Exceptional symmetric variety

What is $\Esix/\Ffour$?

$\Esix/\Ffour$ is what $\SL_3/\SO_3$, the symmetric $3\times3$ matrices of determinant one, becomes when the matrix entries are octonions. These pages build it from the classical models $\SL_2/\SO_2$ (ellipses) and $\SL_3/\SO_3$ (ellipsoids), which you can manipulate by hand.

Setup

The $\Oc$ form an 8-dimensional real algebra that is not associative. Over $\C$, the Albert algebra is

\[\Jalg=\left\{X=\begin{pmatrix}a&z&\bar y\\ \bar z&b&x\\ y&\bar x&c\end{pmatrix}:\ a,b,c\in\C,\ x,y,z\in\Oc_\C\right\},\qquad\dim_\C\Jalg=27,\]

with a cubic $\det:\Jalg\to\C$ and identity $\one$. Two groups act on it:

\[\Esix=\{g\in\GL(\Jalg):\det\circ g=\det\},\ \dim 78,\qquad \Ffour=\{g\in\Esix:g\one=\one\}=\Aut(\Jalg),\ \dim 52 .\]

No parameter varies. There is no family $E_n/F_n$, and no smaller case keeps the non-associativity, so these pages use instead the classical analogue $\SL_3/\SO_3$ (real entries) and the finite list of four algebras $\R,\C,\Ha,\Oc$, which places $\Esix/\Ffour$ at the end of a row of four.

The Concept

Proved on the Proposition page

\[\Esix/\Ffour\;\cong\;\Xe=\{X\in\Jalg:\det X=1\},\qquad g\Ffour\mapsto g\one .\]

It is a of dimension 26 and rank 2, with of type $A_2$, each of multiplicity 8, and it is .

The question it answers: how can a 78-dimensional exceptional group produce the same rank-two root picture as $\SL_3/\SO_3$, and why are the dimensions $26$ and $5$? Answer in one line: $\dim=2+3\mult$, with $\mult=8$ octonionic directions on each wall instead of 1.

One picture

Two identical A2 root diagrams with three positive roots each; on the left each is labelled times 1, on the right times 8. — The plane of diagonal traceless matrices, its three walls and six roots are the same for (a) $\SL_3/\SO_3$ and (b) $\Esix/\Ffour$; only the multiplicity per root changes.

Table 1. The same construction with real and octonionic entries.
	$\SL_3/\SO_3$	$\Esix/\Ffour$
ambient space	$\mathrm{Sym}_3(\C)$, 6	$\Jalg$, 27
$\dim G$, $\dim K$	8, 3	78, 52
$\dim G/K=2+3\mult$	5	26
multiplicity $\mult$	1	8

Map of the pages

Pages · arrows: “is used by”

Goals

Afterwards you can:

draw $\SL_2/\SO_2$ three ways (hyperboloid, disk, ellipses) and say which directions are size and which are angle;
read $\SL_3/\SO_3$ as ellipsoids of fixed volume, move a determinant-one symmetric matrix to $I$, and explain why $\SL_3/\SO_3\cong\Xs{\R}$;
multiply octonion units on the Fano plane, and say what changes when the octonions get complex coefficients;
say what the Albert algebra and its determinant are, and which maps lie in $\Esix$ and in $\Ffour$;
derive the restricted roots of $\Esix/\Ffour$ and explain why each multiplicity is 8;
predict dimension, multiplicity and generic Borel stabilizer for the other members of the $A_2$ row;
prove that $\Esix/\Ffour$ is spherical, and say what is left over ($16=\dim$ of a Borel of $\so_8$);
spot the false reasons: $\rk G-\rk K$, folding, and the word “symmetric”;
say where $\Esix/\Ffour$ appears in physics, and which of those uses are established.

Status and labels

DefinitionProved (cited)Proved hereHeuristic / experimentnumerical checkContextQuestion

Proved here: the classical model; restricted roots and multiplicity 8 (given the Chevalley–Schafer decomposition); $\det$-invariance of the explicit moves; sphericity.
Cited: transitivity of $\Esix$ on $\{\det=1\}$ (Krutelevich, Prop. 12); $\Ffour=\mathrm{Stab}(\one)$ (Yokota); types and dimensions of $\Esix,\Ffour$.
Unverified in the source itself: the Helgason table number; section numbers in Springer–Veldkamp. See References.
The labs check identities on random instances; they are checks, not proofs.

What these pages cost to make

Cumulative tokens (blue, left axis) and dollars at list prices (brown, right axis) over working time; the same total sits top right on every page.

Table 2. Tokens by session.
session	main agent	helper agents	US $

	\(\SL_3/\SO_3\)	\(\Esix/\Ffour\)
ambient space	\(\mathrm{Sym}_3(\C)\), 6	\(\Jalg\), 27
\(\dim G\), \(\dim K\)	8, 3	78, 52
\(\dim G/K=2+3\mult\)	5	26
multiplicity \(\mult\)	1	8