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Translation Dual of a Semifield View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Combinatorial Theory, Series A 115 (2008) 1321–1332 www.elsevier.com/locate/jcta Translation dual of a semifield ✩ Guglielmo Lunardon a, Giuseppe Marino a, Olga Polverino b, Rocco Trombetti a a Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli “Federico II”, I-80126 Napoli, Italy b Dipartimento di Matematica, Seconda Università degli Studi di Napoli, I-81100 Caserta, Italy Received 16 July 2007 Available online 21 March 2008 Communicated by William M. Kantor Abstract In this paper we obtain a new description of the translation dual of a semifield introduced in [G. Lunar- don, Translation ovoids, J. Geom. 76 (2003) 200–215]. Using such a description we are able to prove that a semifield and its translation dual have nuclei of the same order. Combining the Knuth cubical array and the translation dual, we give an alternate description of the chain of twelve semifields in the table of [S. Ball, G.L. Ebert, M. Lavrauw, A geometric construction of finite semifields, J. Algebra 311 (2007) 117–129]. © 2008 Elsevier Inc. All rights reserved. Keywords: Semifield; Nuclei; Spread set 1. Introduction A semifield is an algebra satisfying the axioms of a skew field except (possibly) associativity of multiplication. Semifields coordinatize the projective planes of Lenz–Barlotti class V (semifield planes); two semifields are isotopic if and only if they coordinatize isomorphic planes as proved by A.A. Albert in [2] (for more details on the isotopy relation see [6, Section 3]). A semifield is a skew field if and only if the corresponding plane is Desarguesian. Throughout this paper the term semifield will be always used to denote a finite semifield. ✩ This work was supported by the Research Project of MIUR (Italian Office for University and Research) “Strutture geometriche, combinatoria e loro applicazioni”. E-mail addresses: [email protected] (G. Lunardon), [email protected] (G. Marino), [email protected], [email protected] (O. Polverino), [email protected] (R. Trombetti). 0097-3165/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jcta.2008.02.002 1322 G. Lunardon et al. / Journal of Combinatorial Theory, Series A 115 (2008) 1321–1332 The following subsets of a semifield S are subfields: the left nucleus Nl(S) ={a ∈ S | (ab)c = a(bc), ∀b,c ∈ S}, the middle nucleus Nm(S) ={b ∈ S | (ab)c = a(bc), ∀a,c ∈ S}, the right nucleus Nr (S) ={c ∈ S | (ab)c = a(bc), ∀a,b ∈ S}, the center Z(S) ={a ∈ Nl(S) ∩ Nm(S) ∩ Nr (S) | ab = ba, ∀b ∈ S}. It is straightforward to prove that S is a vector space over each of its nuclei and over its center. A spread S of the projective space PG(2t −1,q)is a partition of the point-set of PG(2t − 1,q) into (t − 1)-dimensional subspaces. It is well known that any spread S of PG(2t − 1,q)defines a translation plane π(S) (see, e.g., [6] or [17] for more details). A spread S is a semifield spread when the corresponding plane is coordinatized by a semifield S. When the plane π(S) is Desar- guesian, we will say that the spread S is Desarguesian. If S is a semifield and π is the projective plane coordinatized by S, then S∗ denotes a semifield which coordinatizes the dual plane of π [12, Section 5.2]. If ω is any polarity of PG(2t − 1,q) and S is a spread of PG(2t − 1,q), then Sω ={Xω | X ∈ S} is called the transpose spread of S and the translation plane π(Sω) is called the trans- pose of π(S). It can be proven that such a plane, up to isomorphisms, does not depend on the choice of the polarity ω. Indeed, if ω and ω are polarities of PG(2t − 1,q)and S is a spread of PG(2t − 1,q), then ωω is a collineation of the space mapping Sω to Sω ; hence by [9, Theo- rem 2.26] π(Sω) and π(Sω ) are isomorphic. When π(S) is isomorphic to a plane coordinatized by a semifield S, then π(Sω) is a semifield plane as well, and we will denote by ST a semifield which coordinatizes it.1 The Knuth cubical array introduced in [12] defines a chain of six semifields starting from a given one. It has been proved in [3] that the six semifield planes arising from the cubical array are coordinatized by S, S∗, ST , S∗T , ST ∗ and S∗T ∗, where ∗ and T are the dual and the transpose operations, respectively. Moreover the nuclei of the six semifields are permuted with fixed rules as proved in [16]. The translation dual S⊥ of a semifield S of dimension at most 2 over its left nucleus has been introduced in [15] generalizing the relationship between semifield flock spreads and translation ovoids of Q(4,q). Hence starting from a semifield of dimension 2 over its left nucleus and applying the Knuth operations and the translation dual, we have a chain of twelve semifields, possibly isotopic. On the other hand, in [5] the authors define, from a given semifield S 2-dimensional over its left nucleus, another one, say T, via a construction2 they introduce; combining this construction with the Knuth operations, another chain of twelve semifields arises (see [5, Table]). In this paper (Theorem 2.2), we prove that T is always isotopic to the semifield (S⊥)∗T ∗ and hence T is a Knuth derivative of the translation dual S⊥ of S. As a consequence, the two chains produce the same semifields up to isotopy. In [13], this property has been proven only for semifields associated with a flock and the author states in Remark 3.1, without a hint of proof, that it is true for all semifields. Also, using the spread sets of linear maps, we prove that the middle and the right nuclei of S⊥ are isomorphic respectively to the middle and the right nuclei of S. Finally, we show that the semifields ST ⊥ and S⊥T are isotopic. 1 Note that ST is just a notation which we use to indicate the semifield, up to isotopy, coordinatizing the transpose of the semifield plane coordinatized by S. 2 The construction introduced in [5] applies to a more general class of semifields. G. Lunardon et al. / Journal of Combinatorial Theory, Series A 115 (2008) 1321–1332 1323 2. The translation dual semifield If S satisfies all the axioms for a semifield, except that it has no identity element, then S is called a pre-semifield. From any pre-semifield, one can naturally construct a semifield which is isotopic to it [12]. Next we recall the construction of the translation dual semifield of a semifield at most 2- dimensional over its left nucleus introduced in [15]. If S has dimension 1 over its left nucleus, ⊥ then S is a field and S is a pre-semifield isotopic to a field. We can suppose that S = (Fqn × Fqn , +, ◦) with Nl(S) = Fqn ×{0}, with center Z(S) = Fq ×{0} and (u, 0) ◦ (x, y) = (ux, uy), for all u ∈ Fqn and (x, y) ∈ Fqn × Fqn . Then for any element (x, y) ∈ Fqn × Fqn there exists a (2 × 2)-matrix Xx,y over Fqn such that (u, v) ◦ (x, y) = (u, v)Xx,y and the set CS ={Xx,y: (x, y) ∈ S} is a spread set of matrices associated with S. Such a set satisfies the following properties: (i) |CS|=q2n; (ii) CS is closed under addition and scalar multiplication over Fq ; (iii) the non-zero elements of CS are non-singular. n Hence CS is an Fq -vector subspace of rank 2n of the vector space M = V(4,q ) of all the 2 × 2 matrices with entries in Fqn . + n n Let Q = Q (3,q ) be the hyperbolic quadric of P(M) = PG(M, Fqn ) = PG(3,q ) defined by det X = 0. Then by (iii), CS defines a so-called Fq -linear set L(S) of P(M) of rank 2n disjoint from Q, precisely S = [ ] ∈ C \{ } L( ) X Fqn : X S 0 , [ ] M F n where X Fqn denotes the 1-dimensional vector subspace of over q defined by the matrix X. We refer to such a set as the Fq -linear set associated with S. Conversely, any Fq -linear set L of rank 2n of P(M), disjoint from the hyperbolic quadric Q, defines a pre-semifield which is iso- topic to a semifield with dimension at most 2 over its left nucleus and whose center contains Fq . Two pre-semifields S1 and S2 are isotopic if the associated semifield spreads are isomorphic and this happens if and only if there exists a semilinear map φ : X ∈ M → AXσ B ∈ M (where φ 3 A and B are two non-singular matrices and σ ∈ Aut(F n )) such that CS = C . The map φ q 2 S1 induces a collineation of P(M) preserving the reguli of Q and hence L(S1) and L(S2) are pro- jectively equivalent under the action of the subgroup G of collineations of P(M) fixing the reguli of Q [15]. Let Trqn/q denote the trace function of Fqn over Fq ;themapTrqn/q (det(X)) defines a quadratic form of M (regarded as Fq -vector space) over Fq . The polar form associated with such a quadratic form is Trqn/q (σ (X, Y )), where 3 Note that in the case of a spread which is not a semifield spread this result is a sufficient but not necessary condition of isomorphism.
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