Billiards and Tilting Characters for $\mathrm{SL}_3$

We formulate a conjecture for the second generation characters of indecomposable tilting modules for $\mathrm{SL}_3$. 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.