Automorphism transitivity on Jordan frames

Statement

Let $J$ be a simple Euclidean Jordan algebra, and let $K=\operatorname{Aut}(J)^{\circ}$ be the identity component of its automorphism group. Then $K$ acts transitively on ordered Jordan frames: for any two frames

(c_1,\ldots,c_r),\qquad(d_1,\ldots,d_r),

there is $k\in K$ satisfying $k(c_i)=d_i$ for every $i$ .

Homogeneous frame space

Fix an ordered frame $\mathbf c=(c_1,\ldots,c_r)$ . Its space of ordered frames is therefore the homogeneous space

K/K_{\mathbf c}, \qquad K_{\mathbf c}=\{k\in K:k(c_i)=c_i\text{ for every }i\}.

Here $K_{\mathbf c}$ is the pointwise stabilizer of all labelled frame entries. This transitivity is what lets the Jordan spectral theorem move a diagonalization from one chosen frame to any other convenient frame.

Ordered, unordered, and connected stabilizers

If the same frame is regarded as the unordered set $C=\{c_1,\ldots,c_r\}$ , its setwise stabilizer is

K_C=\{k\in K:k(C)=C\}.

There is a homomorphism $K_C\to S_r$ recording the permutation of the frame entries, and its kernel is $K_{\mathbf c}$ . Thus a setwise stabilizer may be strictly larger than the pointwise stabilizer even though the acting group $K$ is connected. Moreover, the identity component $(K_C)^{\circ}$ lies in $K_{\mathbf c}$ , but the pointwise stabilizer itself need not be replaced by its identity component without a separate connectedness argument.

For the Albert algebra $H_3(\mathbb O)$ , $K=F_4$ , the pointwise stabilizer of a labelled frame is $\mathrm{Spin}(8)$ , while the setwise stabilizer also contains permutations of the three idempotents. This is the distinction used in the Albert frame-stabilizer theorem.

Scope of the theorem

Simplicity is part of the statement. A general Euclidean Jordan algebra is a direct sum of simple ideals, and its automorphisms must respect the resulting factor structure up to permutations of isomorphic factors. Frame-orbit statements in that setting therefore require keeping track of which primitive idempotents belong to which simple ideal.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994, Chapter IV, Theorem 2.5. Publisher record.
John F. Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapters 1–3. Publisher record.