  We formulate a conjecture for the second generation characters of
indecomposable tilting modules for ${\rm SL}_3$. This gives many new
conjectural decomposition numbers for symmetric groups. Our conjecture can be
interpreted as saying that these characters are governed by a discrete
dynamical system ("billiards bouncing in alcoves"). The conjecture implies that
decomposition numbers for symmetric groups display (at least) exponential
growth.
