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...

A positive integer n is called a CLT number if every group of order n satisfies the converse of Lagrange’s Theorem. In this note, we find all CLT and supersolvable numbers up to 1000. We also formulate some questions about the distribution of these numbers. RÉSUMÉ: Un nombre entier naturel n est appelé nombre CLT si tout groupe d’ordre n satisfait la réciproque du Théorme de Lagrange. Nous déte...

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 G be a simple graph on the vertex set {v1, . . . , vn} with edge set E. Let K be a field. The graphical arrangement AG in K n is the arrangement xi − xj = 0, vivj ∈ E. An arrangement A is supersolvable if the intersection lattice L(c(A)) of the cone c(A) contains a maximal chain of modular elements. The second author has shown that a graphical arrangement AG is supersolvable if and only if ...

We introduce a poset structure on the reduced galleries in a supersolvable arrangement of hyperplanes. In particular, for Coxeter groups of type A or B, we construct a poset of reduced words for the longest element whose Hasse diagram is the graph of reduced words. Using Rambau’s Suspension Lemma, we show that these posets are homotopy equivalent to spheres. We furthermore conjecture that its i...

A result of Roquette 3] states that if D is an absolutely irreducible representation of a p-group G over the eld of complex numbers, then D can be realized in K((g) j g 2 G), where is the character of D and K = Q or K = Q(i) according to whether p 6 = 2 or p = 2. Based on Baum and Clausen's 1] algorithm for computing the irreducible representations of supersolvable groups, we give an elementary...