I’m just beginning to learn about Coxeter groups and at one point the textbook’s author seems to take it for grant that each generator cannot be “reduced” into a product of other generators. Specifically, it seems the following claim is so obvious that it need not be stated:

If $\{s_1, s_2,..., s_n\}$ is the set of generators of a coxeter group $W$, then one cannot have $s_i=s_1s_2...s_{j-1}s_js_{j-1}...s_1$ for some $j$ without $i, j$ being $1$.

So I suspected something like $s_1=s_2s_4s_8$ cannot happen too. I know the question looks really stupid but it seems I’m missing something fundamental.

  • $\begingroup$ This is not obvious -- I suspect it only becomes clear somewhere in Chapter 4 of Björner/Brenti, not earlier. The easiest way to see the claim is if you know that the length function of a parabolic subgroup $W_I$ of $W$ is the restriction of the length function of $W$ (see, e.g., §9.6 of arXiv:math/0208154v2); thus, if $s_i$ was a product of other $s_j$'s, then $s_i$ would have to be a single other $s_j$, which would contradict the (nontrivial) fact that the elements of $S$ are distinct in $W$. $\endgroup$ – darij grinberg Jan 2 at 11:45

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