relabelling

Part IV, The Exceptional Interior · defined in Chapter XV, Spin from the Double Cover

An automorphism of the octonions that permutes the units with signs. There are 1344 of them, every one fixes 1, and each covers a collineation of the Fano plane: the relativity group reaches the fiber through them, up to colour signs.

Plate W.28The eight octonion units graded by

\F_2^3

with the clock

e_7

vertical: a relabelling that fixes the clock keeps the gold line

\{1,e_7\}

and permutes the three blue colour lines through all of

S_3

At one clock the relabellings that cover the clock’s report group and commute with its complex structure number ninety-six. They lie in colour $\SU(3)$ , the stabilizer of the clock’s unit in $G_2$ , and on $\C^3$ they form the complex reflection group $G(4,4,3)$ . They contain no copy of the report group mapping onto it, since every lift of a four-cycle has order eight, so the clock’s relabellings act on the fiber only together with colour sign changes. If the octonion product and its unit are physical, matter’s transports must keep them: a commit is carried exactly to the image commit when the record is relabelled with the product’s signs, and without the signs only 21 of the 1344 do so.

Every relabelling fixes the zero of the grading, so it keeps each observer’s lepton line or carries it to another observer’s, and there is no symmetry between the lepton and a quark to gauge. Across a Coxeter meeting, the relabelling that carries one observer’s lepton plane onto the other’s is unique up to colour.

Proposition(Covariance with the product’s signs) computed

(1) An automorphism $\phi$ over a collineation $g$ satisfies $\phi L_a\phi^{-1}=L_{\phi(e_a)}=s_aL_{g(a)}$ , with $s_a=\pm1$ the product’s sign. If the record relabels each word letter by letter by $g$ and multiplies it by the product of its letters’ signs, every commit $B_a\otimes L_a$ is carried exactly to $B_{g(a)}\otimes L_{g(a)}$ . Without the signs this fails for 1323 of the 1344 automorphisms, all but the 21 that need no sign.

(2) At a clock $c$ , the automorphisms that cover the clock’s report group and act linearly in its frame number 96. Their kernel is four sign changes, acting on the clock’s four complex lines, the lepton line first, as $\mathrm{diag}(1,\pm1,\pm1,\pm1)$ with determinant 1: elements of colour $\SU(3)$ that fix the lepton vector. They contain no copy of the report group mapping onto it, since every lift of a four-cycle has order eight.

(3) Choose for each collineation $g$ and clock $c$ a lift $\phi_{g,c}$ that sends $e_c$ to $+e_{g(c)}$ . Then $\phi_{g,h(c)}\circ\phi_{h,c}=\phi_{gh,c}\circ\chi$ , with $\chi$ a sign change that fixes $e_c$ , for all $168\cdot168\cdot7=197{,}568$ choices of $g$ , $h$ and $c$ : the relativity group acts on the observers’ fibers honestly up to colour signs at the source clock.

As mathematics

An element of $G_2=\Aut(\Oct)$ that is a signed permutation of the basis of units. A relabelling commutes with $L_{e_c}$ exactly when it fixes $e_c$ , and then it lies in $\mathrm{Stab}_{G_2}(e_c)=\SU(3)$ . Over each collineation $g$ , of the 128 realizations of $g$ in $\Spin(7)$ exactly eight preserve the product, the relabellings over $g$ , and exactly two, $\pm A_g$ , preserve the complex structure $J_0=\sum_xL_{e_x}/\sqrt7$ of the table’s light direction. A collineation has a realization keeping both exactly when it lies in the group $F_{21}$ that fixes that light direction.

Here $A_g$ is the orthogonal map, unique up to sign, with $A_gL_{e_x}A_g^{-1}=L_{e_{g(x)}}$ for every unit, and every relabelling over $g$ is $\pm A_gL_{e_{x_1}}\cdots L_{e_{x_k}}$ , where $x_1<\dots<x_k$ are the units it reverses. The maps $\pm A_g$ form a copy of $\SL(2,7)$ .

Its name in another field	Bridge
a signed permutation of the units in $G_2=\Aut(\Oct)$	built
at one clock, the complex reflection group $G(4,4,3)$ inside colour $\SU(3)$	built

A second sense

Chapters I, IV, V, VIII and XIX also relabel the reports: a relabelling there is a permutation of the four reports, an element of the report group $S_4$ , and the sign relabellings $(s,\ell)\mapsto(\varepsilon s,\eta\ell)$ of Chapter I are the displacements. Such a relabelling reaches the fiber only through its lifts to the octonions. Seams uses the word only in the octonionic sense.

Built from: fiberclock
In the volume: XThe Finite Celestial SphereXIIThe Coxeter GraphXIIIThe Level-Seven ShadowXIVWhy OctonionsXVSpin from the Double CoverXVIThe QuartetXVIIITwo Parents of the SkyXXIIILight, Vacuum and Handedness