A crystal structure on the Gross-Hacking-Keel-Kontsevich-basis for SL_n(C)

Mardi, 12. février 2019 - 14:00 - 15:00

Volker Genz (Bochum)


The unipotent radical $U$ of a Borel subgroup of $SL_n(C)$ possesses the structure of a Cluster Variety. Gross-Hacking-Keel-Kontsevich constructed a canonical basis $B$ of the coordinate ring $C[U]$ parametrized by tropical points of the mirror dual Cluster Variety. We introduce on $B$ two crystal structures in the sense of Kashiwara. Extending the Donaldson-Thomas transformation to the frozen vertices we show - up to a conjecture of Goncharov-Shen - that both crystal structures coincide.