# Revision history [back]

### Action of lattice automorphism group on discriminant group

I have a lattice $L$ with automorphism group $G=Aut(L)$. The group $G$ will act on the discriminant group $D=L^*/L$. There will be a normal subgroup $N$ of $G$ which acts trivially on $D$. I need to be able to compute the action of the quotient $G/N$ on $D$. Can anyone provide any hints on how to get sage to do this? I'm new to sage.

### Action of lattice automorphism group on discriminant group

I have a lattice $L$ with automorphism group $G=Aut(L)$. The group action of $G$ will act on $L$ should induce an action on the discriminant group $D=L^*/L$. There $D=L^/L$ such that we have a group homomorphism $\phi: Aut(L) \rightarrow Aut(D)$. The kernel of this map will be a normal subgroup $N$ of $G$ which acts trivially on $D$. $G$. I need to be able to compute the action of the quotient $G/N$ on $D$. In the actual example I am interested in $L$ has rank $20$ and is defined through its Gram Matrix. $L$ is an even lattice. A simpler model of this situation would be to take $L$ to be a root lattice, say the $A_2$ root lattice to be concrete, $Aut(L)$ is the dihedral group $D_6$ arising as the product of the $Z_2$ outer automorphism and the $S_3$ Weyl group of $A_2$. The dual lattice $L^$ is the weight lattice of $A_2$ and the discriminant group $L^*/L$ is $Z_3$ with one non-trivial automorphism, taking $g \rightarrow g^{-1}$. In this simple case everything is computable by hand, but for the case I am interested in I only have $Aut(L)$ presented in terms of $20 \times 20$ matrix generators and computing by hand seems too difficult. Can anyone provide any hints on how to get sage to do this? I'm new to sage.I can compute $Aut(L)$ and $Aut(D)$ using sage, my problem is in figuring out how to determine $N$ and the action of the quotient $G/N$ on $D$.

### Action of lattice automorphism group on discriminant group

I have a lattice $L$ with automorphism group $G=Aut(L)$. The action of $G$ on $L$ should induce an action on the discriminant group $D=L^/L$ such that we have a group homomorphism $\phi: Aut(L) \rightarrow Aut(D)$. The kernel of this map will be a normal subgroup $N$ of $G$. I need to be able to compute the action of the quotient $G/N$ on $D$. In the actual example I am interested in $L$ has rank $20$ and is defined through its Gram Matrix. $L$ is an even lattice. A simpler model of this situation would be to take $L$ to be a root lattice, say the $A_2$ root lattice to be concrete, $Aut(L)$ is the dihedral group $D_6$ arising as the product of the $Z_2$ outer automorphism and the $S_3$ Weyl group of $A_2$. The dual lattice $L^$ is the weight lattice of $A_2$ and the discriminant group $L^*/L$ is $Z_3$ with one non-trivial automorphism, taking $g \rightarrow g^{-1}$. In this simple case everything is computable by hand, but for the case I am interested in I only have $Aut(L)$ presented in terms of $20 \times 20$ matrix generators and computing by hand seems too difficult. Can anyone provide any hints on how to get sage to do this? I can compute $Aut(L)$ and $Aut(D)$ using sage, my problem is in figuring out how to determine $N$ and the action of the quotient $G/N$ on $D$.

### Action of lattice automorphism group on discriminant group

I have a lattice $L$ with automorphism group $G=Aut(L)$. The action of $G$ on $L$ should induce an action on the discriminant group $D=L^/L$ $D=L^\check/L$ such that we have a group homomorphism $\phi: Aut(L) \rightarrow Aut(D)$. The kernel of this map will be a normal subgroup $N$ of $G$. I need to be able to compute the action of the quotient $G/N$ on $D$. In the actual example I am interested in $L$ has rank $20$ and is defined through its Gram Matrix. $L$ is an even lattice. A simpler model of this situation would be to take $L$ to be a root lattice, say the $A_2$ root lattice to be concrete, $Aut(L)$ is the dihedral group $D_6$ arising as the product of the $Z_2$ outer automorphism and the $S_3$ Weyl group of $A_2$. The dual lattice $L^$ $L^\check$ is the weight lattice of $A_2$ and the discriminant group $L^*/L$ is $Z_3$ with one non-trivial automorphism, taking $g \rightarrow g^{-1}$. In this simple case everything is computable by hand, but for the case I am interested in I only have $Aut(L)$ presented in terms of $20 \times 20$ matrix generators and computing by hand seems too difficult. Can anyone provide any hints on how to get sage to do this? I can compute $Aut(L)$ and $Aut(D)$ using sage, my problem is in figuring out how to determine $N$ and the action of the quotient $G/N$ on $D$.

### Action of lattice automorphism group on discriminant group

I have a lattice $L$ with automorphism group $G=Aut(L)$. The action of $G$ on $L$ should induce an action on the discriminant group $D=L^\check/L$ such that we have a group homomorphism $\phi: Aut(L) \rightarrow Aut(D)$. The kernel of this map will be a normal subgroup $N$ of $G$. I need to be able to compute the action of the quotient $G/N$ on $D$. In the actual example I am interested in $L$ has rank $20$ and is defined through its Gram Matrix. $L$ is an even lattice. A simpler model of this situation would be to take $L$ to be a root lattice, say the $A_2$ root lattice to be concrete, $Aut(L)$ is the dihedral group $D_6$ arising as the product of the $Z_2$ outer automorphism and the $S_3$ Weyl group of $A_2$. The dual lattice $L^\check$ is the weight lattice of $A_2$ and the discriminant group $L^*/L$ is $Z_3$ with one non-trivial automorphism, taking $g \rightarrow g^{-1}$. In this simple case everything is computable by hand, but for the case I am interested in I only have $Aut(L)$ presented in terms of $20 \times 20$ matrix generators and computing by hand seems too difficult. Can anyone provide any hints on how to get sage to do this? I can compute $Aut(L)$ and $Aut(D)$ using sage, my problem is in figuring out how to determine $N$ and the action of the quotient $G/N$ on $D$.

### Action of lattice automorphism group on discriminant group

I have a lattice $L$ with automorphism group $G=Aut(L)$. The action of $G$ on $L$ should induce an action on the discriminant group $D=L^\check/L$ $D={L^}/L$ such that we have a group homomorphism $\phi: Aut(L) \rightarrow Aut(D)$. The kernel of this map will be a normal subgroup $N$ of $G$. I need to be able to compute the action of the quotient $G/N$ on $D$. In the actual example I am interested in $L$ has rank $20$ and is defined through its Gram Matrix. $L$ is an even lattice. A simpler model of this situation would be to take $L$ to be a root lattice, say the $A_2$ root lattice to be concrete, $Aut(L)$ is the dihedral group $D_6$ arising as the product of the $Z_2$ outer automorphism and the $S_3$ Weyl group of $A_2$. The dual lattice $L^\check$ $L^$ is the weight lattice of $A_2$ and the discriminant group $L^*/L$ ${L^*}/L$ is $Z_3$ with one non-trivial automorphism, taking $g \rightarrow g^{-1}$. In this simple case everything is computable by hand, but for the case I am interested in I only have $Aut(L)$ presented in terms of $20 \times 20$ matrix generators and computing by hand seems too difficult. Can anyone provide any hints on how to get sage to do this? I can compute $Aut(L)$ and $Aut(D)$ using sage, my problem is in figuring out how to determine $N$ and the action of the quotient $G/N$ on $D$.

### Action of lattice automorphism group on discriminant group

I have a lattice $L$ with automorphism group $G=Aut(L)$. The action of $G$ on $L$ should induce an action on the discriminant group $D={L^}/L$ $D={\tilde L}/L$ such that we have a group homomorphism $\phi: Aut(L) \rightarrow Aut(D)$. The kernel of this map will be a normal subgroup $N$ of $G$. I need to be able to compute the action of the quotient $G/N$ on $D$. In the actual example I am interested in $L$ has rank $20$ and is defined through its Gram Matrix. $L$ is an even lattice. A simpler model of this situation would be to take $L$ to be a root lattice, say the $A_2$ root lattice to be concrete, $Aut(L)$ is the dihedral group $D_6$ arising as the product of the $Z_2$ outer automorphism and the $S_3$ Weyl group of $A_2$. The dual lattice $L^$ $\tilde L$ is the weight lattice of $A_2$ and the discriminant group ${L^*}/L$ ${\tilde L}/L$ is $Z_3$ with one non-trivial automorphism, taking $g \rightarrow g^{-1}$. In this simple case everything is computable by hand, but for the case I am interested in I only have $Aut(L)$ presented in terms of $20 \times 20$ matrix generators and computing by hand seems too difficult. Can anyone provide any hints on how to get sage to do this? I can compute $Aut(L)$ and $Aut(D)$ using sage, my problem is in figuring out how to determine $N$ and the action of the quotient $G/N$ on $D$.