Monday 28th of October 2024, 13:30-14:30 in HG02.052
Philip Schlösser
When are functions special?
Anyone who has had contact with physics for more than just introductory classical mechanics will have heard people talk about special functions. This begs the following question: what is a special function, and what makes them distinct from general functions? Come with me on this journey into the world of representation theory, differential equations, and special functions.
More serious abstract: Let \(G\) be a Lie group and let \(H\), \(A\) be closed subgroups such that \(A\) is Abelian and \(G=HAH\). Let \(χ_1\),\(χ_2\) be 1d representations of \(H\), then a function \(f\) on \(G\) is called \(χ\)-spherical if \(f(hgk)=χ_1(h)χ_2(k) f(g)\) for all \(g\in G\) and \(h,k \in H\). Examples of such functions are appropriate matrix elements of representations of \(G\). A lot of the functions physicists call special arise via this construction, and a lot of their properties can be proven using representation theory.