All linear symmetries of the SU(3) tensor multiplicities

The SU(3) tensor multiplicities are piecewise polynomial of degree 1 in their labels. The pieces are the chambers of a complex of cones. We describe in detail this chamber complex and determine the group of all linear symmetries (of order 144) for these tensor multiplicities. We represent the cells by diagrams showing clearly the inclusions as well as the actions of the group of symmetries and of its remarkable subgroups.