The vibrational modes of simplicial molecules

TL;DR

This paper uses representation theory of the symmetric group to analyze vibrational modes of a simplicial molecule modeled as point masses at simplex vertices connected by springs.

Contribution

It introduces a novel application of symmetric group representation theory to compute vibrational modes of simplicial molecules.

Findings

Derived explicit vibrational mode spectra for simplicial molecules.

Identified symmetry-based degeneracies in vibrational frequencies.

Provided a framework for analyzing similar molecular structures.

Abstract

Consider a "simplicial molecule": $n$ equal point masses placed at the vertices of a regular $(n-1)$-simplex, connected by ${n \choose 2}$ identical springs. We apply the representation theory of the symmetric group $S_n$ to compute its vibrational modes.