Plurisubharmonic Functions on the Octonionic Plane and Spin(9)-Invariant Valuations on Convex Sets

Spin-invariant valuations on the octonionic plane

“…With Adams formula and the algorithm described above, we can decompose Λ k V in irreducible representations as follows : Proof. We first describe the spaces Ω k,l h (SV ) Spin (9) . Let v = (1, 0) T ∈ S 15 ⊂ O 2 .…”

“…A classification of all compact connected Lie groups acting transitively and effectively on S(V ) was obtained by Montgomery-Samelson [37] and Borel [25]. The list contains the series SO(n), U(n), SU(n), Sp(n), Sp(n) · U(1), Sp(n) · Sp(1), (2) and the three exceptional groups G 2 , Spin (7), Spin (9).…”

“…Until now, only partial results have been obtained by Alesker in the case G = Spin (9), acting on a 16-dimensional space by the spin representation. Using the theory of plurisubharmonic functions in quaternionic variables, he constructed a 2-homogeneous Spin(9)-invariant valuation in [9].…”

A Hadwiger-Type Theorem for the Special Unitary Group

“…Some partial results have been obtained by Alesker: several Sp(n) · Sp(1)-invariant valuations on H n were constructed in [6]; a new Spin(9)-invariant valuation on R 16 is given in [9].…”

Valuations on convex functions and convex sets and Monge–Ampère operators

“…. , e 7 are the standard octonionic units (see [5] Since det is a homogeneous polynomial of degree 2, one can define the mixed determinant. This det also satisfies the Sylvester criterion of positive definiteness and the Alexandrov's inequality for mixed determinants [5].…”