Hamiltonian cycles in Cayley graphs of imprimitive complex reflection groups

Generalizing a result of Conway, Sloane, and Wilkes for real reflection groups, we show the Cayley graph of an imprimitive complex reflection group with respect to standard generating reflections has a Hamiltonian cycle. This is consistent with the long-standing conjecture that for every finite group, G, and every set of generators, S, of G the undirected Cayley graph of G with respect to S has a Hamiltonian cycle.Comment: 15 pages, 4 figures; minor revisions according to referee comments, to appear in Discrete Mathematic

Dominant regions in noncrystallographic hyperplane arrangements

For a crystallographic root system, dominant regions in the Catalan hyperplane arrangement are in bijection with antichains in a partial order on the positive roots. For a noncrystallographic root system, the analogous arrangement and regions have importance in the representation theory of an associated graded Hecke algebra. Since there is also an analogous root order, it is natural to hope that a similar bijection can be used to understand these regions. We show that such a bijection does hold for type $H_3$ and for type $I_2(m)$, including arbitrary ratio of root lengths when $m$ is even, but does not hold for type $H_4$. We give a criterion that explains this failure and a list of the 16 antichains in the $H_4$ root order which correspond to empty regions.Comment: 29 pages, 5 figure

Geometry of the Prytz Planimeter

The Prytz planimeter is a simple example of a system governed by a non-holonomic constraint. It is unique among planimeters in that it measures something more subtle than area, combining the area, centroid and other moments of the region being measured, with weights depending on the length of the planimeter. As a tool for measuring area, it is most accurate for regions that are small relative to its length. The configuration space of the planimeter is a non-principal circle bundle acted on by SU(1,1), (isom. to SL(2,R)). The motion of the planimeter is realized as parallel translation for a connection on this bundle and for a connection on a principal SU(1,1)-bundle. The holonomy group is SU(1,1). As a consequence, the planimeter is an example of a system with a phase shift on the circle that is not a simple rotation. There is a qualitative difference in the holonomy when tracing large regions as opposed to small ones. Generic elements of SU(1,1) act on S^1 with two fixed points or with no fixed points. When tracing small regions, the holonomy acts without fixed points. Menzin's conjecture states (roughly) that if a planimeter of length L traces the boundary of a region with area A > pi L^2, then it exhibits an asymptotic behavior and the holonomy acts with two fixed points, one attracting and one repelling. This is obvious if the region is a disk, and intuitively plausible if the region is convex and A >> pi L^2. A proof of this conjecture is given for a special case, and the conjecture is shown to imply the isoperimetric inequality.Comment: AmS-TeX, 23 pages, 12 figures in 2 *.gif files. To appear in Reports on Mathematical Physics. Part of proceedings of Workshop on Non-holonomic Constraints in Dynamics, Univ. of Calgary, Aug. 199

Parabolically induced representations of graded Hecke algebras

We study the representation theory of graded Hecke algebras, starting from scratch and focusing on representations that are obtained with induction from a discrete series representation of a parabolic subalgebra. We determine all intertwining operators between such parabolically induced representations, and use them to parametrize the irreducible representations.Comment: In the second version several new results have been added to prove some claims from the last page of the first version. In the third version the introduction has been extended and we determine the global dimension of a graded Hecke algebr