# What is the name of this lattice?

Suppose we have an atom at every point with integer coordinates in $\mathbb{R}^d$. Take a ($d-1$)-dimensional hyperplane going through $\mathbf{0}$ and orthogonal to $(1,1,1,\ldots)$. What is the name of the lattice formed by atoms in that plane?

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This lattice is used in the image processing paper "Fast High-Dimensional Filtering Using the Permutohedral Lattice" by Adams et al, 2010 (graphics.stanford.edu/papers/permutohedral). According to the caption of their Figure 2, it is the lattice $dA^*_{d-1}$. – Rahul Oct 5 '10 at 21:20
Thanks, that's what I was looking for. Search for "permutohedral" seems to give CS papers exclusively, so I'd guess $A_d$ is the mathematical name – Yaroslav Bulatov Oct 5 '10 at 21:53
Note that the permutohedral lattice is formed by projecting the d-dimensional Cartesian lattice onto your plane, so a priori it is not exactly what you would get by taking a slice. The lattice described in the question could be the inverse of the permutohedral lattice? (as BCC is inverse to FCC) ... that is a wild guess based on the slice-projection theorem – yasmar Oct 5 '10 at 22:51
Oh, as Rahul already mentions the "permutohedral" lattice is $dA^*{d-1}$, presumably the astrisk is significant. – yasmar Oct 5 '10 at 23:05
Yes, it denotes the "dual" of the original lattice. – Robin Chapman Oct 6 '10 at 7:04

It's called $A_{d-1}$. See http://en.wikipedia.org/wiki/Root_system#An .

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To be precise, $A_{d-1}$ doesn't consist of all integer vectors in the hyperplane, only those with zeros as coordinates except for one +1 and one -1. To obtain the whole lattice one can take integer-coefficient linear combinations of the vectors in $A_{d-1}$, but I don't know if that has a name of its own. – Hans Lundmark Oct 6 '10 at 5:38
Hans, the name is "the $A_{d-1}$ lattice" :-) – Robin Chapman Oct 6 '10 at 6:53
@Hans Lundmark: A_{d-1} is an object with more structure than a lattice, namely it comes with a distinguished set of generators. But as far as it refers to the entire root system it refers to all of the integer vectors in the hyperplane in question. – Qiaochu Yuan Oct 6 '10 at 9:44
OK, I take it back. But to my defence, I can say that I was just reading the definition of the $A_n$ root system in the Wikipedia article that you yourself linked to. :-) However, as that article also says, the terminology doesn't seem completely fixed ("some authors omit condition this or that in the definition"), and of course if one talks about the $A_n$ lattice it's clear what is meant, so I shouldn't have been so categorical in my statement. – Hans Lundmark Oct 7 '10 at 8:53