Dowling Lattices Research Articles

Representability of matroids by c-arrangements is undecidable

For a natural number c, a c-arrangement is an arrangement of dimension c subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of c. Matroids arising as normalized rank functions of c-arrangements are also known as multilinear matroids. We prove that it is algorithmically undecidable whether there exists a c such that a given matroid has a c-arrangement representation, or equivalently whether the matroid is multilinear. It follows that certain problems on network coding and secret sharing schemes are also undecidable. In the proof, we encode group presentations in frame matroids of rank three which we call generalized Dowling geometries: the construction is inspired by Dowling geometries of finite groups and by the von Staudt construction. The idea is to construct a reduction from the uniform word problem for finite groups to multilinear representability of matroids. The c-arrangement condition gives rise to some difficulties and their resolution is the main part of the paper.

A note on representing dowling geometries by partitions

A note on representing dowling geometries by partitions

Wonderful Models for Generalized Dowling Arrangements

For any triple given by a positive integer n, a finite group G, and a faithful representation V of G, one can describe a subspace arrangement whose intersection lattice is a generalized Dowling lattice in the sense of Hanlon (Trans. Amer. Math. Soc. 325(1), 1–37, 1991). In this paper we construct the minimal De Concini-Procesi wonderful model associated to this subspace arrangement and give a description of its boundary. Our aim is to point out the nice poset provided by the intersections of the irreducible components in the boundary, which provides a geometric realization of the nested set poset of this generalized Dowling lattice. It can be represented by a family of forests with leaves and labelings that depend on the triple (n,G,V ). We will study it from the enumerative point of view in the case when G is abelian.

A new approach to the r-Whitney numbers by using combinatorial differential calculus

Abstract In the present article we introduce two new combinatorial interpretations of the r-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar G := {y → yxm, x → x}. By specializing m = 1 we obtain also a new combinatorial interpretation of the r-Stirling numbers of the second kind. Again, by specializing to the case r = 0 we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard’s polynomials. Moreover, we recover several known identities involving the r-Dowling polynomials and the r-Whitney numbers using the combinatorial differential calculus. We construct a family of posets that generalize the classical Dowling lattices. The r-Withney numbers of the first kind are obtained as the sum of the Möbius function over elements of a given rank. Finally, we prove that the r-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce [m]-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities

Matroids of gain graphs in applied discrete geometry

A Γ \Gamma -gain graph is a graph whose oriented edges are labeled invertibly from a group Γ \Gamma . Zaslavsky proposed two matroids associated with Γ \Gamma -gain graphs, called frame matroids and lift matroids, and investigated linear representations of them. Each matroid has a canonical representation over a field F \mathbb {F} if Γ \Gamma is isomorphic to a subgroup of F × \mathbb {F}^{\times } in the case of frame matroids or Γ \Gamma is isomorphic to an additive subgroup of F \mathbb {F} in the case of lift matroids. The canonical representation of the frame matroid of a complete graph is also known as a Dowling geometry, as it was first introduced by Dowling for finite groups Γ \Gamma . In this paper, we extend these matroids in two ways. The first one is extending the rank function of each matroid, based on submodular functions over Γ \Gamma . The resulting rank function generalizes that of the union of frame matroids or lift matroids. Another one is extending the canonical linear representation of the union of d d copies of a frame matroid or a lift matroid, based on linear representations of Γ \Gamma on a d d -dimensional vector space. We show that linear matroids of the latter extension are indeed special cases of the first extension, as in the relation between Dowling geometries and frame matroids. We also discuss an attempt to unify the extension of frame matroids and that of lift matroids. This work is motivated by recent results in the combinatorial rigidity of symmetric graphs. As special cases, we give several new results on this topic, including combinatorial characterizations of the symmetry-forced rigidity of generic body-bar frameworks with point group symmetries or crystallographic symmetries and the symmetric parallel redrawability of generic bar-joint frameworks with point group symmetries or crystallographic symmetries.

Associativity in multiary quasigroups: the way of biased expansions

A multiary (polyadic, n-ary) quasigroup is an n-ary operation which is invertible with respect to each of its variables. A biased expansion of a graph is a kind of branched covering graph with an additional structure similar to the combinatorial homotopy of circles. A biased expansion of a circle with chords encodes a multiary quasigroup, the chords corresponding to factorizations, i.e., associative structure. Some but not all biased expansions are constructed from groups (group expansions); these include all biased expansions of complete graphs (with at least four nodes), which correspond to Dowling’s lattices of a group and encode an iterated group operation. We show that any biased expansion of a 3-connected graph (with at least four nodes) is a group expansion, and that all 2-connected biased expansions are constructed by the identification of edges from group expansions and irreducible multiary quasigroups. If a 2-connected biased expansion covers every base edge at most three times, or if every four-node minor that contains a fixed edge is a group expansion, then the whole biased expansion is a group expansion. We deduce that if a multiary quasigroup has a factorization graph that is 3-connected, or if every ternary principal retract is an iterated group isotope, it is isotopic to an iterated group. We mention applications of generalizing Dowling geometries and of transversal designs of high strength.

Nested set complexes of Dowling lattices and complexes of Dowling trees

Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice $\mathcal {Q}_{n}(G)$ (Proc. Internat. Sympos., 1971, pp. 101---115) and of its subposet of the G-symmetric partitions $\mathcal {Q}_{n}^{0}(G)$ which was recently introduced by Hultman ( http://www.math.kth.se/~hultman/ , 2006), together with the complex of G-symmetric phylogenetic trees $\mathcal {T}_{n}^{G}$ . Hultman shows that the complexes $\mathcal {T}_{n}^{G}$ and $\widetilde {\Delta }(\mathcal {Q}_{n}^{0}(G))$ are homotopy equivalent and Cohen---Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov (Selecta Math. (N.S.) 10, 37---60, 2004) shows that in fact $\mathcal {T}_{n}^{G}$ is subdivided by the order complex of $\mathcal {Q}_{n}^{0}(G)$ . We introduce the complex of Dowling trees $\mathcal {T}_{n}(G)$ and prove that it is subdivided by the order complex of $\mathcal {Q}_{n}(G)$ . Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.) 62, 437---468, 2005) shows that, as a simplicial complex, $\mathcal {T}_{n}(G)$ is in fact isomorphic to the Bergman complex of the associated Dowling geometry. Topologically, we prove that $\mathcal {T}_{n}(G)$ is obtained from $\mathcal {T}_{n}^{G}$ by successive coning over certain subcomplexes. It is well known that $\mathcal {Q}_{n}(G)$ is shellable, and of the same dimension as $\mathcal {T}_{n}^{G}$ . We explicitly and independently calculate how many homology spheres are added in passing from $\mathcal {T}_{n}^{G}$ to $\mathcal {T}_{n}(G)$ . Comparison with work of Gottlieb and Wachs (Adv. Appl. Math. 24(4), 301---336, 2000) shows that $\mathcal {T}_{n}(G)$ is intimely related to the representation theory of the top homology of $\mathcal {Q}_{n}(G)$ .

Quasigroup associativity and biased expansion graphs

We present new criteria for a multary (or polyadic) quasigroup to be isotopic to an iterated group operation. The criteria are consequences of a structural analysis of biased expansion graphs. We mention applications to transversal designs and generalized Dowling geometries.

Reconstructing ternary Dowling geometries

We prove that the only geometry of rank n > 4 all of whose proper contractions are ternary Dowling geometries is the ternary Dowling geometry. We use this to prove a stronger version of a conjecture of Joseph Kung and James Oxley.

On Dowling geometries of infinite groups

A finite subgeometry of a Dowling geometry for an infinite group is exhibited, which cannot be embedded in a Dowling geometry for any finite group; this provides a negative answer to a question of Bonin.

On the Homology of the h,k-Equal Dowling Lattice

We define a Dowling lattice generalization of the k-equal partition lattice and the h,k-equal signed partition lattice. We use the theory of lexicographical shellability to show that it has the homotopy type of a wedge of spheres, to describe its Betti numbers, and to give a basis for its homology in terms of labelled trees. For cyclic groups, the h,k-equal Dowling lattice arises as the lattice of intersections of a complex subspace arrangement. We use Whitney homology to compute the cohomology of the complement of this arrangement. Our results generalize and unify previous research.

A characterisation of jointless Dowling geometries

Abstract We use statistics of flats of small rank in order to characterise the jointless Dowling geometries defined by groups of order exceeding three and having rank greater than 3. In particular, we show that if the Tutte polynomial of a matroid is identical to the Tutte polynomial of a jointless Dowling geometry, then the matroid is indeed a jointless Dowling geometry. For rank 3 (and groups of order exceeding 3) this holds only if the order of the group is even.

A characterisation of jointless Dowling geometries

We use statistics of flats of small rank in order to characterise the jointless Dowling geometries defined by groups of order exceeding three and having rank greater than 3. In particular, we show that if the Tutte polynomial of a matroid is identical to the Tutte polynomial of a jointless Dowling geometry, then the matroid is indeed a jointless Dowling geometry. For rank 3 (and groups of order exceeding 3) this holds only if the order of the group is even.

Boolean Packings in Dowling Geometries

An explicit Boolean packing of the Dowling lattice is constructed.

Quasi-Varieties, Congruences, and Generalized Dowling Lattices

Dowling lattices and their generalizations introduced by Hanlon are interpreted as lattices of congruences associated to certain quasi-varieties of sets with group actions. This interpretation leads, by a simple application of Möbius inversion, to polynomial identities which specialize to Hanlon's evaluation of the characteristic polynomials of generalized Dowling lattices. Analogous results are obtained for a few other quasi-varieties.

Modular elements of higher-weight dowling lattices

The higher-weight Dowling lattice L k is the geometric lattice consisting of those subspaces of the vector space V n(q) which have bases of weight k or less, with these subspaces ordered by inclusion. These lattices arise when Crapo and Rota's results on the critical problem are applied to the Fundamental problem of linear coding theory. This paper identifies the modular elements of higher-weight Dowling lattices and applies that analysis to modular complements in L k and to the characteristic polynomial of L k .

Automorphism Groups of Higher-Weight Dowling Geometries

The weight k Dowling geometry M(Ek, n) is the restriction of the projective geometry PG(n − 1, q) to the projective points having k or fewer nonzero coordinate entries. These matroids arise when examining the Fundamental Problem of Linear Coding Theory in the context of the Critical Problem of Matroid Theory. In this paper, we study the group of automorphisms of M(Ek, n) for all weights k ≠ 2. For fixed q ≠ 2, the automorphism groups of the Dowling geometries M(E3, n), M(E4, n), ..., M(En − 1, n) are isomorphic. If q = 2 and k ≤ n − 2, then the automorphism group is either the symmetric group on n elements or the symmetric group on n + 1 elements according to whether k is odd or even. We conclude with remarks on the automorphism groups of weight 2 Dowling geometries M(E2, n).

The Generalized Dowling Lattices

The Generalized Dowling Lattices

The generalized Dowling lattices

In this paper we study a new class of lattices called the generalized Dowling lattices. These lattices are parametrized by a positive integer n n , a finite group G G , and a meet sublattice K K of the lattice of subgroups of G G . For an appropriate choice of K K the generalized Dowling lattice D n ( G , K ) {D_n}(G,K) agrees with the ordinary Dowling lattice D n ( G ) {D_n}(G) . For a different choice of K K , the generalized Dowling lattices are the lattice of intersections of a set of subspaces in complex space. The set of subspaces, defined in terms of a representation of G G , generalizes the thick diagonal in C n {\mathbb {C}^n} . We compute the Möbius function and characteristic polynomial of the lattice D n ( G , K ) {D_n}(G,K) along with the homology of D n ( G , K ) {D_n}(G,K) in terms of the homology of K K . We go on to compute the character of G G wr S n {S_n} acting on the homology of D n ( G , K ) {D_n}(G,K) . This computation provides a nontrivial generalization of a result due to Stanley concerning the character of S n {S_n} acting on the top homology of the partition lattice.

Combinatorial geometries representable over GF(3) and GF(q). II. Dowling geometries

Letq be an odd prime power not divisible by 3. In Part I of this series, it was shown that the number of points in a rank-n combinatorial geometry (or simple matroid) representable over GF(3) and GF(q) is at mostn2. In this paper, we show that, with the exception ofn = 3, a rank-n geometry that is representable over GF(3) and GF(q) and contains exactlyn2 points is isomorphic to the rank-n Dowling geometry based on the multiplicative group of GF(3).