seam over an automorphism

Floor 2, Les sutures · introduced in Chapter 1, Un objet, plusieurs noms

What is a seam after twisting by an automorphism of the group?

A bijection that carries the action of each element to the action of its image under an automorphism of the group; a seam is a seam over the identity, and a bridge refuted for one marking can be built over an outer automorphism.

Plate 2.12The polarity carries each line of the Fano plane to its pole. It is no seam, since the stabilizer of a point fixes no line; it is a seam over the outer automorphism

g\mapsto(g^{\mathsf T})^{-1}

Definition(Seam over an automorphism)

Let $\alpha\in\Aut(G)$ . The twist $Y^\alpha$ of a $G$ -set $Y$ is the $G$ -set with the same underlying set and the action $g\ast y=\alpha(g)y$ ; for a marked set, $(Y_\mu)^\alpha=Y_{\mu\alpha}$ . A seam over $\alpha$ , or a seam after twisting by $\alpha$ , from an incarnation $X$ to an incarnation $Y$ is a $G$ -isomorphism $X\to Y^\alpha$ , that is, a bijection $f$ with $f(gx)=\alpha(g)f(x)$ for all $g$ and $x$ . A seam is a seam over the identity.

The definition makes precise a phrase of the negative space and of the double lives, that a refuted bridge becomes a seam after twisting by the outer automorphism.

Proposition(Seams over automorphisms)

Let $X$ , $Y$ and $Z$ be transitive $G$ -sets and $\alpha,\beta\in\Aut(G)$ .

(a) A seam over $\alpha$ from $X$ to $Y$ exists if and only if $\st(X)=\alpha^{-1}(\st(Y))$ .

(b) If $\alpha(g)=hgh^{-1}$ is inner, then $f$ is a seam over $\alpha$ if and only if $x\mapsto h^{-1}f(x)$ is a seam. So only the class of $\alpha$ in $\operatorname{Out}(G)$ matters.

(c) If $f$ is a seam over $\alpha$ from $X$ to $Y$ and $f'$ a seam over $\beta$ from $Y$ to $Z$ , then $f'\circ f$ is a seam over $\beta\alpha$ . The seams over $\alpha$ from $X$ to $Y$ , if there are any, form a torsor under $\Aut_G(X)$ acting by precomposition.

(d) If $X$ is rigid and its stabilizer class is fixed by $\Aut(G)$ , then for each $\alpha$ there is exactly one seam $\alpha_X$ over $\alpha$ from $X$ to itself. The maps $\alpha_X$ form an action of $\Aut(G)$ on $X$ that extends the action of $G$ , through its inner automorphisms. So $X$ is, in exactly one way, an $\Aut(G)$ -set.

Proof

(a) The stabilizer of $y$ in $Y^\alpha$ is $\alpha^{-1}(G_y)$ ; apply the stabilizer principle to $X$ and $Y^\alpha$ . (b) $h^{-1}f(gx)=h^{-1}hgh^{-1}f(x)=g\,h^{-1}f(x)$ . (c) $f'(f(gx))=f'(\alpha(g)f(x))=\beta(\alpha(g))f'(f(x))$ ; two seams over $\alpha$ differ by the seam $f^{-1}f'$ from $X$ to itself. (d) By (a) a seam over $\alpha$ exists, and by (c) it is unique, since $\Aut_G(X)=1$ ; uniqueness gives $(\beta\alpha)_X=\beta_X\alpha_X$ , and for the inner automorphism by $g$ the map $x\mapsto gx$ is a seam over it.

Example

In the Fano plane, the polarity $\pi$ sending the line $\{v:u\cdot v=0\}$ to the point $u$ satisfies $\pi(gL)=\alpha(g)\pi(L)$ for the outer automorphism $\alpha(g)=(g^{\mathsf T})^{-1}$ : it is a seam over $\alpha$ from the lines to the points. That bridge is refuted over the identity and built over $\alpha$ ; this is the precise sense of a seam after twisting. The twisted symmetries of the Frobenius at two are the inner case: a symmetry $\sigma$ with $\sigma\mu(x)\sigma^{-1}=\mu(cxc^{-1})$ is a seam over $x\mapsto cxc^{-1}$ , and by (b) $\mu(c)^{-1}\sigma$ is a seam.

Example(Two families of algebras

\mathfrak{su}(3)

)

In the Weil representation of $\SL(2,7)$ there are two families $\mathcal F=(\mathfrak f_o)$ and $\mathcal F'=(\mathfrak f'_o)$ of algebras $\mathfrak{su}(3)$ , one algebra over each of the 28 pairs $o$ of points of $\Proj^1(\F_7)$ ; the second belongs to the mirror of the octonion table. Each family is an incarnation of the object of size 28, which is rigid, with stabilizer class fixed by $\Aut(G)\cong\PGL(2,7)$ .

(1) The seam over the identity, $\Phi\colon\mathfrak f_o\mapsto\mathfrak f'_o$ , is unique. (2) An element $m$ of $\PGL(2,7)$ outside $G$ carries $\mathfrak f_o$ onto $\mathfrak f'_{m(o)}$ , so conjugation by $m$ is a seam over the outer automorphism $g\mapsto mgm^{-1}$ , and not a seam. (3) By (d) the object of size 28 is a $\PGL(2,7)$ -set, and conjugation by $m$ is the seam over that automorphism from $\mathcal F'$ to itself composed with $\Phi$ . (4) Over each pair $o$ , $\Phi$ agrees with conjugation by the polarity $\rho_o$ , the improper involution of $\PGL(2,7)$ fixing both points of $o$ ; through the marking $\mu_A$ the 28 elements $\rho_o$ give the 28 polarities of the Fano plane. So the seam over the identity is assembled from 28 seams over outer automorphisms, each correct at its own pair.