new object

Write $\tilde G=\SL(2,7)$ and $\pi\colon\tilde G\to G$ for the map to Möbius transformations, a surjection with kernel $\{\pm I\}$; every $G$-set is a $\tilde G$-set through $\pi$. A $\tilde G$-set is pulled back if $-I$ acts on it trivially, so that it comes from a $G$-set through $\pi$. A transitive $\tilde G$-set that is not pulled back is a new object.

The question comes from the negative space: $\SL(2,7)$ carries spinors that do not descend, since the spin bundle over the seven points is not equivariantly trivial. The new objects are what the double cover does to objects.

(a) $-I$ is the only involution of $\tilde G$. (b) The subgroups of $\tilde G$ containing $-I$ are the 179 preimages $\pi^{-1}(H)$. A subgroup not containing $-I$ has odd order and is mapped by $\pi$ isomorphically onto a subgroup of odd order. For each of the 45 subgroups $H$ of $G$ of odd order, of orders 1, 3, 7 and 21, the elements of odd order of $\pi^{-1}(H)$ form a subgroup $H^\circ$, the odd lift of $H$, the only subgroup of $\tilde G$ mapped isomorphically onto $H$. (c) The 224 subgroups of $\tilde G$ form 19 conjugacy classes: the preimages of the fifteen classes of $G$ and the odd lifts of the classes 1, $C_3$, $C_7$, $7{:}3$.

The new objects of $\tilde G$ are exactly the four sets $\tilde G/H^\circ$, for $H$ in the classes 1, $C_3$, $C_7$, $7{:}3$. On each of them $-I$ acts without fixed points, and the quotient by $-I$ is the object $G/H$.

Their sizes are 336, 112, 48 and 16, with automorphism groups $\tilde G$, $C_4$, $C_6$ and $C_2$. On $\F_7^2$ they are the bases $(v,w)$ with $\det(v,w)=1$; the pairs $(\bar v,\bar w)$ of square classes with $\det(v,w)$ a nonzero square; the nonzero vectors; and the square classes $\{v,2v,4v\}$ of nonzero vectors, whose quotients by $-I$ are the regular object, the ordered pairs of distinct points, the vectors up to sign and the points of $\Proj^1(\F_7)$. An object of $\tilde G$ on which $-I$ acts nontrivially lies over $G/H$ if and only if $|H|$ is odd.

For each new object $\tilde X$ with quotient $X=\tilde X/\{\pm I\}$, the map $\Aut(\tilde X)\to\Aut(X)$ is onto, with kernel $\{1,-I\}$. On the nonzero vectors the automorphisms are the scalars $\F_7^\times\cong C_6$, and the squares $\{1,2,4\}$ map isomorphically onto the automorphism group $C_3$ of the object of size 24: the extension splits. On the object of size 112 the map $\sigma(\bar v,\bar w)=(\bar w,\overline{-v})$ generates the automorphism group and $\sigma^2$ is the action of $-I$; below, $\sigma$ induces the exchange $(a,b)\mapsto(b,a)$ of ordered pairs of points, an involution, and every automorphism lying over it has order 4. On the regular object the automorphisms form $\tilde G$, and $\tilde G\to G$ does not split.

So a cycle of seams whose monodromy on the ordered pairs of points is the exchange has, on the object of size 112, the monodromy $\sigma^{\pm1}$, and going around it twice gives $-I$, not the identity. This is the sign of a spinor, met in a seam system.

Let $\C[X]^-$ be the functions on a $\tilde G$-set $X$ that are odd under $-I$; for a new object it has dimension $|X|/2$, and its constituents are faithful. Then

and $\C[336]^-$ contains each faithful irreducible representation as often as its dimension. The spin bundle lives over the seven points, an object pulled back from $G$, and its sections are $\mathbf 6_\pm\oplus\mathbf 8$; the two smallest faithful representations, $\mathbf 4$ and $\bar{\mathbf 4}$, do not occur there, and are carried by the smallest new object, the sixteen square classes.