We show a method to effectively compute the Wedderburn decomposition and the primitive central idempotents of a semisimple finite group algebra of an abelian-by-supersolvable group G from certain pairs of subgroups of G. In this paper F = Fq denotes a finite field with q elements and G is a finite group of order n such that FG is semisimple, or equivalently (q, n) = 1. The group algebra FG is a...

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...

Let B be a subset of the set of all isomorphism classes of finite groups. We consider the number Fg(x) of positive integers n < x such that all groups of order n lie in B. When ir consists of the isomorphism classes of all finite groups of any of the following types, we obtain an asymptotic formula for Fn(x): cyclic groups, abelian groups, nilpotent groups, supersolvable groups, and solvable gr...

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 finite lattice. We then show how this result can be used to compute and combinatorially explain the Mo bius function in various examples including non-crossing set partitions, shuffle posets, and integer partitions in dominance order. Next we present a ...

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 Euler multiplicity, of a restricted multiarrangement. We compute the Euler multi...

We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, abelian groups, achieves same asymptotic worst-case query complexity as optimal randomized namely~$ \Order(\sqrt{ n}\, )$, where~$n$ is order of group. analogous algorithm non-abelian groups comes within a~$\sqrt{ \log n}$ factor complexity. best known Problem has \emph{expected\/} that sensitive to input,...