The maximal tori theorem and the mackey machinery (2020)
The objective of this project is to document two of the results used in the study of representations of compact Lie groups, these are: The Maximal Tori Theoremand Mackey's Normal Subgroup Analysis or machinery. The first of these results establishes that given a compact Lie group G, every element of G is contained in a maximal torus and that the maximal tori are conjugated to each other. Given T a maximal torus with normalizer N, the quotient group W = N / T is known asthe Weyl group. A consequence of the maximal tori theorem is that the invariantfunctions under conjugation of G are in correspondence with the torus functions invariant under the action of W. This allows us to understand the characters of the representations of G, and, consequently, gives us information about the representations themselves. On the other hand, the second result gives us a way to calculate explicitly the decomposition...