Maybe it is more clear with a picture.fresh_42 said:
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 said:
As i found out it cannot be ##\sqrt{3}## because the lattice of E8 have only half integers and integers coordinates.fresh_42 said:
I think i have found the answer.The translation is √2.fresh_42 said:
The coxeterelement of E9 is a mathematical concept used in the study of Coxeter groups, which are groups generated by reflections in a finite set of hyperplanes.
The translationlength of the coxeterelement of E9 along the coxeteraxis is calculated by measuring the distance between the starting and ending points of the translation, which is determined by the action of the coxeterelement on a given point.
The coxeteraxis is a line in the space where the coxeterelement of E9 acts. It plays a crucial role in determining the translationlength of the coxeterelement, as the translationlength is measured along this axis.
The translationlength affects the coxeterelement of E9 by determining the amount of translation or displacement that occurs when the element is applied to a point. A longer translationlength means a greater displacement along the coxeteraxis.
Yes, the translationlength can be negative in certain cases. This occurs when the coxeterelement of E9 acts on a point in the opposite direction of the coxeteraxis, resulting in a negative displacement along the axis.