Suzuki Coxeter groups proof queries

Question

I am reading a proof over 4.2 in Suzuki group theory I and can't make sense of some parts.

I will just type the proof and then say what my query is.

Statement

Let $(W,S)$ be a Coxeter system. Let $T$ be the set of all elements of $W$ that are conjugate to some generator $s \in S$. Let $X=\{\pm 1 \} \times T$.

For a sequence $\mathfrak{s}=(s_1,\dotsc,s_q)$ define $\mathfrak{t}$ to be the sequence $(t_1,\dotsc,t_q)$ where $$t_i = (s_1\dotsm s_{i-1}) s_i (s_1\dotsm s_{i-1})^{-1} \in T. $$

For any $t \in T$ let $n(\mathfrak{s}, t)$ be the number of indices $j$ such that $t_j=t$.

Define $\eta(\mathfrak{s}, t)=(-1)^{n(\mathfrak{s},t)}$.

Define $U:S \rightarrow \operatorname{sym}(X) $, $s \mapsto U_s $ where $$U_s(\epsilon, t)=(\pm \epsilon, sts) $$ where the signature of the right side of the equation is negative only when $s=t$.

Then the mapping $U $ can be extended to a homomorphism from all of $W$ to $\operatorname{sym}(X)$.

Proof

We have $U_s^2=1$ and so $U_s$ is a permutation of $X$. We just need to show that this map $U$ preserves the relations of the group.

Let $s, s' \in S$ and let $m$ be the order of $ss'$. Then for any $k<m $ we get $$(U_{s'}U_s)^k(\epsilon, t) = (\pm \epsilon, (ss')^{-k}t(ss')^k).$$

Now the sign of $\epsilon $ changes at $k+1$ if and only if $(ss')^{-k}t(ss')^k=s$ or $(ss')^{-k}t(ss')^k = ss's$. (*) In these cases we have either $t=(ss')^{2k}s $ or $t=(ss')^{2k+1}s$. (**) If $t=(ss')^l s $ for some $l < m $ then $t=(ss')^{l+m}s$. So a change of sign occurs twice. Thus $(U_{s'}U_s)^m=1 $ and the mapping preserves the defining relations.

Queries

(*) Firstly, how has he gone from $(ss')^{-k}t(ss')^k = s$ to $t=(ss')^{2k}s $, when surely it should be $t=(ss')^k s (ss')^{-k} $?

(**) Secondly, what does he mean by “the change of sign occurs twice” and how does this imply that $(U_{s'}U_s)^m=1$?

Any help would be greatly appreciated.

Answer 1

For (*), certainly $t$ equals $(s s')^k s(s s')^{-k}$, as you expect; but $s(s s')^{-1}s$ equals $s s'$, so that $(s s')^k s(s s')^{-k}$ equals $(s s')^k(s(s s')^{-1}s)^k s = (s s')^k(s s')^k s$.

For (**), Suzuki is making the point that something happens for every index $k$ such that $t$ equals $(s s')^{2k}s$ or $(s s')^{2k + 1}s$. If $t$ equals $(s s')^l s$ with $l < m$, then we get one such index $k = \lfloor l/2\rfloor$; but we also get another such index, $k = \lfloor(l + m)/2\rfloor$. These can never be equal when $l$ and $m$ are positive integers, and $l$ is less than $m$. Then the observation is that a sign change occurring an even number of times is the same as no sign change occurring at all.