The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the e-multiplicity, of a restricted multiarrangement. We compute the e-multiplicitie...

We show that the order complex of any finite lattice with a chain 0̂ < m1 < · · · < mr < 1̂ of modular elements is at least (r − 2)-connected. In [St], Stanley shows that a finite lattice L with a maximal chain consisting entirely of modular elements is supersolvable and therefore graded and EL-shellable in the sense of [Bj1]. Hence, the order complex ∆(L \ {0̂, 1̂}) has the homotopy type of a wedg...

In this paper we introduce new techniques for the eecient computation of a Fourier transform on a nite group. We use the decomposition of a group into double cosets and a graph theoretic indexing scheme to derive algorithms that generalize the Cooley-Tukey FFT to arbitrary nite groups. We apply our general results to special linear groups and low rank symmetric groups, and obtain new eecient al...

To every affine real arrangement of hyperplanes AR we associate a family of diagrams of spaces over the face poset of the arrangement. We show that any cover of the complement of the complexification of AR is homotopy equivalent to the homotopy colimit of one of the diagrams. More precisely, we show that any cover of the Salvetti complex is isomorphic to the order complex of the poset limit of ...

We produce a one-parameter family of hyperplane arrangements that are counterexamples to the conjecture of Saito that the complexified complement of a free arrangement is K(π , 1) . These arrangements are the restriction of a one-parameter family of arrangements that arose in the study of tilings of certain centrally symmetric octagons. This other family is discussed as well. I. Definitions and...

We introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any nite lattice. We then show how this result can be used to compute and combinatorially explain the Mobius function in various examples including non-crossing set partitions, shu e posets, and integer partitions in dominance order. Next we present a gene...

Tatiana Gateva-Ivanova, American University in BulgariaNoncommutative Grobner Bases in Skew-polynomial RingsWe study a class of graded standard nitely presented quadratic algebrasA over a xed eld K, called binomial skew polynomial rings. Followinga classical tradition (and recent trend), we take a combinatorial approachto study the algebraic properties of A. Our purpose...

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