30, 60, 72, 90
13
170
20, 30, 60
17
290
15, 20, 30, 60, 120, 180
19
362
6, 20, 30, 60, 90, 120, 180
23
530
15, 20, 30, 60, 90, 120, 180, 360
29
842
6, 20, 30, 60, 90, 120, 180, 360
31
962 6, 20, 30, 40, 60, 72, 120, 180, 360
37
1370
20, 30, 60, 90, 120, 180, 360 Table 1
6
V. Abatangelo et al. / Electronic Notes in Discrete Mathematics 40 (2013) 3–7
The second step is to construct the graph Γ whose vertices are the caps with property (i) and adjacency is defined so that any two consistent caps are adjacent. Then the A6 -ovoids are the (maximal) cliques of size p2 + 1. Finally, they are classified up to collineations. The table below shows the number of caps which are G-orbits, the number of A6 -invariant and S6 -invariant ovoids, respectively.
p A6-orbit caps A6 -invariant ovoids S6 -invariant ovoids 5
2
1
0
7
4
2
1
11
14
8
5
13
10
3
2
17
20
4
2
19
39
8
6
23
84
23
6
29
107
7
2
31
177
14
12
37
195
11
6
Table 2
References [1] Biliotti, M., and Korchm´ aros, G., Some finite translation planes arising from A6-invariant ovoids of the Klein quadric, J. Geom. 37 (1990), 29-47. [2] Biliotti M., Jha V., and Johnson N. L., “Handbook of Finite Translation Planes” Pure and Applied Mathematics, 289, Boca Raton: Chapman & Hall/CRC. [3] Conway, J. H.; Kleidman, P. B., and Wilson, R. A., New families of ovoids in O8+ , Geom. Dedicata 26 (1988), 157-170.
V. Abatangelo et al. / Electronic Notes in Discrete Mathematics 40 (2013) 3–7
7
[4] Dempwolff, U., and Guthmann, A., Applications of number theory to ovoids and translation planes, Geom. Dedicata 78 (1999), 201-213. [5] Mason, G., Some translation planes of order 72 which admit SL2 (9), Geom. Dedicata 17 (1985), 297-305. [6] Moorhouse, G. E., Ovoids and translation planes from lattices, Mostly finite geometries (Iowa City, IA, 1996), 123134, Lecture Notes in Pure and Appl. Math., 190, Dekker, New York, 1997. [7] Nakagawa, N., Some translation planes of order 112 which admit SL(2, 9), Hokkaido Math. J. 20 (1991), no. 1, 91-107. [8] Ostrom, T. G., Collineation groups whose order is prime to the characteristic, Math. Z. 156 (1977), 59-71.