The graded Lascoux-Leclerc-Thibon conjecture
In type A, some new algebras introduced recently by Khovanov-Lauda and Rouquier give rise to remarkable Z-gradings on group algebras of symmetric groups and more generally on various cyclotomic Hecke algebras. There are also graded versions of Specht modules for these algebras. In recent work joint with Kleshchev we have computed the q-decomposition numbers of these graded Specht modules. The result, which gives a graded version of the Lascoux-Leclerc-Thibon conjecture, is best expressed in the language of categorification.