n 2 R. SHAW Department of Mathematics, University of Hull, Hull HU6 7RX, United Kingdom 19 August 2005

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1 The Grassmannian G 1;n; has polynomial degree 1 n R. SHAW r.shaw@hull.ac.uk Department of Mathematics, University of Hull, Hull HU6 7RX, United Kingdom 19 August 005 Abstract The Grassmannian G 1;n; of lines of PG(n; ); which is a subset of the points of the projective space PG(N; ); N = n+1 1; is proved to have polynomial degree n 1: 1 Introduction and background The notation and terminology will be the same as in the earlier papers [5] and [6]. In particular, for V n+1 = V (n + 1; ); we deal with the bivector space ^V n+1 ; of dimension n+1 ; with the associated projective space P(^ V n+1 ) = PG(N; ), where N := N n = n+1 1; and with the Grassmannian G 1;n; S := PG (0) (N; ): The natural action of A GL(n + 1; ) upon V N+1 := ^V n+1 is by T A = ^A : a ^ b 7! Aa ^ Ab. Now, for n > 3; the subgroup G(G 1;n; ) of GL(N + 1; ) which stabilizes G 1;n; is the isomorphic image under T of GL(n + 1; ). Consequently the space F (S) of functions S! GF() will be viewed as a GL(n+1; )-space under the action L de ned by (L A f)(x) = f(t 1 A x); A GL(n + 1; ); x S: (1.1) In the following we write N n = n + d n where d n := N n n = n 1: Note therefore, see [5, Theorem 1.5], that F dn = C n ; F n = C dn : (1.) We denote by Q 1;n; F (S) the characteristic function ((G 1;n; ) c ) of the complement (within the set S) of G 1;n; : So the Grassmannian has equation Q 1;n; (x) = 0: The reduced degree of the polynomial function Q 1;n; is termed the polynomial degree of G 1;n; : Much of the material in [5] was concerned with amassing evidence in support of the following conjecture. 1

2 Conjecture A. For all n 3 the Grassmannian G 1;n; has polynomial degree d n = n 1: That is Q 1;n; F dn F dn 1; equivalently, by equation (1.), Q 1;n; C n C n+1 : This conjecture is just the q = special case of a G 1;n;q conjecture in [1]. In the present paper we will settle conjecture A in the a rmative by proving the theorem: Theorem 1.1 For all n 3 the Grassmannian G 1;n; has polynomial degree d n = n 1: That is Q 1;n; F dn F dn 1 (= C n C n+1 ): See section.5 for a brief description of the origins of the present paper. 1.1 Background results Of relevance to the task of proving Theorem 1.1 are the following results from [5]. Concerning R.1, a choice of basis fe 1 ; e ; :::; e n+1 g for V n+1 gives rise to a product basis fe i ^ e j g 1i<jn+1 for V N+1 = ^V n+1 ; and hence to a simplex of reference for PG(N; ): R.1. Denote by F s the set consisting of those functions (X c ) for which X is a face of the simplex of reference of (projective) dimension N s: Then, for 1 r N; F 1 [ F [ ::: [ F r (1.3) is a basis for C N r (and hence is a basis for F r ): R.. Q 1;n; F dn if and only if jx \ G 1;n; j is odd for each d n - at X in PG(N; ): R.3. Q 1;n; F dn if and only if jx \ G 1;n; j is odd for each simplicial at of (projective) dimension d n : R.4. Suppose that there exists a family of d n - ats of PG(N; ) such that (i) f(x c )g X = C dn, and (ii) each X meets G 1;n; in an odd number of points: Then G 1;n; has polynomial degree d n : Commentary. For R.1 see the Simplex Basis Theorem of [5, Theorem 1.8]. (See also [5, Section 1..(iv)].) For R. see [3] or [5, Theorem 1.7]. The result R.3 follows from R.1, R.; cf. [5, C.5 of Theorem.1]. For R.4 see [5, P.3 of Theorem 3.4]. 1. Using simple graphs on n + 1 vertices As in [5] and [6] it helps to make use of certain (simple) graphs = (V n ; E) having vertex set V n := f1; ; ::: ; n+1g and edge set E: Along with we also need its complement = (V n ; E): Using a product basis fe i ^ e j g 1i<jn+1 for V N+1 = ^V n+1 ; a point x = 1i<jn+1 x ij e i ^ e j S has coordinates (x ij ) 1i<jn+1 : To each edge E = ij := fi; jg E of a (simple) graph = (V n ; E) let us associate

3 (i) the coordinate x E := x ij (= x ji ); and hence the hyperplane x E = 0; (ii) the basis element e E := e i ^ e j for V N+1 : Given a graph = (V n ; E) let X E denote the at having coordinate equations x E = 0; each E E; and set Y (E) := hfe E g EE i: Then x Y ( E) if and only if x = P EE x Ee E, that is if and only if x E = 0; for each E E: Hence X E = Y ( E): To each graph = (V n ; E) of size jej = r we will also associate the monomial function m E := EE x E of degree r: One basis for F r is 1 [ [ ::: [ r ; where s = fm E g jej=s ; (1.4) and, setting E := ((X E ) c ); another is the simplex basis F 1 [ F [ ::: [ F r ; where F s = f E g jej=s : (1.5) Often we will be dealing with edge sets E n of size je n j = n; observe that in this case je n j = n+1 n = d n + 1; and so each Y (E n ) = X En is a d n - at. Proof of Theorem 1.1 Our proof of Theorem 1.1 will appeal to R.4 of section 1.1; accordingly it will have the following two stages. Stage 1: we nd a GL(n + 1; )-orbit n of d n - ats of PG(N; ) such that the functions (X c ); X n ; span the whole of F n = C dn : Stage : we show that a at X n meets G 1n; in an odd number of points. In e ecting stage 1 of the proof it helps to consider the transforms L A m E of the monomials m E by various elements A of GL(n + 1; ); see eq. (1.1). In particular suppose that we restrict A to belong to the subgroup G 0 = Sym(n + 1) of GL(n + 1; ) which e ects all permutations of the basis fe 1 ; e ; :::; e n+1 g for V n+1. Then each of the sets r is stable under such L A, and so decomposes into G 0 -orbits, one for each isomorphism class of graphs (V n ; E) of size r:.1 Using transvections Also, for distinct elements a; b V n = f1; ; ::: ; n + 1g the transvection J(a; b) GL(n + 1; ) with e ect J(a; b)e i = e i ; i 6= b; and J(a; b)e b = e a + e b (.1) will be of use. For convenience, put T (a; b) := T J(a;b) and L(a; b) := L J(a;b). Note that if x ^V n+1 then x 0 = T (a; b)x reads in terms of coordinates x 0 ij = x ij ; if i 6= a; j 6= a; and also if ij = ab; x 0 aj = x aj + x bj ; if j 6= b: (.) 3

4 For future use let us note two consequences of this last coordinate transformation. First observe that for some monomials m E of degree r some of the linear combination m E + L(a; b)m E will be of degree r 1: For example, since (x 3 ) = x 3 ; we have x 13 x 3 + L(1; )(x 13 x 3 ) = x 3 : (.3) Secondly consider the monomial m Pn where P n denotes the path P n = f1; 3; 34; ::: n n + 1g (.4) of length jp n j = n: (Caution: in the literature a path of length n is sometimes denoted P n+1 :) It follows from (.) that if Pn 0 1 denotes the path of length n 1 which is obtained from P n by replacing the two edges fi 1; ig and fi; i + 1g by the new edge fi 1; i + 1g then m Pn + L(i; i + 1)m Pn = m En ; where E n = P 0 n 1 [ fi; i + 1g: (.5) By making use of the degree-lowering property of transvections exempli- ed by (.3) one may demonstrate the following: if along with the particular set n of monomials m E of degree n we consider also their transforms L A m E under elements A of GL(n + 1; ) then the resulting set S n of elements of F n generates the whole of F n : Lemma.1 Setting S n := fl A ( n )g AGL(n+1;) = fl A m E g AGL(n+1;);jEj=n ; then S n = F n : Proof. From the basis (1.4) for F n ; the lemma follows once we show that each of the monomials m E 0 of degree je 0 j < n lies in S n : Consider a monomial m E 0 of degree je 0 j = n 1: We need to consider two possibilities for its associated graph (V n ; E 0 ) : (i) there exists a vertex of valency 0; (ii) each of the n + 1 vertices has valency 1: Suppose (i) holds. Then (up to graph isomorphisms) we may suppose that E 0 contains no edge 1i; but does contain the edge 3: Setting E = f13g [ E 0 ; it follows, cf. equation (.3), that m E + L(1; )m E = m E 0: Suppose (ii) holds. In which case there exists a vertex of valency 1 (indeed there exist at least 4 such vertices). So (up to graph isomorphisms) we may suppose that E 0 contains the edge 13; but that 1i = E 0 ; i 6= 3; and also 3 = E 0 : Setting E = f3g [ E 0 ; it follows, cf. equation (.3), that m E + L(1; )m E = m E 0: So we have shown that each monomial m E 0 of degree je 0 j = n 1 lie in S n : From this last, it follows by a similar argument that each monomial m E 0 of degree n lie in S n ; and similarly for the monomials m E 0 of degrees n 3; n 4; :::. 4

5 Remark. The monomial function m E = EE x E depends of course upon our choice of coordinates x E = x ij for V N+1 via our choice of basis B n = fe 1 ; e ; :::; e n+1 g for V n+1 In the following it will help at times to make explicit this dependence upon B n by adopting the fuller notation m E (B n ): If for A GL(n + 1) we write A B n := fae 1 ; Ae ; :::; Ae n+1 g then observe that L A (m E (B n )) = m E ( A B n ): (.6). Stage 1 of the proof We aim to show that stages 1 and can be accomplished by choosing n to be the GL(n + 1; )-orbit of the d n - at X Pn : Lemma.3 (i) The GL(n + 1; )-orbit (X Pn ) of (X c P n ) spans F n : (ii) The GL(n + 1; )-orbit (m Pn ) of the monomial m Pn spans F n : Proof. For the d n - at X En = Y ( E n ) we have, see for example [6, Eq. (.4ii)], (X c E n ) = m En + (terms of degree < n): Consequently the two parts of the lemma are equivalent, and so it su ces to prove part (ii), namely that the orbit (m Pn ) of the particular path monomial m Pn = x 1 x 3 :::x n n+1 ; (.7) of degree n; spans the whole of F n : By Lemma.1 this will follow provided we can show that m En lies in (m Pn ) for a representative E n of each isomorphism class of graph (V n ; E n ) of size je n j = n: We proceed by induction upon n; adopting H n as inductive hypothesis: H n : if je n j = n; then m En (B n ) is a linear combination of path monomials m Pn ( A B n ) for various A GL(n + 1; ): (Recall (.6).) Certainly H holds, since if n = then fx 1 x 3 ; x 13 x 3 ; x 1 x 13 g is a basis for F consisting of path monomials. To show that H n 1 implies H n we consider, for a given m En (B n ); two cases: Case A. There exists a vertex i V n of valency 1. Case B. There exists a vertex i V n of valency 0. The two cases are not mutually exclusive, but together they cover all possibilities for the graph (V n ; E n ); since if case A does not apply then certainly case B does. If case A holds then without loss of generality we may suppose that the vertex n + 1 has valency 1 and is joined to vertex 1; whence m En (B n ) = x 1 n+1 m En 1 (B n 1 ) for some edge set E n 1 of size n 1 which uses the vertex set V n 1 = f1; ; ::: ; ng: By the inductive hypothesis H n 1 the monomial m En 1 (B n 1 ) is a linear combination of path monomials m Pn 1 (Bn 0 1 ) for various bases Bn 0 1 = fe0 1 ; e0 ; :::; e0 ng for V n : Consider one resulting 5

6 term x 1 n+1 m Pn 1 (Bn 0 1 ): Now x 1 n+1 is a linear combination of coordinates x 0 i n+1 with respect to the basis B0 n = Bn 0 1 [ fe n+1g: So, if case A applies, the induction goes through provided we can show that m En lies in (m Pn ) for those edge sets E n just of the kind E n = P n 1 [ fi n + 1g; for some i V n 1 : (.8) That this is so follows from the previous result (.5). If case B holds then without loss of generality we may suppose that the vertex n + 1 has valency 0 and that 1 E n : Consider the edge set En 0 which is obtained from E n by replacing the edge 1 by a new edge f1; n + 1g: Observing that m En = m E 0 n + L(n + 1; )m E 0 n we see that case B reduces to case A since the vertex n + 1 has valency 1 for the edge set En. 0.3 Stage of the proof Lemma.4 The d n - at X Pn meets G 1;n; in an odd number of points. Proof. We adopt the notation and terminology used in [6, Section.3]. So from [6, Lemma.8] we have jx Pn \G 1;n; j = p(p n )+q(p n ); where p(p n ) and q(p n ) are respectively the number of dyads and triads for the graph (V n ; P n ): Setting p n := p(p n ) and q n := q(p n ) the following recurrence relations are easily derived : p n = p n 1 + p n + n 1 1; q n = q n 1 + q n + p n ; (.9) see section.3.1 below. Consequently h n := p n + q n satis es h n = h n 1 + h n + n 1 1; (.10) from which it immediately follows that h n is odd for all n:.3.1 Derivation of the recurrence relations (.9) First take note of the following: (i) each dyad f; g for the graph (V n ; P n ) gives rise to three dyads for (V n ; P n ); namely to f; g; f [ fn + 1g; gg and f; [ fn + 1gg; (ii) dyads for (V n 1 ; P n 1 ) which are not dyads for (V n ; P n ) are of the form f [ fng; g for some dyad f; g for (V n ; P n ); such dyads give rise to two dyads for (V n ; P n ); namely to f[fng; g and f[fn; n+1g; g; (iii) each subset of V n gives rise to the dyad f; fn + 1gg for (V n ; P n ): Since every dyad for (V n ; P n ) arises in one of the mutually exclusive ways (i), (ii), (iii), it follows that p n = 3p n +(p n 1 p n )+ n 1 1; and so we have obtained the rst of the relations (.9): p n = p n 1 + p n + n 1 1: Secondly take note of the following: (i) each triad f; ; g for (V n ; P n ) gives rise to four triads for (V n ; P n ); 6

7 namely to f; ; g; f [ fn + 1g; ; g; f; [ fn + 1g; g and f; ; [ fn + 1gg; (ii) triads for (V n 1 ; P n 1 ) which are not triads for (V n ; P n ) are of the form f [ fng; ; g for some triad f; ; g for (V n ; P n ); such triads give rise to two triads for (V n ; P n ); namely to f[fng; ; g and f[fn; n+ 1g; ; g; (iii) each dyad f; g for the graph (V n ; P n ) gives rise to the triad f; ; fn + 1gg for (V n ; P n ): Since every triad for (V n ; P n ) arises in one of the mutually exclusive ways (i), (ii), (iii), it follows that q n = 4q n + (q n 1 q n ) + p n ; and so we have obtained the second of the relations (.9): q n = q n 1 + q n + p n :.4 Completion of the proof, and a graph theory corollary By R.4 of section 1.1, Theorem 1.1 now follows from Lemmas.3(i) and.4. In [6] conjecture A was translated into an equivalent conjecture in graph theory: see [6, Conjecture.1]. Consequently this graph theory conjecture now becomes a theorem: Theorem.5 If = (V; E) is any nite simple graph such that jej < jvj then p( ) + q( ) is odd. Here p( ) denotes the total number of subgraphs of which are isomorphic to K a;b for some a; b satisfying a b > 0; and q( ) denotes the total number of subgraphs of which are isomorphic to K a;b;c for some a; b; c satisfying a b c > 0:.5 Acknowledgements and history The author thanks Johannes Maks for sending him, in January 004, a copy of the Glynn/Maks/Casse paper [1]. Their paper contained a conjecture about the polynomial degree of the Grassmannian G 1;n;q of lines of PG(n; q): However, for n > 3 and q > no evidence was provided in [1] for the G 1;n;q conjecture. In the binary q = case Glynn, Maks & Casse conjectured that G 1;n; has polynomial degree d n ; but even in this binary case for n > 3 the only evidence given in [1] for the conjecture was the n = 4 result for G 1;4; obtained by Shaw and Gordon in 1994: see [4]. In fact a little more supporting evidence was available, since in 1994 N.A. Gordon obtained by computer the polynomial degree in the further two cases (n; q) = (5; ) and (n; q) = (4; 3): This extra evidence was communicated to Glynn, Maks & Casse in February 004. Provoked by the paucity of supporting evidence for the Glynn/Maks/Casse conjecture, even in the binary case, the present author combined with N.A. Gordon to test the binary conjecture for other values of n: Using some new 7

8 theoretical results, and aided by the computer, we were able to show, in April 004, that the binary conjecture held up also in the cases n = 6 and n = 7: This further evidence for the binary conjecture was eventually included in the paper [5], and a copy of this was sent to Glynn, Maks & Casse in February 005. Towards the end of 004 the author came to the view that in order to prove the binary conjecture, and not merely produce hard-won evidence for it in particular cases, perhaps the best hope was that, for each n; there exists one kind of GL(n + 1; )-orbit n of d n - ats of PG(N; ) such that the functions (X c ); X n ; span the whole of F n = C dn : See [5, Section 4..3]. In a working document of December 004 the author conjectured that the ats X Pn in Lemma.3 were just what were needed. For some low values of n he checked the requisite spanning property by using transvections as in sections.1,., but at this stage failed to do so for general n: However he was able to prove for general n that the d n - ats X Pn had the requisite odd intersection with G 1;n;, as in the above Lemma.4. The proof of this was as above, using the recurrence relations (.9). The author expresses his thanks to N.A. Gordon for carrying out computer checks, in early 005, on the accuracy of the relations (.9). In the spring of 005 the author developed some interesting links with graph theory: see the preprint [6], written prior to the 0th British Combinatorial Conference, Durham (10-15 July, 005). (See Theorem.5 above for one outcome of these links.) In [6, Section.3] it was shown how to compute the intersection numbers jx Pn \ G 1;n; j by simpler combinatorial methods than those employed in [5]. Using these, N.A. Gordon was able to check by computer the binary conjecture in the further case n = 8: Immediately upon returning from Durham the author set about proving for general n his conjecture concerning the d n - ats X Pn ; and he quickly arrived at the proof by induction given above in section.. In early August 005 he sent news of his consequent proof of the binary conjecture to Glynn, Maks & Casse. In return, J.G. Maks replied that they, also in July 005, had been able to prove the binary conjecture, as in their revised document []. An important ingredient of the work in [5] and [6] was frequent appeal to the simplex basis (1.5). The Glynn/Maks/Casse proof also makes crucial use of the simplex basis and, in essence, of result R.3 of section 1.1, see [, Theorem 16, Proof]. Nevertheless the two proofs of the binary conjecture are distinct. Indeed, the Glynn/Maks/Casse proof has two notable virtues: rstly it is of a nicely geometric nature, and secondly it is much shorter than the one in section above. The brevity of their proof arises from their appeal to a certain geometrical lemma, see [, Lemma 15]. With the aid of this lemma they are able to deal with the intersections X E \ G 1;n; for edge sets E of size d n all in the same way, with no need for separate consideration of the di erent isomorphism classes of such edge sets. 8

9 References [1] David G. Glynn, Johannes G. Maks, L.R.A. (Rey) Casse, The polynomial degree of the Grassmannian G(n; 1; q) of lines in nite projective space PG(n; q); preprint (July, 003). [] Revised version of [1], (July, 005) [3] R. Shaw, A characterization of the primals in PG(m; ), Des. Codes Cryptogr. (199), [4] R. Shaw and N.A. Gordon, The lines of PG(4; ) are the points of a quintic in PG(9; ), J. Combin. Theory (A), 68 (1994), [5] R. Shaw and N.A. Gordon, The polynomial degree of the Grassmannian G(1; n; ); to appear in Des. Codes Cryptogr., accessible from: [6] R. Shaw, Grassmann and Segre varieties over GF(): some graph theory links, preprint prior to BCC0, Durham (July 005), accessible from: 9