Translationlength by the coxeterelement of E9 along the coxeteraxis

[reinhard55]

I have searched on Google but i cannot find it.
Does anyone know how long is the translation by the coxerterelement of E9
(which is the affine one point extension of E8) along the coxeteraxis?

 

[fresh_42]

I am not quite sure which number you are looking for. Unfortunately I have no books which especially deal with $E_9$, but the closest I came (K.S. Brown, Buildings) let's me ask: Do you have any doubt that the matrix entry $m_{89}$ of the Coxeter matrix is not $3\,?$

Maybe https://ncatlab.org/nlab/show/E9 or the links on https://ncatlab.org/nlab/show/Kac-Moody+group can help you.

 

[reinhard55]

I am not quite sure which number you are looking for. Unfortunately I have no books which especially deal with $E_9$, but the closest I came (K.S. Brown, Buildings) let's me ask: Do you have any doubt that the matrix entry $m_{89}$ of the Coxeter matrix is not $3\,?$

Maybe https://ncatlab.org/nlab/show/E9 or the links on https://ncatlab.org/nlab/show/Kac-Moody+group can help you.

Maybe it is more clear with a picture.
The picture shows the affine coxetergroup of A2.
The darkgray chamber makes a reflection and a translation by the coxeterelement of affine A2.

My question now is how long is the translationlength by the coxeterelement of E9 (affine E8)?

 

[fresh_42]

We would need the graph for $E_9$, not the simple one of $A_2$. But as the ninth root is basically analogue to the situation of $A_n$ I assume it is $\sqrt{3}$ as well.

 

[reinhard55]

I thought that there already is a result for it but maybe not.I will try to calculate it on myself.
Thanks.

 

[fresh_42]

I thought that there already is a result for it but maybe not.I will try to calculate it on myself.
Thanks.

I found a table for highest weights in terms of a root system for the classical and exceptional groups, but as I said, nothing about $E_9$. But per construction, it should look very similar to the differences between $E_6 \to E_7$ and $E_7 \to E_8$. So once you have those, you should be able to see what changed. As said, the entry of the Coxeter matrix should be $m_{89}=3$.

 

[reinhard55]

I think it must be possible by the rootsystem of E9.
Thanks.

 

[reinhard55]

We would need the graph for $E_9$, not the simple one of $A_2$. But as the ninth root is basically analogue to the situation of $A_n$ I assume it is $\sqrt{3}$ as well.

As i found out it cannot be $\sqrt{3}$ because the lattice of E8 have only half integers and integers coordinates.

 

[fresh_42]

I have found a paper about $E_{10}$. Maybe it helps ($E_9$ on page 6)
https://core.ac.uk/download/pdf/25173425.pdf

 

[reinhard55]

I have found a paper about $E_{10}$. Maybe it helps ($E_9$ on page 6)
https://core.ac.uk/download/pdf/25173425.pdf

I think i have found the answer.The translation is √2.