$E$ is a vector space of dimension $n \geq 2$. $\mathbb{F}=(F_1,F_2,\dots,F_n)$ and $\mathbb{G}=(G_1,G_2,\dots,G_n)$ are two complete flags of $E$.

We say that $(\mathbb{F},\mathbb{G})$ is in position $\sigma \in \mathfrak{S}_n$ if and only if there exists a basis $(e_1,\dots,e_n)$ of $E$ such that for each $(i,j) \in [1,n]^2$, the sub-space $F_i$ is generated by $\{e_1,\dots,e_n\}$ and the sub-space $G_j$ by $\{e_{\sigma(1)},\dots,e_{\sigma(j)}\}$.

Suppose that $(\mathbb{F},\mathbb{G})$ is an ordered pair of complete flags in position $\sigma \in \mathfrak{S}_n$, that $k$ is an integer $1 \leq k \leq n-1$ and that $\sigma^{-1}(k) < \sigma^{-1}(k+1)$. $\mathbb{F^\prime}=(F^\prime_1,F^\prime_2,\dots,F^\prime_n)$ is another complete flag differing from $\mathbb{F}$ only for the sub-space of index $k$, i.e. $F_i=F^\prime_i$ for $i \in [1,n] \setminus \{k\}$ and $F_k \neq F^\prime_k$.

How can we prove that $(\mathbb{F^\prime},\mathbb{G})$ is in position $\tau_k \circ \sigma$ where $\tau_k$ is the transposition which swaps $k$ and $k+1$?

I processed the case $n=2$, $k=1$... but I'm not able to extrapolate to the general case.

$E$ is a vector space of dimension $n \geq 2$. $\mathbb{F}=(F_1,F_2,\dots,F_n)$ and $\mathbb{G}=(G_1,G_2,\dots,G_n)$ are two complete flags of $E$.

We say that $(\mathbb{F},\mathbb{G})$ is in position $\sigma \in \mathfrak{S}_n$ if and only if there exists a basis $(e_1,\dots,e_n)$ of $E$ such that for each $(i,j) \in [1,n]^2$, the sub-space $F_i$ is generated by $\{e_1,\dots,e_i\}$ and the sub-space $G_j$ by $\{e_{\sigma(1)},\dots,e_{\sigma(j)}\}$.

Suppose that $(\mathbb{F},\mathbb{G})$ is an ordered pair of complete flags in position $\sigma \in \mathfrak{S}_n$, that $k$ is an integer $1 \leq k \leq n-1$ and that $\sigma^{-1}(k) < \sigma^{-1}(k+1)$. $\mathbb{F^\prime}=(F^\prime_1,F^\prime_2,\dots,F^\prime_n)$ is another complete flag differing from $\mathbb{F}$ only for the sub-space of index $k$, i.e. $F_i=F^\prime_i$ for $i \in [1,n] \setminus \{k\}$ and $F_k \neq F^\prime_k$.

How can we prove that $(\mathbb{F^\prime},\mathbb{G})$ is in position $\tau_k \circ \sigma$ where $\tau_k$ is the transposition which swaps $k$ and $k+1$?

I processed the case $n=2$, $k=1$... but I'm not able to extrapolate to the general case.

$E$ is a vector space of dimension $n \geq 2$. $\mathbb{F}=(F_1,F_2,\dots,F_n)$ and $\mathbb{G}=(G_1,G_2,\dots,G_n)$ are two complete flags of $E$.

We say that $\mathbb{F}=(F_1,F_2,\dots,F_n)$, $\mathbb{G}=(G_1,G_2,\dots,G_n)$ are in position $\sigma \in \mathfrak{S}_n$ if and only if there exists a basis $(e_1,\dots,e_n)$ of $E$ such that for each $(i,j) \in [1,n]^2$, the sub-space $F_i$ is generated by $\{e_1,\dots,e_n\}$ and the sub-space $G_j$ by $\{e_{\sigma(1)},\dots,e_{\sigma(j)}\}$.

Suppose that $(\mathbb{F},\mathbb{G})$ is an ordered pair of complete flags in position $\sigma \in \mathfrak{S}_n$, that $k$ is an integer $1 \leq k \leq n-1$ and that $\sigma^{-1}(k) < \sigma^{-1}(k+1)$. $\mathbb{F^\prime}=(F^\prime_1,F^\prime_2,\dots,F^\prime_n)$ is another complete flag differing from $\mathbb{F}$ only for the sub-space of index $k$, i.e. $F_i=F^\prime_i$ for $i \in [1,n] \setminus \{k\}$ and $F_k \neq F^\prime_k$.

How can we prove that $(\mathbb{F^\prime},\mathbb{G})$ is in position $\tau_k \circ \sigma$ where $\tau_k$ is the transposition which swaps $k$ and $k+1$?

$E$ is a vector space of dimension $n \geq 2$. $\mathbb{F}=(F_1,F_2,\dots,F_n)$ and $\mathbb{G}=(G_1,G_2,\dots,G_n)$ are two complete flags of $E$.

We say that $(\mathbb{F},\mathbb{G})$ is in position $\sigma \in \mathfrak{S}_n$ if and only if there exists a basis $(e_1,\dots,e_n)$ of $E$ such that for each $(i,j) \in [1,n]^2$, the sub-space $F_i$ is generated by $\{e_1,\dots,e_n\}$ and the sub-space $G_j$ by $\{e_{\sigma(1)},\dots,e_{\sigma(j)}\}$.

Suppose that $(\mathbb{F},\mathbb{G})$ is an ordered pair of complete flags in position $\sigma \in \mathfrak{S}_n$, that $k$ is an integer $1 \leq k \leq n-1$ and that $\sigma^{-1}(k) < \sigma^{-1}(k+1)$. $\mathbb{F^\prime}=(F^\prime_1,F^\prime_2,\dots,F^\prime_n)$ is another complete flag differing from $\mathbb{F}$ only for the sub-space of index $k$, i.e. $F_i=F^\prime_i$ for $i \in [1,n] \setminus \{k\}$ and $F_k \neq F^\prime_k$.

How can we prove that $(\mathbb{F^\prime},\mathbb{G})$ is in position $\tau_k \circ \sigma$ where $\tau_k$ is the transposition which swaps $k$ and $k+1$?

I processed the case $n=2$, $k=1$... but I'm not able to extrapolate to the general case.