Geometric combinatorics of Weyl groupoids

Geometric combinatorics of Weyl groupoids

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of morphisms with fixed target object forms an ortho-complemented meet semilattice. We define the Coxeter complex of a Weyl groupoid with finite root system and show ...

Weyl Groupoids with at Most Three Objects

We adapt the generalization of root systems of the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we completely classify all finite Weyl groupoids with at most three objects. The classification yields that there exist infinitely many “standard”, but only 9 “except...

Lectures in Geometric Combinatorics

The fourteen lectures in this book were prepared for the advanced undergraduate course at the Park City Mathematics Institute on Geometric Combinatorics in July 2004. They begin with the basics of polytope theory with an emphasis on geometry via the theory of Schlegel and Gale diagrams. The lectures lead up to secondary and state polytopes arising from point configurations. These polytopes are ...

Topics in Geometric Combinatorics

Weyl Quantization from Geometric Quantization

In [23] a nice looking formula is conjectured for a deformed product of functions on a symplectic manifold in case it concerns a hermitian symmetric space of non-compact type. We derive such a formula for simply connected symmetric symplectic spaces using ideas from geometric quantization and prequantization of symplectic groupoids. We compute the result explicitly for the natural 2-dimensional...