Thrall's problem for two rows

TL;DR

This paper solves Thrall's problem for two-row partitions by providing a tableau-based formula for the Schur expansion of higher Lie module characters, extending to certain other shapes.

Contribution

It offers a tableau-theoretic description of the Schur expansion for higher Lie modules with two-row partitions, including hook shapes and partitions with specific part restrictions.

Findings

Provides a tableau description involving major index and spin-parity conditions.

Derives formulas for hook shapes and partitions with distinct parts.

Extends results to partitions with parts greater than 2 occurring at most twice.

Abstract

In this paper, we study Thrall's problem for the higher Lie modules $L_\lambda$. Our main result provides a tableau-theoretic description of the Schur expansion of the character of $L_\lambda$ when $\lambda$ has two rows, thereby solving Thrall's problem in this case. This formula is expressed in terms of standard Young tableaux with major index congruence conditions and a spin-parity condition defined through bijections with Yamanouchi domino tableaux. We also obtain tableau formulas for hook shapes and partitions with distinct parts, and these results extend to all partitions in which each part greater than $2$ occurs at most twice.