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.


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.

| cite | improve this answer | |
  • $\begingroup$ 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. $\endgroup$ – Tim Seguine Nov 29 '12 at 15:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.