Super major index and Thrall's problem

Abstract

Thrall's problem asks for the Schur decomposition of the higher Lie modules $\mathcal{L}_\lambda$, which are defined using the free Lie algebra and decompose the tensor algebra as a general linear group module. Although special cases have been solved, Thrall's problem remains open in general. We generalize Thrall's problem to the free Lie superalgebra, and prove extensions of three known results in this setting: Brandt's formula, Klyachko's identification of the Schur--Weyl dual of $\mathcal{L}_n$, and Kr{\'a}skiewicz--Weyman's formula for the Schur decomposition of $\mathcal{L}_n$. The latter involves a new version of the major index on super tableaux, which we show corresponds to a $q,t$-hook formula of Macdonald.