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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.

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up vote 3 down vote accepted

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.

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I fixed the typo. Thanks, I was having trouble picturing the simplexes in the construction using the sublattice of $\mathbb{Z}^{n+1}$. Is there a good word to search for to find this in articles? $A_n$ is a little bit of a dumb thing to type into a search engine. – Tim Seguine Nov 29 '12 at 15:48

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