Transitions between electron-molecule states in electrostatic quantum dots

Intermediate spin states that occur in electrostatic dots in the magnetic field regime just beyond the maximum density droplet are investigated. The 5-electron system is studied with exact diagonalization and group theory. The results indicate that the intermediate spin states are mixed symmetry states with a superposition of 5- and 4-fold electron-molecule configurations. A superposition of 5- and 4-fold correlation functions is found to reproduce the exact mixed symmetry pair correlation function to around 2%.Comment: 4 pages, 3 figure

GIT semistability of Hilbert points of Milnor algebras

Our first result is that a homogeneous form $F$ in $n$ variables is GIT semistable with respect to the natural $SL(n)$-action if and only if the first non-trivial Hilbert point of the associated Milnor algebra is semistable. We also prove that the induced morphism on the GIT quotients is finite, and injective on the locus of stable forms. Our second result is that the associated form of $F$, also known as the Macaulay inverse system of the Milnor algebra of $F$, and which is apolar to the last non-trivial Hilbert point of the Milnor algebra, is GIT semistable whenever $F$ is a smooth form. These two results answer questions of Alper and Isaev from arXiv:1407.6838.Comment: 19 pages, to appear in Math. An