This article is cited in 11 papers

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

George Lusztig^a, Geordie Williamson<sup>b

a</sup> Massachusetts Institute of Technology, Cambridge, MA, USA
^b Sydney University, Sydney, NSW, Australia

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.

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

MSC: 20C20; 17B10; 20C30

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

Language: English

DOI: 10.3842/SIGMA.2018.015

Bibliographic databases:

• 
•