Invariant Functions on Grassmannians

TL;DR

This paper investigates functions on Grassmannians that are invariant under specific orthogonal transformations, providing integral formulas and a Lie-theory-free approach to understanding their structure.

Contribution

It extends invariance results to Grassmann manifolds using bi-Stiefel decomposition, avoiding Lie theory, and derives related integral formulas.

Findings

Derived integral formulas for invariant functions on Grassmannians.

Established a Lie-theory-free method using bi-Stiefel decomposition.

Generalized invariance phenomena to higher-dimensional subspace functions.

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.