# Relationships among lattices U14, C2xG23, A15+ and their Delaunay polytopes

Do you have any references explaining the relationships among the lattices U14, C2xG23 aka Q14, and A15+?

Do you have any references explaining the relationships among these lattices and the 7D packings E7, E7*, 1_33 honeycomb?

U14 aka (E7^2)+: https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/U14.html

C2 x G(2,3) aka Q14: https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/P14.1.html

I would like to understand

(0) what is the construction of any of these lattices U14, C2G23, A15+ from E7, E7* or the 1_33 honeycomb?

(1) what is the construction of A15+ from C2G23 or U14 or vice-versa?

(2) what is the construction of C2G23 from U14 or vice-versa? C2G23 has triple the kissing/coordination number of U14, 252 vs 756.

(3A) U14 and A15+ seem to have only one Delaunay polytope, one kind of hole. Is that correct? I understand that the Delaunay polytopes are in 1:1 correspondence with the holes of the lattice. The U14 and A15+ lattices seem to have only one kind of hole. So these lattices seem to correspond to tesselations of space with only one Delaunay polytope. Is that correct?

If I am correct about that,

what is the relationship between the sole Delaunay polytopes corresponding to the only holes of U14 and A15+? What are the names for these sole polytopes of U14 and A15+, polytopes that tesselate space alone?

Is the sole polytope of A15+ called B15, per this table from Dutour , with 135 vertices not 112 as I have found with my primitive method as listed below? Is there any 15-dimensional lattice or packing associated with this B15 polytope with 135 vertices listed in the table from Dutour above?

(I calculate the holes with a primitive method and I assume that the depth of the hole & its coordination number uniquely identifies the Delaunay polytope for these lattices. I understand that this assumption is not valid for higher-dimensional lattices, like the Leech lattice for which the deepest holes have different Delaunay polytopes, but I assume that the Delaunay polytopes are uniquely identified by their hole (center) depths and coordination numbers for these lattices in 14 and 15D. Is that not correct? Please correct any misunderstanding that I have.)

The lattice C2 x G(2,3) with coordination number 756 seems to be called Q14 also, e.g. https://arxiv.org/abs/math/0505510: I understand that the lattice Q14 discussed in this reference is the same as the lattice C2xG23 because it has the same coordination number, 756.

$$\begin{array}{lll} \mathrm{Lattice} & \mathrm{Coord/kissing \ number} & \mathrm{Delaunay \ polytope(s) \ / \ holes} \\ U14 & 252 & \mathrm{One? \ 128 \ vertices?} \\ C2G23/Q14 & 756 & \mathrm{Two? \ 106 \ vertices?} \\ A15+ & 240 & \mathrm{One? \ \ 112 \ vertices, \ or \ 135?} \end{array}$$

The coordination/kissing numbers are known, in the second column of the table above. But I am not sure about the Delaunay polytopes of these lattices. I have written what I have found about the Delaunay polytopes in the third column. In particular I would like to confirm the Delaunay polytope of A15+. Does it have 112 vertices as I have found, or is it the B15 polytope with 135 vertices? Where can I find more info about the Delaunay polytope(s) of A15+, and the polytope B15 listed in the table from Dutour above? Again is there any 15-dimensional lattice associated with the polytope B15 listed in the table from Dutour above?

(3B) What about the contact polytopes of these lattices? Where can I find more information about these polytopes with 252, 756, and 240 vertices and their relationships?

Here is what I have found about the Delaunay polytopes; please confirm:

U14 and A15+ seem to have one kind of hole; their deepest hole is their only hole. I understand that the Delaunay polytopes of the lattices are 1:1 associated with the holes of the lattices. There seems to be only one kind of hole for A15+ and U14, so A15+ and U14 correspond to tesselations of space with one shape, i.e. these lattices have only one Delaunay polytope. Is that correct?

I have searched for holes of the lattice using increasing numbers of coordination shells. This is what I find:

U14 with 4, 5, 6 coordination shells (85289, 279945, 781065 points):

One Delaunay polytope found: Seems to have 128 vertices. 127 vertices identified with 4 & 5 shells; 128 vertices identified with 6 shells.

C2G23 with 3, 4, or 5 shells (53719, 241207, or 772675 points):

Two Delaunay polytopes found; two depths of holes found. The Delaunay polytope associated with the deepest hole seems to have 106 vertices. 105 vertices found with 3 and 4 shells; 106 vertices found with 5 shells. What is the name of this Delaunay polytope with 106 vertices; what is it called and where can I find more information on it?

A15+ with 4, 5, 6 shells (127127, 450807, 1346487 points):

One Delaunay polytope found, with low confidence. Seems to have 112 vertices. 111 vertices identified with 4 & 5 shells; 112 vertices found with 6 shells. What is this polytope with 112 vertices called & where can I find more info about it? 112 is twice 56, the number of facets in the 7D diplo-simplex, the sole Delaunay polytope of E7*, and the number of 1_22 facets of the polytope 1_32 comprising the 1_33 honeycomb, if I am not mistaken.

So what are the relationships among these lattices U14, C2G23, A15+ and the 7D packings E7, E7*, 1_33 honeycomb and where can I learn more?

-- The A15+ lattice has nearest-neighbor distance sqrt(2) and its holes have radius 1.3228756, seemingly sqrt(7)/2.

-- The U14 lattice has nearest-neighbor distance sqrt(2) and its holes have radius 1.3228756 also.

(3C) Can you provide any reference in which these equalities between the hole depth & NN distance for the lattices U14 and A15+ are discussed?

(3D) Can you provide any reference in which the Delaunay polytopes of U14, C2xG23, or A15+ and their relationships are discussed? Does anything indicate that the associated symmetry groups pertain to the standard model of particle physics?

Thanks, Dan

• The answer for question (0) about $$U_{14}$$ is positive, as described in the Conway-Odlyzko-Sloane paper:

Let $$E_8$$ be the lattice in $$\mathbb R^8$$ with its eight co-ordinates all in $$\mathbb Z$$, or all in $$\mathbb Z + 1/2$$, with even sum and $$E_7$$ be the sub-lattice of $$E_8$$ with $$x_1 + ... + x_8 = 0$$.

Then $$U_{14}$$ is the sum of $$E_7+E_7$$ and the vector $$((\frac14)^6,(-\frac34)^2,(\frac14)^6,(-\frac34)^2)$$.

• The answers for question (0) about $$Q_{14}$$, $$A_{15}^+$$, question (1) and question (2) may be negative, as indicated by the vast differences of their automorphism groups:

For each lattice, I computed the vectors closest to $$0$$ and study the automorphism group (as a subgroup of orthogonal group) of those vectors. This is a supergroup of the true automorphism group.

The results are:

Group Aut(closest vectors)
$$U_{14}$$ $$W(E_7) \wr C_2$$
$$Q_{14}$$ $$C_2 \times G_2(3)$$
$$A_{15}^+$$ $$C_2 \times S_{16}$$

where $$W(E_7)$$ is the Weyl group of $$E_7$$ and $$\wr$$ is the wreath product.

The automorphism group of $$Q_{14}$$ is already computed and it is $$C_2 \times G_2(3)$$.

The definitions given in the Conway-Odlyzko-Sloane paper indicates that its automorphism group contains $$S_{16}$$, so we have $$S_{16} \subset Aut(A_{15}^+) \subset C_2 \times S_{16}$$.

For $$U_{14}$$, I have also computed the automorphism group of vectors with distance $$\sqrt 2$$ and $$\sqrt 3$$ from $$0$$, and it's also $$W(E_7) \wr C_2$$. As the vectors with distance $$\sqrt 2$$ and $$\sqrt 3$$ from $$0$$ generate the whole $$U_{14}$$, it turns out that $$Aut(U_{14})=W(E_7) \wr C_2$$.

EDIT: The answer above also solves (3B) for $$U_{14}$$ and $$A_{15}^+$$: for $$U_{14}$$, the closest vectors are the union of two orthogonal $$2_{31}$$ polytopes; and for $$A_{15}^+$$, the closest vectors are those vectors in $$\mathbb{R}^{16}$$ containing a $$1$$, a $$-1$$ and 14 $$0$$s.