The twenty-eight

Its five incarnations: the pairs of $\Proj^1(\F_7)$; the Sylow 3-subgroups under conjugation; the antiflags of the Fano plane; the bitangents of the Klein quartic; the vertices of the Coxeter graph. The unique seam from the Sylow subgroups sends $P$ to the one point it fixes: the pair of points fixed by $P$, the point and the line fixed by it, the line through its two fixed points on the curve, the vertex it fixes. So the pair $\{0,\infty\}$, the subgroup $\langle z\mapsto2z\rangle$ and the bitangent $x+y+z=0$ are one element.

The two families of $K_4$’s are the two ways the object sees the objects of size 7: the antiflags with a common point, and those with a common line. Which family is the points depends on the marking, and $z\mapsto3z$ exchanges them. Two Coxeter-adjacent bitangents meet at one of the 21 centres of involutions; four bitangents pass through each centre, and the two elements of order 4 fixing it pair them into the two Coxeter edges there. The normalizer of $G$ in the symmetric group of the 28 points has order 336, and in the pairs it is $\PGL(2,7)$.

Further incarnations: the subgroups of order 6; the poles of the bitangents, such as $(1:1:1)$; the hexagons of the Heawood graph; the perfect matchings of $\Proj^1(\F_7)$ in the orbit of $\{01,23,45,6\infty\}$; the 28 ideal edges of Thurston’s congruence link complement; and the pairs of vertices at distance 3 in the link at 2, or of neighbours in the link at 7. The 28 ideal tetrahedra of the same manifold are not this object: that bridge is refuted.

Describe the Coxeter adjacency on the 28 bitangents in the projective geometry of the quartic alone, without naming group elements; and give a conceptual reason why the Coxeter edges are the pairs of Sylow 3-subgroups whose elements of order 3 multiply to elements of order 4.

In the Weil representation of $\SL(2,7)$ two families of algebras $\mathfrak{su}(3)$, one over each pair of points, belong to the octonion table and to its mirror; both are incarnations of this object, so the seam between them is unique, and over each pair it is conjugation by the polarity of the Fano plane that fixes the pair. Transporting the first family between pairs that share a point gives a flat connection on the complex of stars whose holonomy is all of the stabilizer of a pair and acts by inner automorphisms.

At Klein’s lattice the antiflags are vectors: the 28 pairs of vectors of norm 3 are the four diagonals $u_1\pm u_2\pm u_3$ of the cubes at the neighbours of the lattice, and the Coxeter graph is an inner-product graph, with $|h(w,w')|^2$ equal to 7, 4, 2, 1 at Coxeter distance 1, 2, 3, 4. Coxeter neighbours are orthogonal modulo $\sqrt{-7}$, not in the lattice.

In the lattice $\mathrm{Herm}_2(\Z[\omega])$ with the form $\det$, the records $gg^\dagger$, $g\in\SL(2,\Z[\omega])$, form 28 classes modulo $\Gamma(\mathfrak p)$, $\mathfrak p=(3+\omega)$. Splitting a record into its two primitive null vectors and reducing their cusps modulo $\mathfrak p$ is an equivariant bijection from these classes onto the 28 pairs of points of $\Proj^1(\F_7)$: the object of size 28 is a set of classes of rest frames, the edges of the tessellation of $\mathbb H^3$, each the geodesic joining two cusps.

The two families of $K_4$’s are the two ways in which the object of size 28 sees the objects of size 7: the antiflags with a common point, and those with a common line. On the projective line each $K_4$ is four disjoint pairs covering $\Proj^1(\F_7)$. Which family belongs to the points depends on the marking: the map $z\mapsto3z$, outside $G$, induces an outer automorphism and exchanges the two families.