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

Abstract

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.

References in corpus (4)

• Modular representation theory in type A via Soergel bimodules
• Free-monodromic mixed tilting sheaves on flag varieties
• A quantum version of the algebra of distributions of ${\mathrm{SL}}_2$
• On the character of certain modular irreducible representations

Cited by in corpus (4)

• Growth rates of the number of indecomposable summands in tensor powers
• Localized calculus for the Hecke category
• Fractal behavior of tensor powers of the two dimensional space in prime characteristic
• Correction of the Lusztig-Williamson Billiards Conjecture