Topics in Enumerative Group Theory

This is a project at the crossroads of combinatorics and group theory. It consists of two parts: I. Number-Theoretic Aspects Of Hecke Groups And Their Lifts (with possible extension to other virtually free groups) and II. Non-crossing partitions for reflection groups. The first part concerns infinite groups (namely Hecke groups and their lifts, and possibly other finitely generated virtually free groups), the second part concerns finite groups (well-generated complex reflection groups). While there is no direct link between these two families of groups, there do exist several ties which bind these two project parts together. First of all, both parts address important enumerative questions in their respective class of target groups. Moreover, by the very nature of the problems considered, our investigations will require very similar techniques: a structural analysis of the combinatorial properties of these groups leads to objects such as coset diagrams, (combinatorial) maps, and other graph-like objects. To accomplish the enumeration of such objects, recurrence methods and generating function techniques, Lagrange inversion and other tools for solving systems of functional equations, Möbius inversion, etc. will most certainly be needed. Furthermore, the complexity of these problems makes it necessary to accompany, respectively complement, the corresponding considerations by tools from computer algebra. Thus, the solution of the problems we propose to study depends vitally on an interdisciplinary approach. The applicant, Thomas W. Müller (Queen Mary, University of London), and the co-applicant, Christian Krattenthaler (University of Vienna), bring in the complementary expertise necessary for the solution of the proposed problems: a combinatorial group theorist with particular knowledge of analytic, combinatorial, and number-theoretic aspects of groups, and an enumerative combinatorialist with a broad repertoire of combinatorial and analytic techniques, including the use of computer algebra. The power of this collaboration on enumerative questions arising in group and semigroup theory has already been demonstrated in previous work, see [26,27,28] in the list of references of the application. A two years visit as a Lise Meitner Fellow of the applicant would be highly stimulating for this collaboration, and, moreover, it would provide the opportunity to deepen the relations between the applicant and the combinatorics groups in Vienna.