Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal comp...

We present an algorithm to compute a full set of irreducible representations of a supersolvable group G over a finite field K, charK |G|, which is not assumed to be a splitting field of G. The main subroutines of our algorithm are a modification of the algorithm of Baum and Clausen (Math. Comp. 63 (1994), 351–359) to obtain information on algebraically conjugate representations, and an effectiv...

The classical superconformal actions of branes in adS superspaces have a closed form depending on a matrix M quadratic in fermions, as found in hep-th/9805217. One can gauge-fix the local κ-symmetry using the Killing spinors of the brane in the bulk. We show that in such gauges the superconformal actions are simplified dramatically since M = 0 in all cases. The relation between classical and ga...

For any finite Coxeter group W of rank n we show that the order complex of the lattice of non-crossing partitions NC(W ) embeds as a chamber subcomplex into a spherical building of type An−1. We use this to give a new proof of the fact that the non-crossing partition lattice in type An is supersolvable for all n. Moreover, we show that in case Bn, this is only the case if n < 4. We also obtain ...

Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K * M . We call the group M ∩ K/L a c-section of M . And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M . For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L2(p) or Zq, where p and q are primes, ...

in this paper we find systems of subgroups of a finite‎ ‎group‎, ‎which $bbb p$nobreakdash-hspace{0pt}subnormality guarantees supersolvability‎ ‎of the whole group‎.

For a hyperplane arrangement in real vector space, the coefficients of its Poincaré polynomial have many interpretations. An interesting one is provided by Varchenko-Gel'fand ring, which ring functions from chambers to integers with pointwise addition and multiplication. Varchenko Gel'fand gave simple presentation for this along filtration associated graded whose Hilbert series polynomial. We g...

We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and pq, for prime numbers p, q with q|p − 1. We also show that certain twisting deformation of the symmetric group is simple as a Hopf algebra. On the other hand, we p...