Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 015, 22 pages      arXiv:1703.05898      https://doi.org/10.3842/SIGMA.2018.015
Contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics

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

George Lusztig a and Geordie Williamson b
a) Massachusetts Institute of Technology, Cambridge, MA, USA
b) Sydney University, Sydney, NSW, Australia

Received July 18, 2017, in final form February 16, 2018; Published online February 21, 2018

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.

Key words: tilting modules; billiards; $p$-canonical basis; symmetric group.

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