I am a beginner at studying Coxeter group theory, and I am confused with the strong and weak exchange property when reading the book "Combinatorics of Coxeter Groups". Let me first state the properties below:
Let $(W,S)$ be a Coxeter group (or called a Coxeter system).
We define $T=\{wsw^{-1}\mid w\in W, s\in S\}$ and we denote $l(w)$ as the length of a word $w\in W$.
The Strong Exchange Property (SEP) is: (ref. Theorem 1.4.3)
Suppose $w=s_1\cdots s_k\in W$, $(s_i\in S)$ and $t\in T$.
If $l(tw)<l(w)$, then $tw=s_1\cdots\hat{s_i}\cdots s_k$ for some $i\in\{1,\dots,k\}$
The (Weak) Exchange Property (EP) is: (ref. Theorem 1.5.1)
Suppose $w=s_1\cdots s_k\in W$ is a reduced expression and $s\in S$.
If $l(sw)\leq l(w)$, then $sw=s_1\cdots\hat{s_i}\cdots s_k$ for some $i\in\{1,\dots,k\}$
I am quite sure, and the writer also says, that $(SEP)\implies (EP)$. By the definition of $T$, we can see that $S\subset T$, so the statement in $(SEP)$ holds for $t\in S$ as well. However, the writer stressed the possibility of $l(sw)=l(w)$, while this case is not included in $(SEP)$, and this is where I have trouble.
Furthermore, as I am reading different references online, I see that some writers use "$\leq$" in both $(SEP)$ and $(EP)$, and the two properties end up being equivalent. May I know which one is more natural?
Any help is much appreciated!
