UNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA. Semifield spreads

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1 UNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA Semifield spreads Giuseppe Marino and Olga Polverino Quaderni Elettronici del Seminario di Geometria Combinatoria 24E (Dicembre 2007) Dipartimento di Matematica Guido Castelnuovo P.le Aldo Moro, Roma - Italia

2 Semifield spreads Giuseppe Marino and Olga Polverino 1 Introduction Let PG(3, q) be a 3 dimensional projective space over F q. A spread of PG(3, q) is a set of q mutually disjoint lines. Any spread S of PG(3, q) defines via the construction of André/Bruck and Bose a translation plane π(s) of order q 2 (1). A spread S is said to be Desarguesian if π(s) is a Desarguesian plane. Let S be a spread of PG(3, q) and choose homogeneous projective coordinates (X 0, X 1, X 2, X 3 ) in PG(3, q) in such a way that the lines l = {(0, 0, c, d) c, d F q } and l 0 = {(a, b, 0, 0) a, b F q } belong to S. For each line l of S different from l, there is a unique non-singular 2 2 matrix X over F q such that l = l X = {(a, b, c, d) (c, d) = (a, b)x : a, b F q }. The set C S = {X l X S} has the following properties: (i) C S has q 2 elements, (ii) the zero matrix belongs to C S, (iii) X Y is non-singular for all X, Y C S, X Y. Such a set C S is called the spread set associated with S with respect to l 0 and l (see e.g. 3). Conversely, starting from a set C of 2 2 matrices over F q satisfying (i), (ii) and (iii), the set of lines S = {l X : X C} {l } is a spread of PG(3, q) and C S = C. A spread S is a semifield spread with respect to the lines l if C S is closed under sum (see 3); in this case π(s) is a semifield plane of order q 2 and, if it is not Desarguesian, π(s) has kernel isomorphic to F q (i.e., π(s) is coordinatized by a semifield of dimension 2 over its left nucleus). Let π(s) be the semifield plane defined by the semifield spread S of PG(3, q), let C S be the spread set associated with S containing l 0 and l, and let V = V (4, q) denote the vector space of all 2 2 matrices over F q. Since C S is closed under sum and contains the zero matrix, C S defines a vector subspace of V over some subfield of F q. Denote by K the maximal subfield of F q with respect to which C S is a K vector subspace of V, i.e. K is the maximal subfield of F q such that λx C S for any λ K and for any X C S. If π(s) is a non-desarguesian semifield plane, then K is a proper subfield of F q called the center of the semifield plane π(s); equivalently, K is called the center of the semifield spread S. It can be proven that if (F, +, ) is a semifield which coordinatizes the plane π(s), then K is isomorphic to the center K of F (for more details on semifields see 7). From now on suppose that π(s) is a (non-desarguesian) semifield plane and if π(s) has center K, refer to it as a K-semifield plane and to S as a K-semifield spread. Let S be an F q semifield spread of P = ( PG(3, ) q n ) and let C S be the spread set of S with respect a b to the lines l 0 and l. Any matrix M = C c d S defines a point P M of the projective space P = PG(V, F q n) PG(3, q n ). Since M is a non singular matrix, the point P M does not lie on the hyperbolic quadric Q = Q + (3, q) of P with equation X 0 X 3 X 1 X 2 = 0. Also, since C S is an F q vector subspace of V of dimension 2n, the set of points of P defined by the non zero vectors of C S is an F q linear set of P of rank 2n, namely the set L(S) = L CS = { M Fq n : M C S \ {0}}. Hence any F q -semifield spread of P defines an F q -linear set L(S) of P of rank 2n disjoint from Q. 1

3 Conversely, let L be an F q linear set of P of rank 2n, n > 1, with F q maximal subfield of linearity, and assume that L is disjoint from Q. If C is an F q linear subspace of V defining L, then C has dimension 2n over F q, i.e. C = q 2n, C is closed under sum and moreover, since L Q =, each non zero matrix of C is non-singular. This means that C is a spread set of a non-desarguesian semifield spread S of P = PG(3, q n ) with center F q and L is the linear set associated with S, i.e. L = L(S) (see 10). In 2, Cardinali Polverino and Trombetti have proven that if two F q semifield spreads S 1 and S 2 of P = PG(3, q n ) are isomorphic then the associated F q linear sets are isomorphic with respect to the subgroup G of PΓO + (4, q n ) fixing the reguli of the hyperbolic quadric Q of P = PG(3, q n ). Also in 2, by using such a result, semifield spreads of PG(3, q 2 ) with center F q have been classified. In 11 the authors proved that there exist six non isomorphic families F i (i = 0,...,5) of semifield spreads of P = PG(3, q 3 ) with center F q, according to the different configurations of the associated F q linear sets of P = PG(3, q 3 ). In 8 the family F 4 has been partitioned into three non isomorphic subfamilies: F (a) 4, F (b) 4, F(c) 4. In Sections 2 and 3 of these notes we recall some notation and basic results on linear sets of projective spaces and we discuss a number of details on the translation dual which are useful in the subsequent sections. In Section 4 we describe the structure of linear sets associated with semifields belonging to the families F (a) 4, F (b) 4, F(c) 4. Finally, in the last section we pose some open problems related to the classification of the F q semifield spreads of PG(3, q 3 ). 2 Linear sets of a projective space Let Ω = PG(r 1, q n ) = PG(V, F q n), q = p h, p prime, and let L be a set of points of Ω. The set L is said to be an F q linear set of Ω if it is defined by the non zero vectors of an F q vector subspace U of V, i.e., L = L U = { u Fq n : u U \ {0}}. If dim Fq U = t, we say that L has rank t. Any projective subspace of Ω is an F q -linear set, precisely an h-dimensional projective subspace of Ω is an F q linear set of rank n(h+1). Also, the intersection between a projective subspace of Ω and an F q -linear set is an F q -linear set as well, precisely if Λ = PG(W, F q n) is a subspace of Ω and L U is an F q linear set of Ω, then Λ L U is an F q linear set of Λ defined by the F q vector subspace U W and the rank of Λ L U is dim Fq (U W). If L U is an F q -linear set of Ω of rank t, we say that a point P = u Fq n, u U, of L U has weight i in L U if dim Fq ( u Fq n U) = i, and we write ω(p) = i. From the definition of F q linear set and the definition of weight of a point, it is clear that, if x i is the number of points of L U of weight i, we have L U = x 1 + x x n x 1 + (q + 1)x (q n q + 1)x n = q t q + 1. In particular L U q t q + 1. Also, the following property holds true. Property 2.1. (8, Property 3.1 ) A line r of Ω is contained in L U if and only if the rank of the F q linear set L U r is at least n

4 3 The translation dual In the setting of the Introduction, let S be an F q semifield spread of P = PG(3, q n ) with respect to the line l : X 0 = X 1 = 0. Suppose that S contains the line l 0 : X 2 = X 3 = 0, let C S be the spread set associated with S with respect to l and l 0 and let L(S) be the F q -linear set associated with S in P = PG(3, q n ) disjoint from the hyperbolic quadric Q = Q + (3, q n ) : X 0 X 3 X 1 X 2 = 0. If P = PG(V, F q n), regarding V as a 4n dimensional vector space over F q, any point P of P defines an (n 1) dimensional subspace X P of PG(V, F q ) = PG(4n 1, q) and D = {X P : P PG(3, q n )} is a Desarguesian spread ( 1 ) of PG(4n 1, q). The incidence structure whose points are the elements of D and whose lines are the (2n 1) dimensional subspaces of P G(4n 1, q) joining two distinct elements of D, is isomorphic to PG(3, q n ). The pair (PG(4n 1, q), D) is called the F q linear representation of P = PG(3, q n ). If b(x 0, X 1, X 2, X 3 ) = X 0 X 3 X 1 X 2 is the quadratic form which defines the quadric Q and σ b is its polar form, then Tr Fq (b(x 0, X 1, X 2, X 3 )) = 0 defines a non singular hyperbolic quadric ˆQ of PG(4n 1, q) and its polar form is Tr Fq (σ b (X, Y )). Note that if l is a line of P contained in Q, then X l : = P l X P is a maximal singular subspace of ˆQ. Proposition 3.1. The polarity ρ of PG(4n 1, q) associated with ˆQ preserves the linear representation (P G(4n 1, q), D) and induces on P the polarity associated with Q. Proof. The polarity ρ preserves the linear representation (PG(4n 1, q), D) if it maps an element X P of D to a (3n 1)-dimensional subspace of PG(4n 1, q) which is a union of elements of D. Also, ρ induces on P the polarity if it preserves the linear representation (P G(4n 1, q), D) and for any point P of P we have X ρ P = Q P X Q. In order to prove this, let P = v Fq n be a point of P and note that X ρ P = {u V: Tr F q (σ b (u, λv)) = 0 λ F q n} = = {u V: Tr Fq (λσ b (u,v)) = 0 λ F q n}. Since Tr Fq (λσ b (u,v)) = 0 for any λ F q n if and only if σ b (u,v) = 0, we get X ρ P = {u V: σ b(u,v) = 0} = Q P X Q. Since C S is an F q vector subspace of rank 2n of V, it defines in PG(4n 1, q) a (2n 1) dimensional projective subspace L(S) and the F q -linear set L(S) = L CS can be also defined as the set of points P of P such that X P L(S). Applying to L(S) the polarity ρ we get a (2n 1) dimensional projective 1 An (n 1) spread of PG(rn 1, q), with r > 2, is said to be a Desarguesian (or normal) spread if any subspace of PG(rn 1, q) generated by two distinct elements of S is a union of elements of S. 3

5 subspace L(S) ρ of PG(4n 1, q) which defines in P PG(3, q n ) an F q linear set of rank 2n, say L(S), precisely L(S) = {P P : X P L(S) ρ }. From 11, 2.1, Equation (3), it follows that if P and l are a point and a line of P, respectively, then Moreover the following result holds true. dim Fq (P L(S)) = dim Fq (P L(S) ) n; (1) dim Fq (l L(S)) = dim Fq (l L(S) ). (2) Proposition 3.2. The F q linear set L(S) is disjoint from the hyperbolic quadric Q. Proof. We only need to prove that L(S) ρ X P =, for any point P Q, i.e. dim Fq (P L(S) ) = 0 for any point P of Q. Suppose, by way of contradiction, that there exists a point P of Q such that dim Fq (P L(S) ) > 0. By (3) we get dim Fq (P L(S)) n + 1. Let l be one of the lines through P contained in the quadric Q. Since the rank of l (as an F q -linear set) is 2n, the rank of P is 3n and l is contained in P, we get dim Fq (l L(S)) = dim Fq (l (P L(S)) 2n + n + 1 3n = 1, i.e. L(S) l and hence L(S) Q, a contradiction. By Proposition 3.2 and by the arguments of the Introduction, L(S) gives rise to a semifield spread of P PG(3, q n ), called the translation dual of S, that we denote by S. 4 F q semifield spreads of PG(3, q 3 ) An F q semifield spread S of P = PG(3, q 3 ) defines semifields S of order q 6 with left nucleus isomorphic to F q 3 and with center isomorphic to F q, and conversely (see 3). Also, the F q linear set L(S) associated with S in 11 is the same F q linear set L(S) associated with S in the Introduction. In 11, the authors proved that semifields of order q 6 with left nucleus isomorphic to F q 3 and center F q are partitioned into six non isotopic families: F i, i = 0,..., 5 according to the different geometric configurations of the associated F q linear sets; precisely they have proven the following Theorem 4.1. (Marino, Polverino, Trombetti 11, Thm. 4.3) If L is an F q linear set of rank 6 of P = PG(3, q 3 ) (with F q as maximal subfield of linearity), then one of the following configurations occurs: (0) L is a union of either q 2 + q + 1 or q lines of a pencil of P. (1) L is a union of q 2 + q + 1 lines in a plane not belonging to a pencil. (2) L is a union of q 2 + q + 1 lines through a point, not all lines in the same plane. (3) L contains a unique point of weight 2, does not contain any line and is not contained in a plane ( L = q 5 + q 4 + q 3 + q 2 + 1). (4) L contains exactly one line, such a line contains q + 1 points of weight 2 and L is not contained in a plane ( L = q 5 + q 4 + q 3 + 1). 4

6 (5) Any point of L has weight 1 ( L = q 5 + q 4 + q 3 + q 2 + q + 1). Since two F q semifield spreads of PG(3, q n ) are isomorphic if and only if the corresponding semifields are isotopic (see e.g. 7, p.135 for more details on isotopy between semifields), we also have a partition of F q semifield spreads of PG(3, q 3 ) into six non isomorphic families, which we call F i, i = 0,..., 5. In what follows we focus on the family F 4. First of all note that such a family can be partitioned into 3 further classes; indeed Proposition 4.2. (8) Let S be an F q semifield spread of P = PG(3, q 3 ) belonging to F 4, let L(S) be the associated F q linear set of P = PG(3, q 3 ). If r is the unique line of P contained in L(S), one of the following occurs: (a) r L(S) = ; (b) r L(S) = 1; (c) r L(S) = q + 1, where is the polarity induced by Q. From the previous proposition it follows that the family F 4 consists of the following three disjoint non isomorphic subfamilies partitioning it Moreover, F (a) 4 = {S F 4 : r L(S) = }; F (b) 4 = {S F 4 : r L(S) = 1}; F (c) 4 = {S F 4 : r L(S) = q + 1}. Proposition 4.3. (8) An F q semifield spread S belongs to the family F (i) 4 (i {a, b, c}) if and only if its translation dual S belongs to the family F (i) 4 as well. Proof. By Equation (2) a line of P is contained in L(S) if and only if its polar line is contained in L(S). Hence S belongs to F 4 if and only if S does. Also by Eq. (2) the size of r L(S) is the same as r L(S) ; the result follows. Starting from a semifield spread S of a 3 dimensional projective space, another semifield spread, say Ŝ, can be constructed by applying any correlation of the space. The semifield spread Ŝ does not depend, up to isomorphisms, on the chosen correlation and it is called the transpose of S (see e.g. 3). If C S is a spread set associated with S, then a spread set associated with Ŝ is Ct S = {Xt : X C S }, where X t is the transpose matrix of X. Proposition 4.4. Each family F i (i = 0,..., 5) and each subfamily F (j) 4 (j {a, b, c}) is invariant under the transpose operation. Proof. Let S be an F q semifield spread of PG(3, q 3 ) and let C S be an associated spread set. Then the collineation ψ of P with equations X 0 = X 0, X 1 = X 2, X 2 = X 1, X 3 = X 3 maps the linear set L(S) associated with S into the linear set L(Ŝ) associated with Ŝ. This implies that each geometric configuration listed in Theorem 4.1 is preserved under ψ. So S and Ŝ belong to the same family. Also, if S is a semifield of the family F 4 and r is the unique line of P contained in L(S), then r ψ is the unique line contained in L(S) ψ = L(Ŝ) and, since ψ preserves the quadric Q, (r ) ψ = (r ψ ). Hence each of Conditions (a), (b) and (c) of Theorem 4.2 is preserved. 5

7 Here we list the known examples of F q semifield spreads of PG(3, q 3 ). There are two single examples: (i) Semifield spreads associated with the Ganley flock of PG(3, 3 3 ); precisely those semifield spreads whose spread set can be written in the form C S = { ( ) u + t 3 η 1 t + ηt 9 t u : t, u F3 3}, where η is a non square element of F 3 3. (ii) Semifield spreads associated with the Payne Thas ovoid of Q(4, 3 3 ) via the inverse of the Plüker map. A number of examples of F q semifield spreads of PG(3, q 3 ) are associated with the following semifields: (iii) Generalized Dickson semifields with the involved parameters (the corresponding semifield spreads are called Kantor Knuth semifield spreads). (iv) Knuth semifields of type (17) or (19) with the involved parameters (see 3, p. 241). (v) Generalized twisted fields with the involved parameters (see 3, p. 243). (vi) The cyclic semifield spreads with kernel F q 3; i.e. those semifield spreads associated with the cyclic semifields introduced by Jha and Johnson in 6. The families F 0, F 1 and F 2 are completely characterized in 11: S belongs to the family F 0 if and only if S is a Kantor Knuth semifield spread ( (iii) ). S belongs to the family F 1, if and only if q = 3 and S is associated with the Payne Thas ovoid of Q(4, 3 3 ) ( (ii) ). S belongs to the family F 2, if and only if q = 3 and S is associated with the Ganley flock of PG(3, 3 3 ) ( (i) ). So far, examples of F q semifield spreads of PG(3, q 3 ) belonging to the family F 3 are not known. The F q semifield spreads of PG(3, q 3 ) associated with semifields (iv) and (v) belong to the family F 5. From the above list it follows that the only known examples of semifield spreads which belong to F 4 are the cyclic ones (for further details about cyclic semifields see 7, Chapter 27). More precisely, Theorem 4.5. (8) Any cyclic semifield spread S of PG(3, q 3 ) with center F q belongs to the family F (c) 4. Here we give an alternative proof of this result based on the following ( ) u 0 Lemma 4.6. Let F q be the field of scalar 2 2 matrices over F q ; i.e. F q = { : u F 0 u q }. If F is any field of 2 2 matrices over F q 3 isomorphic to F q 2 and containing F q, then { } u + At Bt F = : u, t F q, where A, B, C, D are elements of F q 3 such that A + D, BC AD F q and the polynomial X 2 + (A + D)X + AD BC is F q irreducible. Moreover, if α is any element of F then ( ) α q u + Dt Bt =. Ct u + At ( ) A B Proof. Let M = be an element of F \ F C D q. Any element of F has the form ( ) ( ) ( ) ( ) α 0 β 0 A B α + βa βb + =, 0 α 0 β C D βc α + βd 6

8 with α, β F q. Since the field F is closed under the product, straightforward computations show that A + D F q and BC AD F q. Moreover, since F is a field, any matrix of F is non singular, i.e. the polynomial X 2 +(A+D)X+AD BC is F q irreducible. Let now α be any element of F. Since X 2 + (A + D)X + AD BC is F q irreducible, then also X 2 (A + D)X + AD BC is F q irreducible. Let λ be one root in F q 2 of X 2 (A + D)X + AD BC; then λ F q 2 \ F q and hence {λ, 1} is an F q basis of F q 2. Also, the map ( ) u + At Bt ψ : F λt + u F q 2 is an isomorphism between F and F q 2 = F q (λ). Hence, since λ q = (A + D) λ, if ( ) u + At Bt α =, with u, t F q, then ( ) α q = ψ 1 (λ q u t + u) = + At Bt Ct u + Dt, where u = u + t(a + D) and t = t; the result follows. Now, we are able to prove Theorem 4.5. Proof. (Theorem 4.5) If S is a cyclic semifield spread of PG(3, q 3 ), 6 dimensional over the center, then its spread set with respect to the lines l and l 0 consists of the matrices Iα + Tβ + T 2 γ, with α, β, γ F (3) where F is a field of matrices isomorphic to F q 2, T is a 2 2 matrix over F q 3 such that αt = Tα q for any α F and such that T induces a non linear collineation of PG(2, q 2 ) without fixed points (i.e., (T, F) is an irreducible pair, see 6). Since S is an F q semifield spread, we may assume that F contains the field of scalar matrices over F q and by Lemma 4.6 { } u + At Bt F = : u, t F q, where A+D, AD BC F q and X 2 +(A+D)X +AD BC is F q irreducible. Such a field is contained in the field P of matrices (isomorphic to F q 6) { } u + At Bt P = : u, t F q 3. If T = x0 x 1, with x x 2 x i F q 3, by αt = Tα q and by Lemma 4.6 we get 3 u + At Bt x0 x 1 x0 x = 1 u + Dt Bt x 2 x 3 x 2 x 3 Ct u + ta 7

9 for all u, t in F q and this implies the following conditions: Hence, then T 2 = x 3 = x 0, x 1 = (D A)x 0 Bx 2. C T = So the matrices of the spread set C S are uα + At α Bt α Ct α u α + Dt α + x0 (D A)x 0 Bx 2 C x 2 x 0, x (D A) C x 0x 2 B C x x (D A) C x 0x 2 B C x2 2 x0 (D A)x 0 Bx 2 C x 2 x 0 x (D A) C x 0x 2 B C x x (D A) C x 0x 2 B C x2 2 uβ + At β Bt β Ct β + u β + Dt β uγ + At γ Ct γ Bt γ u γ + Dt γ where u ρ, t ρ F q, ρ {α, β, γ}. The points of L(S) with u β = t β = 0, belong to the line r of P = PG(3, q 3 ) with equations { CX1 BX 2 = 0 BX 0 + (D A)X 1 BX 3 = 0, i.e. r L(S) is an F q linear set of rank 4, hence r L(S). Also, L(S) contains the q + 1 points {(u α + At α, Bt α, Ct α, u α + Dt α ) : u α, t α F q } of L(S) of weight 2. The line r has equations { X0 + X 3 = 0 DX 0 CX 1 BX 2 + AX 3 = 0 and L(S) r consists of the points with u α = t α = u γ = t γ = 0. Since (D A)x 0 Bx 2 x0 uβ + At C β Bt β x 2 x 0 Ct β u β + Dt β is a scalar multiple of T only when t β = 0, then we have L(S) r = q + 1 and hence S belongs to the family F (c) 4. Corollary 4.7. The translation dual of any cyclic semifield spread of PG(3, q 3 ) with center F q belongs to the family F (c) 4. Proof. The result follows from Proposition 4.3 and Theorem In 8, it has been proven that any semifield spread of PG(3, q 3 ) with center F q retaining the same properties on the nuclei of cyclic semifield spreads is isomorphic to a cyclic semifield spread of PG(3, q 3 ). Precisely, Theorem 4.8. (8) Any F q semifield spread of PG(3, q 3 ) whose associated semifield has middle and right nuclei both isomorphic to F q 2 is isomorphic to a cyclic semifield spread of PG(3, q 3 ) and hence it belongs to the family F (c) 4. 8

10 5 Open problems In 4, examples of semifield spreads belonging to the family F (c) 4 with right and middle nuclei both of order q have been constructed for any values of q. Moreover in 5 examples of semifield spreads belonging to the family F (a) 4 have been exhibited for any odd q, whereas just one example of semifield spread belonging to F (b) 4 has been shown for q = 3. Here below we present some open problems: Is the family F 3 not empty? Are there semifield spreads belonging to the family F (b) 4 when q 3? Are there semifield spreads belonging to the family F (a) 4 when q is even? Are there semifield spreads belonging to the family F 5 isomorphic neither to a Knuth semifield spread nor to a semifield spread associated with a Generalized twisted field? References 1 R.H. Bruck, R.C. Bose, The construction of translation planes from projective spaces, J. Algebra, 1 (1964), I. Cardinali, O. Polverino, R. Trombetti: Semifield planes of order q 4 with kernel F q 2 and center F q, European J. Combin., 27 (2006), P. Dembowski: Finite Geometries, Springer Verlag, Berlin, G.L. Ebert, G. Marino, O. Polverino, R. Trombetti: Infinite families of new semifields, submitted. 5 G.L. Ebert, G. Marino, O. Polverino, R. Trombetti: On semifields of order q 6, in preparation. 6 V. Jha, N.L. Johnson: Translation Planes of large dimension admitting nonsolvable groups, J. Geom., 45 (1992), N.L. Johnson, V. Jha, M. Biliotti : Handbook of Finite Translation Planes, Boca Raton, FL: Chapman & Hall/CRC. xix, 861 p., N.L. Johnson, G. Marino, O. Polverino, R. Trombetti: Semifield spreads of PG(3, q 3 ) with center F q, Finite Fields Appl., in press, available online 8 May N.L. Johnson, G. Marino, O. Polverino, R. Trombetti: On a generalization of cyclic semifields, submitted. 10 G. Lunardon: Translation ovoids, J. Geom., 76 (2003), G. Marino, O. Polverino, R. Trombetti: On F q linear sets of PG(3, q 3 ) and semifields, J. Combin. Theory Ser. A, 114 (2007),

11 12 J.A. Thas: Generalized quadrangles and flocks of cones, European J. Combin., 8 (4) (1987), G. Marino Dipartimento di Matematica e Applicazioni Università degli Studi di Napoli Federico II I Napoli, Italy giuseppe.marino@unina.it O. Polverino Dipartimento di Matematica Seconda Università degli Studi di Napoli I Caserta, Italy olga.polverino@unina2.it, opolveri@unina.it 10