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

[摘要] 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.

[关键词] tilting modules;billiards;p-canonical basis;symmetric group