Connectivity of spaces of finite unit-norm tight frames

摘要

We show that the spaces of finite unit norm tight frames are connected, which verifies a conjecture first appearing in Dykema and Strawn (2006). Our central technique involves continuous liftings of paths from the polytope of eigensteps (see Cahill et al., 2012), or Gelfand-Tsetlin patterns, to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this result to show that spaces of FUNTFs are irreducible in the algebro-geometric sense.