I'm trying to find information about a specific lattice, which is proving difficult since I am not sure what its standard name is.


Consider the regular $n$-Simplex embedded in $\mathbb{R}^n$ with one of the $(n+1)$ vertices centered at the origin. Does the lattice generated by the position vectors of the remaining vertices have a name? Is this the root lattice $A\_n$?


Any help would be greatly appreciated.



Help with (Coxeter?) lattice identification.
The regular $n$-simplex has $n+1$ vertices. Other than that typo, you are right. 


The lattice $A\_n$ can be embedded into $\mathbb{Z}^{n+1}$, where $\mathbb{R}^{n+1}$ has the standard norm, and $A\_n$ is the rank $n$ sublattice where the coordinates sum to zero.


Now, inside $\mathbb{Z}^{n+1}$, the $n+1$ basis vectors are clearly the vertices of a regular simplex. Translating $e\_{n+1}$ to the origin gives another regular simplex with vertices at $0$, $e\_1-e\_{n+1}$, $e\_2-e\_{n+1}$, ..., $e\_n-e\_{n+1}$. The lattice this simplex generates is $A\_n$. Your question is simply asking about this construction without mentioning the ambient $n+1$ space.


