For each ﬁnite ﬁeld $F$ of prime order there is a constant $c$ such that every 4-connected matroid has at most $c$ inequivalent representations over $F$. We had hoped that this would extend to all ﬁnite ﬁelds, however, it was not to be. The $(m,n)$-mace is the matroid obtained by adding a point freely to $M(K_{m,n})$. For all $n \geq 3$, the $(3,n)$-mace is 4-connected and has at least $2n$ representations over any ﬁeld $F$ of non-prime order $q \geq 9$. More generally, for $n \geq m$, the $(m,n)$-mace is vertically $(m+1)$-connected and has at least $2n$ inequivalent representations over any ﬁnite ﬁeld of non-prime order $q\geq m^m$

Geelen, J. (Jim)

Gerards, A.M.H. (Bert)

Whittle, G. (Geoff)

English

CWI's Institutional Repository

Journal of Combinatorial Theory, Series B 100 (2010) 740–743Contents lists available at ScienceDirectJournal of Combinatorial Theory,Series Bwww.elsevier.com/locate/jctbOn inequivalent representations of matroids over non-primefieldsJim Geelen a,1, Bert Gerards b, Geoff Whittle ca Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canadab CentrumWiskunde & Informatica, Amsterdam, The Netherlandsc School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealanda r t i c l e i n f o a b s t r a c tArticle history:Received 2 April 2009Available online 16 August 2010Keywords:MatroidsInequivalent representationsConnectivityFor each finite field F of prime order there is a constant csuch that every 4-connected matroid has at most c inequivalentrepresentations over F. We had hoped that this would extendto all finite fields, however, it was not to be. The (m,n)-maceis the matroid obtained by adding a point freely to M(Km,n).For all n  3, the (3,n)-mace is 4-connected and has at least2n representations over any field F of non-prime order q  9.More generally, for n  m, the (m,n)-mace is vertically (m + 1)-connected and has at least 2n inequivalent representations overany finite field of non-prime order qmm .© 2010 Elsevier Inc. All rights reserved.1. IntroductionFor every finite field F of non-prime order  9, we show that there are 4-connected matroidshaving arbitrarily many inequivalent representations over F. This is in striking contrast to the situa-tion for fields of prime order. Earlier we proved that, for each finite field F of prime order, there isa constant c such that every 4-connected matroid has at most c inequivalent representations over F;see [1]. This dichotomy came as a surprise to us. In fact, we fully expected that the proof of the resultfor prime fields would extend routinely to the non-prime case. When we came to write the proof ourillusions had an awkward encounter with reality. After a time, and an increasingly desperate flurry ofemail correspondence, we were left staring glumly at the culprits.The (m,n)-mace is the matroid obtained by adding a point freely to M(Km,n), where Km,n is thecomplete bipartite graph with colour-classes of size m and n respectively. Let M be an (m,n)-mace,1 This research was partially supported by grants from the Natural Sciences and Engineering Research Council of Canada andthe Marsden Fund of New Zealand.0095-8956/$ – see front matter © 2010 Elsevier Inc. All rights reserved.doi:10.1016/j.jctb.2010.08.001J. Geelen et al. / Journal of Combinatorial Theory, Series B 100 (2010) 740–743 741with m  n. Since M(Km,n) is vertically m-connected, M is vertically (m + 1)-connected. Since Km,nhas no triangles, the (3,n)-mace is 4-connected for each n 3.We assume that the reader is familiar with matroid representation. We mostly use the notationand definitions of Oxley [4]. In this paper, however, we say that two representations are equivalentif one can be obtained from the other by elementary row operations and column scaling. We do notallow the use of field automorphisms.For a prime-power q = pk we definemq ={(k − 1)(p − 1) + 1, p  3,k − 1, p = 2.Theorem 1.1. Let F be a finite field of order q = pk where p is prime and k  2. If m mq, then the (m,n)-mace has at least 2n inequivalent representations over F.The following result is an immediate corollary of Theorem 1.1.Corollary 1.2. For any finite field F of non-prime order q 9 and any integer c, there is a 4-connected matroidwith at least c inequivalent representations over F.Corollary 1.2 leaves two exceptional fields, namely GF(4) and GF(8). Kahn [2] proved that all 3-connected matroids have at most two inequivalent representations over GF(4). We conjecture thatthere is an integer c such that every 4-connected matroid has at most c inequivalent representationsover GF(8).Let F+ be the additive group of a finite field F. Subgroups of F+ play an important role in repre-senting a mace (which is why maces do not cause difficulties for fields of prime order). The followinglemma is an immediate consequence of Lemma 2.3 which is proved in the next section.Lemma 1.3. If Γ is a subgroup of F+ and there exists a sequence (α1, . . . ,αm−1) of elements in F such that nonon-empty subsequence sums to an element of Γ , then the (m,n)-mace has at least (|Γ | − 1)n inequivalentrepresentations over F.Suppose that |F| = pk for some prime p and integer k; thus F+ = Zkp . To obtain matroids withmany inequivalent representations from Lemma 1.3, we require |Γ | 3. For p  3 we choose Γ = Zpand, for p = 2, we choose Γ = Z22. We would like to know the length of the longest sequence ofelements in F such that no non-empty subsequence sums to an element of Γ ; we denote this numberby m(Γ )− 1. Note that m(Γ ) is the smallest integer such that any sequence of m(Γ ) elements in thequotient group F+/Γ has a non-empty subsequence that adds to zero; this is exactly the Davenportconstant of the group F+/Γ . Note that F+/Γ = Ztp where t = k − 1 when p  3, and t = k − 2 whenp = 2. The Davenport constant of Ztp is known to be t(p− 1)+ 1; see Olsen [3]. Therefore m(Γ ) =mqand, hence, Theorem 1.1 is a consequence of Lemma 1.3.Kahn [2] conjectured that for each finite field F there is a constant c such that every 3-connectedmatroid has at most c inequivalent representations over F. Kahn’s conjecture was refuted by Oxley,Vertigan, and Whittle [5] for all fields of order  7. Our results show that Kahn’s conjecture cannotbe revived by replacing 3-connected by vertically k-connected for any fixed integer k. However webelieve that maces are, in some sense, as bad as it gets.Conjecture 1.4. For any finite field F of non-prime order q there exists an integer c such that every (mq + 2)-connected matroid has at most c inequivalent representations over F.2. Extensions of graphic matroidsLet F be a field and let G = (V , E) be a graph with V = {1, . . . ,n}. Now let A be the signed-incidence matrix of G; that is, the rows of A are indexed by vertices, the columns are indexed by742 J. Geelen et al. / Journal of Combinatorial Theory, Series B 100 (2010) 740–743edges, and, for an edge e = i j, with i < j, the i-th entry is 1, the j-th entry is −1, and all otherentries are zero. For α ∈ FV , we let M(G,α) denote the matroid represented by [A,α] over F. ThusM(G,α) is a single element extension of M(G); we call the new element t . The following matrixshows [A,α] in the case that G = K3,3.e14 e15 e16 e24 e25 e26 e34 e35 e36 t1 1 1 0 0 0 0 0 0 α10 0 0 1 1 1 0 0 0 α20 0 0 0 0 0 1 1 1 α3−1 0 0 −1 0 0 −1 0 0 α40 −1 0 0 −1 0 0 −1 0 α50 0 −1 0 0 −1 0 0 −1 α6For S ⊆ V , we let α(S) =∑(αi: i ∈ S). The following facts are straightforward.(i) If M = MF(B), for some matrix B over F, and M \ t = M(G), then B is equivalent to [A,α] forsome α ∈ FV .(ii) If G is 2-connected and α,α′ ∈ FV are non-zero vectors, then [A,α] and [A,α′] are equivalent ifand only if α is a scalar multiple of α′ .(iii) For α ∈ FV , the element t is spanned by E in M(G,α) if and only if, for each component H of G ,we have α(V (H)) = 0.The next lemma follows immediately from (iii).Lemma 2.1. Let G = (V , E) be a connected graph and let α ∈ FV with α(V ) = 0. Then t is freely placed inM(G,α) if and only if there is no subgraph F of G such that F is not connected, and α(V (H)) = 0 for eachcomponent H of F .We let the bipartition in Km,n be ({1, . . . ,m}, {m+ 1, . . . ,m+n}). The following result is a special-ization of Lemma 2.1 to extensions of M(Km,n).Lemma 2.2. Let α ∈ Fm+n with α({1, . . . ,m + n}) = 0. Then M(Km,n,α) is the (m,n)-mace if and only ifαi = 0 for each i ∈ {1, . . . ,m + n}, and α(S) = 0 for each S ⊆ {1, . . . ,m + n} with 1 |S ∩ {1, . . . ,m}| <mand 1 |S ∩ {m + 1, . . . ,m + n}| < n.Lemma 2.3. Let Γ be a subgroup of F+ . Let α1, . . . ,αm−1 be a sequence of elements in F such that nonon-empty subsequence adds to an element of Γ , let αm+1, . . . ,αm+n ∈ Γ − {0}, and define αm so thatα({1, . . . ,m + n}) = 0. Then M(Km,n,α) is the (m,n)-mace.Proof. Certainly α1, . . . ,αm+n are all non-zero. Suppose that there exists S ⊆ {1, . . . ,m+n} such that1 |S ∩ {1, . . . ,m}| <m, 1 |S ∩ {m + 1, . . . ,m + n}| < n, and α(S) = 0. By possibly replacing S with{1, . . . ,m + n} − S we may assume that m /∈ S . Since α(S) = 0, we have α(S ∩ {1, . . . ,m − 1}) =−α(S ∩ {m + 1, . . . ,m + n}) ∈ Γ . This gives a non-empty subsequence of (α1, . . . ,αm−1) that sums toan element of Γ . This contradiction completes the proof. 3. A more general constructionFor nm > 3, the (m,n)-mace is vertically (m+ 1)-connected but is not even 5-connected. In thissection we briefly sketch a proof of the following theorem.Theorem 3.1. Let F be a finite field of non-prime order q. For any positive integer c, there is an (mq + 1)-connected matroid that has at least c inequivalent representations over F.J. Geelen et al. / Journal of Combinatorial Theory, Series B 100 (2010) 740–743 743The result holds for q = 4 (consider 2-sums of copies of U2,4), so we may assume that q = 4.Let m = mq , g = mq + 1, and n = log2 c. There exists an m-regular, internally (m + 1)-connectedgraph H with girth at least g . Let u ∈ V (H) and let v1, . . . , vm be the neighbours of u. Now constructa graph G by starting with m new vertices x1, . . . , xm and n copies (H1, . . . , Hn) of H−u and, for eachi ∈ {1, . . . ,m}, adding edges connecting xi to each of the n copies of vi . Note that G is m-connectedand has girth at least g .Let w ∈ V (H) − {u} and let xm+1, . . . , xm+n be the n copies of w in G . Let X = {x1, . . . , xn+m}. It isstraightforward to show that there is a unique matroid M(G; X) obtained by extending M(G) by anelement e so that, for each subgraph F of G , e is spanned by E(F ) if and only there is a componentof F that contains X . Moreover, since g =m + 1, the matroid M(G; X) is (m + 1)-connected.There is a subgroup Γ of F+ with |Γ | 3 and elements α(x1), . . . ,α(xm−1) ∈ F such that no non-empty subsequence of (α(x1), . . . ,α(xm−1)) adds to an element of Γ . For each i ∈ {m+ 1, . . . ,m+ n},let α(xi) ∈ Γ − {0}. Now define α(xm) so that α(x1) + · · · + α(xm+n) = 0 and let α(y) = 0 for ally ∈ V (G) − {x1, . . . , xm+n}. We let M(G,α) denote the extension of M(G) as described in the pre-vious section. It is easy to verify that M(G; X) = M(G,α). There are 2n  c different choices of(α(xm+1), . . . ,α(xm+n)), which proves Theorem 3.1.References[1] J. Geelen, G. Whittle, Inequivalent representations of matroids over prime fields, in preparation.[2] J. Kahn, On the uniqueness of matroid representations over GF(4), Bull. Lond. Math. Soc. 20 (1988) 5–10.[3] J.E. Olsen, A combinatorial problem on finite abelian groups, I, J. Number Theory 1 (1961) 8–10.[4] J.G. Oxley, Matroid Theory, Oxford University Press, New York, 1992.[5] J. Oxley, D. Vertigan, G. Whittle, On inequivalent representations of matroids over finite fields, J. Combin. Theory Ser. B 67(1996) 325–343.

On inequivalent representations of matroids over non-prime ﬁelds

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