twisting element

Let $Y_\mu$ be a marked set and $\sigma$ a bijection of $Y$ with $\sigma\circ\mu(x)\circ\sigma^{-1}=\mu(cxc^{-1})$ for all $x\in G$ and some $c\in G$. The element $c$ is the twisting element of $\sigma$, and $\Phi=\mu(c)^{-1}\circ\sigma$ is its equivariant twist.

So a symmetry that normalizes the group gives two objects: an element of $G$, and an automorphism of the incarnation.

$\Phi$ is an automorphism of $Y_\mu$. If $G$ has trivial centre, $c$ is unique. If $\sigma$ fixes a point $y$, then $\Phi(y)=c^{-1}y$.

In the language of powers: if $c$ normalizes a cyclic stabilizer $\langle x\rangle$ with power $k$, then $\Phi$ has power $k^{-1}$ on the corresponding incarnation.

Let $\sigma_2$ be the automorphism $\zeta\mapsto\zeta^2$ of $\Q(\zeta_7)$, the Frobenius at 2, acting on points of $\Proj^2$ coordinatewise.

(a) $\sigma_2(\rho(x))=\rho(h^{-1}xh)$ for every $x\in G$; the twisting element $c=h^{-1}$ satisfies $cgc^{-1}=g^2$, so it has power 2.

(b) $\Phi=\rho(h)\circ\sigma_2$ is an automorphism of every incarnation in the quartic that $\sigma_2$ preserves. On the flexes $\Phi=\tau$, the flex-tangent map, of power 4; on the bitangents, the centres and the flex triangles $\Phi$ is the identity.

(c) At a prime above 2 the 24 flexes reduce to the 24 points of the quartic over $\F_8$, and the arithmetic Frobenius $v\mapsto v^2$, twisted by $\rho(h)$, is again $\tau$.

(d) The vectors fixed by this twisted Frobenius form a three-dimensional $\F_2$-space $V_0\subset\F_8^3$, stable under $G$. The two primes above 2 give two Fano planes $\Proj(V_0)$, whose points have stabilizers of class $S_4^b$ and $S_4^a$ respectively.

(e) For each point $p$ of the quartic over $\F_8$, $f\mapsto f(p)$ is an $\F_8$-coordinatization of the dual plane $\Proj(V_0^*)$; sending $p$ to it is a seam onto the 24 coordinatizations, and it carries $\tau$ to post-composition with the Frobenius of $\F_8$.

Under every seam between incarnations of the object of size 24 in the Klein quartic and in the Fano plane coordinatized by $\F_8$, the vertex link of the octonion completion, the $G$-equivariant Frobenius automorphisms correspond: both equal $\tau$, of power 4. The twisting elements, $h^{-1}$ for the quartic and the Frobenius for the completion, both have power 2.

So there is no incoherence. The relation between $\tau$, of power 4, and a Frobenius of power 2 is the general relation $\Phi(y)=c^{-1}y$ between a twisting element and its equivariant twist, and it takes the same form in both theories. The flex-tangent map $\tau$, a fact of the projective geometry of the quartic, is the Frobenius at 2 made equivariant.

On the double cover $\tilde G=\SL(2,7)$, acting through its Weil representation, the Galois automorphism $\zeta\mapsto\zeta^2$ carries $W(x)$ to $W(cxc^{-1})$ both for $c=d_4$ and for $c=d_3=-d_4$, the two lifts of the twisting element $h^{-1}$; $d_4$ has order 3 and is the odd lift, and $d_3$ has order 6. The twist is the identity on the sixteen vectors for $d_4$ and $-1$ for $d_3$. On the new object of size 48, the nonzero vectors of $\F_7^2$, the automorphisms lying over $\tau$ are the scalings by 2, of order 3, and by $5=-2$, of order 6, whose cube is $-I$.

So $\tau^3=-I$ is not forced. The twisting element is determined only up to the centre $\{\pm I\}$ of $\tilde G$, which is the deck transformation; the odd lift, the only lift of the same order as $h^{-1}$, gives the lift of $\tau$ of order 3, and the lift of $\tau$ of order 6, with $\tau^3=-I$, is the choice of the twisting element of even order.