Invariant Functions on Grassmannians

Abstract

It is known, that every function on the unit sphere in $\bbr^n$, which is invariant under rotations about some coordinate axis, is completely determined by a function of one variable. Similar results, when invariance of a function reduces dimension of its actual argument, hold for every compact symmetric space and can be obtained in the framework of Lie-theoretic consideration. In the present article, this phenomenon is given precise meaning for functions on the Grassmann manifold $G_{n,i}$ of $i$-dimensional subspaces of $\bbr^n$, which are invariant under orthogonal transformations preserving complementary coordinate subspaces of arbitrary fixed dimension. The corresponding integral formulas are obtained. Our method relies on bi-Stiefel decomposition and does not invoke Lie theory.

11 pages

References in corpus (1)

• The Lower Dimensional Busemann-Petty Problem for Bodies with the Generalized Axial Symmetry

Cited by in corpus (1)

• Erdelyi-Kober integrals on the cone of positive definite matrices and Radon transforms on Grassmann manifolds