Wavefronts and reflection groups

O. P. Shcherbak

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 43 (1988) no. 3, pp. 149-194

Résumé

Typical singularities of wave fronts and ray systems passing by smooth obstacle in $3$-space are described in the article. These singularities turn out to be connected with noncristallographic Coxeter groups $I_2(5)$, $H_3$, $H_4$. Proofs are based on the detail in­vestigation of the discriminants of these groups by their inclusion into cristallographic ones $A_4$, $D_6$, $E_8$ correspondently. Besides, there is given a geometrical description of some singularities of bicaustics in collisionless flows of particles. It is based on inclu­sions of Coxeter groups $A_1^\mu$, $D_\mu$, $D_4$ into $B_\mu$, $G_\mu$, $F_4$ as normal subgroups. The article contains a wide table matherial on neutral stratification of discriminants of reflection groups. 32 refs.