Let $\mu$ be a partition of a given number $n\in\mathbb{N}$. Denote $S^\mu_p$ as the Specht module with respect to $\mu$ over a given field $F$ with char$(F)=p$, where $p\neq0$ is a prime number. Let $\mu'$ be the conjugate partition of $\mu$.
I am interested in the question:
"If $\mu'\neq\mu$ and $p>2$, is it possible that $S^{\mu'}_p$ and $S^\mu_p$ have the same composition factors?"
Note that a property says that $(S^{\mu'}_p)^*\cong S^\mu_p\otimes S^{(1^n)}_p$, where $(S^{\mu'}_p)^*$ is the dual of $S^{\mu'}_p$, and $(1^n)$ denotes the partition $(1,1,\cdots,1)$ of $n$ (for example, $(1^5)=(1,1,1,1,1)$). So my question is equivalent to:
"If $\mu'\neq\mu$ and $p>2$, is it possible that $S^\mu_p\otimes S^{(1^n)}_p$ and $S^\mu_p$ have the same composition factors?"
It is clear that they have the same composition factors when $p=2$. How about $p>2$? I guess they can not equal when $p>2$, but I have no idea to prove it.
Thank you for your suggestion!