Let $(W,S)$ be a Coxeter system and assume that $|S|$ is finite. Certainly, $W$ is generated by $|S|$ simple reflections. My question is: Can $W$ be generated by fewer reflections? (Including non-simple reflections.)
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1$\begingroup$ Maybe see mathoverflow.net/questions/56767. As an answer there points out, the Coxeter group $A_1\times A_2$ has 3 generators and is isomorphic to $G_2$ with 2 generators (but I don't know if these are reflections viewed in the first Coxeter system). $\endgroup$– Sam HopkinsCommented Aug 5 at 12:53
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$\begingroup$ Thank you for your reply. I mean the reflections with respect to (W,S). $\endgroup$– user46809Commented Aug 5 at 13:06
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7$\begingroup$ If by reflections you mean elements with a codimension one fixed space, then the answer is obviously no, because the codimension of the fixed space of the group generated can only go down by at most one with each new reflection. $\endgroup$– Dave BensonCommented Aug 5 at 14:23
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3$\begingroup$ This is a good question, but I want to be sure what you mean. Here is a precise question: Let $(W,S)$ be a Coxeter system, let $T$ be the set of elements conjugate in $W$ to elements of $S$. (Equivalently, make the standard geometric representation of $W$ and let $T$ be the set of elements of $W$ that act with codimension-1 fixed space in that representation.) Is there a subset of $T$ strictly smaller than $|S|$ that generates $W$? If that is indeed what you mean by your question, then Dave Benson's simple argument answers it. If that's not what you mean, please fill us in. $\endgroup$– Nathan ReadingCommented Aug 5 at 16:09
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2$\begingroup$ Thank you, I see. By reflection, I mean the element in $T$. To summarize, $W$ may be generated by fewer involutions (but not reflections). $\endgroup$– user46809Commented Aug 5 at 22:39
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