Tetrads of lines spanning PG(7, 2)

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1 Tetrads of lines spanning PG(7, 2) Ron Shaw, Neil Gordon, Hans Havlicek Abstract Our starting point is a very simple one, namely that of a set L 4 of four mutually skew lines in PG(7, 2). Under the natural action of the stabilizer group G(L 4 ) < GL(8, 2) the 255 points of PG(7, 2) fall into four orbits ω 1, ω 2, ω 3, ω 4, of respective lengths 12, 54, 108, 81. We show that the 135 points ω 2 ω 4 are the internal points of a hyperbolic quadric H 7 determined by L 4, and that the 81-set ω 4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G 81 = (Z3 ) 4 of G(L 4 ). There are 40 subgroups = (Z 3 ) 3 of G 81, and each such subgroup H < G 81 gives rise to a decomposition of ω 4 into a triplet {R H, R H, R H} of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S 3(2) in PG(7, 2). This ties in with the recent finding that each S = S 3 (2) in PG(7, 2) determines a distinguished Z 3 subgroup of GL(8, 2) which generates two sibling copies S, S of S. MSC2010: 51E20, 05B25, 15A69 Key words: Segre variety S 3 (2); line-spread; invariant polynomials 1 Introduction We work for most of the time over F 2 = GF(2), and so we can then identify a projective point x PG(n 1, 2) with the nonzero vector x V (n, 2). In fact we will be dealing with vector space dimension n = 8, and we will start out from a(ny) direct sum decomposition V 8 = V a V b V c V d (1) of V 8 := V (8, 2) into 2-dimensional spaces V a, V b, V c, V d. For h {a, b, c, d} we will write V h = {u h ( ), u h (0), u h (1), u h (2)}, with u h ( ) = 0. (The reason for this labelling of the four element of V h is that in a later section we wish to use 0, 1, 2 as the elements of the Galois field F 3.) So PV 8 = PG(7, 2) is the span of the four projective lines L h := PV h = {u h (0), u h (1), u h (2)}, h {a, b, c, d}. (2) It is surprising, but gratifying, that from such a simple starting point so many interesting and intricate geometrical aspects quickly emerge, as we now describe. 1

2 1.1 H 7 -tetrads of lines in PG(7, 2) For v h V h let (v a, v b, v c, v d ) := v a v b v c v d denote a general element of V 8. Setting U ijkl := (u a (i), u b (j), u c (k), u d (l)) then the 255 points of PG(7, 2) are {U ijkl i, j, k, l {, 0, 1, 2}, ijkl }. (3) First observe that the subgroup G(L 4 ) of GL(8, 2) which preserves the direct sum decomposition (1), and hence the foregoing tetrad of lines, has the semi-direct product structure L 4 := {L a, L b, L c, L d } (4) G(L 4 ) = N Sym(4), where N := GL(V a ) GL(V b ) GL(V c ) GL(V d ), and where Sym(4) = Sym({a, b, c, d}). Hence G(L 4 ) = = 31, 104. The G(L 4 )-orbits of points are easily determined. In addition to the weight wt(p) = wt B (p) of a point p PG(7, 2) with respect to a basis B for V 8, let us also define its line-weight lw(p) as follows: lw(u ijkl ) = r whenever precisely r of i, j, k, l are in {0, 1, 2}. Then the 255 points of PG(7, 2) clearly fall into just four G(L 4 )-orbits ω 1, ω 2, ω 3, ω 4, where ω r = {p PG(7, 2) lw(p) = r}. (5) The lengths of these orbits are accordingly ω 1 = 12, ω 2 = ( ) = 54, ω 3 = ( 4 3) 3 3 = 108, ω 4 = 3 4 = 81. Next take note that there is a unique Sp(8, 2)-geometry on V 8 := V (8, 2), given by a non-degenerate alternating bilinear form B, such that the subspaces V a, V b, V c, V d are hyperbolic 2-dimensional spaces which are pairwise orthogonal. If B = {e i } i {1,2,3,4,5,6,7,8} is any basis such that V 8 = V a V b V c V d = e 1, e 8 e 2, e 7 e 3, e 6 e 4, e 5 (6) then the symplectic product x y := B(x, y) is determined by its values on basis vectors: and so has the coordinate expression e 1 e 8 = e 2 e 7 = e 3 e 6 = e 4 e 5 = 1, e i e j = 0 for other values of i, j, (7) x y = (x 1 y 8 + x 8 y 1 ) + (x 2 y 7 + x 7 y 2 ) + (x 3 y 6 + x 6 y 3 ) + (x 4 y 5 + x 5 y 4 ). Perhaps less obvious is the fact that the tetrad (4) also determines a particular non-degenerate quadric Q in PG(7, 2). For, as we now show, such a quadric Q is uniquely determined by the two conditions 2

3 (i) it has equation Q(x) = 0 such that the quadratic form Q polarizes to give the foregoing symplectic form B : Q(x + y) + Q(x) + Q(y) = x y; (ii) the 12-set of points P(L 4 ) := ω 1 = L a L b L c L d PG(7, 2) supporting the tetrad L 4 is external to Q. For it follows from (i) that the terms of degree 2 in Q must be P 2 (x) = x 1 x 8 + x 2 x 7 + x 3 x 6 + x 4 x 5, and then the eight conditions Q(e i ) = 1 entail that the linear terms in Q must be P 1 (x) = 8 i=1 x i, so that Q(x) = P 2 (x) + P 1 (x) = x 1 x 8 + x 2 x 7 + x 3 x 6 + x 4 x 5 + u x, (8) where u := 8 i=1 e i. Further Q in (8) is seen to satisfy also the four conditions Q(e i + e j ) = 1, ij {18, 27, 36, 45}, so indeed Q(p) = 1 for all p ω 1. Theorem 1. The quadric Q is a hyperbolic quadric H 7 ; moreover H 7 = ω 2 ω 4. Proof. There exist just two kinds, E 7 and H 7, of non-degenerate quadrics in PG(7, 2). An elliptic quadric E 7 has 119 points and a hyperbolic quadric H 7 has 135 points; see [7, Theorem 5.21], [9, Section 2.2]. Since Q is uniquely determined, its internal points must be a union of the G(L 4 )-orbits ω 2, ω 3, ω 4, of respective lengths 54, 108, 81. So the only possibility is that Q is a hyperbolic quadric H 7 = ω 2 ω 4, having = 135 points. (So we will term such a tetrad L 4 of lines in PG(7, 2) a H 7 -tetrad.) Corollary 2. G(L 4 ) is a subgroup of the isometry group G(Q) = O + (8, 2) < Sp(8, 2) of the hyperbolic quadric H 7. Remark 3. In fact G(L 4 ) is a maximal subgroup of O + (8, 2) = O + 8 (2) 2; see [3, p. 85], where it is recorded as S 3 wr S G(L 4 )-invariant polynomials The tetrad L 4 = {L a, L b, L c, L d } determines the following G(L 4 )-invariant sets of flats in PG(7, 2) : (i) four 5-flats: L a, L b, L c, L a, L b, L d, L a, L c, L d, L b, L c, L d ; (ii) six 3-flats: L a, L b, L a, L c, L a, L d, L b, L c, L b, L d, L c, L d ; (iii) four 1-flats: L a, L b, L c, L d. Let (i) F hkl = 0 be the quadratic equation of the 5-flat L h, L k, L l, (ii) F hk = 0 be the quartic equation of the 3-flat L h, L k and (iii) F h = 0 be the sextic equation of the line L h. (See [11, Lemma 2].) Consequently the tetrad L 4 determines the G(L 4 )-invariant polynomials Q 2, Q 4, Q 6, of respective degrees 2, 4, 6, defined as follows: (i) Q 2 = F abc + F abd + F acd + F bcd, (ii) Q 4 = F ab + F ac + F ad + F bc + F bd + F cd, (iii) Q 6 = F a + F b + F c + F d. 3

4 Theorem 4. The 81-set ω 4 has the sextic equation Q ω4 (x) = 0, where Q ω4 := Q 6 + Q 4 + Q 2. Proof. Setting ψ Q := {p PG(7, 2) Q(p) = 0}, the last entry in the following table follows from the three preceding entries. Q(p) if p Q deg Q ω 1 ω 2 ω 3 ω 4 ψ Q ψ Q Q ω 2 ω Q ω 3 ω Q ω 2 ω 3 ω Q 2 + Q 4 + Q ω 4 81 Remark 5. Of course Q 2 = 0 is, see Eq. (8), the H 7 quadric Q of Theorem 1. Also Q 4 was denoted Q 4 in [11, Theorem 17] and Q 6 was denoted Q 6 in [11, Example 20]. The sextic terms in Q 6 are readily found, since Q 6 = Π i 1,8 (1 + x i ) + Π i 2,7 (1 + x i ) + Π i 3,6 (1 + x i ) + Π i 4,5 (1 + x i ). Consequently, in terms of the sextic monomials x j x k := Π i/ {j,k} x i, we see that Q 6 = x 1 x 8 + x 2 x 7 + x 3 x 6 + x 4 x 5 + (terms of degree < 6). (9) Remark 6. A sextic polynomial Q determines, via complete polarization, an alternating multilinear form 6 V 8 F 2, and hence an element b 6 V8 = 2 V 8. (See [10, Section 1.1].) Since Q 6 is G(L 4 )-invariant, and since there is a unique nonzero G(L 4 )-invariant element of 2 V8 = Alt( 2 V 8, F 2 ), namely B in Eq. (7), it follows, in the case Q = Q 6, that b must be the 2 V 8 image of B 2 V8, namely b = e 1 e 8 + e 2 e 7 + e 3 e 6 + e 4 e 5. (10) It follows from (10) that the sextic terms in Q 6 must be the four monomials in (9). So the result (9) could in fact have been deduced in this alternative manner. From the foregoing it is not too difficult to find by hand the explicit coordinate form of the sextic polynomial Q ω4. In fact we used Magma, see [2], to obtain the result below. At times writing 1 = 8, 2 = 7, 3 = 6, 4 = 5, let us 4

5 define the following polynomials: P 1 = 1 i 8 x i, P 2 = 1 i<j 8 x ix j, P 3 = 1 i<j<k 8 x ix j x k, P 4 = 1 k<l 4 x kx k x l x l, P 4 = x m x m P mm, where P mm = P 5 = 1 m 4 1 k<l 4 m / {k,k,l,l } x k x k x l x l x m, k<l, l k k,l / {m,m } x k x l, P 6 = 1 k<l<m 4 x kx k x l x l x m x m = x 1 x 8 + x 2 x 7 + x 3 x 6 + x 4 x 5. Then, assisted by Magma, we found that Q ω4 = P 6 + P 5 + P 4 + P 4 + P 3 + P 2 + P 1. 2 The eight distinguished spreads {L ijk 85 } i,j,k {1,2} Next we show that the partial spread L 4 of four lines determines a privileged set of eight extensions to a complete spread L 85 of 85 lines in PG(7, 2). To this end, for each h {a, b, c, d} let us choose that element ζ h GL(V h ) of order 3 which effects the cyclic permutation (u h (0)u h (1)u h (2)) of the points of L h. Consider the eight Z 3 -subgroups {Z ijk } i,j,k {1,2} of G(L 4 ) defined by Z ijk = A ijk, where A ijk := (ζ a ) i (ζ b ) j (ζ c ) k ζ d. (11) When working using the basis B we will make the following choices for the four ζ h in (11): ζ a : e 1 e 8 e 1 + e 8, ζ b : e 7 e 2 e 2 + e 7, ζ c : e 3 e 6 e 3 + e 6, ζ d : e 5 e 4 e 4 + e 5. (12) We will also choose the u h (0) so that U 0000 is the unit point u of the basis B. Since (A ijk ) 2 + A ijk + I = 0, each Z ijk acts fixed-point-free on PG(7, 2) and gives rise to a spread L ijk 85 of lines in PG(7, 2), with a point p PG(7, 2) lying on the line L ijk (p) := {p, A ijk p, (A ijk ) 2 p} L ijk 85. (13) Note that if in (11) one or more of the ζ h is replaced by the identity element I h GL(V h ) then, although a Z 3 -subgroup of G(L 4 ) which preserves the lines is generated, it is not fixed-point-free on PG(7, 2). So there exist precisely eight extensions of L 4 to a Desarguesian spread of 85 lines in PG(7, 2). Observe that in the case where p is the unit vector u := 8 i=1 e i of the basis B then the eight lines (13) are distinct: for, using ijkl as shorthand for e i + e j + e k + e l, they are 5

6 explicitly L 111 (u) = {u, 1357, 2468}, L 122 (u) = {u, 1256, 3478}, L 212 (u) = {u, 5678, 1234}, L 221 (u) = {u, 2358, 1467}, L 222 (u) = {u, 2568, 1347}, L 211 (u) = {u, 3578, 1246}, L 121 (u) = {u, 1235, 4678}, L 112 (u) = {u, 1567, 2348}. (14) Lemma 7. For p PG(7, 2) the eight lines L ijk (p) are distinct if and only if p ω 4. Proof. We have in (14) just seen that the eight lines are distinct for the point u ω 4, and hence for all p ω 4. Consider a point p = (0, v b, v c, v d ) of line-weight 3. Since A 1jk p = A 2jk p, and so L 1jk (p) = L 2jk (p), the lines L ijk (p) coincide in pairs. Similarly for other points p ω 3. For a point p ω 2 of line-weight 2 the analogous reasoning shows that only two of the lines L ijk (p) are distinct. And of course if p ω 1, that is if p L h for some h {a, b, c, d}, then L ijk (p) = L h for all eight values of ijk. Recall that on a H 7 quadric there exist two systems of generators, see [8, Section 22.4], elements of either system being solids (3-flats). Consequently it follows from the next theorem that the foregoing eight Z 3 -subgroups of G(L 4 ) divide naturally into two sets of size four, namely Z and Z where Z = {Z 111, Z 122, Z 212, Z 221 }, Z = {Z 222, Z 211, Z 121, Z 112 }. (15) Theorem 8. For p ω 4 let Π(p) denote the flat spanned by the four lines L ijk (p), ijk {111, 122, 212, 221}, and let Π (p) denote the flat spanned by the four lines L ijk (p), ijk {222, 211, 121, 112}. Then Π(p) and Π (p) are generators of H 7 which moreover belong to different systems. Proof. The flat Π(p) = L 111 (p), L 122 (p), L 212 (p), L 221 (p) for the point p = (v a, v b, v c, v d ) ω 4 is seen, upon using (ζ h ) 2 + ζ h = I h, to consist of the nine points L 111 (p) L 122 (p) L 212 (p) L 221 (p) of line-weight 4 together with the following six points of line-weight 2: (v a, v b, 0, 0), (v a, 0, v c, 0), (v a, 0, 0, v d ), (0, 0, v c, v d ), (0, v b, 0, v d ), (0, v b, v c, 0). (16) By Theorem 1, for each p ω 4, the flat Π(p) is in fact a solid on the quadric H 7. Similarly the same applies to the flat Π (p), whose six points of line-weight 2 moreover coincide with those of Π(p). So Π(p) Π (p) is the isotropic plane consisting of p = (v a, v b, v c, v d ) together with the six points (16). Consequently, see [8, Theorem , Corollary], for each p ω 4 the generators Π(p) and Π (p) belong to different systems. 6

7 3 The normal subgroup G 81 of G(L 4 ) Let Z h denote that Z 3 subgroup of G(L 4 ) which fixes pointwise each of the three lines L 4 \ L h, h {a, b, c, d}. Then clearly the elementary abelian group G 81 := Z a Z b Z c Z d = Z3 Z 3 Z 3 Z 3 is a normal subgroup of N = GL(V a ) GL(V b ) GL(V c ) GL(V d ) and also of G(L 4 ) = N Sym(4). Observe that ω 4 is a single G 81 -orbit. One easily sees that G 81 is equally well the direct product Z 111 Z 122 Z 212 Z 221 of the four members of Z, and also the direct product Z 222 Z 211 Z 121 Z 112 of the four members of Z. Consider now any Z 3 Z 3 Z 3 subgroup H < G 81. If G 81 = H H H denotes the decomposition of G 81 into the cosets of H then we define subsets R := R H, R := R H, R := R H of ω 4 by R = {hu, h H}, R = {h u, h H }, R = {h u, h H }. (17) In particular R = R H is the orbit of u under the action of the group H. Each such subgroup H < G 81 gives rise to a decomposition ω 4 = R R R of ω 4 into a triplet of 27-sets. As we will now demonstrate, the study of such triplets is greatly simplified by viewing G 81 in a GF(3) light. 3.1 A GF(3) view of G 81 For i, j, k, l F 3 = GF(3) = {0, 1, 2} define A ijkl := (ζ a ) i (ζ b ) j (ζ c ) k (ζ d ) l. Note that if i, j, k {1, 2} then A ijk1 = A ijk, as previously defined in (11). In the following we will view ijkl as shorthand for the element (i, j, k, l) (F 3 ) 4. Since A σ A τ = A σ+τ, σ, τ (F 3 ) 4, (18) observe that A : σ A σ, σ (F 3 ) 4, is an isomorphism mapping the additive group (F 3 ) 4 onto the multiplicative group G 81. Now the orbit of any point p ω 4 under the action of the group G 81 is the whole of ω 4. In particular this is so for the unit point u := 8 i=1 e i of the basis B. Consequently the 81-set ω 4 is in bijective correspondence with (F 3 ) 4 as given by the map θ u : (F 3 ) 4 ω 4 defined by θ u (σ) = p σ := A σ u, σ (F 3 ) 4. (19) Observe that the choices made in (2) and (12) imply that θ u (ijkl) = U ijkl for all ijkl (F 3 ) 4. In the GF(3) space V (4, 3) = (F 3 ) 4 we will chiefly employ the basis B ε := {ε 1, ε 2, ε 3, ε 4 }, where ε 1 = 1000, ε 2 = 0100, ε 3 = 0010, ε 4 = 0001, (20) 7

8 and then write a general element ξ = 4 r=1 ξ rε r V (4, 3) as ξ = ξ 1 ξ 2 ξ 3 ξ 4. We denote the weight of ξ (F 3 ) 4 in the basis B ε by wt ε (ξ). Let us now study subgroups of G 81 by viewing them in the light of their corresponding subspaces in the vector space V (4, 3) = (F 3 ) 4. A Z 3 subgroup of G 81 is of the form {I, A σ, A 2σ } for some non-zero σ V (4, 3). So G 81 contains 40 subgroups = Z 3 which are in bijective correspondence with the 40 points of the projective space PG(3, 3) = PV (4, 3). If we denote by Ξ Ξ the following eight elements of (F 3 ) 4 : Ξ : α = 1111, β = 1221, γ = 2121, δ = 2211, Ξ : α = 2221, β = 2111, γ = 1211, δ = 1121, (21) then observe that A α, A β,..., A δ are the respective generators of the eight Z 3 -subgroups Z 111, Z 122,..., Z 112 Z Z considered in (15). Now under the action by conjugacy of G(L 4 ) on G 81 the particular 4-set {Z a, Z b, Z c, Z d } = { A ε1, A ε2, A ε3, A ε4 } of Z 3 subgroups is fixed, whence T ε := { ε 1, ε 2, ε 3, ε 4 } is a G(L 4 )-distinguished tetrahedron of reference in PG(3, 3). Consequently take note that the eight Z 3 subgroups { A ρ } ρ Ξ Ξ considered in (15) are picked out as the only Z 3 subgroups A ρ of G 81 for which wt ε (ρ) = 4. Next let us consider subgroups H = Z 3 Z 3 Z 3 of G 81. Theorem 9. The normal subgroup G 81 < G(L 4 ) contains precisely 40 subgroups H = Z 3 Z 3 Z 3. These fall into four conjugacy classes C 0, C 1, C 2, C 3 of G(L 4 ), of respective sizes 8, 16, 12, 4. Proof. Each subgroup A ρ, A σ, A τ = Z 3 Z 3 Z 3 arises as {A λ λ V 3 := ρ, σ, τ } from a corresponding projective plane PV 3 = ρ, σ, τ in PG(3, 3). Now there exist precisely 40 planes in PG(3, 3), and these fall into four kinds P 0, P 1, P 2, P 3, where P r denotes those planes in PG(3, 3) which contain precisely r of the vertices ε i of the tetrahedron of reference T ε. There are 8 planes of kind P 0, namely those with one of the 8 equations ξ 4 = c 1 ξ 1 + c 2 ξ 2 + c 3 ξ 3, c 1, c 2, c 3 {1, 2}. (22) Similarly we see that there are, respectively, 16, 12, 4 planes of kinds P 1, P 2, P 3. The theorem now follows, since planes of the same kind are seen to correspond to conjugate Z 3 Z 3 Z 3 subgroups. Finally let us consider subgroups H = Z 3 Z 3 of G 81. Such a subgroup A ρ, A σ arises from a corresponding line ρ, σ PG(3, 3), and so we need to classify lines with respect to the G(L 4 )-distinguished basis B ε. If n w points of a line L PG(3, 3) have weight w, w {1, 2, 3, 4}, with respect to the basis B ε then we will say that L has weight pattern π ε (L) = (n 1, n 2, n 3, n 4 ). 8

9 Theorem 10. The normal subgroup G 81 < G(L 4 ) contains precisely 130 subgroups = Z 3 Z 3. These fall into seven conjugacy classes K 1,..., K 7 of G(L 4 ), of respective sizes 6, 24, 16, 12, 16, 48, 8. Proof. Each subgroup A ρ, A σ = Z 3 Z 3 arises from a corresponding line ρ, σ PG(3, 3), and there exist precisely 130 lines in PG(3, 3). With respect to the G(L 4 )-distinguished tetrahedron of reference T ε the 130 lines L are of seven kinds Λ 1,..., Λ 7 as described in the following table. π e (L) Λ i L Λ 1 (2, 2, 0, 0) 6 L Λ 2 (1, 1, 2, 0) 24 L Λ 3 (0, 3, 1, 0) 16 L Λ 4 (0, 2, 0, 2) 12 L Λ 5 (1, 0, 1, 2) 16 L Λ 6 (0, 1, 2, 1) 48 L Λ 7 (0, 0, 4, 0) 8 (23) The theorem now follows since lines of the same kind correspond to conjugate Z 3 Z 3 subgroups. 3.2 The 27-set denizens of ω 4 It follows from Theorem 9 that the 81-set ω 4 is populated by 40 triplets {R, R, R } of 27-set denizens, as in (17), and that these triplets, and the 120 denizens of ω 4, can be classified into four kinds C 0, C 1, C 2, C 3. One of our aims is to show that precisely eight of these triplets are triplets of Segre varieties S 3 (2). So it helps to remind ourselves at this point about certain aspects of a Segre variety S = S 3 (2) in PG(7, 2), and to relate our present concerns to those in [4], [5] and [11]. First of all, S defines a (27 3, 27 3 ) configuration, each of the 27 points of S lying on precisely 3 lines S, namely three of the 27 generators of S. Moreover the stabilizer group G S of S contains as a normal subgroup a group A 1, A 2, A 3 = Z 3 Z 3 Z 3 which acts transitively on the 27 points of S, the three generators of S through a point p S being the lines L r (p) := {p, A r p, (A r ) 2 p}, r = 1, 2, 3. (24) Here A r satisfies (A r ) 2 + A r + I = 0, each Z 3 group A r acting fixed-point-free on PG(7, 2). Further, as noted in [11, Theorem 5], S determines a distinguished Z 3 -subgroup W which also acts fixed-point-free on PG(7, 2), the distinguished tangent, see [11, Section 2.1], at p S being the line {p, W p, W 2 p}. Moreover, see [11, Section 4.2], under the action of the distinguished Z 3 -subgroup W the Segre variety S gave rise to a triplet {S, S = W (S), S = W 2 (S)} of Segre varieties. In [4, p. 82] (although without proof and using a different notation), [5, Proposition 5] and [11, Section 4.1] the five G S -orbits O 1, O 2, O 3, O 4, O 5 of 9

10 points were described, with O 5 = S and O 4 = S S. These are related to the four G(L 4 )-orbits (5) in the following simple manner: ω 1 = O 1, ω 2 = O 2, ω 3 = O 3, ω 4 = O 4 O 5 = S S S. (25) So ω 4 is a single orbit under the action of the group A 1, A 2, A 3, W = (Z 3 ) 4, this last thus being the group G 81 in our present context. Lemma 11. If p ω 4 and λ (F 3 ) 4, λ 0000, then L λ p := {p, A λ p, A 2λ p} is a line in ω 4 if and only if ±λ Ξ Ξ. Proof. We already know, see (13), that L λ p is a line if λ Ξ Ξ or if λ Ξ Ξ. Also, as noted after equation (21), if ±λ / Ξ Ξ then wt ε (λ) < 4, and so λ = mnrs where at least one of m, n, r, s is 0. For example, suppose λ = mnr0, where m, n, r F 3. Then (I + A λ + A 2λ )U ijkl = U l 0. Remark 12. A partial affine space, see [1, p. 35] or [6, p. 794], is an affine space from which some parallel classes have been removed. For example, the affine space on (F 3 ) 4 turns into a partial affine space if we consider only affine lines with a direction vector ±λ Ξ Ξ and restrict the parallelism of (F 3 ) 4 to the set of those lines. Lemma 11 shows that ω 4 arises as the point set of an isomorphic partial affine space in the following way: The lines in ω 4 are of the form L λ p with ±λ Ξ Ξ. Two lines are parallel if they belong to the same distinguished spread L ijk 85. Theorem 13. A triplet of 27-sets {R H, R H, R H } in (17) which arises from a (Z 3 ) 3 subgroup H = {A λ λ V 3 } will consist of Segre varieties S 3 (2) if and only if the projective plane P = PV 3 PG(3, 3) is of kind P 0. So the 81-set ω 4 contains precisely 24 copies of a Segre variety S 3 (2). Proof. Since each point of a Segre S = S 3 (2) lies on precisely three generators of S, see (24), it follows from the preceding lemma that in order for R H to be a S 3 (2) the subgroup H must be of the form A λ, A µ, A ν for precisely three element λ, µ, ν Ξ Ξ. But a straightforward check shows that planes of the kinds P 0, P 1, P 2, P 3 contain, respectively, precisely 3, 2, 4, 0 points λ with λ Ξ Ξ. So only in the eight cases where the Z 3 subgroup H < G 81 is of kind C 0 can R H be Segre variety S 3 (2). By (25) for one such subgroup a Segre variety arises and, by Theorem 9, the same holds for the remaining subgroups of kind C 0. Remark 14. It is easy to check that if all three of λ, µ, ν are in Ξ, or if all three are in Ξ, then P( λ, µ, ν ) is a plane in PG(3, 3) of kind P 0, thus accounting for all eight planes of kind P 0. But if λ, µ, ν are split 2, 1 or 1, 2 between Ξ, Ξ, then we see that H = A λ, A µ, A ν contains a fourth Z 3 subgroup A ρ with ρ Ξ Ξ : for example, observe results such as α, β, α = α, β, α, β, and α, β, γ = α, β, γ, δ. (26) So such a (Z 3 ) 3 subgroup H is of kind C 2, not C 0, and R H gives rise to a (27 4, 36 3 ) configuration in contrast to the (27 3, 27 3 ) configuration arising from a S 3 (2). 10

11 It was proved in [11, Theorem 18] that a Segre variety S 3 (2) in PG(7, 2) has a sextic equation. In fact, from our results in Section 1.2 we can deduce that if S is any of the 24 copies of a Segre variety S 3 (2) in ω 4 then S has a sextic equation of the form Q S (x) = 0 where, for some polynomial F S of degree < 6, Q S = x 1 x 8 + x 2 x 7 + x 3 x 6 + x 4 x 5 + F S. (27) For recall from (25) that ω 4 = O 4 O 5 where O 5 = S. Now, see [11, Theorem 17], the 201-set (O 4 ) c := ω 1 ω 2 ω 3 S has a quartic equation, say F S = 0, and, see Theorem 4, O 4 S has sextic equation Q 6 + Q 4 + Q 2 = 0. It follows that S has equation Q 6 + Q 4 + Q 2 + F S = 0 which, see (9), is of the form (27). 3.3 Non-Segre triplets in ω 4 When the authors first considered 27-sets such as R αβγ = {A ρ u} ρ α,β,γ they were briefly misled into thinking that R αβγ was a Segre S 3 (2). Upon discovering their error, see (26), they decided that all the deceitful non-segre 27-sets in ω 4 should be termed rogues. In this terminology we can summarize our foregoing results as follows. The 81-set ω 4 is populated by 120 denizen 27-sets which occur as 40 triplets {R, R, R } such that R R R = ω 4. Of these triplets eight are of Segre varieties S 3 (2), sixteen are of rogues of kind C 1, twelve are of rogues of kind C 2 and four are of rogues of kind C 3. Since a Segre variety S 3 (2) spans PG(7, 2), the next two theorems confirm in a more vivid manner that rogues of kinds C 2 and C 3 are not Segre varieties. Theorem 15. Suppose that a 27-set R ω 4 is a rogue of kind C 2. Then R is a 5-flat in PG(7, 2). Proof. Six of the twelve planes of kind P 2 are those having equations of the kind ξ r = ξ s and the other six are those having equations of the kind ξ r = 2ξ s. Consider a plane P = PV 3 in the first six, say with equation ξ 3 = ξ 4. Then, see (3), the subset θ u (V 3 ) consists of the 27 points R := {U ijkk i, j, k F 3 = {0, 1, 2}}. (28) Now any two elements a, b of a line L h L 4 satisfy a b = 1 if a b and a b = 0 if a = b. So the three points of the line L R := {U 00, U 11, U 22 } L c, L d are perpendicular to every point of R. However this is not the case for any other point of PG(7, 2), and so R is the 5-flat (L R ). A plane P = PV 3 in the second six, say with equation ξ 3 = 2ξ 4 = ξ 4, can be treated similarly, with the subset θ u (V 3 ) consisting of the 27 points R := {U ijkk i, j, k F 3 = {0, 1, 2}}, (29) 11

12 where, for k F 3, k denotes k(= 2k). If R is as in (29) then we see that R is the 5-flat (L R ) where L R := {U 00, U 12, U 21 } L c, L d. Remark 16. Consider the triplet {R, R, R } of kind C 2 which contains R as in (28). Then R = A µ (R) and R = A 2µ (R) for suitable µ (F 3 ) 4, for example µ = Consequently L R := {U 01, U 12, U 20 }, L R := {U 02, U 10, U 21 }. Similarly, in the case of the triplet {R, R, R } of kind C 2 which contains R as in (29) we see that L R := {U 01, U 10, U 22 }, L R := {U 02, U 11, U 20 }. So the three lines {L R, L R, L R } are a regulus in the 3-flat L c, L d, and the three lines {L R, L R, L R } are the opposite regulus, the 9-set supporting the two reguli being that hyperbolic quadric H 3 in the 3-flat L c, L d which has L c and L d as its two external lines. Of course similar considerations apply to all twelve of the planes in PG(3, 3) of kind P 2. Thus each of the six pairs of lines in L 4 gives rise to a pair of opposite reguli and hence to a set of = 36 lines L ω 2. Each such L gives rise to a rogue R of kind C 2, namely to R = L ω 4. Theorem 17. Suppose that a 27-set R ω 4 is a rogue of kind C 3. Then R is a 6-flat in PG(7, 2). Proof. By Theorem 9 there are four triplets in ω 4 of kind C 3, which arise from the four planes ξ r = 0, r {1, 2, 3, 4}. Consider the plane P = PV 3 with equation ξ 4 = 0. Then, see (3), it gives rise to the following triplet of 27-sets R := {U ijk0 }, R := {U ijk1 }, R := {U ijk2 }, i, j, k F 3 = {0, 1, 2}. It quickly follows that R, R and R are 6-flats, namely R = U 0, R = U 1, R = U 2, where {U 0, U 1, U 2 } = L d. Of course the other three triplets in ω 4 of kind C 3 are associated in a similar way with the other three lines L a, L b, L c L 4. 4 Intersection properties 4.1 Introduction If 1 and 2 are any two distinct triplets of 27-set denizens of ω 4 note that N ( 1, 2 ) := {R 1 R 2 : R 1 1, R 2 2 } (30) is an ennead of 9-sets which provides a partition of ω 4. For suppose that H 1 and H 2 are two (Z 3 ) 3 subgroups of G 81 whose orbits in ω 4 yield the triplets 1 12

13 and 2. Then the (Z 3 ) 2 subgroup H = H 1 H 2 yields one member {hu, h H} of the ennead of 9-sets (30), the other members of the ennead being the other orbits of H in ω 4. Since the origin of the present research arose from our interest in Segre varieties S 3 (2) in PG(7, 2), let us at least look at the different kinds of intersection S R of a Segre variety S ω 4 with another 27-set denizen R of ω 4. Such an intersection we will term a section of the Segre S. Recall from Theorem 13 that a Segre variety S ω 4 arises as S H := {hu, h H} from a (Z 3 ) 3 subgroup H < G 81 which is of class C 0, being the image, under the isomorphism A in (18), of a 3-dimensional subspace V 3 such that the projective plane P = PV 3 PG(3, 3) is of kind P 0. Lemma 18. Suppose that P = PV 3 PV (4, 3) is a projective plane of kind P 0. Then the 13 lines L P fall into three G(L 4 )-orbits: (i) 3 lines of kind Λ 4 ; (ii) 6 lines of kind Λ 6 ; (iii) 4 lines of kind Λ 3. Proof. Without loss of generality we may, see Remark 14, consider the particular plane P = PV βγδ, where V βγδ = β, γ, δ V (4, 3) = (F 3 ) 4. Observe that the 13 lines in the plane P are as follows: (i) the 3 lines P β, γ, P β, δ, P γ, δ of weight pattern (0, 2, 0, 2); (ii) the 6 lines P β, γ ± δ, P γ, β ± δ, P δ, β ± γ of weight pattern (0, 1, 2, 1); (iii) the 4 lines P β ± γ, β ± δ of weight pattern (0, 3, 1, 0). Hence, see (23), the stated result holds. Equivalently expressed, the thirteen Z 3 Z 3 subgroups of A β, A γ, A δ < G 81 comprise: (i) three of class K 4 (ii) six of class K 6 (iii) four of class K Sections of a Segre variety S ω 4 Without loss of generality we may consider the particular Segre variety S βγδ := S H where H = A β, A γ, A δ : S βγδ = θ u (V βγδ ), where V βγδ = β, γ, δ (F 3 ) 4. In detail the 27 elements of V βγδ are: T δ T δ (31) In the display (31) the rows of each 9-set are orbits of T β and the columns of each 9-set are orbits of T γ, where T λ denotes the translation which maps 13

14 µ (F 3 ) 4 to λ + µ (F 3 ) 4 ; of course (T λ ) 2 = T 2λ. Incidentally observe from (31) that, in conformity with (22), V βγδ is that 3-dimensional subspace of V (4, 3) having the equation ξ 1 + ξ 2 + ξ 3 + ξ 4 = 0. Using the basis B, with ζ h as in (12), we see, in the shorthand notation of Eq. (14), that S βγδ thus consists of the following 27 points: u u 78u u 34u A δ u 38u 58u 137u 246u 23u 468u 157u A δ u 47u 67u 248u 135u 14u 357u 268u. (32) Observe that each 9-set in (32) is a Segre variety S 2 (2), whose generators are the rows and columns in the display, these being orbits of, respectively, A β and A γ. Acting upon S βγδ with a Z 3 subgroup of G 81 which is not in A β, A γ, A δ, for example with A α, will produce the siblings S βγδ, S βγδ of S βγδ, these siblings being the images under θ u of the two affine subspaces in V (4, 3) = (F 3 ) 4 which are translates of V βγδ. Recalling the proof of Lemma 18, let us make the following choices of representatives for the three kinds of 2-dimensional subspaces V 2 V βγδ : (i) β, γ, (ii) δ, β + γ, (iii) β γ, β δ. For the choice (i) the nine elements of V 2 = β, γ are those in the first 9-set in (31). The corresponding section θ u (V 2 ) of S βγδ is the first of the three S 2 (2) varieties in (32). For the choice (ii) the nine elements of V 2 = δ, β + γ satisfy ξ 1 = ξ 2 and are those underlined in: T δ T δ We will term the resulting section θ u (V 2 ) of the S 3 (2) a 3-generator set: it consists of three parallel generators of S 3 (2) which meet a perpendicular S 2 (2) in three points no two of which lie on the same generator of the S 2 (2). Finally the nine elements of V 2 = β γ, β δ satisfy ξ 4 = 0 and are those underlined in: T δ T δ We will term the resulting section θ u (V 2 ) of the S 3 (2) a fan: Definition 19. A subset F of nine points of a S 3 (2) is a fan (= f ar-apart nine) if no two points of F lie on the same generator. (So if F is a fan for a S 3 (2) then the 3 generators through each of the 9 points of F account for all 3 9 = 27 generators of S 3 (2).).. 14

15 We may summarize the foregoing as follows. Theorem 20. A section of a Segre variety S 3 (2) in ω 4 is either (i) a S 2 (2), or (ii) a 3-generator set, or (iii) a fan. 4.3 Hamming distances and troikas In the GF(3) space V (4, 3) = (F 3 ) 4 we have been employing the basis B ε := {ε 1, ε 2, ε 3, ε 4 }, see (20). Using this basis it helps at times to make use of the associated Hamming distance hd ε (ρ, σ) between two elements σ, τ (F 3 ) 4, as defined by hd ε (ρ, σ) = wt ε (ρ σ). Remark 21. If ρ, σ are belong to the same row in (21) observe that hd ε (ρ, σ) = 2, while if ρ, σ belong to different rows then hd ε (ρ, σ) is odd. The next lemma demonstrates that some aspects of orthogonality in the GF(2) space PG(7, 2) can be neatly dealt with in GF(3) terms. Lemma 22. Two points p ρ, p σ ω 4 are orthogonal or non-orthogonal according as hd ε (ρ, σ) is even or odd. Proof. For two points u h (i), u h (j) L h we have u h (i) u h (j) = 1 + δ ij. Hence if ρ = ijkl and σ = i j k l it follows that p ρ p σ = δ ii + δ jj + δ kk + δ ll, whence the stated result. A description of the different sections of a Segre variety S ω 4 can sometimes be helped by the use of the alternative basis B Ξ := {β, γ, δ, α} for V (4, 3) = (F 3 ) 4. Here, as in (21), β = 1221, γ = 2121, δ = 2211, α = The change of basis equations are therefore: β = ε 1 ε 2 ε 3 + ε 4, γ = ε 1 + ε 2 ε 3 + ε 4, δ = ε 1 ε 2 + ε 3 + ε 4, α = ε 1 + ε 2 + ε 3 + ε 4, ε 1 = β γ δ + α, ε 2 = β + γ δ + α, ε 3 = β γ + δ + α, ε 4 = β + γ + δ + α. (33) Observe that the chosen ordering of the elements of the basis B Ξ results in the change of basis matrix M having the simple properties M t = M = M 1. So in V (4, 3) we now have available the Hamming distance in the basis B Ξ, namely hd Ξ (ρ, σ) := wt Ξ (ρ σ), ρ, σ (F 3 ) 4, where wt Ξ (λ) denotes the weight of an element λ V (4, 3) in the basis B Ξ. Lemma 23. wt Ξ (λ) = 2 wt ε (λ) = 2, wt Ξ (λ) = 3 wt ε (λ) = 3, wt Ξ (λ) = 1 wt ε (λ) = 4, wt Ξ (λ) = 4 wt ε (λ) = 1. (34) 15

16 Proof. The results (34) follow immediately from (33). Let us also define the Hamming distance hd(p ρ, p σ ) between two points p ρ, p σ ω 4 to be hd(p ρ, p σ ) := hd Ξ (ρ, σ). Observe that this definition does not depend upon the choice of the point u used in the bijective correspondence θ u in (19), due to the invariance of hd Ξ under translations: hd Ξ (λ + ρ, λ + σ) = hd Ξ (ρ, σ). Suppose that we confine our attention to points p, p,... on a particular Segre variety in ω 4, say S = S βγδ in (32). Then observe that hd(p, p ) = d, d {1, 2, 3}, provided that p can be obtained from p only by the use of at least d of the generating subgroups A β, A γ, A δ of S. In particular distinct points p and p lie on the same generator of S if and only if hd(p, p ) = 1. This Hamming distance on S appears also in [5]. Definition 24. (i) A troika on a given S 3 (2) variety S in ω 4 is a set of three points of S which are hd Ξ = 3 apart. (ii) The centre of a troika t = {p 1, p 2, p 3 } is the point c(t) = p 1 + p 2 + p 3. For example, the three points u, 246u, 357u S βγδ in (32) are hd Ξ = 3 apart and so form a troika. Alternatively this follows from (34) since 0000, 0111, 0222 in (31) are hd ε = 3 apart. Theorem 25. (i) A fan F for a Segre variety S in ω 4 can be uniquely expressed as the union F = t t t of a triplet of troikas. (ii) The three troikas t, t, t in F share the same centre, say c F. (iii) Moreover a fan F for S determines uniquely a triplet T = {F, F, F } of fans such that F F F = S. Further, L(T ) := {c F, c F, c F } is one of the lines of the H 7 -tetrad L 4. Proof. Suppose that S = θ u (V 3 ) and that F S is a fan which contains u. (i) So F = θ u (V 2 ), V 2 V 3, where the projective line L = PV 2 is of kind Λ 3, with line pattern (0, 3, 1, 0). Consequently L has a unique point λ with wt(λ) = 3. Hence the element 0000 (F 3 ) 4 has the unique extension V 1 := {0000, λ, 2λ} to a 3-set of elements of V 2 which are Hamming distance 3 apart. So the point u belongs to a unique troika t F, namely t := θ u (V 1 ) = {u, A λ u, A 2λ u}. If V 2 = λ, µ, the two translates T µ (V 1 ), T 2µ (V 1 ) of V 1 in V 2 yield two other troikas t = A µ t, t = A 2µ t, giving rise to the claimed unique decomposition F = t t t. (ii) Since wt(λ) = 3, precisely one of the coordinates of λ in the basis B ε is zero. First suppose λ satisfies ξ 4 = 0. Then for PV 2 = P( λ, µ ) to be of kind Λ 3 the element µ must also satisfy ξ 4 = 0. So a point p F must be of the form U ijk0. Hence that troika {p, A λ p, A 2λ p} F which contains p has centre c = (I + A λ + (A λ ) 2 )U ijk0 = U 0. (35) So the same point c = U 0, which lies on the line L d L 4, is the centre of each of the three troikas in F. Of course, if instead λ satisfies ξ i = 0 for i = 1, 2, 3 16

17 then the analogous reasoning shows that the common centre of the three troikas in F is U 0 L a, U 0 L b or U 0 L c, according as i = 1, 2 or 3. (iii) We are dealing with S = θ u (V 3 ) where V 3 is of the form V 3 = V 2 ν, V 2 = λ, µ, and where may choose ν to have wt ε = 4. The fan F = θ u (V 2 ) determines a triplet T = {F, F, F } of fans, where F = A ν (F) and F = A 2ν (F), such that F F F = S. Moreover if c F = U 0 as in (35) then c F = A ν c F, c F = A 2ν c F will be the other two points U 1, U 2 of the line L d. Similarly for the other three cases considered in (ii) above, where {c F, c F, c F } is one of the other lines of the tetrad L 4. In the paper [11] the fact that a Segre variety S = S 3 (2) in PG(7, 2) determines a distinguished tetrad L 4 of lines which span PG(7, 2) only emerged rather late, see [11, Section 4.1]. One of the motivations for the present paper was to come to a clearer understanding of the relationship between S and L 4. This can now be achieved: see the next theorem, where it is shown how to obtain the same tetrad L 4 from any of the 24 copies of a Segre variety S 3 (2) in the 81-set ω 4. Theorem 26. A Segre variety S in ω 4 determines precisely four triplets T i, i (1, 2, 3, 4) of fans; further the resulting four lines L i := L(T i ) are the four lines of the H 7 -tetrad L 4. Proof. A Segre variety in ω 4 is of the form S = θ u (V 3 ) where the projective plane P = PV 3 PG(3, 3) is of kind P 0. Now, see (31), there are precisely 8 elements of V 3 of weight 3, and these form 4 pairs, say {±λ 1 }, {±λ 2 }, {±λ 3 }, {±λ 3 }. Consequently the element 0000 V 3 has precisely 4 extensions, namely {0000, λ i, λ i }, i = 1, 2, 3, 4, to a 3-set of elements of V 3 which are Hamming distance 3 apart. Hence the point u = θ u (0000) lies in precisely 4 troikas and, by Theorem 25(i), in precisely 4 fans F i, i = 1, 2, 3, 4. By Theorem 25(iii) each of the resulting 4 triplets T i = {F i, F i, F i } of fans determines a line L(T i) of the tetrad L 4. Further, from the proof of Theorem 25(iii), we see that these lines are distinct. 4.4 Future research We are of the opinion that some further investigation of the denizens of ω 4, and of their interactions, should prove worthwhile. Moreover such an investigation should not be confined to the 27-set denizens arising from Theorem 9, since at least some of the 9-set denizens of ω 4 arising from Theorem 10 deserve attention. In particular the 9-sets in ω 4 which arise from those lines in the table (23) which have weight pattern (0, 0, 4, 0) are certainly noteworthy. For suppose that V (2, 3) V (4, 3) is such that L = PV (2, 3) is of kind Λ 7, and so wt ε (ρ) = 3 for every nonzero element ρ V (2, 3). It follows that any pair of distinct elements ρ, σ of V (2, 3) are Hamming distance 3 apart, and hence, by Lemma 22, the points p ρ, p σ are non-perpendicular: p ρ p σ = 1. In this easy manner we have constructed a 9-cap N = {p ρ } ρ V (2,3) on the quadric H 7. Moreover under the 17

18 action on N of G 81 we will obtain a partition of the 81-set ω 4 into an ennead of quadric 9-caps. References [1] A. Blunck and A. Herzer, Kettengeometrien, Shaker, Aachen, [2] W. Bosma, J. Cannon and C. Playoust, The MAGMA algebra system I: The user language, J. Symbol. Comput., 24 (1997), [3] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, [4] D.G. Glynn, T.A. Gulliver, J.G. Maks and M.K. Gupta, The Geometry of Additive Quantum Codes, available online: [5] H. Havlicek, B. Odehnal and M. Saniga, On invariant notions of Segre varieties in binary projective spaces, Des. Codes Cryptogr., 62 (2012), [6] A. Herzer, Chain geometries, in F. Buekenhout, ed., Handbook of Incidence Geometry, pp , Elsevier, Amsterdam, [7] J.W.P. Hirschfeld, Projective Geometries over Finite Fields, 2nd editition, Clarendon, Oxford, [8] J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford University Press, Oxford, [9] R. Shaw, Finite geometry, Dirac groups and the table of real Clifford algebras, pp , in R. Ablamowicz and P. Lounesto, eds., Clifford Algebras and Spinor Structures, Kluwer Acad. Pubs., Dordrecht, [10] R. Shaw, The psi-associate X # of a flat X in PG(n, 2), Des. Codes Cryptogr., 45 (2007), [11] R. Shaw, N. Gordon and H. Havlicek, Aspects of the Segre variety S 1,1,1(2), Des. Codes Cryptogr., 62 (2012), Ron Shaw, Centre for Mathematics, University of Hull, Hull HU6 7RX, UK r.shaw@hull.ac.uk Neil Gordon, Department of Computer Science, University of Hull, Hull HU6 7RX, UK n.a.gordon@hull.ac.uk Hans Havlicek, Institut für Diskrete Mathematik und Geometrie, Technische Universität, Wiedner Hauptstraße 8 10/104, A-1040 Wien, Austria havlicek@geometrie.tuwien.ac.at 18