# Branching rule for Specht modules over Kazhdan-Lusztig basis

Suppose $$\lambda\vdash n$$ is a partition and $$S^\lambda$$ is the associated irreducible representation of $$S_n$$. As we know from the branching rule, we have an isomorphism of $$S_{n-1}$$ modules $$S^\lambda\cong\bigoplus_{\mu}S^{\mu}$$ where $$\mu$$ are the partitions obtained by deleting an outer corner from $$\lambda$$.

In particular, I am concerned with the case when $$S^\lambda$$ is spanned by basis elements $$\{B_Q\mid Q\in\text{SYT}(\lambda)\}$$ of the Kazhdan-Lusztig basis (with the relevant left-action of the generators). Can the isomorphism from the branching rule be realised in such a way that preserves the KL-basis in some form? That is, if $$\phi$$ is a realisation of the isomorphism, then $$\phi\cdot B_Q=\sum_{\mu}\sum_{P\in\text{SYT}(\mu)}z_pB_P$$ for some constants $$z_P$$. What can we say about these constants $$z_P$$?

For example, when $$\lambda$$ is a rectangular partition (and so $$S^\lambda\cong S^\mu$$), then the isomorphism can be realised as $$B_Q\mapsto B_{d(Q)}$$, where $$d(Q)$$ is the tableau obtained by deleting the $$n$$-box from $$Q$$. I am unsure as to the theory for arbitrary partitions.

Any help would be appreciated!