Super major index and Thrall's problem

TL;DR

This paper extends Thrall's problem to free Lie superalgebras, generalizing key results and introducing a new super tableau major index linked to Macdonald's $q,t$-hook formula.

Contribution

It generalizes Thrall's problem to superalgebras and proves extensions of three known results, including a new super tableau major index and its connection to Macdonald polynomials.

Findings

Extension of Brandt's formula to superalgebras

Identification of Schur--Weyl dual in the super setting

Introduction of a super tableau major index linked to Macdonald's formula

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.