Let’s look back over what we’ve done.
After laying down some definitions on reflections, we defined a root system $latex Phi$ as a collection of vectors with certain properties. Specifically, each vector is a point in a vector space, and it also gives us a reflection of the same vector space. Essentially, a root system is a finite collection of such vectors and corresponding reflections so that the reflections shuffle the vectors among each other. Our project was to classify these configurations.
The flip side of seeing a root system as a collection of vectors is seeing it as a collection of reflections, and these reflections generate a group of transformations called the Weyl group of the root system. It’s one of the most useful tools we have at our disposal through the rest of the project.
To get a perspective on the classification, we defined the category of…
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