Geometric group theory | Metric geometry

Weyl distance function

In combinatorial geometry, the Weyl distance function is a function that behaves in some ways like the distance function of a metric space, but instead of taking values in the positive real numbers, it takes values in a group of reflections, called the Weyl group (named for Hermann Weyl). This distance function is defined on the collection of chambers in a mathematical structure known as a building, and its value on a pair of chambers a minimal sequence of reflections (in the Weyl group) to go from one chamber to the other. An adjacent sequence of chambers in a building is known as a gallery, so the Weyl distance function is a way of encoding the information of a minimal gallery between two chambers. In particular, the number of reflections to go from one chamber to another coincides with the length of the minimal gallery between the two chambers, and so gives a natural metric (the gallery metric) on the building. According to , the Weyl distance function is something like a geometric vector: it encodes both the magnitude (distance) between two chambers of a building, as well as the direction between them. (Wikipedia).

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Related pages

Metric space | Empty product | Coxeter complex | Hermann Weyl | Building (mathematics) | Weyl group | Reflection (mathematics) | Group (mathematics)