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!

  • $\begingroup$ $l(sw)=l(w)$ is not possible. That's a consequence of the fact that all of the defining relators in Coxeter groups have even length. $\endgroup$
    – Derek Holt
    Commented Jun 13 at 7:36
  • $\begingroup$ 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. $\endgroup$
    – capoocapoo
    Commented Jun 14 at 14:44
  • $\begingroup$ 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! $\endgroup$
    – capoocapoo
    Commented Jun 14 at 14:47


You must log in to answer this question.

Browse other questions tagged .