# Relationship between the Strong and Weak Exchange Property of Coxeter Groups

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 $$l(tw), then $$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!

• $l(sw)=l(w)$ is not possible. That's a consequence of the fact that all of the defining relators in Coxeter groups have even length. Commented Jun 13 at 7:36
• Thanks! I think you have made it clear to me. I have actually mixed up a bit with the properties. The $(EP)$ can be defined for an arbitrary group $W$ and a generating set $S\subset W$ where $s^2=e$ $\forall s\in S$. My problem should be proving that such a pair $(W,S)$ is a Coxeter system $\iff$ it satisfies $(EP)$. So to prove $(\implies)$, the case when $l(sw)=l(w)$ is eliminated as $(W,S)$ is assumed to be a Coxeter system. Commented Jun 14 at 14:44
• However, for an arbitrary group $W$ and such a generating set $S\subset W$ mentioned in my last comment, could anyone give some example when $l(sw)=l(w)$ can really happen? I have been struggling to find one. Thanks a lot! Commented Jun 14 at 14:47