Timeline for Generators of a Coxeter group
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 5 at 22:39 | comment | added | user46809 | Thank you, I see. By reflection, I mean the element in $T$. To summarize, $W$ may be generated by fewer involutions (but not reflections). | |
| Aug 5 at 16:09 | comment | added | Nathan Reading | 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. | |
| Aug 5 at 14:46 | history | edited | YCor | CC BY-SA 4.0 |
edited tags, fixed capitals
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| Aug 5 at 14:23 | comment | added | Dave Benson | 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. | |
| Aug 5 at 14:17 | comment | added | LSpice | The answer @SamHopkins references (by @TomDeMedts). Sadly, as @SamHopkins suspected, at least one natural choice of isomorphism doesn't identify the two $\mathsf G_2$ reflections with reflections in $\mathsf A_1 \times \mathsf A_2$; namely, if I write $\{a_1\}$ for $\mathsf A_1$, $\{a_2, a_3\}$ for $\mathsf A_2$, and $\{g_\text s, g_\text l\}$ for $\mathsf G_2$, then an isomorphism is given by $g_\text s \mapsto a_1 a_2$ and $g_\text l \mapsto a_3$. | |
| Aug 5 at 13:57 | history | edited | LSpice | CC BY-SA 4.0 |
Deleted "Expecting your reply. Thank you."
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| Aug 5 at 13:06 | comment | added | user46809 | Thank you for your reply. I mean the reflections with respect to (W,S). | |
| Aug 5 at 12:53 | comment | added | Sam Hopkins | 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). | |
| Aug 5 at 12:49 | history | asked | user46809 | CC BY-SA 4.0 |