Beyond Rouquier partitions

Abstract

We obtain closed formulas, in terms of Littlewood-Richardson coefficients, for the canonical basis elements of the Fock space representation of $U_v({\widehat{\mathfrak{sl}}}_e)$ which are labelled by partitions having 'locally small' $e$-quotients and arbitrary $e$-cores. We further show that, upon evaluation at $v=1$, this gives the corresponding decomposition numbers of the $q$-Schur algebras in characteristic $l$ (where $q$ is a primitive $e$-th root of unity if $l\ne e$ and $q=1$ otherwise) whenever $l$ is greater than the size of each constituent of the $e$-quotient.

17 pages. This replaces the earlier version entitled 'Some decomposition numbers of q-Shcur algebras