The smallest singular values of the icosahedral group

TL;DR

This paper investigates the smallest singular values of the icosahedral group within the context of Dunkl operators, revealing unique properties and determining positivity intervals for its irreducible representations.

Contribution

It provides a detailed analysis of the smallest singular values for the icosahedral group, highlighting differences from symmetric groups and explicitly determining positivity intervals for all irreducible representations.

Findings

The positivity interval for the Gaussian form is explicitly determined for each irreducible representation of the icosahedral group.

The symmetric property of the interval around zero, observed in symmetric groups, does not hold for the icosahedral group.

Counterexamples are provided showing the non-symmetry of the positivity interval in the icosahedral case.

Abstract

For any finite reflection group $W$ on $\mathbb{R}^{N}$ and any irreducible $W$-module $V$ there is a space of polynomials on $\mathbb{R}^{N}$ with values in $V$. There are Dunkl operators parametrized by a multiplicity function, that is, parameters associated with each conjugacy class of reflections. For certain parameter values, called singular, there are nonconstant polynomials annihilated by each Dunkl operator. There is a Gaussian bilinear form on the polynomials which is positive for an open set of parameter values containing the origin. When $W$ has just one class of reflections and $\dim V>1$ this set is an interval bounded by the positive and negative singular values of respective smallest absolute value. This interval is always symmetric around $0$ for the symmetric groups. This property does not hold in general, and the icosahedral group $H_{3}$ provides a counterexample. The interval for positivity of the Gaussian form is determined for each of the ten irreducible representations of $H_{3}$.