Thrall's problem for two rows

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.