Cayley form, comass, and triality isomorphisms

Cayley 4-form Comass and Triality Isomorphisms

The Cayley 4-form ωCa is of fundamental importance both in the calibration theory of R. Harvey and H. B. Lawson [HL82], who calculated its comass ‖ωCa‖ using the non-associative algebra of the Cayley numbers, and in exceptional holonomy of type Spin(7) studied by R. Bryant [Br87] and D. Joyce [Jo00]. The Cayley form is self-dual and the triality property of SO(8) relates the representation on s...

Cayley Form

Following an idea of Dadok, Harvey and Morgan, we apply the triality property of Spin(8) to calculate the comass of selfdual 4-forms on R. In particular, we prove that the Cayley form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio (equation (1.5)) is SO(8)-conjugate to the Cayley form. We also use triality to prove that the stabilizer in SO(8) of the Cayley for...

Automorphism groups and isomorphisms of Cayley digraphs

Isomorphisms of Cayley graphs on nilpotent groups

Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G1;S1) and Cay(G2;S2) are connected Cayley graphs of finite valency on two nilpot...

On Isomorphisms of Finite Cayley Graphs

A Cayley graph Cay(G, S) of a group G is called a CI-graph if whenever T is another subset of G for which Cay(G, S) ∼= Cay(G, T ), there exists an automorphism σ of G such that Sσ = T . For a positive integer m, the group G is said to have the m-CI property if all Cayley graphs of G of valency m are CI-graphs; further, if G has the k-CI property for all k ≤ m, then G is called an m-CI-group, an...