aa r X i v : . [ m a t h - ph ] O c t ON MINKOWSKI SPACE AND FINITE GEOMETRY
MARKO OREL
Abstract.
The main aim of this interdisciplinary paper is to characterizeall maps on finite Minkowski space of arbitrary dimension n that map pairsof distinct light-like events into pairs of distinct light-like events. Neitherbijectivity of maps nor preservation of light-likeness in the opposite direction,i.e. from codomain to domain, is assumed. We succeed in in many cases,which include the one with n divisible by 4 and the one with n odd and ≥ Introduction
When Einstein introduced special relativity [28] and derived the Lorentz trans-formation that relates the coordinates between two frames of reference, he assumedthat this transformation is an affine map, that is, a sum of an additive and aconstant map. It was later proved by Aleksandrov [7], that this assumption is re-dundant and that the constancy of speed of light, which is equal in both framesof reference, is sufficient. More precisely, he proved that any bijective map Φ on4-dimensional Minkowski space-time, for which a pair of events are light-like if andonly if their Φ-images are light-like, is up to translation and dilation a Lorentztransformation, so an affine map. Maps characterized by Aleksandrov are some-times referred as bijective maps that preserve the speed of light in both directions(cf. [47]). A very similar characterization was independently obtained by Zee-man [73], who assumed in addition that causality is preserved. A generalization ofAleksandrov theorem for n -dimensional ( n ≥
3) Minkowski spaces was proved byhimself [6] and by Borchers and Hegerfeldt [17]. Hua observed that pairs of dis-tinct events ( ct , x , y , z ) and ( ct , x , y , z ) in 4-dimensional Minkowski space-time are light-like if and only if their corresponding complex hermitian matrices (cid:2) ct + x y + iz y − iz ct − x (cid:3) and (cid:2) ct + x y + iz y − iz ct − x (cid:3) are adjacent , that is, their difference is of rankone [37, Chapter 5]. Bijective maps that preserve adjacency in both directions arecharacterized by fundamental theorems of geometry of matrices (see the book [70]),so Hua was able to use such a theorem for 2 × that the maps that preserve adjacency are bijective, and the adjacency was assumedto be preserved in just one directions. In the language of Minkowski space-time,their result applied on 2 × R of real numbers isinfinite.Perhaps the first to consider an approximation between structures over the fieldof real numbers and structures over a very large finite field were astronomers Kus-taanheimo [44, 45] and J¨arnefelt [41]. Motivated by some alternative theories inparticle physics, Lorentz transformations over finite fields were considered by manyauthors: see for example Coish [24], Shapiro [67], Ahmavaara [1–4], Yahya [72],Joos [42], Beltrametti and Blasi [12, 13]. Recently, Foldes [29] derived an approx-imation result between Lorentz transformations over real numbers and Lorentztransformations over a finite field. Blasi et al [15, Proposition 2] obtained an ana-log of Aleksandrov’s result for 4-dimensional Minkowski space over finite fields ofprime cardinality. Lester [48] generalized this theorem to more general spaces thatinclude n -dimensional ( n ≥
3) Minkowski spaces over finite fields. Recently, theauthor characterized maps on 4-dimensional Minkowski space over a finite field,that map pairs of distinct light-like events into pairs of distinct light-like events(see [57, Theorem 4.9] and [55]). No bijectivity was assumed and the ‘speed oflight’ was assumed to be preserved in just one directions. However, it turned outthat both bijectivity of maps and the preservation of the ‘speed of light’ in theopposite direction is obtained automatically. In particular, there is a fundamentaldifference with the real case mentioned above, since there do not exist maps withthe image contained in a set of pairwise light-like events. Nonexistence of suchmaps is not trivial, and several tools from graph theory were applied to prove it.It is the aim of this paper to obtain an analog of [57, Theorem 4.9] on n -dimensional case, that is, to characterize maps on n -dimensional Minkowski spaceover a finite field that map pairs of distinct light-like events into pairs of distinctlight-like events. It should be immediately emphasized that unlike the usual case,when generalizing certain problems from small number of dimensions to higher di-mensional objects requires just some minor modifications in the proofs, here thesituation is quite different. Firstly, and less importantly, the technique used inthe proof of [57, Theorem 4.9] translated the problem of characterizing maps thatpreserve the speed of light on 4-dimensional Minkowski space to the problem ofcharacterizing maps that preserve adjacency on 2 × n -dimensional Minkowski space can be different as the one obtained in 4 dimensions.Nevertheless, a completely analogous result as in 4 dimensions will be obtained for n ≡ n ≥
9, and for some other special choices of space dimen-sion and field cardinality. Surprisingly, we will see that for the rest of the cases the
N MINKOWSKI SPACE AND FINITE GEOMETRY 3 question whether there exist nonbijective maps that map pairs of distinct light-likeevents into pairs of distinct light-like events (and these maps include those withthe image contained in a set of pairwise light-like events) is related to a well knownproblem from finite geometry, namely to existence of ovoids in orthogonal polarspaces. This problem remains unsolved in general, despite an extensive research inthis area performed in the last few decades. Perhaps the quickest way, for a non-specialist, to realize the amount of research done that is related to the existence ofovoids, is to check the MathSciNet database [51], which currently contains almost150 articles with the word ‘ovoid’ or ‘ovoids’ in the title (excluding those, where‘ovoid’ means something else).The paper is organized as follows. In Section 2 and Section 3 we recall severalresults from graph and matrix theory, respectively, which are used in the sequel.Section 4, where we investigate the core of an affine polar graph, is the fundamentalpart of the paper. It is divided in three subsections. In the first we prove that anaffine polar graph is either a core or its core is a complete graph. The other twosubsections explore which of the two possibilities actually occur. Second subsectionis based on spectral graph theory, while in the third subsection tools from finitegeometry, which concern ovoids in orthogonal polar spaces, are used. Results fromSection 4 are applied in Section 5, where we classify mappings on finite Minkowskispace that map pairs od distinct light-like events into pairs of distinct light-likeevents.Beside the main results of the paper, that is, Theorem 3, Theorem 2, and The-orem 1, we prove several auxiliary results in the process that may be interestingon its own. In particular, Proposition 4 (ii) may be useful in graph theory, whileProposition 6 together with Theorem 2 may be interesting from finite geometricperspective. 2.
Graph theory
All graphs in this paper are finite undirected with no loops and multiple edges.Such a graph
Γ is a pair (
V, E ) formed by the vertex set V = V (Γ), which is finite,and the edge set E = E (Γ), which contains some unordered pairs of distinct vertices.The complement Γ of graph Γ is a graph with the same vertex set V (Γ) = V (Γ)and edges defined by { u, v } ∈ E (Γ) , u = v ⇐⇒ { u, v } / ∈ E (Γ) , u = v. A graph Γ ′ is a subgraph of graph Γ if V (Γ ′ ) ⊆ V (Γ) and E (Γ ′ ) ⊆ E (Γ). A subgraphΓ ′ is induced by the set U ⊆ V (Γ) if U = V (Γ ′ ) and E (Γ ′ ) = (cid:8) { u, v } ∈ E (Γ) : u, v ∈ U (cid:9) . Graph Γ is connected if for every pair of distinct vertices u, v ∈ V (Γ),there is a path that connects them, i.e., there are vertices u = v , v , . . . , v t − , v t = v such that { v i − , v i } ∈ E (Γ) for all i . The length of the shortest path between u and v , that is, the minimal value t is the distance between u and v , which isdenoted by d ( u, v ). The diameter of a connected graph equals max u,v ∈ V (Γ) d ( u, v ).The neighborhood of vertex u ∈ V (Γ) is the subgraph that is induced by the set { v ∈ V (Γ) : d ( u, v ) = 1 } . The closed neighborhood of vertex u ∈ V (Γ) is thesubgraph that is induced by the set { v ∈ V (Γ) : d ( u, v ) ≤ } .A graph homomorphism between two graphs Γ and Γ is a map Φ : V (Γ ) → V (Γ ) such that the following implication holds for any v, u ∈ V (Γ ):(1) { u, v } ∈ E (Γ ) = ⇒ { Φ( u ) , Φ( v ) } ∈ E (Γ ) . MARKO OREL
In particular, Φ( u ) = Φ( v ) for any edge { u, v } . A bijective homomorphism for whichthe converse implication of (1) also holds, is a graph isomorphism . If Γ = Γ , a ho-momorphism is an endomorphism and an isomorphism is an automorphism . Sincethe graphs considered are finite, an automorphism is the same as bijective endo-morphism (cf. [35, Observation 2.3]). A graph Γ is a core if all its endomorphismsare automorphisms. If Γ is a graph, then its subgraph Γ ′ is a core of Γ if it is a coreand there exists some homomorphism Φ : Γ → Γ ′ . Any graph has a core, which isalways an induced subgraph and unique up to isomorphism [32, Lemma 6.2.2]. If Γ ′ is a core of Γ, then there exists a retraction Ψ : Γ → Γ ′ , that is, a homomorphism,which fixes the subgraph Γ ′ . In fact, if Φ : Γ → Γ ′ is any homomorphism, thenthe restriction Φ | Γ ′ is an automorphism, since Γ ′ is a core. Hence, the composi-tion Ψ := (Φ | Γ ′ ) − ◦ Φ is a retraction onto Γ ′ . A graph is regular of valency k , ifthe neighborhood of arbitrary vertex consists of k vertices. A graph Γ is vertex-transitive if for every pair of vertices u, v ∈ V (Γ) there exists some automorphismΦ of Γ such that Φ( u ) = v . A vertex-transitive graph is always regular. An arc ofa graph is just a directed edge, that is, an ordered pair ( u, v ) of adjacent vertices.A graph Γ is arc-transitive if for every pair of arcs ( u , v ) and ( u , v ) there ex-ists some automorphism Φ of Γ such that ( u , v ) = (cid:0) Φ( u ) , Φ( v ) (cid:1) . If a graph isvertex-transitive, then its core is vertex-transitive as well (cf. [32, Theorem 6.13.1]).Analogous result with the same proof holds for arc-transitivity [32, p. 128, Exer-cise 5].A complete graph K t on t vertices is a graph such that every pair of distinctvertices form an edge. Complete graphs are cores. A clique in a graph Γ is a subset K ⊆ V (Γ) that induces a complete subgraph. A maximum clique is a clique of thelargest possible cardinality, which is referred as the clique number ω (Γ) of graphΓ. A set I ⊆ V (Γ) is independent if none of the pairs of vertices in I form anedge. The independence number α (Γ) is the cardinality of the largest independentset in Γ. Since cliques and independent sets swap their roles in the complement ofa graph, we have(2) ω (Γ) = α (Γ) and α (Γ) = ω (Γ) . It is well known that, for a vertex-transitive graph,(3) ω (Γ) α (Γ) ≤ | V (Γ) | , where | V (Γ) | denotes the number of vertices in the graph (cf. [9, Corollary 3.11,p. 1471]). Moreover, the following holds. Lemma 1. [23, p. 148] If the core of a vertex-transitive graph Γ is a completegraph, then ω (Γ) α (Γ) = | V (Γ) | . The lexicographic product of graphs Γ and Ξ is the graph Γ[Ξ] whose vertex setis V (Γ) × V (Ξ), and for which { ( g , h ) , ( g , h ) } is an edge of Γ[Ξ] precisely if { g , g } ∈ E (Γ), or g = g and { h , h } ∈ E (Ξ). It was proved in [30, Theorem 1]that(4) α (cid:0) Γ[Ξ] (cid:1) = α (Γ) α (Ξ) . Since Γ[Ξ] = Γ[Ξ] (cf. [36, p. 57]), it follows from (2) and (4) that(5) ω (cid:0) Γ[Ξ] (cid:1) = ω (Γ) ω (Ξ) . Given a graph Γ, we use χ (Γ) to denote its chromatic number, that is, thesmallest integer m for which there exists a vertex m -coloring, i.e., a map ϕ : V (Γ) → N MINKOWSKI SPACE AND FINITE GEOMETRY 5 { x , x , . . . , x m } on the vertex set, which maps into a set of cardinality m andsatisfies ϕ ( u ) = ϕ ( v ) whenever { u, v } is an edge. Obviously, χ (Γ) ≥ ω (Γ). Equalityholds if and only if the core of Γ is a complete graph (cf. [23, 33]).Let G be a finite group and S a subset of G that is closed under inverses and doesnot contain the identity. The Cayley graph
Γ = Cay(
G, S ) is the graph with G asits vertex set, two vertices g and h being joined by an edge if and only if g − h ∈ S .For any fixed g ∈ G , the map h gh is an automorphism of Γ, so Cayley graphsare vertex-transitive. The following result, which claims that a stronger statementthan the one in Lemma 1 holds for a special type of Cayley graphs, was alreadyproved in unpublished notes [31] (see also the thesis [62]). We provide a prooffor reader’s convenience. Moreover, the proof contains a construction of a graphhomomorphisms onto a maximum clique (7), which will be relevant in Section 5. Proposition 1.
Let
Γ =
Cay ( G, S ) be a Cayley graph such that the inverting map g g − is an automorphism of Γ . Then the following is equivalent: (i) a core of Γ is a complete graph, (ii) a core of Γ is a complete graph, (iii) ω (Γ) α (Γ) = | V (Γ) | .Remark . It is easy to see that the map g g − is a graph automorphism ifand only if the Cayley graph is normal (cf. [61]). In particular, this is satisfied forgraphs over abelian groups. However, there exist Cayley graphs over non-abeliangroups that are normal. An example of such is obtained if G = GL n ( F q ) is thegeneral linear group over a finite field F q with q elements and S ⊆ G is a subsetof matrices with determinant x or x − , where x is a fixed generator of the cyclicmultiplicative group F q \{ } . Remark . There exist vertex-transitive graphs such that ( i ) and ( ii ) are not equiv-alent. An example of such is the graph Q +5 (5) (see Section 4 for the definition). Infact, it follows from [23, Theorem 3.5] and [60, Theorem 4.1] that the core of Q +5 (5)is complete, while the core of Q +5 (5) is not complete, as it follows from [23, Theo-rem 3.5] and [63, Theorem 6(b)]. In fact, Q +5 (5) is a core. Remark . A graph is a
CIS graph if every its maximal clique and maximal inde-pendent set (according to inclusion) intersect. A characterization of CIS circulantswas obtained in [18, Theorem 3]. Subsequently, the result was generalized for arbi-trary CIS vertex-transitive graphs [27]. In particular, it was proved that equation( iii ) from Proposition 1 is satisfied for any such graph. Hence, any CIS normalCayley graph has a complete core.
Proof of Proposition 1.
Since (2) holds and a complement of a vertex-transitivegraph is vertex-transitive, both ( i ) and ( ii ) implies ( iii ) by Lemma 1.Assume now that ( iii ) holds. Let K and I be a clique of size ω (Γ) and anindependent set of size α (Γ), respectively. We claim that(6) G = K · I := { ki : k ∈ K , i ∈ I} . By ( iii ) it suffices to show that k i = k i for ( k , i ) = ( k , i ). So assumethat k i = k i . Since I is an independent set and the inverting map is a graphautomorphism, it follows that k − k = i i − / ∈ S . Since K is a clique, we deducethat k = k , and consequently i = i . Hence, (6) holds and any g ∈ G can beuniquely written as a product g = k g i g , k g ∈ K , i g ∈ I . MARKO OREL
We claim that the map G → K , defined by(7) g k g , is a homomorphism between Γ and a complete subgraph induced by K , which istherefore a core of Γ, so ( i ) holds. Since K is a clique, it suffices to show thatadjacent vertices are not mapped in the same vertex. This certainly holds, since k g = k g implies g − g = i − g i g / ∈ S , which means that g and g are not adjacent.Similarly we see that the map G → I , defined by g i g , is a homomorphismbetween Γ and a complete subgraph induced by I , which is therefore a core of Γ,so ( ii ) holds. (cid:3) The spectrum of graph Γ consists of the eigenvalues (and their multiplicities) ofits adjacency matrix , i.e., the | V (Γ) | × | V (Γ) | real binary matrix with 1 at position( i, j ) if the i -th and j -th vertices are adjacent and 0 otherwise (order of verticesis arbitrary). The next lemma is the well known Hoffman upper bound for theindependence number (cf. [22, Theorem 3.5.2]). Lemma 2. If Γ is a regular graph with nonzero valency, then (8) α (Γ) ≤ | V (Γ) | · − λ min λ max − λ min , where λ max and λ min are the largest and smallest eigenvalue of Γ , respectively. Ifan independent set I meet this bound, then every vertex not in I is adjacent toprecisely − λ min vertices of I . Let G be a finite abelian group. A character ξ of G is a group homomorphism ξ : G → C \{ } , that is, a nowhere-zero complex map that satisfies ξ ( gh ) = ξ ( g ) ξ ( h )for all g, h ∈ G . It is well known, that there exist precisely | G | distinct characters(cf. [49, Theorem 5.5]). Moreover, they determine the spectrum of Cayley graphson G . Lemma 3. (cf. [22, Subsection 1.4.9]). If G is a finite abelian group, then theeigenvalues of Cay ( G, S ) are precisely the values P s ∈ S ξ ( s ) , where ξ ranges overall characters of G . Matrix theory
From now on, if not stated otherwise, F q is a finite field with q elements, where q = p k is a power of an odd prime p . The number of nonzero squares and thenumber of non-squares in F q both equal q − . More precisely, for any non-square d ∈ F q , the field is of the form(9) F q = { , x , . . . , x q − , dx , . . . , dx q − } for some nonzero x , . . . , x q − ∈ F q (cf. [68, Theorem 6.20]). Let n ≥ GL n ( F q ) denote the set of all n × n invertible matrices with coefficientsin F q . Similarly let SGL n ( F q ) ⊆ GL n ( F q ) be the subset of all symmetric matrices,that is, the elements of SGL n ( F q ) satisfy A ⊤ = A and det( A ) = 0, where ⊤ anddet are the transpose and the determinant of a matrix, respectively. The set of allcolumn vectors of dimension n , i.e. n × F q ,is denoted by F nq . Its elements are written in bold style, like x . N MINKOWSKI SPACE AND FINITE GEOMETRY 7
We now recall some auxiliary results from matrix theory. The first lemma is wellknown and it calculates the determinant of a rank–one perturbation of an invertiblematrix.
Lemma 4. (cf. [66, Chapter 14]) Let A ∈ GL n ( F q ) and x , y ∈ F nq . Then det( A + xy ⊤ ) = (det A ) · (1 + y ⊤ A − x ) . Corollary 1.
Let a := ( a , . . . , a n ) ⊤ ∈ F nq , where a = 0 . Then, (10) det a a a a a . . . a a n a a a a a a . . . a a n a ... ... . . . ... a a n a a a n a . . . a n a = a ⊤ a a . Proof.
Matrix in (10) equals I n − + a bb ⊤ , where I n − is the ( n − × ( n − b = ( a , . . . , a n ) ⊤ . The proof ends by Lemma 4. (cid:3) Lemma 5.
Let A ∈ GL n − ( F q ) , x ∈ F n − q , and a ∈ F q . Matrix (11) (cid:18) a x ⊤ x A (cid:19) is singular if and only if a = x ⊤ A − x in which case there is P ∈ GL n ( F q ) suchthat (12) (cid:18) x ⊤ A − x x ⊤ x A (cid:19) = P ⊤ (cid:18) A (cid:19) P. Proof. If a = x ⊤ A − x , then it is straightforward to check that matrix (11) hasinverse (cid:18) A − (cid:19) + 1 a − x ⊤ A − x (cid:18) − A − x (cid:19) (cid:18) − A − x (cid:19) ⊤ . For a = x ⊤ A − x , the matrix (11) equals (12), where P = (cid:18) A − x I n − (cid:19) , so it is singular. (cid:3) Lemma 6 can be found in [69, Theorem 6.8]. It can be also deduced fromcelebrated Witt’s Theorem (cf. [40, Theorem 8, p. 167] or [8, Theorem 3.9]), byusing a similar procedure as in [56, Lemma 2.3].
Lemma 6.
Let A ∈ SGL n ( F q ) and let Q , Q be two n × m matrices with coeffi-cients in F q such that rank Q = m = rank Q . Then there is P ∈ GL n ( F q ) suchthat P ⊤ AP = A and P Q = Q if and only if Q ⊤ AQ = Q ⊤ AQ . Corollary 2.
Let A ∈ SGL n ( F q ) , x , x ∈ F nq , and assume a , a , d ∈ F q arenonzero, where d is a non-square. If either x ⊤ i A x i = a i for i = 1 , or x ⊤ i A x i = da i for i = 1 , , then there is P ∈ GL n ( F q ) such that P x = a a x and P ⊤ AP = A. MARKO OREL
Proof.
Let Q = x and Q = a a x . Then Q ⊤ AQ = Q ⊤ AQ , so the result followsfrom Lemma 6. (cid:3) Corollary 3.
Let A ∈ SGL n ( F q ) . For i = 1 , assume that x i , y i ∈ F nq satisfy x ⊤ i A x i = 0 = y ⊤ i A y i and x ⊤ i A y i = 0 . Then there exist P ∈ GL n ( F q ) and nonzero α ∈ F q such that P ⊤ AP = A , P x = x and P y = α y .Proof. Define α := x ⊤ A y x ⊤ A y . Let Q and Q be n × x , y and x , α y as their columns, respectively. It follows from the assumptions that rank Q = 2 =rank Q , so the proof ends by Lemma 6. (cid:3) Polar spaces
The main purpose of this section is to investigate what is a core of an affine polargraph
V O εn ( q ) (see below for the definition). In the first subsection we start byrecalling few more definitions and properties of finite fields and related structures.We end it with the result that shows that the core of V O εn ( q ) is either complete orthe graphs itself is a core.4.1. Preliminaries and auxiliary results.
Recall that q = p k is a power of anodd prime p . Let F p := { , , , . . . , p − } ⊆ F q be the prime subfield, that is, F p = { x ∈ F q : x p = x } . The trace map Tr : F q → F p , defined by Tr( x ) := x + x p + . . . + x p k − , is F p -linear and surjective (cf. [49, Theorem 2.23]), while(13) Tr( x ) = 0 ⇐⇒ x = y p − y for some y ∈ F q (cf. [49, Theorem 2.25]). Moreover, any F p -linear map ψ : F q → F p is of the form(14) ψ ( x ) = Tr( yx )for some y ∈ F q (cf. [49, Theorem 2.24]). Lemma 7 follows immediately from [49,Theorems 6.26 and 6.27]. Lemma 7.
Let b ∈ F q and A ∈ SGL n ( F q ) . Then (cid:12)(cid:12) { x ∈ F nq : x ⊤ A x = b } (cid:12)(cid:12) = ( q n − + v ( b ) q n − η (cid:0) ( − n det( A ) (cid:1) if n is even ,q n − + q n − η (cid:0) ( − n − b det( A ) (cid:1) if n is odd . Here, the integer valued maps v : F q → Z and η : F q → Z are defined as follows: v (0) := q − , v ( b ) := − if b = 0 , η (0) := 0 , η ( d ) := 1 if d is a nonzero square,and η ( d ) = − if d is a non-square in F q . Let h x i be the one-dimensional vector subspace in F nq that is spanned by nonzerocolumn vector x ∈ F nq . For A ∈ SGL n ( F q ), the set {h x i : x ⊤ A x = 0 , x = 0 } isa quadric . If n is odd, then the quadric is parabolic and has q n − − q − elements byLemma 7. If n is even, then, by Lemma 7, the quadric has either ( q n/ − q n/ − +1) q − or ( q n/ +1)( q n/ − − q − elements. In the first case the quadric is hyperbolic , while inthe second case it is elliptic . Every quadric represents the vertex set of a pointgraph of an orthogonal polar space , where two distinct elements h x i and h y i of aquadric form an edge if and only if(15) x ⊤ A y = 0 . N MINKOWSKI SPACE AND FINITE GEOMETRY 9
Two graphs constructed from two different matrices
A, A ′ ∈ SGL n ( F q ) are iso-morphic, provided that the two corresponding quadrics are of the same type. Thegraphs obtained in this way are denoted by Q n − ( q ), Q + n − ( q ), and Q − n − ( q ) ifthe quadric is parabolic, hyperbolic, and elliptic, respectively. In the general case,when it is not specified, which of the three types is meant, we write Q εn − ( q ).Graph Q εn − ( q ) is vertex-transitive (cf. [23]). Maximum cliques in a point graph ofan orthogonal polar space are formed by maximal totally isotropic subspaces , alsoreferred as generators , which consist of(16) s := q r − q − h x i , . . . , h x s i that satisfy x ⊤ j A x k = 0 for all j, k . Here r is theWitt index (cf. [23, 65]), that is,(17) r = n − in parabolic case , n in hyperbolic case , n − , so the clique numbers equal(18) ω (cid:0) Q n − ( q ) (cid:1) = q ( n − / − q − , ω (cid:0) Q + n − ( q ) (cid:1) = q n/ − q − , ω (cid:0) Q − n − ( q ) (cid:1) = q n/ − − q − . In the literature devoted to polar spaces (cf. [65]), there is a usual assumption forWitt index to be at least two, since the case r < Q εn − ( q ) is defined for all n ≥
2. So, Q − ( q ) is an empty graph (i.e., a graph without vertices), while Q +1 ( q ), Q ( q ), and Q − ( q ) are graphs with 2, q + 1, and q + 1 isolated vertices (i.e., there are no edges),respectively. Affine polar graphs are defined similarly as point graphs of orthogonal polarspaces (cf. [19]). Here, the vertex set equals F nq ( n ≥
2) and two distinct columnvectors x and y form an edge if and only if(19) ( x − y ) ⊤ A ( x − y ) = 0 , where A ∈ SGL n ( F q ) is a fixed matrix. Two graphs constructed from two differ-ent matrices A, A ′ ∈ SGL n ( F q ) are isomorphic, provided that the two quadricsdefined by A and A ′ are of the same type. Graphs obtained in this way aredenoted by V O n ( q ), V O + n ( q ), and V O − n ( q ), if the corresponding quadric is par-abolic, hyperbolic, and elliptic, respectively. In the general case, when it is notspecified, which of the three types is meant, we write V O εn ( q ). Observe that V O εn ( q ) is the Cayley graph Cay( G, S ) for the additive group G := ( F nq , +) andthe set S := { x ∈ F nq \{ } : x ⊤ A x = 0 } , so it is a vertex-transitive graph. If x ⊤ A x = 0 = y ⊤ A y , then equations (15) and (19) are equivalent. Hence, by iden-tifying h x i with any nonzero multiple of x , we deduce that Q εn − ( q ) is (isomorphicto) a subgraph in V O εn ( q ). Moreover, the following holds. Lemma 8.
Let N be the neighborhood of any vertex in V O εn ( q ) . Then N is iso-morphic to lexicographic product Q εn − ( q )[ K q − ] .Proof. If we consider the graph
V O − ( q ), then N and Q − ( q )[ K q − ] are emptygraphs, hence isomorphic. Assume now that V O εn ( q ) = V O − ( q ). Since V O εn ( q )is a vertex-transitive graph, we may assume that N is the neighborhood of the zerovertex, that is, the subgraph induced by the set { x ∈ F nq \{ } : x ⊤ A x = 0 } . The complete graph K q − can be viewed as the graph on the vertex set F q \{ } , wheredistinct scalars are adjacent. Fix nonzero x , . . . , x t ∈ F nq such that h x i , . . . , h x t i are precisely all distinct vertices of Q εn − ( q ). Then, the map ( h x i i , a ) a x i , with a ∈ F q \{ } , is the desired isomorphism from Q εn − ( q )[ K q − ] onto N . (cid:3) Corollary 4 is well known. It can be easily deduced also from Lemma 8.
Corollary 4.
The clique numbers of affine polar graphs equal (20) ω (cid:0) V O n ( q ) (cid:1) = q ( n − / , ω (cid:0) V O + n ( q ) (cid:1) = q n/ , ω (cid:0) V O − n ( q ) (cid:1) = q n/ − . Proof.
Let N be the neighborhood of the zero vector in V O εn ( q ), K a maximumclique in V O εn ( q ), and x ∈ K arbitrary. Then K − x := { x − x : x ∈ K} isa maximum clique, which is contained in the closed neighborhood N of the zerovector. Hence, ω (cid:0) V O εn ( q ) (cid:1) = |K| = |K − x | = ω (cid:0) N (cid:1) = ω ( N ) + 1 . The proof ends by Lemma 8, (5), and (18). (cid:3)
Note that any clique of size (20) is a (totaly isotropic) vector space or its trans-lation.
Lemma 9.
Graph
V O εn ( q ) is arc-transitive.Proof. Let ( x , y ) and ( x , y ) be two arcs, that is, ( y i − x i ) ⊤ A ( y i − x i ) = 0 and y i = x i for i = 1 ,
2, where A is the defining matrix (19). If we evaluate Lemma 6at Q i := y i − x i , we obtain P ∈ GL n ( F q ) such that P ⊤ AP = A and(21) P ( y − x ) = y − x . Then all three mapsΦ ( z ) := z − x , Φ ( z ) := P z , Φ ( z ) := z + x are automorphisms of V O εn ( q ), so the same holds for their composition Φ := Φ ◦ Φ ◦ Φ , which is given by Φ( z ) = P z − ( P x − x ). By (21) it satisfies Φ( x ) = x and Φ( y ) = y , so the graph is arc-transitive. (cid:3) It is well known that graphs
V O − n ( q ) and V O + n ( q ) are strongly regular [39, C. . ± ] (see [20] for the definition of a strongly regular graph). So, except forthe graph V O − ( q ), which is formed by q isolated vertices, all other graphs areconnected with diameter 2 (cf. [20, p. 4]). The same holds for graph V O n ( q ) asshown below. Proposition 2.
Let n ≥ be odd. Graph V O n ( q ) is connected with diameter 2.Proof. Since various defining invertible matrices A in (19) produce isomorphicgraphs, we may assume that A = diag(1 − d, d, − , . . . , −
1) is a diagonal ma-trix, where d ∈ F q is a non-square. Obviously, the diameter is at least two. Let x , y ∈ F nq be two non-adjacent vertices, that is, ( x − y ) ⊤ A ( x − y ) = 0. By (9),there is nonzero a ∈ F q such that ( x − y ) ⊤ A ( x − y ) ∈ { da , a } . To deduce thatthe graph is connected and the diameter equals two, we need to find z ∈ F nq that isadjacent to both x and y . We separate two cases. N MINKOWSKI SPACE AND FINITE GEOMETRY 11
Case 1 . Let ( x − y ) ⊤ A ( x − y ) = da . Let e i = (0 , . . . , , , , . . . , ⊤ be the i -thstandard vector. Define w := e and u := ( e + e + e ). Since w ⊤ A w = d = d · ,Corollary 2 shows that there is P ∈ GL n ( F q ) such that P ( y − x ) = a w and P ⊤ AP = A . Define z := x + a P − u . Since A = ( P − ) ⊤ AP − and y − x = a P − w , we easily see that ( z − x ) ⊤ A ( z − x ) = 0 = ( y − z ) ⊤ A ( y − z ), that is, z isadjacent to x and y . Case 2 . Let ( x − y ) ⊤ A ( x − y ) = a . The proof is the same as in Case 1 with theonly exception that here w := e + e . (cid:3) In determining the core of
V O εn ( q ) we need to consider few of the elementarycases, with Witt index less than 2, separately. Proposition 3.
Let Γ be one of the graphs V O − ( q ) , V O +2 ( q ) , or V O ( q ) . Thenstatements ( i ) , ( ii ) , and ( iii ) from Proposition 1 hold.Proof. By Proposition 1 it suffices to show ( iii ). The graph
V O − ( q ) is formed by q isolated vertices, so ω (cid:0) V O − ( q ) (cid:1) = 1, α (cid:0) V O +2 ( q ) (cid:1) = q , and ( iii ) holds.If Γ = V O +2 ( q ), then we may assume that the defining matrix A in (19) equals (cid:18) − (cid:19) . Then, { (0 , x ) ⊤ : x ∈ F q } is an independent set. In fact, (cid:0) (0 , x ) ⊤ − (0 , y ) ⊤ (cid:1) ⊤ A (cid:0) (0 , x ) ⊤ − (0 , y ) ⊤ (cid:1) = − ( x − y ) = 0for all distinct x, y ∈ F q , so α (Γ) ≥ q . Since | V (Γ) | = q and ω (Γ) = q byCorollary 4, (3) implies ( iii ).In the case Γ = V O ( q ) we may assume that A = − d , where d ∈ F q is non-square. Then { (0 , x, y ) ⊤ : x, y ∈ F q } is an independent set,since (cid:0) (0 , x , y ) ⊤ − (0 , x , y ) ⊤ (cid:1) ⊤ A (cid:0) (0 , x , y ) ⊤ − (0 , x , y ) ⊤ (cid:1) == − ( x − x ) + d ( y − y ) = 0for (0 , x , y ) ⊤ = (0 , x , y ) ⊤ . So α (Γ) ≥ q . We proceed in the same way as in thecase Γ = V O +2 ( q ). (cid:3) It is not so rare, that the core of a graph is complete. In fact, in [33, Theorem 4.1]it was shown that the core of a connected regular graph, with the automorphismgroup acting transitively on pairs of vertices at distance two, is either complete orthe graph itself is a core. Though the automorphism group of
V O εn ( q ) does nothave this property in general, we will infer from Corollary 3 that a particular orbitof such an action is large enough to obtain a result for V O εn ( q ) that is analogousto [33, Theorem 4.1]. Lemma 10.
The graph
V O εn ( q ) is either a core or its core is a complete graph on ω (cid:0) V O εn ( q ) (cid:1) vertices. Proof.
For graph
V O − ( q ) the result is proved in Proposition 3, so we may assumethat V O εn ( q ) is connected. Let A be its defining matrix (19). For x = y we write x ∼ y if equation (19) is satisfied, that is, if vertices x and y are adjacent in V O εn ( q ).Let Γ ′ be a core of V O εn ( q ). Assume that Γ ′ is neither complete nor the wholegraph. Since any endomorphism maps a clique of size ω (cid:0) V O εn ( q ) (cid:1) to a clique of size ω (cid:0) V O εn ( q ) (cid:1) , Γ ′ must contain such a clique K . Since V O εn ( q ) is a connected graph,so is its core. Since Γ ′ is not complete, we deduce that there is u ∈ V (Γ ′ ) \K and v ∈ K such that u ∼ v .Now, since V O εn ( q ) is connected and Γ ′ = V O εn ( q ), there exists w outside Γ ′ that is adjacent to some v in Γ ′ . Let Ψ be any retraction of V O εn ( q ) onto Γ ′ . Thenit maps w to some neighbor u of v in Γ ′ . By Lemma 9, V O εn ( q ) is arc-transitive,so from Section 2 we know that Γ ′ is arc-transitive as well. Consequently, thereis an automorphism Φ ′ of Γ ′ that maps the arc ( u , v ) to arc ( u , v ), so thereexists a maximum clique K := Φ ′ ( K ) in Γ ′ that contains v , while u / ∈ K .Define x := u − v and y := w − v . Then x ∼ ∼ y and x ≁ y , so x ⊤ A x = 0 = y ⊤ A y and x ⊤ A y = 0. Since the clique K := K − v is ofmaximum size and x = u − v / ∈ K , there is y ∈ K such that y ≁ x . Since0 = v − v ∈ K , it follows that y ⊤ A y = ( y − ⊤ A ( y −
0) = 0, so we deducethat x ⊤ A y = 0. By Corollary 3 there exists P ∈ GL n ( F q ) and nonzero α ∈ F q such that(22) P ⊤ AP = A,P x = x , and P y = α y . Since the clique K is of maximum size and containsthe zero vector, it is a (totally isotropic) vector space. Since it contains y , itfollows that α y ∈ K . Consequently, α y + v ∈ K . By (22), the map Φ( x ) := P x − ( P v − v ) is an automorphism of V O εn ( q ). Moreover, it satisfies Φ( u ) = u and Φ( w ) = α y + v . Hence, the restriction of the composition Φ ′′ := Ψ ◦ Φ − toΓ ′ is an endomorphism of Γ ′ that is not bijective, since Φ ′′ ( u ) = u = Φ ′′ ( α y + v ).This contradicts the fact that Γ ′ is a core. (cid:3) Spectrum.
As we shall see, the spectrum will provide us partial answer re-garding the (non)completeness of the core of affine polar graphs.Since the graphs
V O − n ( q ) and V O + n ( q ) are strongly regular with known param-eters, their spectrum is easy to derive, well known, and mentioned already in thesurvey paper [39, p. 375, C. . ± ] (see also [19]). Lemma 11. (cf. [19, 39]) Let n ≥ be even. The eigenvalues λ i and their multi-plicities m λ i of the elliptic and hyperbolic affine polar graphs are as follows: λ = ( q n − − q n + 1) m λ = 1 V O − n ( q ) λ = q n − − m λ = q n − ( q − q n + 1) λ = − q n + q n − − m λ = ( q n − − q n + 1) λ = ( q n − + 1)( q n − m λ = 1 V O + n ( q ) λ = q n − q n − − m λ = ( q n − + 1)( q n − λ = − q n − − m λ = q n − ( q − q n − .Remark . Eigenvalues λ , λ , λ are distinct, except in the case of a graph V O − ( q ),where λ = 0 = λ and m λ = 0. N MINKOWSKI SPACE AND FINITE GEOMETRY 13
Remark . Let M Γ be the set of all real symmetric | V (Γ) | × | V (Γ) | matrices M ,which have 1 at ( i, j )-th entry, whenever i = j or i -th and j -th vertex are not adja-cent. The Lov´asz’s ϑ -function of a graph Γ is defined as ϑ (Γ) = inf M ∈M Γ λ max ( M ),where λ max ( M ) is the largest eigenvalue of M [22]. As proved in [50], for vertex-transitive graphs α (Γ) ≤ ϑ (Γ) ≤ −| V (Γ) | λ min (Γ) λ max (Γ) − λ min (Γ) and ϑ (Γ) ϑ (Γ) = | V (Γ) | hold,where λ max (Γ) and λ min (Γ) are the largest and the smallest eigenvalue of Γ, re-spectively. If Γ = V O + n ( q ), then Corollary 4 implies that α (Γ) = ω (Γ) = q n .Since eigenvalues of Γ are easily computed from the eigenvalues of its complement(cf. [22, p. 4]), which are given in Lemma 11, we deduce that −| V (Γ) | λ min (Γ) λ max (Γ) − λ min (Γ) = q n .Therefore ϑ (cid:0) V O + n ( q ) (cid:1) = q n and consequently ϑ (cid:0) V O + n ( q ) (cid:1) = q n .Distinct eigenvalues of V O n ( q ) are described in the last column in the charactertable in [11, Case 3, p. 6129] (the table was firstly computed in [46, Theorem 2], butit contained a misprint, as observed in [11]). These eigenvalues are λ = q n − − λ = q n − − λ = −
1, and λ = − q n − −
1. We were not able to find theirmultiplicities in the literature. One possible strategy to compute them is to usethe intersection matrices in [46, Section 3.7] and apply the procedure describedin [20, p. 46] and [21, Proposition 2.2.2.a]. However, since q and n are general (andnot given fixed numbers), it seems that this tactic is too complicated. Thereforewe apply a strategy from [58], which will derive the multiplicities and recomputethe eigenvalues λ , λ , λ , λ . The following result was essentially observed alreadyin [58, Proof of Theorem 1.3] though not written in a such generality. Proposition 4.
Let V be a finite dimensional vector space over F q . (i) The | V | characters of the group ( V, +) are precisely the maps (23) ξ ( v ) := e πi Ψ( v ) /p , where Ψ ranges over all F p -linear maps Ψ : V → F p . (ii) Let
Γ =
Cay ( V, S ) be a Cayley graph, where S is closed under multiplica-tion by scalars in F p . Then the eigenvalues of Γ are precisely the values (24) |△| p − | S | p − , where △ := { s ∈ S : Ψ( s ) = 0 } and Ψ ranges over all F p -linear maps Ψ : V → F p .Remark . The statement and proof of Proposition 4 is valid for q odd or even. Proof. ( i ) Obviously, the maps (23) are characters of ( V, +), that is, maps thatsatisfy ξ ( u + v ) = ξ ( u ) ξ ( v ). Since the number of all characters and the num-ber of all F p -linear maps Ψ : V → F p both equal | V | = p dim p V (cf. [49, Theo-rem 5.5]), where dim p V is the dimension of V as a vector space over F p , it suf-fices to show that distinct Ψ and Ψ generate distinct characters. So assumethat e πi Ψ ( v ) /p = e πi Ψ ( v ) /p for all v . Then there exist integers k ( v ) such that2 πi Ψ ( v ) /p = 2 πi Ψ ( v ) /p + 2 πik ( v ), that is, Ψ ( v ) = Ψ ( v ) + pk ( v ). Hence,Ψ = Ψ (mod p ).( ii ) By Lemma 3 and ( i ), the eigenvalues of Γ are precisely the values(25) X s ∈ S e πi Ψ( s ) /p , where Ψ ranges over all F p -linear maps Ψ : V → F p . For such map Ψ let △ := { s ∈ S : Ψ( s ) = 0 } . Then, for s ∈ S \△ , we have X a ∈ F p \{ } e πi Ψ( a s ) /p = X a ∈ F p \{ } e πia Ψ( s ) /p = p − X j =1 e πij/p = p − X j =0 e πij/p − − . Consequently, (25) equals X s ∈△ e πi Ψ( s ) /p + X s ∈ S \△ e πi Ψ( s ) /p = |△| + | S \△| · ( − p − |△| + (cid:0) | S | − |△| (cid:1) · ( − p − , which is the same as (24). (cid:3) Corollary 5.
Let n ≥ be odd. The eigenvalues λ i and their multiplicities m λ i ofthe parabolic affine polar graph are as follows: V O n ( q ) λ = q n − − m λ = 1 λ = q n − − m λ = ( q − q n − + q n − ) λ = − m λ = q n − − λ = − q n − − m λ = ( q − q n − − q n − ) .Proof. Graph
V O n ( q ) is a Cayley graph for the additive group ( F nq , +) and theset S = { x ∈ F nq \{ } : x ⊤ A x = 0 } , where we may assume that A = I is theidentity matrix. The eigenvalues of V O n ( q ) are, by Proposition 4, precisely thevalues (24), where △ := { s ∈ S : Ψ( s ) = 0 } and Ψ ranges over all F p -linear mapsΨ : F nq → F p . Any such map is of the form Ψ( x ) = P nj =1 ψ j ( x j ) for some F p -linearmaps ψ j : F q → F p , where x = ( x , . . . , x n ) ⊤ . By (14), there are scalars a j ∈ F q such that ψ j ( x j ) = Tr( a j x j ). Since the trace map is additive, we deduce that(26) Ψ( x ) = Tr( a ⊤ x ) , where a = ( a , . . . , a n ) ⊤ . Obviously, the map (26) is F p -linear for any a ∈ F nq , andtwo distinct a , a ∈ F nq generate two distinct maps.To proceed, consider the map x x p − x on F q . It is p -to-1, since in a field ofcharacteristic p the equivalence x p − x = y p − y ⇐⇒ ( x − y ) p = x − y ⇐⇒ x − y ∈ F p holds. Consequently, the map x x p − x attains q/p distinct values d , . . . , d q/p ,where one of them, say d q/p , is zero. Equivalence (13) implies thatTr( x ) = 0 ⇐⇒ x ∈ { d , . . . , d q/p } , so (26) shows that(27) △ = (cid:8) x ∈ F nq \{ } : x ⊤ x = 0 and a ⊤ x ∈ { d , . . . , d q/p } (cid:9) . Given a ∈ F nq and b ∈ F q let Ω a b := { x ∈ F nq \{ } : x ⊤ x = 0 and a ⊤ x = b } .If b, c ∈ F q are nonzero, then | Ω a b | = | Ω a c | , since x cb x is a bijection betweenthe two sets. By Lemma 7, | S | = q n − −
1. In fact, n is odd and we need toexclude the zero vector. On the contrary, S equals the disjoint union S b ∈ F q Ω a b , so N MINKOWSKI SPACE AND FINITE GEOMETRY 15 | S | = ( q − | Ω a | + | Ω a | , that is, | Ω a | = | S |−| Ω a | q − . Therefore (27) implies that |△| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q/p [ j =1 Ω a d j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( q/p − | Ω a | + | Ω a | = q/p − q − (cid:0) | S | − | Ω a | (cid:1) + | Ω a | = q − q/pq − | Ω a | + q/p − q − | S | , an the eigenvalue (24) equals(28) q | Ω a | − | S | q − q | Ω a | − q n − + 1 q − . We need to compute | Ω a | for each a = ( a , . . . , a n ) ⊤ ∈ F nq . The second of the fol-lowing two cases splits in three subcases, for a total of four distinct eigenvalues (28). Case 1 . Let a = 0. Then Ω a = S , and the eigenvalue (28) equals λ := q n − − Case 2 . Let Let a = 0. Without lost of generality, we may assume that a = 0.Then(29) Ω a = ( x ∈ F nq \{ } : x = − P nj =2 a j x j a , ( x , . . . , x n ) A ( x , . . . , x n ) ⊤ = 0 ) , where A is the matrix in (10) with determinant a ⊤ a a . Case 2a . Assume that ( − ( n − / a ⊤ a is a nonzero square in F q . By Lemma 7, | Ω a | = q n − + q n − − q n − −
1, so the eigenvalue (28) equals λ := q n − − F q are squares, Lemma 7 implies that there are m λ := ( q − q n − + q n − ) vectors a ∈ F nq that satisfy the assumption of Case 2a. Case 2b . Assume that ( − ( n − / a ⊤ a is non-square in F q . By Lemma 7, | Ω a | = q n − − q n − + q n − −
1, so the eigenvalue (28) equals λ := − q n − −
1. Sincehalf of nonzero elements in F q are non-squares, Lemma 7 implies that there are m λ := ( q − q n − − q n − ) vectors a ∈ F nq that satisfy the assumption of Case 2b. Case 2c . Assume that a ⊤ a = 0. Since a = 0, there is j ≥ a j = 0.Without lost of generality, we may assume that a = 0. Then A is singular, butits lower-right ( n − × ( n −
2) block A is invertible, since, by Corollary 1, itsdeterminant equalsdet A = a + a + a + . . . + a n a = a ⊤ a − a a = − a a = 0 . By Lemma 5 there is P ∈ GL n − ( F q ) such that A = P ⊤ (cid:18) A (cid:19) P. Hence, (29) and the bijective transformation ( y , · · · , y n ) ⊤ := P ( x , · · · , x n ) ⊤ yield | Ω a | = (cid:12)(cid:12)(cid:8) ( x , . . . , x n ) ∈ F n − q : ( x , . . . , x n ) A ( x , . . . , x n ) ⊤ = 0 (cid:9)(cid:12)(cid:12) − q · (cid:12)(cid:12)(cid:8) ( y , . . . , y n ) ∈ F n − q : ( y , . . . , y n ) A ( y , . . . , y n ) ⊤ = 0 (cid:9)(cid:12)(cid:12) − . Note that we have subtracted 1 for the zero vector. By Lemma 7, applied at A ,we deduce that | Ω a | = q n − −
1, so the eigenvalue (28) equals λ := −
1. ByLemma 7 there are m λ := q n − − a ∈ F nq that satisfy the assumption ofCase 2c. (cid:3) Remark . It follows from Corollary 5 that
V O n ( q ) is a Ramanujan graph (if q isodd), that is, a regular graph that satisfy | λ | ≤ √ λ − λ = ± λ , excluding the ones with the largest absolute value (this was observed alreadyin [11, Theorem 3.3.(i)]). Ramanujan graphs are good expanders and precisely thoseregular graphs, for which their Ihara zeta function satisfies an analog of Riemannhypothesis (cf. [53]). Remark . It follows from Corollary 5 that graph
V O n ( q ) ( n ≥
3) has 4 distincteigenvalues. Therefore it follows from [14, Proposition 21.2] that
V O n ( q ) is not adistance-regular graph, since its diameter equals two by Proposition 2. We referto [20] for the definition of a distance-regular graph. Remark . A similar method, as the one used in the proof of Corollary 5, can beused to prove Lemma 11. Hence, by [32, Lemma 10.2.1], we can easily deduce (thewell known fact) that graphs
V O + n ( q ) and V O − n ( q ) are strongly regular, since theyhave precisely three distinct eigenvalues (unless we consider the graph V O − ( q )).The spectrum and Lemma 10 provide us first major result on the core of anaffine polar graph. Theorem 1.
Let n ≥ be even and q be odd. Then V O − n ( q ) is a core. In particular,the core of V O − n ( q ) and the core of the complement V O − n ( q ) is not a complete graph,and ω (cid:0) V O − n ( q ) (cid:1) α (cid:0) V O − n ( q ) (cid:1) < (cid:12)(cid:12) V (cid:0) V O − n ( q ) (cid:1)(cid:12)(cid:12) .Proof. By Proposition 1 and Lemma 10 it suffices to show that(30) α (cid:0) V O − n ( q ) (cid:1) < q n +1 , since (cid:12)(cid:12) V (cid:0) V O − n ( q ) (cid:1)(cid:12)(cid:12) = q n and ω (cid:0) V O − n ( q ) (cid:1) = q n − by Corollary 4. By Lemma 2and Lemma 11, we deduce the inequality α (cid:0) V O − n ( q ) (cid:1) ≤ q n · − ( − q n + q n − − q n − − q n + 1) − ( − q n + q n − − q n +1 if n >
2, so (30) holds. (cid:3)
The reader may observe that in the hyperbolic and parabolic case the proofabove fails. In fact, we will see in the next subsection that an analog of Theorem 1for hyperbolic and parabolic affine polar graphs is false in some cases. There arehowever some cases, where such an analogous result is correct, but the spectrumalone seems to not provide enough information to prove it.4.3.
Ovoids.
In this subsection we investigate the (non)completeness of the coreof
V O εn with certain geometrical tools. Therefore we assume the usual assumptionfrom finite geometry that the Witt index r in (17) is at least two, that is, n ≥ n ≥
4, and n ≥
6, if the affine polar graph is parabolic, hyperbolic, and elliptic,respectively.An ovoid of an orthogonal polar space is a subset of the vertex set V (cid:0) Q εn − ( q ) (cid:1) meeting every generator in exactly one point. If an ovoid O exists, then its car-dinality is well known (cf. [65]). In fact, it can be deduced by computing the N MINKOWSKI SPACE AND FINITE GEOMETRY 17 cardinality of the set { ( x , U ) : x ∈ O , U is a generator that contains x } , whichequals |O| · g = g , where g is the number of all generators and g is the numberof all generators that contain a particular point. If s is the number of points in agenerator, then g · s = | V (cid:0) Q εn − ( q ) (cid:1) | · g , so(31) |O| = gg = | V (cid:0) Q εn − ( q ) (cid:1) | s . Hence, the following lemma is deduced from (16).
Lemma 12. (cf. [65]) Let r ≥ . Assume that an ovoid O exists in Q εn − ( q ) . Then |O| equals q n − + 1 , q n − + 1 , and q n + 1 if the orthogonal polar space is parabolic,hyperbolic, and elliptic, respectively. A partial ovoid PO is a subset of the vertex set V (cid:0) Q εn − ( q ) (cid:1) meeting everygenerator in at most one point. If g ′ and g ′′ are the number of generators thatcontain one and zero points of PO , respectively, then g = g ′ + g ′′ . Moreover, asimilar calculation as in (31) shows that |PO| = g ′ g . So if a partial ovoid hascardinality (31), then it is an ovoid, since g ′′ = 0. Since Q εn − ( q ) is arc-transitive(cf. [23]), any pair of its adjacent vertices lie in some generator. Hence, partialovoids are precisely the independent sets in Q εn − ( q ). Consequently, the followingis true. Corollary 6.
Let r ≥ . If α (cid:0) Q εn − ( q ) (cid:1) is the same as the cardinality |O| fromLemma 12, then the corresponding independent set is an ovoid. Most of the observations above about ovoids, partial ovoids, and independentsets were already used in the proof of [23, Theorem 3.5].The ‘No-Homomorphism Lemma’ [5, 35, 36] states that if there exists some ho-momorphism Φ : Γ ′ → Γ between two graphs, where Γ is vertex-transitive, then(32) α (Γ ′ ) | V (Γ ′ ) | ≥ α (Γ) | V (Γ) | . In particular, (32) holds if Γ ′ is a subgraph in Γ. In Proposition 5 (i) we essentiallyrewrite the proof of [35, Lemma 3.3] to deduce a strict inequality in (32) if Γ is anaffine polar graph and Γ ′ is a closed neighborhood of a vertex. Consequently, inpart (ii) the sizes of independent sets in V O εn ( q ) provide us lower bounds for thesize of the largest partial ovoid in orthogonal polar space. Proposition 5. (i)
Let N be the closed neighborhood of any vertex in V O εn ( q ) . Then (33) α ( N ) | V ( N ) | > α ( V O εn ( q )) V (cid:0) V O εn ( q ) (cid:1) . (ii) Let r ≥ . Then α (cid:0) Q n − ( q ) (cid:1) > q · α ( V O n ( q )) ,α (cid:0) Q + n − ( q ) (cid:1) > q n + q − q n +1 · α ( V O + n ( q )) ,α (cid:0) Q − n − ( q ) (cid:1) > q n − q + 1 q n +1 · α ( V O − n ( q )) . Proof. (i) Since
V O εn ( q ) is vertex-transitive, we may assume that N is the closedneighborhood of the zero vertex, that is, the subgraph induced by the set { x ∈ F nq : x ⊤ A x = 0 } , where A is the matrix that defines the polar space.Let Γ = V O εn ( q ) and Γ ′ = N . Let J (Γ) denote the family of all independentsets in Γ of size α (Γ). Since Γ is vertex-transitive, any its vertex lies in the samenumber, say m , of members of J (Γ). If we count the elements of the set (cid:8) ( x , I ) : x ∈ V (Γ) , I ∈ J (Γ) contains x (cid:9) in two different ways, we deduce that(34) | V (Γ) | · m = |J (Γ) | · α (Γ) . Obviously, |I ∩ V (Γ ′ ) | ≤ α (Γ ′ ) for all I ∈ J (Γ). Fix one independent set I ∈ J (Γ)and x ∈ I . Then I − x := { x − x : x ∈ I } ∈ J (Γ). Since it contains the zerovector, its other elements are not in V (Γ ′ ), that is, | ( I − x ) ∩ V (Γ ′ ) | = 1 < α (Γ ′ ).Consequently,(35) X I∈J (Γ) |I ∩ V (Γ ′ ) | < α (Γ ′ ) · |J (Γ) | . Since P I∈J (Γ) |I ∩ V (Γ ′ ) | = | V (Γ ′ ) | · m , we deduce (33) from (34) and (35).(ii) Let N and N be the neighborhood and the closed neighborhood of the zerovector in V O εn ( q ), respectively. Since the zero vector is adjacent to all vertices in N , we have α ( N ) = α ( N ). By Lemma 8, (4), and (i) it follows that α ( Q εn − ( q )) = α ( Q εn − ( q )) · α ( K q − ) = α ( N )= α ( N ) > | V ( N ) | V (cid:0) V O εn ( q ) (cid:1) · α ( V O εn ( q )) . Now, V (cid:0) V O εn ( q ) (cid:1) = q n , while to compute | V ( N ) | we need to multiply the cardi-nality of the quadric by | F q \{ }| = q −
1, and then add 1 for the zero vector. Theresult follows. (cid:3)
In [23, Theorem 3.5] it was determined (in terms of existence of ovoids, spreads,and partitions into ovoids), when is the core of Q εn − ( q ) or its complement complete.Recall from Remark 2 that the result was not (necessary) symmetric for the graphand its complement. Moreover, in the proof it was observed that ω (cid:0) Q εn − ( q ) (cid:1) α (cid:0) Q εn − ( q ) (cid:1) = V (cid:0) Q εn − ( q ) (cid:1) if and only if an ovoid exists in the polar space (provided that r ≥ Theorem 2.
Let r ≥ and let V O εn ( q ) be parabolic or hyperbolic. Then thestatements (i)-(v) are equivalent: (i) ω (cid:0) V O εn ( q ) (cid:1) α (cid:0) V O εn ( q ) (cid:1) = (cid:12)(cid:12) V (cid:0) V O εn ( q ) (cid:1)(cid:12)(cid:12) , (ii) a core of V O εn ( q ) is a complete graph, (iii) a core of V O εn ( q ) is a complete graph, (iv) V O εn ( q ) is not a core, (v) χ (cid:0) V O εn ( q ) (cid:1) = ω (cid:0) V O εn ( q ) (cid:1) .If any of statements (i)-(v) is true, then (vi) the polar space of Q εn − ( q ) has an ovoid. Recall that equivalence of statements ( i ) − ( iii ) is proved in Proposition 1, since V O εn ( q ) is a Cayley graph over an abelian group. Equivalence between ( ii ) and( iv ) is proved in Lemma 10. We know the equivalence between ( ii ) and ( v ) from N MINKOWSKI SPACE AND FINITE GEOMETRY 19
Section 2. So Theorem 2 demands just the proof of implication ( i ) ⇒ ( vi ). Weprovide two different proofs. Proof 1.
Assume that ( i ) holds. Then α ( V O εn ( q )) = q n ω (cid:0) V O εn ( q ) (cid:1) , where ω (cid:0) V O εn ( q ) (cid:1) is computed in Corollary 4.In the parabolic case we deduce that α (cid:0) Q n − ( q ) (cid:1) > q ( n − / from Proposition 5.Since α (cid:0) Q n − ( q ) (cid:1) is an integer, it follows that α ( Q n − ( q )) ≥ q ( n − / + 1. From (3)and (18) we deduce that α ( Q n − ( q )) ≤ q ( n − / + 1, so α ( Q n − ( q )) = q ( n − / + 1.By Corollary 6, the polar space has an ovoid.In the hyperbolic case we similarly deduce that α (cid:0) Q + n − ( q ) (cid:1) > q n/ − + 1 − q , so α (cid:0) Q + n − ( q ) (cid:1) ≥ q n/ − + 1. The same procedure as in the parabolic case showsthat the polar space has an ovoid. (cid:3) Proof 2.
Assume that ( i ) holds. Then, both for parabolic and hyperbolic case,Corollary 5 and Lemma 11, together with Corollary 4, imply that equality is at-tained in (8). Let K and I be a clique of size ω (cid:0) V O εn ( q ) (cid:1) and independent set ofsize α (cid:0) V O εn ( q ) (cid:1) , respectively. Choose arbitrary k ∈ K and i ∈ I . Then K ′ := K − k and I ′ := I − i are clique and independent set of the same size as K and I , re-spectively. Recall from the proof of Proposition 1, that { k ′ + I ′ : k ′ ∈ K ′ } is apartition of the vertex set V (cid:0) V O εn ( q ) (cid:1) into independent sets of size α (cid:0) V O εn ( q ) (cid:1) .Since 0 = 0 + 0 ∈ I ′ , we see that 0 / ∈ k ′ + I ′ for arbitrary nonzero k ′ ∈ K ′ , soby Lemma 2 there are precisely − λ min elements x , . . . , x − λ min ∈ k ′ + I ′ , that areadjacent to 0. That is, if A is the defining matrix (19), then x ⊤ j A x j = 0 ( j = 1 , . . . , − λ min ) , x ⊤ j A x k = −
12 ( x j − x j ) ⊤ A ( x j − x k ) = 0 ( j = k ) . Since, by Corollary 5 and Lemma 11, − λ min equals q ( n − / + 1 and q n/ − + 1in parabolic and hyperbolic case, respectively, we deduce from Corollary 6 that h x i , . . . , h x − λ min i form an ovoid in Q εn − ( q ). (cid:3) Recall from Theorem 1 that ( i ) − ( v ) do not hold in the elliptic case. Neitherdoes ( vi ) if n ≥ i ) ⇒ ( vi ) above fails, so the spectral tools used to prove Theorem 1 are indeedneeded.It is our concern, whether the statement ( i ) − ( v ) in Theorem 2 are true or not.Certainly, if ( vi ) does not hold, then neither do the statements ( i ) − ( v ). However,whether ( vi ) is true or not is still not known in general, despite a huge amount ofthe research in this area in last few decades. We now briefly list the main resultsof what is known and refer to a recent survey [26] for more details.The parabolic polar space of Q n − ( q ) does not posses ovoids if n ≥ n = 5 (cf. [59, 3.4.1.(i)]). If n = 7, the (non)existence of ovoids in parabolicspace seems to be still an open problem, though there are some partial results of a mixed type, that is, ovoids do not exist if q > q is a power of 3 (cf. [63, Theorem 17]).The hyperbolic polar space of Q + n − ( q ) is even more mysterious. Existence ofovoids is known if n = 4 or n = 6 (cf. [65]). If n = 8, the existence is known for prime q [25, 52], for q = p k with k odd and p ≡ q isa power of a prime p that satisfies the inequality p n − > (cid:0) n + p − n − (cid:1) − (cid:0) n + p − n − (cid:1) [16].This is the case, for example, if n = 10 and p = 3, or n = 12 and p ∈ { , } .By [63, Theorem 8], nonexistence of ovoids in Q + n − ( q ) implies nonexistence ofovoids in Q + n +1 ( q ), so there are no ovoids in Q + n − ( q ) if n ≥ p = 3 or n ≥ p ∈ { , } . It is believed by many mathematicians that there are no ovoids in Q +9 ( q )(and consequently in Q + n − ( q ), n ≥
10) for general q (cf. [26]).We do not know, whether statement ( vi ) from Theorem 2 implies statements( i ) − ( v ) or not. Nevertheless, to check whether ( vi ) or ( i ) − ( v ) is true seems to beof a similar difficulty, as indicated by the following result together with Theorem 2. Proposition 6. (i)
Let n ≥ be odd and assume that Q n − ( q ) has an ovoid. Then the state-ments ( i ) − ( v ) in Theorem 2 are true for V O n − ( q ) and V O + n − ( q ) . (ii) Let n ≥ be even and assume that Q + n − ( q ) has an ovoid. Then thestatements ( i ) − ( v ) in Theorem 2 are true for V O + n − ( q ) . Observe that the proof below shows how to construct a maximum indepen-dent set that satisfies statement ( i ) in Theorem 2 from an ovoid, for affine polargraphs/orthogonal polar spaces in question. Proof.
We may assume that the defining matrix A (19) is(36) . . . in the parabolic case and (cid:18) B − (cid:19) , in the hyperbolic case, where B is ( n − × ( n −
1) matrix of the form (36). Then,for x = ( x , . . . , x n ) ⊤ , x ⊤ A x equals(37) x n +12 + 2 n − X i =1 x i x n − i in parabolic case and(38) x n − x n + 2 n − X i =1 x i x n − − i . in hyperbolic case. Assume that O = {h x i , . . . , h x t i , h x t +1 i} is an ovoid in Q εn − ( q ),where t equals q n − and q n − in the parabolic and hyperbolic case, respectively, N MINKOWSKI SPACE AND FINITE GEOMETRY 21 as it follows from Lemma 12. Then x ⊤ i A x i = 0 ( i = 1 , . . . , t + 1) , (39) x ⊤ i A x i = 0 ( i = i ) , (40)since an ovoid is also an independent set in graph Q εn − ( q ). Obviously, there is i ∈ { , . . . , t + 1 } and j ≤ n such that j -th component of x i is nonzero and j = n +12 in parabolic case and j / ∈ { n, n } in hyperbolic case. To simplify writingswe can assume that i = t + 1 and j = 1. Let a i be the first component of x i , so a t +1 = 0. Let nonzero scalars b , . . . , b t be such that(41) b i x ⊤ i A x t +1 = 1 ( i ≤ t ) , and denote y i := b i x i − a i b i a t +1 x t +1 . Then I = { y , . . . , y t } is an independent set in V O εn ( q ), since (39)-(41) imply that( y i − y i ) ⊤ A ( y i − y i ) == (cid:16) b i x i − b i x i + a i b i − a i b i a t +1 x t +1 (cid:17) ⊤ A (cid:16) b i x i − b i x i + a i b i − a i b i a t +1 x t +1 (cid:17) = ( b i x i − b i x i ) ⊤ A ( b i x i − b i x i )+ 2 a i b i − a i b i a t +1 · ( b i x i − b i x i ) ⊤ A x t +1 + (cid:16) a i b i − a i b i a t +1 (cid:17) x ⊤ t +1 A x t +1 = − b i b i · x ⊤ i A x i + 0 + 0 = 0for i = i . We next define vectors z , . . . , z t ∈ F n − q and a ( n − × ( n −
2) matrix A ′ as follows. In the parabolic case, we obtain z i by deleting the first and thelast component of y i . Similarly, A ′ is obtained from A by deleting the first/lastrow/column. In the hyperbolic case we delete the first and ( n − y i and first/( n − A to obtain z i and A ′ . Since the firstcomponent of y i is zero by the construction, it follows from the forms (37) and (38)that y ⊤ i A y i = z ⊤ i A ′ z i for all i and i , so I ′ = { z , . . . , z t } is an independentset of size t in graph V O εn − ( q ), which is defined by matrix A ′ . So for this affinepolar graph the statement ( i ) in Theorem 2 is satisfied. This proves the claim ( ii )and half of claim ( i ) from Proposition 6. Now, if I ′ is an independent set of size t = q n − in V O n − ( q ), then, by extending its members with zero value as thelast component, we obtain an independent set { ( z ⊤ , ⊤ : z ∈ I ′ } of size q n − in V O + n − ( q ), where the defining matrix equals (cid:18) A ′ − (cid:19) . This completes the proof of claim ( i ). (cid:3) As it was mentioned above, it is of great interest in finite geometry to knowwhether Q ( q ) and Q + n − ( q ), n ≥ q . If onewants to show that there are no ovoids, then, by Proposition 6, it is sufficient toshow that the statement ( i ) in Theorem 2 is not true for V O ( q ) or V O +6 ( q ) and V O + n − ( q ), respectively. For some values of q this might be easier, since there areless dimensions to consider.Recall that there are ovoids in Q ( q ), q arbitrary; Q (3 k ); Q +7 ( q ), q prime or q = p k with k odd and p ≡ graphs satisfy statements ( i ) − ( v ) in Theorem 2: V O +4 ( q ) , q odd ,V O (3 k ) , k arbitrary ,V O +6 (3 k ) , k arbitrary ,V O +6 ( q ) , q odd prime ,V O +6 ( q ) , q = p k , p and k odd , p ≡ . In some cases of affine polar graphs of Witt index <
2, independent sets that attainsthe equality in statement ( i ) in Theorem 2, i.e. those with cardinality(42) (cid:12)(cid:12) V (cid:0) V O εn ( q ) (cid:1)(cid:12)(cid:12) ω (cid:0) V O εn ( q ) (cid:1) , were already constructed in the proof of Proposition 3. Below we present fewexamples with Witt index ≥ Example . Consider the graph
V O +4 ( q ), where A = diag(1 , − d, − d,
1) is the defin-ing matrix (19) and d ∈ F q is a non-square. Then I = n ( x, y, , ⊤ : x, y, ∈ F q o isan independent set. In fact, if u i = ( x i , y i , , ⊤ for i = 1 ,
2, then( u − u ) ⊤ A ( u − u ) = ( x − x ) − d ( y − y ) = 0for u = u . The size of I equals q , i.e., (42).Independent sets from Example 2 and Example 3 are constructed by applyingthe technique described in the proof of Proposition 6 to Thas-Kantor ovoids of Q (3 k ). The model used to represent the ovoid is from [71]. Example . Consider the graph
V O ( q ), where q = 3 k for some k ≥
1, and thedefining matrix (19) equals(43) A = −
10 0 0 − − − . Let d ∈ F q be a non-square and let I = n ( x, y, z, x y − dy − xz, − d − x + xy + yz ) ⊤ : x, y, z ∈ F q o . If u i = ( x i , y i , z i , x i y i − dy i − x i z i , − d − x i + x i y i + y i z i ) ⊤ for i = 1 ,
2, then a directcomputation with a use of equality 3 = 0 shows that( u − u ) ⊤ A ( u − u ) == − d − (cid:16) ( x − x ) − d ( y − y ) (cid:17) + (cid:16) ( z − z ) + ( x y − x y ) (cid:17) , which equals zero only if u = u , so I is an independent set of size (42). Example . Consider the graph
V O +6 ( q k ), where q = 3 k for some k ≥
1, and thedefining matrix equals(44) (cid:18) A − (cid:19) , N MINKOWSKI SPACE AND FINITE GEOMETRY 23 where A is the matrix in (43). Let I ′ ⊆ F q be formed from the set I in Example 2by extending its members with 0 in the last entry, that is, I ′ = { ( x ⊤ , ⊤ : x ∈ I} .Then I ′ is an independent set of size (42). Example . Let − F q , or equivalently q = p k , p and k both odd,with p ≡ Q +7 ( q ) [43] and we can use the technique from the proof of Proposition 6to construct an independent set in V O +6 ( q ) of size (42). We treat only the casewhen − − A = − − − be the defining matrix and let I = n ( x, y, xz + yw, z, yz − xw, w ) ⊤ : y, z, w ∈ F q , x = − ( z + w ) / o . If u i = ( x i , y i , x i z i + y i w i , z i , y i z i − x i w i , w i ) ⊤ for i = 1 ,
2, then by evaluating bothsides of the equation( u − u ) ⊤ A ( u − u ) = (cid:16) x + x + z z + w w (cid:17) + (cid:16) y − y + z w − z w (cid:17) , with a use of equality x i = − ( z i + w i ) /
2, we see that they are equal indeed. ByLemma 7, the sum of two squares is zero only if both squares are zero, so we seethat the expression above is zero only if u = u . Hence, I is an independent setof size (42).Bijective maps which preserve cones on nonsingular non-anisotropic metric spacesof dimension ≥
3, with a symmetric bilinear form, were characterized in [48]. Inparticular, this result characterizes the automorphisms of affine polar graph
V O εn ( q )for q odd and n ≥
3. Hence, for the elliptic case this theorem characterizes also allgraph endomorphisms, as it follows from Theorem 1. The same holds for parabolicand hyperbolic case if the polar space related to Q εn − ( q ) has no ovoids, as it followsfrom Theorem 2. This characterization even simplifies a bit if q is a prime or if thedefining matrix A in (19) has entries that are fixed by all automorphisms of thefield F q (for example, if entries are just 0, 1, − Finite Minkowski spaces
In what follows we assume that − ∈ F q is not a square, which is equivalent to q ≡ F q = { , x , . . . , x q − , − x , . . . , − x q − } . In the literature devoted to some alternative theories in particle physics that arebased on geometry over finite fields (cf. [1–4,12,13,15,24,67]), squares x , . . . , x q − and non-squares − x , . . . , − x q − are interpreted as ‘positive’ and ‘negative’ num-bers, respectively. Though the product of two ‘positive’ or two ‘negative’ numbersis a ‘positive’ number and the product of a ‘positive’ and a ‘negative’ number is‘negative’, this analogy with real numbers breaks down with addition, since a sumof two squares is not necessarily a square. However, it was shown in [44] that, forcertain prime values q , the elements 1 , , , . . . , N are all squares for certain N , sothe sum of two such elements is a square if it does not exceed N . Consequently, arelation a > b def ⇐⇒ a − b is ‘positive’becomes transitive for a large subset of consecutive numbers. There are infinitenumber of such prime values q and value N can be made arbitrary large by select-ing sufficiently large q (see [1,24,44] for more details). By comparing the differencebetween two consecutive numbers in 1 , , , . . . , N with the ‘smallest observablelength’ one can consider an approximation between real and finite field coordi-nates, and consequently an approximation between large subsets of mathematicalstructures over a large finite field and corresponding structures over real numbers(cf. [1]). It is not the purpose of this section to take a ‘philosophical point ofview’, which would be in favor of either finite or real geometry to be used as amodel for some physical theory. Our research provides a better understanding ofthe mathematical background in finite case, and shows some of the fundamentalgeometric differences between finite and real Minkowski space that go beyond theapproximation of coordinates.A finite Minkowski space of dimension n is the vector space F nq equipped withthe inner product(47) ( x , y ) := x y − x y − · · · − x n y n , where x := ( x , . . . , x n ) ⊤ and y := ( y , . . . , y n ) ⊤ . Two such events x and y are light-like if ( x − y , x − y ) = 0. A map Φ : F nq → F nq that maps light-like events intolight-like events, that is, (cid:0) Φ( x ) − Φ( y ) , Φ( x ) − Φ( y ) (cid:1) = 0 whenever ( x − y , x − y ) = 0,is said to preserve the speed of light . Observe that the product (47) can be simplywritten as ( x , y ) = x ⊤ M y for the diagonal matrix M := diag(1 , − , . . . , − ∈ SGL n ( F q ). We say that matrices L and K , of size n × n , are Lorentz and anti-Lorentz , if(48) ( L x , L y ) = ( x , y ) and ( K x , K y ) = − ( x , y )holds for all x , y ∈ F nq , respectively. The two conditions in (48) are equivalent to L ⊤ M L = M and K ⊤ M K = − M , respectively. By applying the determinant tothe last equation we see that anti-Lorentz matrices do not exist for odd n , sincethe opposite would imply that (det K ) = ( − n = −
1, which contradicts ourassumption that − N MINKOWSKI SPACE AND FINITE GEOMETRY 25 a , b ∈ F q such that a + b = −
1, so the block diagonal matrix a b − b a . . . a b − b a , is anti-Lorentz. This differs from the real case, where the Sylvester’s law of inertiaforbids the existence of anti-Lorentz matrices for n ≥ n ≥ − preserve the speed of light, were classified in [48].Actually, this result considered more general spaces. It was proved that any suchmap is of the form Φ( x ) = L ( x ) + Φ(0), where bijective map L : F nq → F nq satisfy (cid:0) L ( x ) , L ( y ) (cid:1) = d τ (cid:0) ( x , y ) (cid:1) , L ( x + y ) = L ( x ) + L ( y ), and L ( a x ) = τ ( a ) L ( x ) forall x , y ∈ F nq and a ∈ F q . Here, d ∈ F q is a fixed nonzero scalar and τ : F q → F q is a field automorphism. Now, given a field automorphism σ and a column vector x or a matrix A , let x σ and A σ denote the vector/matrix that is obtained byapplying σ entry-wise. Then, the map x
7→ L ( x ) τ − is linear, i.e., a multiplicationby some invertible matrix P , so L ( x ) = ( P x ) τ for all x . Since the entries of M arejust 1, −
1, and 0, which are fixed by field automorphisms, we see that M τ = M .Consequently,( x τ ) ⊤ ( P τ ) ⊤ M P τ y τ = (cid:0) L ( x ) , L ( y ) (cid:1) = d τ (cid:0) ( x , y ) (cid:1) = d ( x τ ) ⊤ M τ y τ = ( x τ ) ⊤ ( d M ) y τ for all x and y , so ( P τ ) ⊤ M P τ = d M . By (46), d = ± x for some nonzero x ,so the matrix x − P τ is either Lorentz or anti-Lorentz, and Φ( x ) = x ( x − P τ ) x τ +Φ(0). Hence the main theorem in [48] simplifies into the following lemma in thecase of a finite Minkowski space. Lemma 13. (cf. [48]) Let n ≥ . A bijective map Φ : F nq → F nq satisfies the rule ( x − y , x − y ) = 0 ⇐⇒ (cid:0) Φ( x ) − Φ( y ) , Φ( x ) − Φ( y ) (cid:1) = 0 if and only if it is of the form Φ( x ) = aL x τ + x (49) or Φ( x ) = aK x τ + x ( n is even ) , (50) where = a ∈ F q and x ∈ F nq are fixed, τ is a field automorphism of F q thatis applied entry-wise to x , while L and K are Lorentz and anti-Lorentz matricesrespectively.Remark . Recall that in the case q is a prime, the identity map is the uniqueautomorphism τ . More generally, if q = p k , where p is a prime, then the fieldautomorphisms are precisely the maps a a p j for 0 ≤ j ≤ k − We define a finite Minkowski graph M n ( q ) as the affine polar graph with M asthe defining matrix A in (19). Since det M = ( − n − , we deduce from Lemma 7that M n ( q ) = V O n ( q ) if n is odd ,V O + n ( q ) if n ≡ ,V O − n ( q ) if n ≡ . Observe that maps that are characterized in Lemma 13 are precisely the automor-phisms of M n ( q ). We are now able to state and prove the main theorem of thispaper. Theorem 3.
Let q ≡ mod and n ≥ . Assume that one of the followingconditions is satisfied: n ≡ mod ,n ≥ is odd ,n = 7 and q > is a prime ,n = 7 and Q n − ( q ) does not have an ovoid ,n ≡ mod and Q + n − ( q ) does not have an ovoid . A map
Φ : F nq → F nq satisfies the rule (51) ( x − y , x − y ) = 0 , x = y = ⇒ (cid:0) Φ( x ) − Φ( y ) , Φ( x ) − Φ( y ) (cid:1) = 0 , Φ( x ) = Φ( y ) if and only if it is of the form (49) or (50), where the form (50) is possible only foreven n .For general n ≥ , if there is a map Φ that satisfies the rule (51) and is neitherof the form (49) nor (50), then there exists a map that satisfies the rule (51) andhas ω (cid:0) M n ( q ) (cid:1) pairwise light-like events as its image.Remark . For n = 4 a different proof of Theorem 3 was already presented by theauthor in the arXiv version of [57]. For many dimensions, which include n = 4, thefinite and the real Minkowski space differ in an essential way, since in the real casethere always exist non-bijective maps Φ : R n → R n that satisfy the condition (51).In fact, since the field R is infinite, there is an injection f : R n → R , so the mapΦ( x ) = (cid:0) f ( x ) , f ( x ) , , . . . , (cid:1) ⊤ satisfies (51), since its image consists of pairwiselight-like events. Proof of Theorem 3.
Maps that satisfy the rule (13) are precisely the endomorphismsof graph M n ( q ). If n ≡ M n ( q ) = V O − n ( q ) is a core by Theorem 1, soany its endomorphism is an automorphism, and hence characterized in Lemma 13.If n is odd or n ≡ M n ( q ) equals V O n ( q ) or V O + n ( q ), respectively.If the corresponding orthogonal polar space does not have an ovoid, then M n ( q ) isa core by Theorem 2 and the result follows as in the elliptic case. Recall that thereare no ovoids in Q n − ( q ) if n ≥ n = 7 and q > n ≥
4, if there is a map obeying the rule (51) and not of the form (49)-(50), then it is a graph endomorphisms but not an automorphism, so, by Lemma 10,the core of M n ( q ) is a complete graph on ω (cid:0) M n ( q ) (cid:1) vertices. Consequently thereexists an endomorphism with this complete graph as its image. (cid:3) Recall from Proposition 1 that in the case that α (cid:0) M n ( q ) (cid:1) = V (cid:0) M n ( q ) (cid:1) ω (cid:0) M n ( q ) (cid:1) N MINKOWSKI SPACE AND FINITE GEOMETRY 27 holds, then there is a map that satisfies the rule (51) and has image formed by ω (cid:0) M n ( q ) (cid:1) pairwise light-like events. In fact, if K and I are maximum clique andmaximum independent set, respectively, then the map (7) has this property. Thismap is of the form Φ( x ) = x , where x ∈ K and x ∈ I are the unique vectorsfrom these sets that satisfy x = x + x . Below we construct few examples of suchmaps. The cases n = 2 and n = 3 are trivial, while other maps are constructedfrom the independent sets in Examples 2, 3, 4. Example . If n = 2, we can choose K = { ( x , x ) ⊤ : x ∈ F q } and I = { (0 , x ) ⊤ : x ∈ F q } as the clique and independent set in M ( q ), respectively. The decom-position x = ( x , x ) ⊤ = ( x , x ) ⊤ + (0 , x − x ) ⊤ ∈ K + I implies that the mapΦ( x ) = ( x , x ) ⊤ satisfies the rule (51).If n = 3, we can choose K = { ( x , x , ⊤ : x ∈ F q } and I = { (0 , x , x ) ⊤ : x , x ∈ F q } , which induce the map Φ( x ) = ( x , x , ⊤ , where x = ( x , x , x ) ⊤ . Example . Let n = 5 and q = 3 k , where k is odd, so that − P ⊤ M P = A , where A is the matrix (43) and(52) P = − − − with P − = − −
10 1 − − − . It follows from Example 2 that P n(cid:0) x , x , x , f ( x , x , x ) , g ( x , x , x ) (cid:1) ⊤ : x , x , x ∈ F q o , with f ( x , x , x ) = x x + x − x x and g ( x , x , x ) = x + x x + x x , is anindependent set in M ( q ). Similarly, P { (0 , , , x , x ) ⊤ : x , x ∈ F q } is a clique.From the decomposition P x = P ( x , x , x , x , x ) ⊤ = P (0 , , , x − f ( x , x , x ) , x − g ( x , x , x )) ⊤ + P ( x , x , x , f ( x , x , x ) , g ( x , x , x )) ⊤ , we deduce the map Φ( P x ) = P (0 , , , x − f ( x , x , x ) , x − g ( x , x , x )) ⊤ . If( y , y , y , y , y ) ⊤ := P x , that is, x = P − y = ( − y − y , y − y + y , − y + y , y + y + y , − y + y ) ⊤ , then we obtain mapΦ( y ) = P y + y + y − f ( − y − y , y − y + y , − y + y ) − y + y − g ( − y − y , y − y + y , − y + y ) that satisfies the rule (51) in explicit form. Example . Let n = 6 and q = 3 k , where k is odd. Then the matrix (44) equals Q ⊤ M Q , where Q = (cid:18) P
00 1 (cid:19) and P is the matrix in (52). It follows from Example 3 that Q n(cid:0) x , x , x , f ( x , x , x ) , g ( x , x , x ) , (cid:1) ⊤ : x , x , x ∈ F q o , where f, g are defined in Example 6, is an independent set in M ( q ). The set Q { (0 , , x , x , x , x ) ⊤ : x , x , x ∈ F q } is a clique. We can use the decomposition Q x = Q (0 , , x , x − f ( x , x , x − x ) , x − g ( x , x , x − x ) , x ) ⊤ + Q ( x , x , x − x , f ( x , x , x − x ) , g ( x , x , x − x ) , ⊤ , and proceed in the same way as in Example 6, to obtain mapΦ( y ) = Q y y + y + y − f ( − y − y , y − y + y , − y + y − y ) − y + y − g ( − y − y , y − y + y , − y + y − y ) y that satisfies the rule (51). Example . Let n = 6 and q = p k , where k is odd and p ≡
11 (mod 12). Notethat, by Chinese remainder theorem and Law of quadratic reciprocity, this is justa compact way of saying that − k is odd, and p ≡ a , b , c be suchthat a + b = − c = 3. Then P ⊤ M P = A , where A is the matrix (45) and P = −
10 0 a − a b b b − b − a − a c with P − = c − a − b
00 0 a b
00 1 0 − b a − − b a . It follows from Example 4 that P n(cid:0) f ( x , x ) , x , g ( x , x , x ) , x , h ( x , x , x ) , x (cid:1) ⊤ : x , x , x ∈ F q o , where f ( x , x ) = − ( x + x ) / ,g ( x , x , x ) = f ( x , x ) x + x x ,h ( x , x , x ) = x x − f ( x , x ) x , N MINKOWSKI SPACE AND FINITE GEOMETRY 29 is an independent set in M ( q ). Similarly, P { ( x , c x , x , , x , ⊤ : x , x , x ∈ F q } is a clique. We can use the decomposition P x = P x − f ( x , x ) c x − c f ( x , x ) x − g (cid:0) x − c x + c f ( x , x ) , x , x (cid:1) x − h (cid:0) x − c x + c f ( x , x ) , x , x (cid:1) + P f ( x , x ) x − c x + c f ( x , x ) g (cid:0) x − c x + c f ( x , x ) , x , x (cid:1) x h (cid:0) x − c x + c f ( x , x ) , x , x (cid:1) x , and proceed in the same way as above, to obtain mapΦ( y ) = P y c − f (cid:0) s, t (cid:1) y − c f (cid:0) s, t (cid:1) y − a y − b y − g (cid:16) y − y + c f (cid:0) s, t (cid:1) , s, t (cid:17) y − b y + a y − h (cid:16) y − y + c f (cid:0) s, t (cid:1) , s, t (cid:17) that satisfies the rule (51). Here, s = s ( y , y , y ) := y + a y + b y ,t = t ( y , y , y ) := − y − b y + a y . Recall that there exist a map Φ : F q → F q that satisfy the rule (51) and hasthe image formed by pairwise light-like events, whenever q is a prime of the form q ≡ q of thisform, we were not able to construct such a map explicitly, since the Conway etal/Moorhouse ovoids in Q +7 ( q ) [25, 52], which guarantee the existence of such amap, are not parameterized. Acknowledgements.
The author is thankful to Andries E. Brouwer for pointinghim to reference [39]. He would also like to thank Gabriel Verret for pointing himto reference [61], which subsequently led to the observation that Proposition 1 hasalready been proved in [31].
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