Testing isomorphism between tuples of subspaces

TL;DR

This paper develops theoretical foundations and practical algorithms for determining isomorphism between tuples of subspaces under group actions, with applications to Grassmannian and graph isomorphism problems.

Contribution

It introduces polynomial-time algorithms for testing subspace tuple isomorphism without permutations, expanding computational tools in this area.

Findings

Polynomial-time algorithms for subspace isomorphism testing

Matlab implementations demonstrating practical applicability

Complexity results relating to graph isomorphism

Abstract

Given two tuples of subspaces, can you tell whether the tuples are isomorphic? We develop theory and algorithms to address this fundamental question. We focus on isomorphisms in which the ambient vector space is acted on by either a unitary group or general linear group. If isomorphism also allows permutations of the subspaces, then the problem is at least as hard as graph isomorphism. Otherwise, we provide a variety of polynomial-time algorithms with Matlab implementations to test for isomorphism. Keywords: subspace isomorphism, Grassmannian, Bargmann invariants, $H^\ast$-algebras, quivers, graph isomorphism