The Carter–Payne homomorphism conjecture for Specht modules

Let $\lambda$ and $\mu$ be partitions of $n$ forming a Carter–Payne pair with parameters $(a,z,\gamma)$, and assume $a=1$, with $z$ the length of $\lambda$. Let $\nu$ be the partition of $n+\gamma$ defined by

$$\nu_r=\begin{cases} \lambda_r+\gamma,&r=1,\\ \lambda_r,&\text{otherwise}, \end{cases}$$

write $\nu=(\nu_1^{b_1},\nu_2^{b_2},\ldots,\nu_s^{b_s})$ with $\nu_1>\nu_2>\cdots>\nu_s>0$, and set $B_i=\sum_{k=1}^i b_k$ and $c_r=\nu_r-B_r$. Define

$$L_{\lambda\mu}=\prod_{r=1}^{s-1}\prod_{j=1}^{\gamma}\bigl(L_{n+j}-[c_r]\bigr).$$

The Carter–Payne homomorphism conjecture. If $\gamma<e$, then right multiplication by $L_{\lambda\mu}$ induces a non-zero $\mathscr{H}_n$-homomorphism

$$S^\lambda\longrightarrow S^\mu.$$

The preceding construction already gives an $\mathscr{H}_n$-homomorphism, but its non-vanishing is not clear in general. The claim extends the known case $\gamma=1$, for which non-zero homomorphisms can be obtained using the stated corollary or the work of Ellers and Murray.