kernel

The kernel of a description $d\colon X\to Y$ at a point $x_0$ is $K_d(x_0)=\{\gamma\in\Gamma:\ d(\gamma x_0)=d(x_0)\}=\Gamma_{d(x_0)}$, the stabilizer of the image of $x_0$; what $d$ forgets at $x_0$ is the fibre of $d$ through it.

(a) If $\varphi\colon\Gamma\to G$ is a surjective homomorphism, regarded as a description of the regular $\Gamma$-set $\Gamma$ by the $\Gamma$-set $G$, its kernel at 1 is $\ker\varphi$. (b) For a point $y$ of a $\Gamma$-set, the orbit description $\gamma\mapsto\gamma y$ has kernel $\Gamma_y$ at 1; so the stabilizer principle says that an object is the regular set with the kernel of its orbit description divided out. (c) If $d\colon X\to Y$ is a description between transitive $\Gamma$-sets with $\Gamma_{x_0}=H$ and $\Gamma_{d(x_0)}=L$, then $H\le L$, the kernel at $x_0$ is $L$, and what $d$ forgets at $x_0$ is a copy of $L/H$.

These are classical, and no novelty is claimed for them. What the definition adds is a single word for three situations that the following results treat alike.

Let $\varphi\colon\Gamma\to G$ be a surjective homomorphism of finite groups with kernel $N$. A transitive $\Gamma$-set $\Gamma/\hat H$ comes from a $G$-set through $\varphi$ if and only if $N$ acts on it trivially, if and only if $N\le\hat H$. If $N$ is central of prime order, then $N$ acts on every transitive $\Gamma$-set either trivially or without fixed points.

For the double cover $\SL(2,7)\to\PSL(2,7)$ the kernel is $\{\pm I\}$, and the objects it cannot see are the four new objects, those on which the kernel acts without fixed points.

Let $\omega$ be a primitive cube root of unity, $\mathcal O=\Z[\omega]$ and $\mathfrak p=(3+\omega)$, a prime of norm 7. The reduction $\PSL(2,\mathcal O)\to\PSL(2,7)$ is a description with kernel $\Gamma(\mathfrak p)$. Let $\PSL(2,\C)$ act on the Hermitian $2\times2$ matrices by $X\mapsto gXg^*$, preserving $\det X=t^2-x^2-y^2-z^2$, and put $p=3+\omega$,

(a) The images of $t_1$, $t_2$, $t_3$ lie in $\Gamma(\mathfrak p)$. (b) The quadratic forms invariant under $t_1$ and $t_2$ are spanned by $\det$ and the square of the lower diagonal entry. (c) The quadratic forms invariant under $t_1$, $t_2$ and $t_3$ are the multiples of $\det$. Hence the quadratic forms invariant under $\Gamma(\mathfrak p)$ are the multiples of the Minkowski form.

For each imaginary unit $e_x$ of the octonion table, the description of the derivation algebra $\mathfrak g_2$ by its effect on $e_x$, $D\mapsto D(e_x)$, maps $\mathfrak g_2$ onto the 6-dimensional orthogonal complement of $\mathrm{span}(1,e_x)$. Its kernel is the algebra $\mathfrak{su}(3)_x$, of dimension 8, which commutes with left multiplication by $e_x$.