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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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    Yes, it denotes the "dual" of the original lattice.2010-10-06

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

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    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.2010-10-07