Families of Symmetries and the Hydrogen Atom

Symmetries of the hydrogen atom and algebraic families

We show how the Schrödinger equation for the hydrogen atom in two dimensions gives rise to an algebraic family of Harish-Chandra pairs that codifies hidden symmetries. The hidden symmetries vary continuously between $SO(3)$, $SO(2,1)$ and the Euclidean group $O(2)\ltimes \mathbb{R}^2$. We show that solutions of the Schrödinger equation may be organized into an algebraic family of Harish-Chandra modules. Furthermore, we use Jantzen filtration techniques to algebraically recover the spectrum of the Schrödinger operator. This is a first application to physics of the algebraic families of Harish-C…

The equivariant Ehrhart theory of polytopes with order-two symmetries

We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant $H^*$-polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycles and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period …

双收缩与代数族

Dual contractions and algebraic families

本文研究对称李代数的双收缩，通过定义复化中的双实形式，将原收缩与其双收缩作为单个代数族的实纤维，建立几何框架。