Timeline for Order from Coxeter-Dynkin diagram
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Mar 16, 2021 at 19:25 | comment | added | Sam Hopkins | Regarding determining the length of the longest element: for Weyl groups, $\ell(w_0)$ is the number of positive roots, which is $\frac{hr}{2}$ where $r$ is the rank and $h$ the Coxeter number. We have $h=a_1+\cdots+a_r+1$ where the $a_i$ are as in my answer. | |
| Mar 16, 2021 at 16:20 | comment | added | LSpice | Steinberg's monograph is such a wonderful resource. The result you quote is Theorem 1.25. | |
| Mar 16, 2021 at 15:49 | comment | added | Nathan Reading | The problem, of course, is that we have to know $\ell(w_0)$. (Note that we cannot just set $q=1$ and use this as a recursion on the order, because then, if $|S|$ is even, the order of $W$ vanishes from the equation. The length of $w_0$ can be determined using exponents, for example, but if we're going to find those anyway, there is a non-recursive formula for the Poincaré series (as noted in my comment on the question). | |
| Mar 16, 2021 at 15:41 | history | edited | Nathan Reading | CC BY-SA 4.0 |
Backing off a claim.
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| Mar 16, 2021 at 15:36 | history | answered | Nathan Reading | CC BY-SA 4.0 |