Let $w \in S_n$ and $inv(w) = \{(i,j): i,j \in \{1,\ldots,n\}, i<j, w(i)>w(j)\}$ the inversion set of $w$. Let ${\bf i}=(i_1,\ldots,i_m)$ be a sequence such that $s_{i_1}\cdots s_{i_m}$ is a reduced expression of $w$. There is a bijection $f_{\bf i}: \{1,\ldots,m\} \to inv(w)$ given by $f_{\bf i}(k) = s_{i_1}s_{i_2} \cdots s_{i_k} \cdots s_{i_2}s_{i_1}$.

Theorem: Let $w \in S_n$. Suppose that $i<j<k$, $(i,j),(i,k),(j,k) \in inv(w)$, $i,j,k \in \{1, \ldots, n\}$, $inv(w)=m$, ${\bf i} = (i_1, \ldots, i_m)$, $s_{i_1} \cdots s_{i_m}$ is a reduced expression of $w$. Then there is some $r \in \{1,\ldots,m-2\}$ such that $f_{\bf i}(r) = (i,j)$, $f_{\bf i}(r+1) = (i,k)$, $f_{\bf i}(r+2) = (j,k)$.

Are there some reference about this theorem? Thank you very much.

Let $w \in S_n$ and $inv(w) = \{(i,j): i,j \in \{1,\ldots,n\}, i<j, w(i)>w(j)\}$ the inversion set of $w$. Let ${\bf i}=(i_1,\ldots,i_m)$ be a sequence such that $s_{i_1}\cdots s_{i_m}$ is a reduced expression of $w$. There is a bijection $f_{\bf i}: \{1,\ldots,m\} \to inv(w)$ given by $f_{\bf i}(k) = s_{i_1}s_{i_2} \cdots s_{i_k} \cdots s_{i_2}s_{i_1}$.

Theorem: Let $w \in S_n$. Suppose that $i<j<k$, $(i,j),(i,k),(j,k) \in inv(w)$, $i,j,k \in \{1, \ldots, n\}$, $inv(w)=m$, ${\bf i} = (i_1, \ldots, i_m)$, $s_{i_1} \cdots s_{i_m}$ is a reduced expression of $w$. Then there is some $r \in \{1,\ldots,m-2\}$ such that $f_{\bf i}(r) = (i,j)$, $f_{\bf i}(r+1) = (i,k)$, $f_{\bf i}(r+2) = (j,k)$.

Theorem (corrected version): Let $w \in S_n$. Suppose that $i<j<k$, $(i,j),(i,k),(j,k) \in inv(w)$, $i,j,k \in \{1, \ldots, n\}$, $inv(w)=m$. Then there is a reduced expression $s_{i_1} \cdots s_{i_m}$ of $w$ and some $r \in \{1,\ldots,m-2\}$ such that $f_{\bf i}(r) = (i,j)$, $f_{\bf i}(r+1) = (i,k)$, $f_{\bf i}(r+2) = (j,k)$, where ${\bf i} = (i_1, \ldots, i_m)$.

Are there some reference about this theorem? Thank you very much.

Let $w \in S_n$ and $inv(w) = \{(i,j): i,j \in \{1,\ldots,n\}, i<j, w(i)>w(j)\}$ the inversion set of $w$. Let ${\bf i}=(i_1,\ldots,i_m)$ be a sequence such that $s_{i_1}\cdots s_{i_m}$ is a reduced expression of $w$. There is a bijection $f_{\bf i}: \{1,\ldots,m\} \to inv(w)$ given by $f_{\bf i}(k) = s_{i_1}s_{i_2} \cdots s_{i_k} \cdots s_{i_2}s_{i_1}$.

Theorem: Suppose that $i<j<k$, $i,j,k \in \{1, \ldots, n\}$, $w \in S_n$, $inv(w)=m$, ${\bf i} = (i_1, \ldots, i_m)$, $s_{i_1} \cdots s_{i_m}$ is a reduced expression of $w$. Then there is some $r \in \{1,\ldots,m-2\}$ such that $f_{\bf i}(r) = (i,j)$, $f_{\bf i}(r+1) = (i,k)$, $f_{\bf i}(r) = (j,k)$.

Are there some reference about this theorem? Thank you very much.

Let $w \in S_n$ and $inv(w) = \{(i,j): i,j \in \{1,\ldots,n\}, i<j, w(i)>w(j)\}$ the inversion set of $w$. Let ${\bf i}=(i_1,\ldots,i_m)$ be a sequence such that $s_{i_1}\cdots s_{i_m}$ is a reduced expression of $w$. There is a bijection $f_{\bf i}: \{1,\ldots,m\} \to inv(w)$ given by $f_{\bf i}(k) = s_{i_1}s_{i_2} \cdots s_{i_k} \cdots s_{i_2}s_{i_1}$.

Theorem: Let $w \in S_n$. Suppose that $i<j<k$, $(i,j),(i,k),(j,k) \in inv(w)$, $i,j,k \in \{1, \ldots, n\}$, $inv(w)=m$, ${\bf i} = (i_1, \ldots, i_m)$, $s_{i_1} \cdots s_{i_m}$ is a reduced expression of $w$. Then there is some $r \in \{1,\ldots,m-2\}$ such that $f_{\bf i}(r) = (i,j)$, $f_{\bf i}(r+1) = (i,k)$, $f_{\bf i}(r+2) = (j,k)$.

Are there some reference about this theorem? Thank you very much.