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

## TL;DR

This paper proposes a conjecture linking tilting module characters for SL_3 to a billiards-like dynamical system, predicting exponential growth in symmetric group decomposition numbers.

## Contribution

It introduces a novel conjecture connecting algebraic characters to a billiards model, providing new insights into decomposition number growth.

## Key findings

- Conjecture relates tilting characters to billiard dynamics in alcoves.
- Predicts exponential growth in symmetric group decomposition numbers.
- Provides new conjectural decomposition numbers for symmetric groups.

## Abstract

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.