$\mathrm{Spin}(9)$-invariant valuations on the octonionic plane

TL;DR

This paper computes the dimensions of invariant valuation spaces under Spin(9) on the octonionic plane and constructs a specific valuation on Riemannian manifolds, advancing understanding in octonionic geometry and valuation theory.

Contribution

It provides explicit dimension formulas for Spin(9)-invariant valuations and constructs a notable valuation on Riemannian manifolds, linking differential forms, octonionic geometry, and representation theory.

Findings

Dimensions of invariant valuation spaces are explicitly computed.

A new valuation on Riemannian manifolds is constructed.

The work connects differential forms, octonionic geometry, and valuation theory.

Abstract

The dimensions of the spaces of $k$-homogeneous $\mathrm{Spin}(9)$-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular interest is constructed which yields, as a special case, an element of $\mathrm{Val}_2^{\mathrm{Spin}(9)}$.