Return to Revisions

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:

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 $Aut(U_{14})=W(E_7) \wr C_2$.