The smallest singular values of the icosahedral group (original) (raw)

For any finite reflection group W on R N and any irreducible Wmodule V there is a space of polynomials on R N with values in V . There are Dunkl operators parametrized by a multiplicity function, that is, parameters asscociated with each conjugacy class of reflections. For cerfain 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 .