Proof of Lemma 26, p. 17 — permutations of S(n−1) are invariant under the partial transposition ′ over the last subsystem

Let $V(\tau)$ denote the permutation operator of $\tau\in S(n)$ on $(\mathbb C^d)^{\otimes n}$, and let $X'$ be the partial transpose of $X$ on the last system. Regard $S(n-1)$ as the permutations of $S(n)$ fixing the last system. Then for all $\rho,\rho'\in S(n-1)$ and $\tau\in S(n)$:

1. permutation operators of $S(n-1)$ are invariant under partial transposition,

2. they pass through the partial transposition,

This is the fact used in the proof of Lemma 26 to rewrite $\Pi_b V(f_{ab}^{-1}(\sigma_{ab}))V'(n\ n-1)\Pi_a^{-1}$ as $V'(\pi_b\circ f^{-1}_{ab}(\sigma_{ab})\circ(n\ n-1)\circ\pi_a^{-1})$ (eq. (82)); it is what makes $\mathcal A$ invariant under left and right multiplication by $S(n-1)$.

Formalization Note. The paper's $n$ is $k+2$ here; $S(n-1)$ embeds into $S(n)$ by liftPerm (k+1) 1. Statement (2) is the general form of the step in eq. (82); the case of eq. (82) is $\rho=\pi_b\circ f^{-1}_{ab}(\sigma_{ab})$, $\tau=(n\ n-1)$, $\rho'=\pi_a^{-1}$.