Subspace Arrangements Defined by Products of Linear Forms

The vanishing ideal of an arrangement of linear subspaces ina vector space is considered, and the paper investigates whenthis ideal can be generated by products of linear forms. A combinatorialconstruction (blocker duality) is introduced which yields suchgenerators in cases with a great deal of combinatorial structure,and examples are presented that inspired the work. A constructionis given which produces all elements of this type in the vanishingideal of the arrangement. This leads to an algorithm for decidingif the ideal is generated by products of linear forms. Genericarrangements of points in P² and lines in P³ are also considered.     

Cohomology of Real Diagonal Subspace Arrangements via Resolutions

We express the cohomology of the complement of a real subspace arrangement of diagonal linear subspaces in terms of the Betti numbers of a minimal free resolution. This leads to formulas for the cohomology in some cases, and also to a cohomology vanishing theorem valid for all arrangements.     

Cluster-Tilted Algebras of Type D n

Let H be a hereditary algebra of Dynkin type D _n over a field k and 𝒞 _H be the cluster category of H. Assume that n ≥ 5 and that T and T′ are tilting objects in 𝒞 _H . We prove that the cluster-tilted algebra Γ = End_{𝒞 H} (T)^op is isomorphic to Γ′ = End_{𝒞 H} (T′)^op if and only if T = τ ⁱ T′ or T = στ ^j T′ for some integers i and j, where τ is the Auslander–Reiten translation and σ is the automorphism of 𝒞 _H defined in Section 4.     

Torus Actions, Equivariant Moment-Angle Complexes, and Coordinate Subspace Arrangements

We show that the cohomology algebra of the complement of a coordinate subspace arrangement in the m-dimensional complex space is isomorphic to the cohomology algebra of the StanleyReisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) After that we calculate the latter cohomology algebra by means of the standard Koszul resolution of a polynomial ring. To prove these facts, we construct a homotopy equivalence (equivariant with respect to the torus action) between the complement of a coordinate subspace arrangement and the moment-angle complex defined by a simplicial complex. The moment-angle complex is a certain subset of the unit polydisk in the m-dimensional complex space invariant with respect to the action of the m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere; otherwise, the complex has a more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the EilenbergMoore spectral sequence. We also relate our results with well-known facts in the theory of toric varieties and symplectic geometry. Bibliography: 23 titles.     

Schensted-Type Correspondences and Plactic Monoids for Types B n and D n

We use Kashiwara's theory of crystal bases to study plactic monoids for U _q(so _2n+1) and U _q(so _2n ). Simultaneously we describe a Schensted type correspondence in the crystal graphs of tensor powers of vector and spin representations and we derive a Jeu de Taquin for type B from the Sheats sliding algorithm.     

Arrangements of n Points whose Incident-Line-Numbers are at most n/2

We consider a set X of n noncollinear points in the Euclidean plane, and the set of lines spanned by X, where n is an integer with n ≥ 3. Let t(X) be the maximum number of lines incident with a point of X. We consider the problem of finding a set X of n noncollinear points in the Euclidean plane with t(X) ￡ ?n/2 ?{t(X) \le \lfloor n/2 \rfloor}, for every integer n ≥ 8. In this paper, we settle the problem for every integer n except n = 12k + 11 (k ≥ 4). The latter case remains open.     

Recurrences for Eulerian Polynomials of Type B and Type D

We introduce new recurrences for the type B and type D Eulerian polynomials, and interpret them combinatorially. These recurrences are analogous to a well-known recurrence for the type A Eulerian polynomials. We also discuss their relationship to polynomials introduced by Savage and Visontai in connection to the real-rootedness of the corresponding Eulerian polynomials.     

Arrangements of hyperplanes with property D

We denote by the complement of the complexification of a real arrangement of hyperplanes. It is known that there is a certain technical property, called property D, on real arrangements of hyperplanes such that: if a real arrangement of hyperplanes is simplicial then has property D, and if has property D then is aK(, 1) space. Our main goal is to prove that: if has property D then is simplicial. We also prove that a quasi-simplicial arrangement is always simplicial.     

On Minimal Generating Sets of E(D n ), A(D n) and I(D n ) with Even n

Let Dn be the dihedral group of order 2n. Denote by E(Dn) (resp. A(Dn), I(Dn)) the distributively generated nearring generated by the set of all endomorphisms (resp. automorphisms, inner automorphisms). In this paper, we determine for each one of the above three nearrings a minimal (additive) generating set. For E(Dn), this set contains the identity mapping and four other endomorphisms; for A(Dn), the identity mapping, one outer automorphism and one inner automorphisms; and for I(Dn), the identity mapping and two inner automorphisms.     

Finding Specified Sections of Arrangements: 2D Results

Given a configuration C of geometric objects in R ² (called the input configuration), a target configuration T of geometric objects in R ¹, and a class S of allowable sectioning lines we consider in this paper many variations on the following problem: ‘Is there a line S∈S such that the section S∩C is equivalent by rigid motion to the target T?’     

Some properties of basis invariants of the symmetry groupsG(m,p,n), B n m,D n m

We study the properties of the basis invariants of the symmetry groups of the complex polytope and the generalized n-cube γ _n ^m , as well as its subgroups D _n ^m . We give an explicit construction of all the basis invariants of even degree of these groups. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

On Subspace Lattices I. Closedness Type Properties and Tensor Products

We study properties of subspace lattices related to the continuity of the map Lat and the notion of reflexivity. We characterize various “closedness” properties in different ways and give the hierarchy between them. We investigate several properties related to tensor products of subspace lattices and show that the tensor product of the projection lattices of two von Neumann algebras, one of which is injective, is reflexive.     

C n and D n Very-Well-Poised 10 φ 9 Transformations

In this paper we derive multivariable generalizations of Bailey's classical terminating balanced very-well-poised _{10 9} transformation. We work in the setting of multiple basic hypergeometric series very-well-poised on the root systems A _n , C _n , and D _n . Following the distillation of Bailey's ideas by Gasper and Rahman [11], we use a suitable interchange of multisums. We obtain C _n and D _{n 10 9} transformations combined with A _n , C _n , and D _n extensions of Jackson's _{8 7} summation. Milne and Newcomb have previously obtained an analogous formula for A _n series. Special cases of our _{10 9} transformations include several new multivariable generalizations of Watson's transformation of an _{8 7} into a multiple of a _{4 3} series. We also deduce multidimensional extensions of Sears' _{4 3} transformation formula, the second iterate of Heine's transformation, the q -Gauss summation theorem, and of the q -binomial theorem. August 28, 1996. Date revised: September 8, 1997.     

Gauss–Man i n Conne ction s for Arrangements, I E igenvalues

We construct a formal connection on the Aomoto complex of an arrangement of hyperplanes, and use it to study the Gauss–Manin connection for the moduli space of the arrangement in the cohomology of a complex rank one local system. We prove that the eigenvalues of the Gauss–Manin connection are integral linear combinations of the weights which define the local system.     

D n -normal semigroups of transformations

Given a subgroup G of the symmetric group S _n on n letters, a semigroup S of transformations of X _n is G-normal if G _S =G, where G _S consists of all permutations h∈S _n such that h ⁻¹ fh∈S for all f∈S. A semigroup S is G-normax if it is a maximal semigroup in the set of all G-normal semigroups. In 1996, I. Levi showed that the alternating group A _n can not serve as the group G _S for any semigroup of total transformations of X _n . In 2000 and 2001, I. Levi, D.B. McAlister and R.B. McFadden described all A _n -normal semigroups of partial transformations of X _n . Also, in 1994, I. Levi and R.B. McFadden described all S _n -normal semigroups. In this paper, we show that the dihedral group D _n may serve as the group G _S for semigroups of transformations of X _n . We characterize a large class of D _n -normax semigroups and describe certain D _n -normal semigroups.     

Arrangements and Ranking Patterns

In the unidimensional unfolding model, given m objects in general position on the real line, there arise 1 + m(m − 1)/2 rankings. The set of rankings is called the ranking pattern of the m given objects. Change of the position of these m objects results in change of the ranking pattern. In this paper we use arrangement theory to determine the number of ranking patterns theoretically for all m and numerically for m ≤ 8. We also consider the probability of the occurrence of each ranking pattern when the objects are randomly chosen. Received March 5, 2005