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Is the $F_4$ lattice (i.e. 4-dimensional body-centered hypercubic lattice, spanned by the simple roots of $F_4$) bipartite? And if so, what is a good explicit partition of the vertices?

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What do you mean by a lattice being "bipartite"?$$ $$BTW, as a lattice in $4$-dimensional space $F_4$ is isomorphic with $D_4$ (the $F_4$ roots are the vectors of norm $2$ and $4$ in $D_4$). – Noam D. Elkies Jan 27 '12 at 2:43
Take $\mathbb{Z}^4$. Its points are vertices of an infinite graph, whose edges join the nearest neighbor vertices. This is a standard hypercubic lattice. It is bipartite in the sense that the vertices can be labelled by two colours, say 'red' and 'black', such that every edge only connects red--black (no connection red--red or black--black exists). Just like the vertices in the 3D crystalline form of sodium chloride can be labelled 'Na' and 'Cl' and only Na--Cl bonds exist. If we take the 4D body-centered hypercubic lattice, or $F_4$ lattice, can we label its vertices in a similar way? – helvio Jan 27 '12 at 3:38
up vote 12 down vote accepted

No such labeling can exist because this lattice contains triangles such as $\lbrace(0,0,0,0), \phantom.(2,0,0,0), \phantom.(1,1,1,1)\rbrace$. This is predictable from the root diagram, which contains the $A_2$ root system. [Note that to identify this "body-centered hypercubic lattice" with a root lattice we must use the norm $\langle x,x\rangle = \frac12(x_1^2+x_2^2+x_3^2+x_4^2)$ so that the minimal vectors satisfy $\langle r,r \rangle = 2$.]

Indeed the nearest-neighbor graph of this lattice is not even $3$-colorable, because it has $4$-cliques such as the regular tetrahedron $$ T = \lbrace(0,0,0,0), \phantom.(2,0,0,0), \phantom.(1,1,1,1), \phantom.(1,1,1,-1)\rbrace. $$ This, too, is predictable from the root diagrams once we know that the $F_4$ lattice is the same as the $D_4$ lattice (the $F_4$ roots are vectors of norm $2$ and $4$ in $D_4$): the root diagram of $D_4$ contains $A_3$. In general a root lattice has $n$-dimensional simplices spanned by roots if and only if it contains $A_n$.

There does exist a $4$-coloring: the vectors of norm $4$ (the long $F_4$ roots) span a sublattice $L_0$ of index $4$ isometric with $2^{1/2} D_4$, and we can assign each of its cosets a different color.

In fact this $4$-coloring is unique up to color permutation. To prove this, first observe that each of the vertices of $T$ must have a different color. The same is true for the tetrahedron obtained from $T$ by replacing $(2,0,0,0)$ by $(0,2,0,0)$, so these two minimal vectors must have the same color. They differ by a long root of $F_4$, and all the long roots are equivalent under lattice isometries. Hence any two vectors that differ by a long root have the same color, whence each coset of $L_0$ must have constant color, QED.

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Thank you for your answer. It is very clear. :) – helvio Jan 27 '12 at 12:10

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