The smallest singular values of the icosahedral group

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 $H_{3}$.