Affine Coxeter groups have a natural presentation as reflection groups on some affine space. Hence the set R of all its reflections, that is all conjugates of its standard generators, is a natural (infinite) set of generators. Computing the reflection length of an element in an affine Coxeter group means that one wants to determine the length of a minimal presentation of this element with respect to R. In joint work with Joel Brewster Lewis, Jon McCammond and T. Kyle Petersen we were able to provide a simple formula that computes the reflection length of any element in any affine Coxeter group. In this talk I would like to explain this formula, give its simple uniform proof and allude to the geometric intuition behind it.
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Talks 2018
<< 20172 entries found
 January

25. January17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraPetra Schwer, KIT Karlsruhe: Reflection length in affine Coxeter groups

18. January17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraFlorian Eisele, University of London: A counterexample to the first Zassenhaus conjecture
There are many interesting problems surrounding the unit group U(RG) of the ring RG, where R is a commutative ring and G is a finite group. Of particular interest are the finite subgroups of U(RG). In the seventies, Zassenhaus conjectured that any u in U(ZG) is conjugate, in the group U(QG), to an element of the form +/g, where g is an element of the group G. This came to be known as the "Zassenhaus conjecture". In recent joint work with L. Margolis, we were able to construct a counterexample to this conjecture. In this talk I will give an introduction to the various conjectures surrounding finite subgroups of U(RG), and how they can be reinterpreted as questions on the (non)existence of certain R(GxH)modules, where H is another finite group. This establishes a link with the representation theory of finite groups, and I will explain how, plocally, our example is made up of certain ppermutation modules.