On bases of some simple modules of symmetric groups and Hecke algebras

Abstract

We consider simple modules for a Hecke algebra with a parameter of quantum characteristic $e$. Equivalently, we consider simple modules $D^{\lambda }$, labelled by $e$-restricted partitions $\lambda$ of $n$, for a cyclotomic KLR algebra $R_n^{{\Lambda }_0}$ over a field of characteristic $p\ge 0$, with mild restrictions on $p$. If all parts of $\lambda$ are at most $2$, we identify a set ${\mathsf{DStd}}_{e,p}(\lambda )$ of standard $\lambda$-tableaux, which is defined combinatorially and naturally labels a basis of $D^{\lambda }$. In particular, we prove that the $q$-character of $D^{\lambda }$ can be described in terms of ${\mathsf{DStd}}_{e,p}(\lambda )$. We show that a certain natural approach to constructing a basis of an arbitrary $D^{\lambda }$ does not work in general, giving a counterexample to a conjecture of Mathas.

Final version, to appear in Transform. Groups

References in corpus (3)

• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Graded decomposition numbers for cyclotomic Hecke algebras
• Generalised column removal for graded homomorphisms between Specht modules

Cited by in corpus (1)

• Specht modules labelled by hook bipartitions I