Heun Function - Symmetries

Symmetries

Heun's equation has a group of symmetries of order 192, isomorphic to the Coxeter group of the Coxeter diagram D₄, analogous to the 24 symmetries of the hypergeometric differential equations obtained by Kummer. The symmetries fixing the local Heun function form a group of order 24 isomorphic to the symmetric group on 4 points, so there are 192/24 = 8 = 2 × 4 essentially different solutions given by acting on the local Heun function by these symmetries, which give solutions for each of the 2 exponents for each of the 4 singular points. The complete list of 192 symmetries was given by Maier (2007) using machine calculation. Several previous attempts by various authors to list these by hand contained many errors and omissions; for example, most of the 48 local solutions listed by Heun contain serious errors.