Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames

2013/11/19 by Jameson Cahill, Dustin G. Mixon, Cahill, Jameson +3 · 3 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Neurofibromatosis and Schwannoma Cases #Protein Tyrosine Phosphatases

paper · pdf · doi:10.48550/arxiv.1311.4748

openalex publication_date 2013/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we settle a long-standing problem on the connectivity of spaces of finite unit norm tight frames (FUNTFs), essentially affirming a conjecture first appearing in [Dykema and Strawn, 2003]. Our central technique involves continuous liftings of paths from the polytope of eigensteps 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, and also that generic FUNTFs are full spark.

Cited by

• Simply connectedness of spaces of tight frames
• Optimization and the Topology of Spaces of Parseval Frames
• Connections Between Frames with Rational Eigensteps and Semistandard Young Tableaux

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