Geometrically distinct solutions given by symmetries of variational problems with the $O(N)$-symmetry

TL;DR

This paper explains a group-theoretic method for finding multiple geometrically distinct solutions in variational problems with $O(N)$-symmetry, showing that the number of solutions grows exponentially with dimension $N$.

Contribution

It provides a detailed correspondence between subgroup classes, orthogonal flags, and partitions of $N$, demonstrating exponential growth in solutions.

Findings

Number of solutions grows exponentially with $N$

Spaces invariant under different groups are linearly independent

Method improves upon previous solutions growth rates

Abstract

For variational problems with $O(N)$-symmetry the existence of several geometrically distinct solutions had been shown by use of group theoretic approach in previous articles. It was done by a crafty choice of a family $H_i \subset O(N)$ subgroups such that the fixed point subspaces $E^{H_i} \subset E$ of the action in a corresponding functional space are linearly independent, next restricting the problem to each $E^{H_i}$ and using the Palais symmetry principle. In this work we give a thorough explanation of this approach showing a correspondence between the equivalence classes of such subgroups, partial orthogonal flags in $\mathbb{R}^N$, and unordered partitions of the number $N$. By showing that spaces of functions invariant with respect to different classes of groups are linearly independent we prove that the amount of series of geometrically distinct solutions obtained in this way grows exponentially in $N$, in contrast to logarithmic, and linear growths of earlier papers.