Billiards and Tilting Characters for SL₃

Abstract

We formulate a conjecture for the second generation characters of indecomposable tilting modules for SL₃. 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.