We give an explicit graded cellular basis of the sl3-web algebra K_S. In order to do this, we identify Kuperberg's basis for the sl3-web space W_S with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weight of flows on webs and the q-degree of foams. We use this to give a “foamy” version of Hu and Mathas graded cellular basis of the cyclotomic Hecke algebra which turns out to be a graded cellular basis of the sl3-web algebra K_S. We restrict ourselves to the sl3 case here, but our approach should, up to the combinatorics of sln-webs, work for all n>1.
Journal of Algebraic Combinatorics, 2014, Vol 40, Issue 4, p. 1001-1076