Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames
CCONNECTIVITY AND IRREDUCIBILITY OF ALGEBRAICVARIETIES OF FINITE UNIT NORM TIGHT FRAMES
JAMESON CAHILL, DUSTIN G. MIXON, AND NATE STRAWN
Abstract.
In this paper, we settle a long-standing problem on the connectiv-ity of spaces of finite unit norm tight frames (FUNTFs), essentially affirminga conjecture first appearing in [14]. Our central technique involves contin-uous liftings of paths from the polytope of eigensteps (see [9]) to spaces ofFUNTFs. After demonstrating this connectivity result, we refine our analysisto 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 thealgebro-geometric sense, and also that generic FUNTFs are full spark. Introduction
Background.
Frame theory began with the definition of frames by Duffin andSchaeffer [12], and today, frames provide a rich source of redundant representationsand transformations. A frame is a collection of vectors { f n } n ∈ I in a Hilbert space H for which there exists strictly positive constants A and B satisfying A (cid:107) x (cid:107) H ≤ (cid:88) n ∈ I |(cid:104) x, f n (cid:105) H | ≤ B (cid:107) x (cid:107) H for all x ∈ H , where (cid:104)· , ·(cid:105) H is the inner product on H which induces the norm (cid:107) · (cid:107) H . We call the frame tight if we can take A = B . If the index set I is finite,then H ∼ = F d , where F = R or C , and if (cid:107) f n (cid:107) H = 1, we say that the frame is unitnorm . If the frame is finite, unit norm, and tight, we call it a finite unit norm tightframe (FUNTF) . To put it simply, a FUNTF is the collection of column vectorsin a matrix whose row vectors are orthogonal with equal norm, and whose columnvectors each have unit norm.Much of the early work on frames focused on infinite-dimensional frames: Fourierframes [12], Gabor frames [4], and wavelet frames [15]. In recent years, finite frameshave been studied more rigorously because of their applications (for example, inwireless telecommunications [24] and sigma-delta quantization [3]). While applica-tions for frames abound, some of the most basic questions concerning frames remainunresolved.1.2. The Frame Homotopy Problem.
The sets of real and complex FUNTFsof N vectors in d dimensions are denoted F R N,d and F C N,d respectively. The
Frame Homotopy Problem asks for which N and d these spacesare path-connected. Speculation on all the possible pairs of N and d for which Mathematics Subject Classification.
Primary 42C15, 47B99; Secondary 14M99.
Key words and phrases. frame theory, real algebraic geometry. a r X i v : . [ m a t h . F A ] J a n JAMESON CAHILL, DUSTIN G. MIXON, AND NATE STRAWN path-connecivity holds was first formally enunciated in Conjectures 7.6 and 7.7 of[14], but the problem itself was first posed by D. R. Larson in a NSF Research Ex-periences for Undergraduates Summer Program in 2002. Though there are a largenumber of degrees of freedom in the spaces of FUNTFs, it has been surprisinglydifficult to analytically construct anything but the simplest paths through thesespaces. Moreover, many of these spaces have singularities which complicate theanalysis of their geometry.The first step forward for the homotopy problem was shown in [14]. The identi-fication of FUNTFs in R with closed planar chains having links of length one inTheorem 3.3 of [2] made it possible to obtain the connectivity result of [14] usingconnectivity results for these chains [19]. Such connectivity results were studiedearlier because of the relevance with the well-studied problem of robotic motionplanning. However, the analogue of the characterization in R is not so simple.This is because the identification in R is essentially obtained by the identificationof the circle S with the real projective space RP via the map( x , x ) (cid:55)−→ (cid:18) x x x x x x (cid:19) . In R , the sphere S is not identifiable with RP . Closed chains are still releveantin this higher dimensional situation, but the configuration space is a subset ofproducts of RP embedded into the space of 3 × d vectors in d complex dimensions. Aside from this work, a lack oftechniques for constructing paths has basically made it impossible to move forwardon the problem over the last decade.1.3. Main Results.
In this paper, we completely resolve the Frame HomotopyProblem. In particular, we establish the following theorems:
Theorem 1.1.
The space F C N,d is path-connected for all d and N satisfying d ≥ and N ≥ d . Theorem 1.2.
The space F R N,d is path-connected for all d and N satisfying d ≥ and N ≥ d + 2 . The proofs of these results are distinct since the unitary group U ( d ) is connectedand the orthogonal group O ( d ) is not. This makes the proof of Theorem 1.1 sub-stantially simpler. However, it is clear that Theorem 1.2 implies Theorem 1.1 ifany complex FUNTF is path-connected to some real FUNTF. While our methodsmay be used to produce such a reduction, we establish the results independently.Let G F N,d denote the set of rank d orthogonal projections on F N with all diagonalentries equal to d/N . There is a quotient map from F F N,d to G F N,d which preservesconnectivity (see [14]), so we also solve the main problem presented in [16] as acorollary of Theorems 1.1 and 1.2:
Corollary 1.3.
The space G C N,d is path-connected for all N and d satisfying N ≥ d ≥ G R N,d is path-connected for all N and d satisfying d ≥ N ≥ d + 2. ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 3
For N and d relatively prime, it was shown in [14] that F F N,d has no singularitiesfor F = R or F = C , and hence Theorems 1.1 and 1.2 immediately imply thatsuch an F F N,d is an irreducible real algebraic variety. This raises the interestingpossibility that the singularities in a general space of FUNTFs might be due tointersections of numerous irreducible components in F F N,d . Indeed, for F R , , this isevidently the case given the extensive analysis of this space in [14]. However, byrefining our connectivity result, we show that this is simply not the case in general,and that F R , is exceptional in this regard. Theorem 1.4. F C N,d is an irreducible real algebraic variety for all N and d sat-isfying N ≥ d ≥ . F R N,d is an irreducible real algebraic variety for all N and d satisfying N ≥ d + 2 ≥ , except when N = 4 and d = 2 . This result tells us that the singularities of spaces of FUNTFs either result fromthe space folding in on itself or developing cusps. Characterizing the local geometryaround these singularities remains an open problem.
Problem 1.5.
Describe the local geometry around singular points of F F N,d .Finally, we use these methods to show that generic FUNTFs are full spark (see[1]), i.e., they have the property that every subcollection of d frame elements islinearly independent: Theorem 1.6.
A generic frame in F C N,d is full spark for every d and every N ≥ d .A generic frame in F R N,d is full spark for every d and every N ≥ d . Organization.
In Section 2, we fix our notation and provide the backgroundon necessary frame theory concepts. In Section 3, we establish our key technicaltool, a lifting lemma for paths in spaces of eigensteps. The proofs of Theorems 1.1and 1.2 are demonstrated in Section 4. In Section 5 we discuss the real algebraicgeometry of the FUNTF varieties. In particular, we introduce many definitions andresults from real algebraic geometry, we determine the dimension of each F F N,d asa real algebraic variety, we exhibit a dense subset of nonsingular points on thesevarieties, and we show that connectivity of this dense subset implies that F F N,d isan irreducible real algebraic variety. In Section 6 we show when these dense subsetsin the F F N,d form a path-connected set, thereby demonstrating when F F N,d is anirreducible real algebraic variety. In Section 7, we conclude with the consequencesof our path-connectivity results, including the proofs of Theorems 1.4 and 1.6.2.
Prelimaries and Notation
In general, we work with vectors in F d , where F is either the set R of real numbersor the set C of complex numbers. We assume that the inner product on F d is thestandard (symmetric or Hermitian) inner product. The set M F N,d consists of all d by N matrices (thought of as lists of columns) with entries in F . For X ∈ M C N,d ,we let (cid:60) ( X ) ∈ M R N,d and (cid:61) ( X ) ∈ M R N,d denote the matrices obtained by taking thereal and imaginary parts the entries, respectively.We let I k denote the k × k identity matrix, k denotes the vector in R k withentries all equal to 1, and denotes a block of zero entries whose dimensions shouldbe inferred from context. For any k × k matrix A , we use diag( A ) to denote thevector in F k with entries equal to the diagonal entries of A in order, and for anyvector v ∈ F k , we let diag( v ) denote the k × k matrix with diagonal entries coinciding JAMESON CAHILL, DUSTIN G. MIXON, AND NATE STRAWN with the entries of v and off-diagonal entries equal to zero. For a given collectionof vectors { v i } i ∈ I ⊂ F k , we let span { v i } i ∈ I denote the linear span of the collection,and we use span ⊥ { v i } i ∈ I to denote the orthogonal complement of the linear spanin F k . We shall sometimes use { e n } dn =1 to denote the standard orthonormal basisof F k .For k ≥
1, we use U ( k ) to denote the Lie group of k by k unitary matrices, O ( k ) denote the Lie group of k by k orthogonal matrices, and SO ( k ) denotesthe special orthogonal matrices (the matrices whose columns consist of positively-oriented orthonormal bases for R k ). If W is a subspace of R k , we may use SO ( W )to denote the restriction of SO ( k ) to W , and we note that there is a canonicalembedding SO ( W ) (cid:44) → SO ( k ).For any integers m and n with m ≤ n we let [ m, n ] denote the integers { m, m +1 , . . . , n } . We also abide by the convention that [ m, n ] = ∅ if n < m . Furtherassuming that m ≥ m ] denote the first m nonzero positive integers andwe let [ m ] denote [ m ] ∪ { } .2.1. Frame Theory.
We let F F N,d denote the space of finite unit norm tight frames (FUNTFs) consisting of N vectorsin F d . Given an F ∈ F F N,d , we shall interchangeably identify F with the indexedcollection { f n } n ∈ [ N ] and the d × N matrix of columns (cid:0) f f · · · f N (cid:1) . With aslight abuse of notation, if F = { f n } k n =1 and G = { g n } k n =1 are two ordered collec-tions of vectors, we identify F ∪ G with the matrix (cid:0) f · · · f k g · · · g k (cid:1) .Exploiting this identification, the matrix F F ∗ is called the frame operator of F .Eigensteps [9] are the key tool used to derive our central technical lemmas. Weshall let Λ N,d denote the space of FUNTF eigensteps , or sequences λ = { λ n ; i } i ∈ [ d ] ,n ∈ [ N ] of nonincreasing sequences λ n satisfying(i) λ i = 0 for all i ∈ [ d ](ii) λ N ; i = N/d for all i ∈ [ d ](iii) λ n (cid:118) λ n +1 for all n ∈ [ N − (cid:80) i λ n ; i = (cid:80) i λ n +1; i for all n ∈ [ N − a (cid:118) b means that the interlacing inequalities a d ≤ b d and b i +1 ≤ a i ≤ b i for i ∈ [ d − N,d is a convex polytope, and hence it is path-connected. We letint(Λ
N,d ) denote the relative interior of this polytope, which is characterized byProposition 2.1. If λ satisfied the conditions of Proposition 2.1, we say that itsatisfies strict interlacing . We also use ∂ Λ N,d = Λ
N,d \ int(Λ N,d ) to denote therelative boundary of Λ
N,d . Proposition 2.1. If λ ∈ Λ N,d , then λ ∈ int(Λ N,d ) if and only if λ n ; i < λ n +1; i for n ∈ [ i − , N − d + i − i ∈ [ d ] and λ n +1; i +1 < λ n ; i for n ∈ [ i, N − d + i − i ∈ [ d − ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 5
Proof.
In Theorem 3.2 of [17], a point (cid:98) λ ∈ Λ N,d is constructed to satisfy these strictinequalities, and these inequalities constitute the maximal list of inequalities whichmay hold without equality. (cid:3)
For a given frame F ∈ F F N,d , we shall let Λ( F ) denote the eigensteps associatedwith F . That is, { } ∪ {{ λ n ; i ( F ) } i ∈ [ d ] } n ∈ [ N ] is the set of eigenvalues (countingmultiplicity and nonincreasing in the index i ) of the operator k (cid:88) n =1 f n f ∗ n , which is referred to as the k th partial frame operator of F . Note that Λ : F F N,d → Λ N,d is a well defined mapping, but it is not injective so there can be many framesthat have the same eigensteps. We shall sometime refer to Λ as the eigensteps map.A frame F = { f n } n ∈ [ N ] is said to be orthodecomposable (OD, pronounced “odd”)if there is a nontrivial disjoint partition of [ N ] into S and T such thatspan { f n } n ∈ S = span ⊥ { f n } n ∈ T . As we shall soon see, the set of OD frames contain the singular points of the FUNTFvarieties, and the nonorthodecomposable (NOD, pronounced “nod”) frames arecontained in the set of nonsingular points of the FUNTF varieties.The correlation network was introduced in [21] to provide a simple characteri-zation of OD frames, and certain arguments dramatically simplify when they areexpressed in terms of the correlation networks. The correlation network (now some-times known as the frame graph) of a frame γ ( F ) = ( V, E ) is the undirected graphwith vertex set V = [ N ] such that ( i, j ) ∈ E if and only if (cid:104) f i , f j (cid:105) (cid:54) = 0. Proposition 2.2 (Lemma 3.2.5 in [21]) . A frame F is NOD if and only if γ ( F ) isconnected.Given a frame F ∈ F F N,d , we define the spark of F (written spark( F )) to be thesize of the smallest linearly dependent subset of F . Note that spark( F ) ≤ d + 1; ifspark( F ) = d +1 then we say that F is full spark . Observe that a frame is full sparkif and only if the following statement holds: If k < d and W ⊆ F d is a subspace ofdimension k , then | W ∩ F | ≤ k . Therefore it follows that any orthodecomposableframe with N > d is not full spark.Our final essential ingredient is the Naimark complement of a Parseval frame [18].A frame F for F d is said to be a Parseval frame if F F ∗ = I d (which is equivalent to A = B = 1 for the optimal frame bounds A and B – see the first chapter of [11]).This is equivalent to F ∗ F being an orthogonal projection. A frame G for F N − d iscalled a Naimark complement of a Parseval frame F if F ∗ F + G ∗ G = I N . Naimarkcomplements preserve many important properties of frames. Proposition 2.3.
Suppose F and G are Naimark complementary Parseval frames.Then(i) F is equal norm if and only if G is equal norm,(ii) F is OD if and only if G is OD, and(iii) F is full spark if and only if G is full spark.Here, (i) follows from considering the diagonal entries of F ∗ F and G ∗ G , (ii)follows from Proposition 2.2 since γ ( F ) = γ ( G ), while (iii) is far less obvious (seeTheorem 4(iii) in [1] for the essential ingredients of the proof). Noting that a JAMESON CAHILL, DUSTIN G. MIXON, AND NATE STRAWN
FUNTF is a scalar multiple of a Parseval frame, a series of rescalings allows usto extend the notion of Naimark complements to FUNTFs. The following is animportant consequence of the Naimark complement:
Proposition 2.4 (Corollary 7.3 in [14]) . When
N > d , F F N,d is connected if andonly if F F N,N − d is connected.With Proposition 2.3(ii), it is easy to imitate the argument for the above resultto get a similar result for NOD frames. Proposition 2.5.
When
N > d , the set of NOD frames in F F N,d is connected ifand only if the set of NOD frames in F F N,N − d is connected.3. Lifting paths in Λ N,d to F F N,d
This section provides the technical lemmas involving eigensteps that we willexploit throughout the remainder of the paper. The main idea behind these lemmasis that paths in the eigensteps polytope Λ
N,d can be lifted to paths of frames in F F N,d in such a way that the eigensteps of each frame in the frame path are givenby the corresponding point in the eigensteps path.In order to construct these paths, we first show that Theorem 7 of [9] greatlysimplifies for the frames F F N,d when considering eigensteps on the relative interior ofΛ
N,d . We now state the simplified result and present an example of the simplifiedstructure in Figure 1.
Theorem 3.1 (c.f. Theorem 7 of [9]) . Suppose λ ∈ int(Λ N,d ) , set κ ( n ) = n − N + d + 1 for n ≥ N − d , and define the d by N − matrices v ( λ ) and w ( λ ) , and thesequence of d by d matrices W n ( λ ) for n ∈ [ N − by fixing the entries v n ; i ( λ ) = (cid:104) − (cid:81) j ∈ [ n +1] ( λ n ; i − λ n +1; j ) (cid:81) j ∈ [ n +1] \{ i } ( λ n ; i − λ n ; j ) (cid:105) / for n ∈ [ d − , i ∈ [ n + 1] (cid:104) − (cid:81) j ∈ [ d ] ( λ n ; i − λ n +1; j ) (cid:81) j ∈ [ d ] \{ i } ( λ n ; i − λ n ; j ) (cid:105) / for n ∈ [ d, N − d ] , i ∈ [ d ] (cid:104) − (cid:81) j ∈ [ κ ( n ) ,d ] ( λ n ; i − λ n +1; j ) (cid:81) j ∈ [ κ ( n ) ,d ] \{ i } ( λ n ; i − λ n ; j ) (cid:105) / for n ∈ [ N − d + 1 , N − , i ∈ [ κ ( n ) , d ]0 otherwise w n ; i ( λ ) = (cid:104) − (cid:81) j ∈ [ n +1] ( λ n +1; i − λ n ; j ) (cid:81) j ∈ [ n +1] \{ i } ( λ n +1; i − λ n +1; j ) (cid:105) / for n ∈ [ d − , i ∈ [ n + 1] (cid:104) − (cid:81) j ∈ [ d ] ( λ n +1; i − λ n ; j ) (cid:81) j ∈ [ d ] \{ i } ( λ n +1; i − λ n +1; j ) (cid:105) / for n ∈ [ d, N − d ] , i ∈ [ d ] (cid:104) − (cid:81) j ∈ [ κ ( n ) ,d ] ( λ n +1; i − λ n ; j ) (cid:81) j ∈ [ κ ( n ) ,d ] \{ i } ( λ n +1; i − λ n +1; j ) (cid:105) / for n ∈ [ N − d + 1 , N − , i ∈ [ κ ( n ) , d ]0 otherwise and ( W n ) j ; i ( λ ) = v n ; i ( λ ) w n ; j ( λ ) λ n +1; j − λ n ; i when n ∈ [ i − , N − d + i − ∩ [ j − , N − d + j − when i = j and n (cid:54)∈ [ i − , N − d + i − otherwise If U is a d by d unitary matrix, and V n is a sequence of diagonal d by d unitarymatrices for n ∈ [ N − , then the sequence { f n } Nn =1 ⊂ F d defined by (1) f = u (the first column of U ) (2) f n +1 = U n V n v n and U n +1 = U n V n W n for n ∈ [ N − is such that F = (cid:0) f f · · · f N (cid:1) ∈ F F N,d . O NN E CT I V I T YAN D I RR E D U C I B I L I T Y O FF UN T F S P A C E S n v n ( λ ) w n ( λ ) W n ( λ )1 (cid:113) − ( λ − λ )( λ − λ ) λ − λ (cid:113) − ( λ − λ )( λ − λ ) λ − λ (cid:113) ( λ − λ )( λ − λ ) λ − λ (cid:113) ( λ − λ )( λ − λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ
00 0 1 (cid:113) − ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:113) − ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:113) − ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:113) ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:113) ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:113) ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ (cid:113) − ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:113) − ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:113) − ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:113) ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:113) ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) (cid:113) ( λ − λ )( λ − λ )( λ − λ )( λ − λ )( λ − λ ) v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ (cid:113) − ( λ − λ )( λ − λ ) λ − λ (cid:113) − ( λ − λ )( λ − λ ) λ − λ (cid:113) ( λ − λ )( λ − λ ) λ − λ (cid:113) ( λ − λ )( λ − λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ v ( λ ) w ( λ ) λ − λ Figure 1.
The v n , w n , and W n of Theorem 3.1 when N = 6 and d = 3. JAMESON CAHILL, DUSTIN G. MIXON, AND NATE STRAWN
Proof.
Using the notation from Theorem 7 of [9], the index sets I n ⊂ [ d ] and J n ⊂ [ d ] are defined as follows: • I n ⊆ [ d ] consists of the indices i satisfying the inclusion critera:(1) λ n ; i < λ n ; j for all j < i ;(2) and the multiplicity of λ n ; i as a value in { λ n ; j } dj =1 exceeds its multi-plicity as a value in { λ n +1; j } dj =1 . • J n ⊆ [ d ] consists of the indices i satisfying the inclusion critera:(1) λ n +1; i < λ n +1; j for all j < i ;(2) and the multiplicity of λ n +1; i as a value in { λ n +1; j } dj =1 exceeds itsmultiplicity as a value in { λ n ; j } dj =1 .The following remarks should help demystify the role of I n and J n , and how theyare expected to simplify on int(Λ N,d ).(1) The sets I n are attempting to identify the eigenspaces of the partial frameoperators (cid:80) ni =1 f i f ∗ i that are “shrinking” as n increases. This “shrinking”is effectively tracked by considering if the multiplicity of a particular eigen-value is smaller at the next step, where eigenvalues not in the next spectrumare assigned multiplicity 0.(2) The sets J n are attempting to identify the eigenspaces of the partial frameoperators that are “growing” as n increases. This “growth” is effectivelytracked by considering if the multiplicity of a particular eigenvalue is largerthan it was at the last step, where eigenvectors not in the previous spectrumare assigned multiplicity 0.(3) On int(Λ N,d ), the strict interlacing inequalities essentially imply that suc-cessive spectra of the partial frame operators will be disjoint if we excludethe eigenvalues 0 and Nd . Thus, I n = J n = [ d ] except for when n < d (whenthe eigenspace corresponding to the eigenvalue 0 is shrinking as n increases)or n > N − d (when the eigenspace corresponding to the eigenvalue Nd isgrowing as n increases).We will now show that I n = { , . . . , n + 1 } if n < d { , . . . , d } if d ≤ n ≤ N − d { κ ( n ) , . . . , d } if N − d < n ≤ N − J n = I n if n ≤ N − d { , n − N + d + 2 , . . . , d } if N − d < n < N − { } if n = N − λ ∈ int(Λ N,d ). We shall do this by demonstrating that i ∈ [ d ] belongs to I n if and only if n ∈ [ i − , N − d + i − i ∈ [ d ] belongs to J n if and only if n ∈ [ i − , N − d + i −
2] or i = 1. To do so, we make extensive use of the strictinterlacing inequalities of Proposition 2.1.We observe that λ n ;1 < λ n ; j and λ n +1;1 < λ n +1; j holds vacuously for all j < i >
1, we have that λ n ; i ≤ λ n − i − < λ n ; i − for all n − ∈ [ i − , N − d + i −
2] with n − ≥ λ n +1; i ≤ λ n ; i − < λ n +1; i − for all n ∈ [ i − , N − d + i −
2] with n ≥ . ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 9 by (2.1). Using the fact that the entries of λ n are ordered for each n , we havethat the first inclusion criterion for I n is satisfied for all index pairs ( n ; i ) satisfying i = 1 or i ∈ [2 , d ] and n ∈ [ i − , N − d + i − J n is satisfied for all index pairs ( n ; i ) satisfying i = 1 or i ∈ [2 , d ] and n ∈ [ i − , N − d + i −
2] with n ≥
0. Also note that these inequalities and the factthat the entries of λ n are ordered for each n imply that λ n ; d has multiplicity 1 in { λ n ; j } dj =1 for n ≥ d − I n . If n < i −
1, then λ n ; i = λ n ; n − =0, so the first inclusion criterion fails. If n ∈ [ N − d + i, N − λ n ; i = Nd , andsince Nd has multiplicity n − ( N − d ) in { λ n ; j } dj =1 but multiplicity n + 1 − ( N − d )in { λ n +1; j } dj =1 , the second inclusion criterion fails for these pairs of indices. Notethis also excludes the cases where i = 1 but n ≥ N − d + 1.The remaining step is to show that the pairs ( n ; i ) satisfying i ∈ [ d ] and n ∈ [ i − , N − d + i −
1] satisfy the second inclusion criterion for I n . If n = i − λ n ; i = 0. Noting that 0 has multiplicity d − i in λ n and multiplicity d − i − λ n +1 , we conclude that the second inclusion criterion is satisfied forthese cases. Finally, for all i ∈ [ d −
1] and n ∈ [ i, N − d + i − λ n ; i +1 ≤ λ n +1; i +1 < λ n ; i hold by (2.2). Coupling this with our note about the multiplicity of λ n ; d , we getthat the multiplicity of λ n ; i in λ n is 1 for all n ∈ [ i, N − d + i − λ n are ordered, and so the bounds λ n +1; i +1 < λ n ; i < λ n +1; i for all n ∈ [ i, N − d + i − i ∈ [ d −
1] and λ n ; d < λ n +1; d for all n ∈ [ d, N − λ n ; i in λ n +1 is 0 for all n ∈ [ i, N − d + i − I n consists of all i ∈ [ d ] satisfying n ∈ [ i − , N − d + i − J n , and we begin by excludingindices. If i ∈ [2 , d ] and n ∈ [ N − d + i − , N − n + 1 ∈ [ N − d + i, N ]and hence λ n +1; i = Nd . But then λ n +1;1 = Nd , and the first inclusion criteria for J n fails for these indices. On the other hand, if n = i − i ∈ [2 , d ], we know that λ n +1; n +2 = 0. Since 0 has multiplicity d − n − λ n +1 and multiplicity d − n in λ n ; j , the second inclusion criterion for J n fails for these indices. We now proceed toshow that the indices ( n ; i ) satisfying i = 1 or i ∈ [2 , d ] and n ∈ [ i − , N − d + i − J n .First note that if n ∈ [ N − d, N − λ n +1;1 = Nd has multiplicity n +1 − ( N − d ) in λ n +1 and multiplicity n − ( N − d ) in λ n , and therefore the secondinclusion criterion for J n is satisfied by these indices. Now observing that λ n +1; i +1 ≤ λ n ; i < λ n +1; i for all n ∈ [ i − , N − d + i − i ∈ [ d − λ n +1 gives us that λ n +1;1 hasmultiplicity 1 in λ n +1 , and further observing that λ n +1; i ≤ λ n ; i − < λ n +1; i − for all n ∈ [ i − , N − d + i − when i ∈ [2 , d ], we have that λ n +1; i has multiplicity 1 in λ n +1 for all n ∈ [ i − , N − d + i −
2] when i ∈ [ d ]. The inequalities λ n ;1 < λ n +1;1 for all n ∈ [1 , N − d − λ n together imply that λ n +1;1 has multiplicity 0in λ n for all n ∈ [0 , N − i ∈ [2 , d ], we have that λ n ; i < λ n +1; i < λ n ; i − for n ∈ [ i − , N − d + i − λ n together imply that λ n +1; i has multiplicity 0in λ n for these indices. Thus, the second inclusion criterion for J n is satisfied forall index pairs ( n ; i ) satisfying i ∈ [ d ] and n ∈ [ i − , N − d + i − J n consists of the index 1 along with any i ∈ [2 , d ] satisfying n ∈ [ i − , N − d + i − I n and J n , we recall the definitions of the permu-tations π I n and π J n from Theorem 7 of [9]: • First, recall that a permutation π on a set [ d ] is said to be increasing on A ⊆ [ d ] if π ( i ) ≤ π ( j ) for all i, j ∈ A with i ≤ j . • π I n is the unique permutation that is increasing on both of the sets I n and[ d ] \ I n , and such that π I n ( I n ) = [ |I n | ]. • Similarly, π J n is the unique permutation that is increasing on both of thesets J n and [ d ] \ J n , and such that π J n ( J n ) = [ |J n | ].Under our hypothesis, we note that the associated permutation matrices Π I n andΠ J n defined in Theorem 7 of [9] satisfy Π I n = Π J n = I d when n ≤ N − d ,Π I n = (cid:18) I N − n I n − N + d (cid:19) when N − d < n < N , Π J n = I N − n − I n − N + d when N − d < n < N −
1, and Π J N − = I d .We now claim that v n ( λ ) = Π T I n (cid:18) v n (cid:19) , w n ( λ ) = Π T I n (cid:18) w n (cid:19) , and W n ( λ ) = Π T I n (cid:18) W n I k (cid:19) Π I n for all n ∈ [ N − v n , w n , and W n defined in Theorem 7 of [9] are thecounterparts the v n ( λ ), w n ( λ ), and W n ( λ ) that we have defined.For n ≤ d −
1, we observe that v n ; i ( λ ) = (cid:115) − (cid:81) j ∈ [ n +1] ( λ n ; i − λ n +1; j ) (cid:81) j ∈ [ n +1] \{ i } ( λ n ; i − λ n ; j ) = (cid:115) − (cid:81) j ∈J n ( λ n ; i − λ n +1; j ) (cid:81) j ∈I n \{ i } ( λ n ; i − λ n ; j ) = v n ; i if i ∈ [ n + 1], and v n ; i ( λ ) = 0 for i ∈ [ d ]. For n ∈ [ d, N − d ], the equivalence isobvious. If n ∈ [ N − d +1 , N − v n ; i ( λ ) = 0 for i ∈ [ n − N + d ] = π I n ([ d ] \I n ). ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 11
Noting that λ n +1;1 = λ n +1; κ ( n ) = Nd if n ∈ [ N − d + 1 , N − v n ; i ( λ ) = (cid:115) − (cid:81) j ∈ [ κ ( n ) ,d ] ( λ n ; i − λ n +1; j ) (cid:81) j ∈ [ κ ( n ) ,d ] \{ i } ( λ n ; i − λ n ; j )= (cid:115) − (cid:81) j ∈{ }∪ [ n − N + d +2 ,d ] ( λ n ; i − λ n +1; j ) (cid:81) j ∈ [ κ ( n ) ,d ] \{ i } ( λ n ; i − λ n ; j )= (cid:115) − (cid:81) j ∈J n ( λ n ; i − λ n +1; j ) (cid:81) j ∈I n \{ i } ( λ n ; i − λ n ; j )= v n ; π I n ( i ) if n < N −
1, and v N − d ( λ ) = (cid:112) − ( λ N − d − λ N ; d ) = (cid:112) − ( λ N − d − λ N ;1 ) = v n, .The identity for w n ( λ ) follows similarly, and the identity for W n ( λ ) follows fromthe identities for v n ( λ ) and w n ( λ ).Now, in the inductive definition of f n in Theorem 7 of [9], U n +1 = U n V n Π T I n (cid:18) W n I k ( n ) (cid:19) Π J n , for some function k ( n ), and the update used for our theorem is effectively U n +1 = U n V n Π T I n (cid:18) W n I k ( n ) (cid:19) Π I n = U n V n Π T I n (cid:18) W n I k ( n ) (cid:19) Π J n Π T J n Π I n . For n ≤ N − d , these definitions are identical because Π T J n Π I n = I d . For n >N − d , the application of Π T J n Π I n simply permutes the basis of eigenvectors for theeigenspace of the ( n + 1)th partial frame operator corresponding to the eigenvalue Nd . Since the next vector f n +2 must be orthogonal to this subspace, it follows thatthe f n +2 of both sequences coincides. This completes the proof. (cid:3) Lemma 3.2.
Given any F ∈ F F N,d such that µ = Λ( F ) ∈ int(Λ N,d ) , there is acontinuous map θ : int(Λ N,d ) → F F N,d such that θ (Λ( F )) = F and Λ( θ ( λ )) = λ forall λ ∈ int(Λ N,d ) .Proof. Applying Theorem 7 of [9] to F , we obtain unitary matrices U and { V n } N − n =1 that can be used to recover the individual columns of F from µ and the matrices v ( µ ), w ( µ ), and W n ( µ ) from Theorem 3.1.Note that v ( λ ), w ( λ ) and W n ( λ ) are all continuous as functions of λ ∈ int(Λ N,d )because their entries are all radicals of positive-valued rational functions whosedenominators are never zero on the domain of interest. Hence, the vector-valuedfunctions defined by f ( λ ) = u , U ( λ ) = U , f n +1 = U n ( λ ) V n v n ( λ ), and U n ( λ ) = U n ( λ ) V n W n ( λ ) are all continuous as well. Consequently, the matrix-valued function θ ( λ ) = (cid:0) f ( λ ) f ( λ ) · · · f N ( λ ) (cid:1) is continuous and takes values in F F N,d . Finally, the converse part of Theorem 7 of[9] ensures that θ ( µ ) ∈ F F N,d satisfies Λ( θ ( µ )) = µ . (cid:3) Lemma 3.3.
Suppose F ∈ F F N,d with N ≥ d + 2 , assume that µ = Λ( F ) is in theinterior of Λ N,d , fix ν ∈ Λ N,d , and define the linear path (cid:96) : [0 , → Λ M,d by (cid:96) ( t ) =(1 − t ) µ + tν . Then there is a continuous lifting (or frame path), (cid:101) (cid:96) : [0 , → F F N,d such that ( λ ◦ (cid:101) (cid:96) )( t ) = (cid:96) ( t ) for all t ∈ [0 , . Proof. F satisfies the hypothesis of Lemma 3.2, so we may obtain a lift θ : int(Λ N,d ) →F F N,d satisfying θ ( µ ) = F and Λ( θ ( λ )) = λ for all λ ∈ int(Λ N,d ). Thus, we maydefine the lift of (cid:96) by (cid:101) (cid:96) = θ ◦ (cid:96) : [0 , → F F N,d and verify that (cid:101) (cid:96) (0) = F . If ν is alsoon the interior, it is clear that the this path extends continuously to all of [0 , ν lies on the boundary, then some interlacing inequalities becomeequalities ( ν n ; i = ν n +1; i or ν n ; i = ν n +1; i +1 for some n and i ) and the definitionsof v n ( λ ), w n ( λ ), and W n ( λ ) in Lemma 3.2 appear to involve undefined quantitiesbecause of the terms in the denominators. First, we show that these problematicterms cancel along this path.We shall explicitly show that the problematic denominators cancel for the indices n ∈ [ d, N − d ]. When n ∈ [ d ] or n ∈ [ N − d + 1 , N − i ∈ [ d ] and note that the set of all j such that ν n ; i = ν n ; j ( ν n +1; i = ν n +1; j for w ( λ )) forms an interval [ k , k ] ⊂ [ d ] because these valuesare ordered. The interlacing conditions on ν then imply that ν n +1; k = ν n ; i for all k ∈ [ k , k − h j ( t ) = [(1 − t ) µ n ; i + tν n ; i − (1 − t ) µ n +1; j − tν n +1; j ]and g j ( t ) = [(1 − t ) µ n ; i + tν n ; i − (1 − t ) µ n ; j − tν n ; j ]for all j ∈ [ d ], note that • h k ( t ) = (1 − t )( µ n ; i − µ n +1; k ) for all k ∈ [ k , k − • g k ( t ) = (1 − t )( µ n ; i − µ n ; k ) for all k ∈ [ k , k ]; • if k (cid:54)∈ [ k , k ], then g k ( t ) (cid:54) = 0 for all t ∈ [0 ,
1] since µ ∈ int(Λ N,d ) and ν n ; i (cid:54) = ν n ; k ; • and v n ; i ( (cid:96) ( t )) = (cid:113) − (cid:81) dj =1 h j ( t ) / (cid:81) j (cid:54) = i g j ( t ) for t ∈ [0 , v n ; i ( (cid:96) ( t )) = (cid:118)(cid:117)(cid:117)(cid:116) − (cid:81) dj =1 h j ( t ) (cid:81) j (cid:54) = i g j ( t )= (cid:115) − (cid:81) j ∈ [ k ,k − h j ( t ) (cid:81) j (cid:54)∈ [ k ,k − h j ( t ) (cid:81) j ∈ [ k ,k ] \{ i } g j ( t ) (cid:81) j (cid:54)∈ [ k ,k ] g j ( t )= (cid:115) − (cid:81) j ∈ [ k ,k − (1 − t )( µ n ; i − µ n +1; j ) (cid:81) j (cid:54)∈ [ k ,k − h j ( t ) (cid:81) j ∈ [ k ,k ] \{ i } (1 − t )( µ n ; i − µ n ; j ) (cid:81) j (cid:54)∈ [ k ,k ] g j ( t )= (cid:115) − (cid:81) j ∈ [ k ,k − ( µ n ; i − µ n +1; j ) (cid:81) j (cid:54)∈ [ k ,k − h j ( t ) (cid:81) j ∈ [ k ,k ] \{ i } ( µ n ; i − µ n ; j ) (cid:81) j (cid:54)∈ [ k ,k ] g j ( t ) , and the denominator term (cid:81) j ∈ [ k ,k ] \{ i } ( µ n ; i − µ n ; j ) (cid:81) j (cid:54)∈ [ k ,k ] g j ( t ) is nonzero forall t ∈ [0 , v ◦ (cid:96) continuously extends to all of [0 , w ◦ (cid:96) continuously to all of [0 , ν n +1; j = ν n ; i for some i and j , then let the [ k , k ] ⊂ [ d ] denote the set ofall j (cid:48) satisfying ν n ; i = ν n ; j (cid:48) and let [ l , l ] ⊂ [ d ] denote the set of all j (cid:48) satisfying ν n +1; j = ν n +1; j (cid:48) , and note that i ∈ [ k , k ] and j ∈ [ l , l ]. Setting b k ( t ) = (1 − t ) µ n +1; i + tν n +1; i − (1 − t ) µ n ; k − tν n ; k ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 13 and a k ( t ) = (1 − t ) µ n +1; i + tν n +1; i − (1 − t ) µ n +1; k − tν n +1; k , we observe that( W n ) j ; i ( (cid:96) ( t )) = v n ; i ( (cid:96) ( t )) w n ; j ( (cid:96) ( t ))(1 − t ) µ n +1; j + tν n +1; j − (1 − t ) µ n ; i − tν n ; i = v n ; i ( (cid:96) ( t )) w n ; j ( (cid:96) ( t ))(1 − t )( µ n +1; j − µ n ; i ) . This last numerator is (cid:115) − (cid:81) k ∈ [ l ,l ] h k ( t ) (cid:81) k (cid:54)∈ [ l ,l ] h k ( t ) (cid:81) k ∈ [ k ,k ] \{ i } h k ( t ) (cid:81) k (cid:54)∈ [ k ,k ] h k ( t ) (cid:115) (cid:81) k ∈ [ k ,k ] b k ( t ) (cid:81) k (cid:54)∈ [ k ,k ] h k ( t ) (cid:81) k ∈ [ l ,l ] \{ j } a k ( t ) (cid:81) k (cid:54)∈ [ l ,l ] a k ( t ) . Similar to our previous considerations, we have that • h k ( t ) = (1 − t )( µ n ; i − µ n +1; k ) for all k ∈ [ l , l ] since ν n +1; k = ν n +1; j = ν n ; i • b k ( t ) = (1 − t )( µ n +1; j − µ n ; k ) for all k ∈ [ k , k ] since ν n ; k = ν n ; i = ν n +1; j and therefore the (1 − t ) term has an exponent of ( k − k ) + ( l − l ) in thenumerator under the radical, and an exponent of ( k − k ) + ( l − l ) − − t ) under the radical,which ultimately cancels with the (1 − t ) term in the denominator of ( W n ) j ; i ( (cid:96) ( t )).We observe that all the denominator terms are now nonzero for all t ∈ [0 , W n ( (cid:96) ( t )) continuously extends to all of [0 , v ( (cid:96) ( t )), w ( (cid:96) ( t )), and the W n ( (cid:96) ( t )), ourdefinition of θ ◦ (cid:96) extends continuously to [0 , θ ◦ (cid:96) ( t ) ∈ F F N,d for all t ∈ [0 ,
1] since F N,d is compact. Moreover,continuity gives us that Λ( θ ( (cid:96) ( t )) = (cid:96) ( t ) for all t ∈ [0 , (cid:101) (cid:96) ( t ) is a continuous lifting of (cid:96) ( t ). (cid:3) Connectivity of F F N,d
We now prove the connectivity of real algebraic varieties of complex FUNTFsusing the path lifting argument of the previous section.
Proof of Theorem 1.1.
The connectivity result is well known if N = d or N = d +1,so we only consider the cases N ≥ d + 2 where the interior of Λ N,d is not empty.Let F and G belong to F C N,d . We consider two cases. First, suppose that Λ( F )belongs to the interior of Λ N,d . By Lemma 3.3, we may continuously connect F to a G (cid:48) in F C N,d with Λ( G ) = Λ( G (cid:48) ). By Theorem 7 of [9], we have that the onlydifference between G and G (cid:48) is the choice of U and the V n . However, since theunitary matrices are connected and products of connected sets are connected, thereare continuous paths connecting U (cid:48) to U and each V (cid:48) n to V n . These then inducea continuous path from G (cid:48) to G . By traversing this path after the path providedby Lemma 3.3, we successfully connect F to G by a continuous path. On the otherhand, if F and G both belong to the boundary of Λ N,d , we may choose H fromthe interior of Λ N,d and connect F to H and H to G in the manner previouslydescribed. Traversing these paths in order produces the desired path. (cid:3) In the complex case, the inverse image of a point of Λ
N,d under the map λ isconnected because the V n ’s from Theorem 7 of [9] are block diagonal with unitaryblocks and this set is connected (it is a product of unitary groups which are eachindividually connected). In the real case, this inverse image has at least two disjoint connected components. This makes the proof much more delicate, and we shallnow require access to some simple continuous motions to circumvent the ostensibleobstructions. Perhaps the most obvious motion that a FUNTF may undergo isthe “spinning” of an orthogonal pair of frame elements inside of their span. Thefollowing lemma generalizes this kind of continuous motion. Lemma 4.1.
Let G = { g n } N n =1 ⊂ R d be a FUNTF for W = span { g n } N n =1 , andsuppose H = { h n } N n =1 ⊂ R d is such that F = G ∪ H ∈ F R N,d . Let U ∈ SO ( W ) (cid:44) →SO ( U ) and set G (cid:48) = { U g n } N n =1 . Then F (cid:48) = G (cid:48) ∪ H ∈ F R N,d , and there is a there isa continuous motion in F R N,d connecting F to F (cid:48) .Proof. It is well known that SO ( W ) is connected, so there is continuous path u :[0 , → SO ( W ) such that u (0) = I d and u (1) = U . Defining F ( t ) = { u ( t ) g n } N n =1 ∪ H , we observe that (cid:107) u ( t ) g n (cid:107) = (cid:107) g n (cid:107) for all t ∈ [0 ,
1] and F ( t ) F ( t ) ∗ = N (cid:88) n =1 u ( t ) g n g ∗ n u ( t ) ∗ + N (cid:88) n =1 h n h ∗ n = u ( t ) (cid:32) N (cid:88) n =1 g n g ∗ n (cid:33) u ( t ) ∗ + N (cid:88) n =1 h n h ∗ n = u ( t ) (cid:18) N dim( W ) P W (cid:19) u ( t ) ∗ + N (cid:88) n =1 h n h ∗ n = N dim( W ) P W + N (cid:88) n =1 h n h ∗ n = N (cid:88) n =1 g n g ∗ n + N (cid:88) n =1 h n h ∗ n = F F ∗ = I d . Therefore F ( t ) ∈ F R N,d for all t ∈ [0 , F ( t ) is a continuous path with F (0) = F and F (1) = F (cid:48) . (cid:3) The next important motion involves “swapping” vectors inside a FUNTF thatis a union of two FUNTFs.
Lemma 4.2.
Let G ∈ F R N ,d and H ∈ F R N ,d and set F = (cid:0) G H (cid:1) ∈ F R N,d where N = N + N . For any N by N permutation matrix Π , there is a continuous pathfrom F to F Π in F R N,d .Proof.
Without loss of generality, we simply show that such paths exist for per-mutation matrices arising from transpositions since any permutation is a productof transpositions and successive applications of transpositions may be obtained byconcatenating paths.Suppose Π represents the transposition of the m th column of G = (cid:0) g · · · g N (cid:1) with the n th column of H (cid:0) h · · · h N (cid:1) in F . Pick any U ∈ SO ( d ) such that U g m = h n , note that the span of the vectors in G is all of R d , and use Lemma4.1 to continuously connect to (cid:0) U G H (cid:1) . Now, the columns in
U G with indices in[ N ] \ m form a FUNTF with the n th column of H . Applying Lemma 4.1 to thisFUNTF and the rotation U ∗ , we arrive at F (cid:48) = F Π. ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 15
If the pairs are from the same collection (say G ), we first choose a third “chap-erone” vector from H , and then run three continuous paths swapping vectors fromopposite collections. Let g m , g m , and h n denote the pair of vectors in G and thechaperone in H , respectively. Furthermore, let Π denote the permutation matrixwhich transposes m -th columns with ( N + n )th columns by left multiplication,let Π denote the permutation matrix which transposes the m -th columns with( N + n )th column by left multiplication, and note that Π = Π Π Π is the per-mutation matrix which transposes the m -th column with m -th column by leftmultiplication. By the previous arguments, the application of Π , Π , and thenΠ again may each be induced using continuous paths, and therefore concatenatingthese paths in the proper order may be used to induce a continuous path in F R N,d from F to F Π. (cid:3) Finally, we need paths which induce negations of target vectors.
Lemma 4.3.
Let G ∈ F R N ,d and H = ∈ F R N ,d and set F = (cid:0) G H (cid:1) ∈ F R N,d where N = N + N . For any N by N diagonal matrix D with diagonal entries from theset {− , } , there is a continuous path from F to F D in F R N,d .Proof.
We note that D factors into a product of diagonal matrices with one diagonalentry equal to − N − F , and since we may always concatenate continuouspaths to get another continuous path, we therefore need only produce a continuouspath that negates a single vector of F to obtain the full result.Without loss of generality, suppose we are attempting to negate g ∈ G . The firststep in the construction of this path is to apply Lemma 4.1 to position a chaperone h ∈ H so that g and U h are orthogonal. Then we may apply Lemma 4.1 tosimultaneously rotate g and U h to U h and − g respectively. All that is nowrequired is to transpose the columns so that g and U h arrive at − g and U h respectively. This motion is obtained from Lemma 4.2. Finally, we use Lemma 4.1to return H to its original position. (cid:3) Proof of Theorem 1.2.
We shall show the result using induction on N inside of aninduction on d , with the following induction structure:(i) For d = 2, the result was shown in [14] for all N ≥ d >
2, we have that connectivity of F R N (cid:48) ,d (cid:48) for all N (cid:48) ≥ d (cid:48) + 2 when d (cid:48) < d implies connectivity of F R N,d when d + 2 ≤ N < d .(iii) For d >
2, we prove connectivity independently for N = 2 d, d + 1 , and2 d +2, and then show that connectivity in the case of N (cid:48) implies connectivityin the case of N (cid:48) + d + 1 (which becomes relevant for showing the cases N = ( d + 2) + d + 1 = 2 d + 3 and beyond). Case ( N = 2 d ): Let F and G belong to F R N,d . Let µ denote the eigenstepsfor the frame consisting of two successive copies of the standard orthonormal basis.Using Lemma 3.3, we connect both F and G to F (cid:48) and G (cid:48) in int(Λ N,d ), and thenconnect F (cid:48) and G (cid:48) to H and H (cid:48) with Λ( H ) = Λ( H (cid:48) ) = µ . This implies that H and H (cid:48) are both a union of two orthonormal bases. We shall now show any two unionsof orthonormal bases may be continuously connected in F R d,d .Since orthonormal bases are permutation equivalent to positively oriented or-thonormal bases, we apply Lemma 4.2 to continuously connect our starting union of two successive orthonormal bases to a union of two successive, positively ori-ented orthonormal bases. Two applications of Lemma 4.1 then connects this unionof positively oriented orthonormal bases to the union of two successive standardorthonormal bases. Case ( N = 2 d + 1 , d + 2 ): Let µ denote any eigensteps for a frame F suchthat { f n } d +1 n =1 ∈ F R d +1 ,d and { f n } Nn = d +1 ∈ F R N − d − ,d . That is, F is a union of twotight subframes. In particular, if N = 2 d + 1, then F is the union of a members of F R d +1 ,d and an orthonormal basis. If N = 2 d + 2, then F is a union of two membersof F F d +1 ,d . Just as in the N = 2 d case, we now use Lemma 3.3 to connect any given F and G to H and H (cid:48) with eigensteps µ and note that H, H (cid:48) ∈ F R d +1 ,d × F R N − d − ,d .We now show that H and H (cid:48) are connected by a path in F R N,d to complete thedemonstration of this case.The central obstruction to connecting F and G in F R d +1 ,d × F R N − d − ,d is that F R d +1 ,d has 2 d connected components as shown in [14] (alternatively, one may ob-serve that the these frames are the Naimark complements of ( d +1)-length sequencesof 1s and − F to any connectedcomponent in F R d +1 ,d × F R N − d − ,d . Lemma 4.1 then finalizes the connectivity resultin these cases. Case ( N implies N + d + 1 ): We simply connect the frame with N + d + 1members to a frame such that the first d + 1 vectors form a member of F R d +1 ,d .The remaining N vectors form a member of F R N,d and hence we may generalize ourprevious arguments to connect to a frame whose first d +1 vectors form a particularmember of F R d +1 ,d . Finally, the connectivity of F R N,d implies that the last N vectorsin the frame may be connected to a particular member of F R N,d , and hence we haveshown connectivity of F R N + d +1 ,d . Case ( N (cid:48) ≥ d (cid:48) + 2 ≥ for all d (cid:48) < d implies d + 2 ≤ N < d ): Suppose N and d satisfy d + 2 ≤ N < d , and let N (cid:48) = N and d (cid:48) = N − d . Then N (cid:48) > d (cid:48) + 2and our induction step implies F R N (cid:48) ,d (cid:48) is connected. Thus, by Proposition 2.4, F R N,d is connected. (cid:3) Real Algebraic Geometry of F F N,d
A subset V ⊆ R k is called a real algebraic variety if there is a set of polynomials { p i } i ∈ I ⊆ R [ x , ..., x k ] such that V = { x ∈ R k : p i ( x ) = 0 for every i ∈ I } , where R [ x , ..., x k ] is the ring of polynomials in the variables x , ..., x k with real coefficients.We also let (cid:104){ p i } i ∈ I (cid:105) denote the ideal in R [ x , . . . , x k ] generated by { p i } i ∈ I . Givenan ideal I ∈ R [ x , ..., x k ], we let V ( I ) denote the set of points x for which p ( x ) = 0for all p ∈ I . Letting R [( x i,j ) ( i,j ) ∈ [ d ] × [ N ] ] denote the polynomials defined on theentries of real N by d matrices, we define the polynomials p R i ( X ) = Nd − N (cid:88) j =1 x i,j q R j ( X ) =1 − d (cid:88) i =1 x i,j r i,j ( X ) = N (cid:88) k =1 x k,i x k,j , ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 17 we set Π R N,d = { p R i } di =1 ∪ { q R j } N − j =1 ∪ { r i,j } i The set of NOD frames in F F N,d is semi-algebraic. Proof. Note that ∗ i,j in the definition of a semi-algebraic set may also be “ (cid:54) =”because this set is encoded by p i,j ( x ) < − p i,j ( x ) < 0. For any nonemptyproper subset A ⊂ [ N ], let A c = [ N ] \ A denote the complement, and define thepolynomials u R A ( X ) = (cid:88) n ∈ A (cid:88) m ∈ A c (cid:32) d (cid:88) k =1 x k,n x k,m (cid:33) and u C A ( X, Y ) = (cid:88) n ∈ A (cid:88) m ∈ A c (cid:32) d (cid:88) k =1 x k,n x k,m + y k,n y k,m (cid:33) + (cid:32) d (cid:88) k =1 x k,n y k,m − x k,n y k,m (cid:33) . Note that u R A ( F ) = 0 (and u C A ( (cid:60) ( F ) , (cid:61) ( F )) = 0 for the real and imaginary partsof F ∈ F C N,d ) if and only if span { f n } n ∈ A ⊥ span { f m } m ∈ A c . Therefore, the NODframes of F R N,d are exactly the intersection of the sets d (cid:92) i =1 { X ∈ M R N,d : p R i ( X ) = 0 } , N − (cid:92) j =1 { X ∈ M R N,d : q R i ( X ) = 0 } , (cid:92) ≤ i A real semi-algebraic set V induces an ideal I ( V ) ⊂ R [ x , . . . , x k ] which consistsof all polynomials that vanish on V . If I ⊂ R [ x , . . . , x k ] is an ideal, it is importantto note that I ( V ( I )) = √ I , where √ I is the radical of the ideal I , and √ I (cid:54) = I ingeneral. The dimension of a real semi-algebraic set V (denoted dim( V )) is definedto be the dimension of the ring of polynomials P ( V ) = R [ x , . . . , x k ] / I ( V ), which isequal to the maximal length of chains of prime ideals in P ( V ). While the dimensionof a real semi-algebraic set may be difficult to compute directly from this defintion,the following proposition shows that the dimension is the usual dimension whenthe real semi-algebraic is also a manifold. Proposition 5.2 (c.f. Proposition 2.8.14 of [5]) . Let V ⊂ R k be a semi-algebraicset which is a C ∞ submanifold of dimension d in R k . Then the dimension of V asa semi-algebraic set is also d .Combining this proposition, Proposition 5.1, and Corollary 4.9 of [14], we obtainthe following proposition. Proposition 5.3. The subset of NOD frames of F R N,d has dimension ( N − d − d − 1) as a real semi-algebraic set, and the subset of NOD frames in F C N,d hasdimension 2 d ( N − d ) + d − N + 1 as a real semi-algebraic set.In order to define nonsingular points on F F N,d , we actually need their dimensionas real algebraic varieties. The first step is to demonstrate that the NOD framesare actually dense in F F N,d . Proposition 5.4. The subset of NOD frames in F F N,d is dense in F F N,d . ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 19 Proof. Note that a frame in F ∈ F F N,d is either OD or NOD, so we need onlyshow that there is a NOD frame arbitrarily close to any OD frame. Given anOD frame F ∈ F F N,d , partition it into maximal NOD subsets. An orthonormal set { u i } ki =1 may be obtained by choosing a single vector from each of maximal NODsubsets. Note that for any orthonormal set { u i } ki =1 and any ε > 0, there is anotherorthonormal set { (cid:101) u i } ki =1 ⊂ span { u i } ki =1 such that (cid:80) ki =1 (cid:107) u i − (cid:101) u i (cid:107) < ε , and suchthat (cid:104) u i , (cid:101) u j (cid:105) (cid:54) = 0 for all i, j = 1 , . . . , k . Replacing the vectors { u i } ki =1 with { (cid:101) u i } ki =1 in our original frame to obtain (cid:101) F , we see that (cid:101) F ∈ F F N,d by Lemma 4.1 and weclaim that (cid:101) F is also a NOD frame for ε sufficiently small. To prove this last step,note that the correlation network of (cid:101) F retains all the edges from the correlationnetwork of the old frame since (cid:107) u i − (cid:101) u i (cid:107) < ε for ε small implies (cid:104) f k , (cid:101) u i (cid:105) (cid:54) = 0 if (cid:104) f k , u i (cid:105) (cid:54) = 0 whenever f k comes from a finite list. Then the fact that (cid:104) u i , (cid:101) u j (cid:105) (cid:54) = 0for i (cid:54) = j implies that (cid:104) f k , (cid:101) u j (cid:105) (cid:54) = 0 whenever (cid:104) f k , u i (cid:105) (cid:54) = 0. Thus, γ ( (cid:101) F ) retains all theconnected components of γ ( F ), but also includes edges between all these connectedcomponents. Consequently, γ ( (cid:101) F ) is connected and hence (cid:101) F is NOD. Finally, notethat (cid:107) F − (cid:101) F (cid:107) < ε by construction. (cid:3) Proposition 5.4 gives us that the closure of the subset of NOD frames in F F N,d is all of F F N,d . Proposition 2.8.8 of [5] then gives us the dimension of F F N,d as areal-algebraic set has the same dimension as the real-algebraic set of NOD framesin F F N,d . Proposition 5.5. As real algebraic sets, the dimensions of F R N,d and F C N,d are( N − d − d − 1) and 2 d ( N − d ) + d − N + 1, respectively.The definition of a nonsingular point of a real algebraic set requires more alge-braic background than is necessary for the rest of this paper. The interested readeris referred to Definition 3.3.9 of [5]. For our purposes, we only need the followingcharacterization. Proposition 5.6 (c.f. Proposition 3.3.10 of [5]) . Let V ⊂ R k be a real algebraicvariety of dimension d . Then x ∈ V is nonsingular in dimension d if and only ifthere exist k − d polynomials p , . . . , p k − d ∈ I ( V ) and an open neighborhood U of x in R k such that(1) V ∩ U = V ( (cid:104) p , . . . , p k − d (cid:105) ) ∩ U (2) and the Jacobian ∂p dx ( x ) · · · ∂p dx k ( x )... . . . ... ∂p k − d dx ( x ) · · · ∂p k − d dx k ( x ) has rank k − d .We now claim that any NOD frame F ∈ F F N,d is nonsingular. Since the FUNTFvarieties are locally manifolds around NOD frames by Corollary 4.9 of [14], thereis an open neighborhood U of F in M F N,d such that F R N,d ∩ U = V ( I F N,d ) ∩ U, and by the characterization of the tangent spaces of F R N,d obtained in [21], we knowthat the null space of the Jacobian of π F N,d has dimension dim( F F N,d ). Proposition 5.7. The NOD frames in F F N,d are nonsingular in dimension dim( F F N,d ),and hence the singular points of F F N,d are contained in set of OD frames of F F N,d .Proposition 5.7 brings up the possibility that orthodecomposability may notfully characterize the algebraic singularities of F F N,d , which provides an interestingproblem for future consideration. Problem 5.8. Either show that F ∈ F F N,d is a singularity of F F N,d if and only if F is OD, or exhibit an OD frame in F F N,d is not a sigularity of F F N,d .By defining closed sets to be real algebraic varieties, we get a topology on F k called the real Zariski topology (note that this is different from the usual Zariskitopology on C k ). For a subset V ⊆ F k we use the notation Z ( V ) to denote theclosure of V in this topology, that is, Z ( V ) is the smallest variety containing V .We will also use the real Zariski topology of a real algebraic variety V ⊆ F k tomean the subspace topology of the real Zariski topology of F k . Note that any setwhich is closed in the real Zariski topology is also closed in the standard topology,but the converse of this is far from true.A variety V ⊆ F k is called irreducible if we cannot write V = V ∪ V where V and V are proper subvarieties of V . A variety is called nonsingular if it hasno singular points. If a variety is reducible and path-connected, then any point inthe intersection of two irreducible components is not nonsingular. Therefore, if avariety is path-connected and nonsingular, then it must be irreducible. When N and d are not relatively prime, it is a simple exercise to construct OD frames in F F N,d , and the possibility that these are singular points means that irreducibilitydoes not follow immediately from connectedness. In our case, Proposition 5.9 givesus a way forward. Proposition 5.9. Suppose V is a real algebraic variety such that(i) the set of nonsingular points of V is path-connected, and(ii) the set of nonsingular points is dense in V (in the standard topology).Then V is an irreducible real algebraic variety. Proof. Let V denote the set of nonsingular points of V . We first claim that (i)implies Z ( V ) is irreducible. To see this, suppose to the contrary that there aretwo subvarieties of Z ( V ), say V and V , such that V ∪ V = Z ( V ) and thereexists x ∈ V ∩ V and y ∈ V ∩ V . Then any path in V connecting x and y must pass through V ∩ V (since Z ( V ) \ ( V ∩ V ) = ( Z ( V ) \ V ) ∪ ( Z ( V ) \ V )is disconnected). Overall, we have that V ∩ V intersects V nontrivially, but thiscontradicts the fact that components must intersect at singular points; this factfollows from Proposition 3.3.10 of [5].Next, we apply (ii) to get V = V , where bar denotes closure in the standardtopology. Since any Zariski closed set is also closed in the standard topology, wefurther have V = V ⊆ Z ( V ). Moreover, since V ⊆ V and V is Zariski closed,the reverse containment also holds: Z ( V ) ⊆ V . As such, V = Z ( V ) and so V isirreducible by the previous paragraph. (cid:3) The converse of Proposition 5.9 is false (see Figure 5 for counterexamples). Inour situation, we note that the argument of Proposition 5.4 indicates that, for anyOD frame F ∈ F F N,d , there is a continuous path γ : [0 , → F F N,d such that γ ( t )is a NOD frame for any t ∈ [0 , 1) and γ (1) = F . Moreover, the NOD frames are ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 21 dense in F F N,d and are a subset of the nonsingular points of F F N,d , and hence thenonsingular points of F F N,d are dense in F F N,d . Consequently, the following lemmathen follows from Proposition 5.9. Lemma 5.10. If the NOD frames in F F N,d form a connected set, then F F N,d is anirreducible real algebraic variety. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (a) y = x ( x + 1) (cid:230)(cid:230) (cid:45) (cid:45) (cid:45) (cid:45) (b) x + y = x + y Figure 2. Real algebraic counterexamples to the converse ofProposition 5.9. (a) An irreducible variety which violates (i); re-moving the singularity at the origin would disconnect the non-singular points into three pieces. (b) An irreducible variety whichviolates (ii); the nonsingular points form a curve which is not densein the variety, since the variety also contains an isolated point (asingularity).Given a subset of Euclidean space V ⊆ R k , we say W ⊆ V is generic (in V ) if W contains an open and dense set in the topology induced on V by the standardtopology on R k . If V is an irreducible variety over the complex numbers then anyZariski-open subset of V is generic, however this is not necessarily the case forirreducible real algebaric varieties. Indeed, the variety shown in Figure 5(b) is anexample of an irreducible variety that contains a Zariski-open set that is not densein the standard topology; namely, the complement of the origin. Nevertheless, usingthe same hypotheses from Proposition 5.9, we can demonstrate the following: Proposition 5.11. Let V be a real algebraic variety such that the nonsingularpoints of V form a dense connected subset. If U is another real algebraic variety,then either V \ U is empty or generic in V . Proof. Equivalently, we show that either V ⊂ U or V ∩ U is nowhere dense in V in the topology on V induced by the standard topology. If V ∩ U is nowhere densein V in the relative topology, then the statement holds, so suppose that V ∩ U isnot nowhere dense in V in the relative toplogy. Because V ∩ U is closed and notnowhere dense, this means that V ∩ U contains a nonempty open subset Q ⊂ V inthis relative topology. Now, let U denote an open cover of the nonsingular points of V so that for each W ∈ U , there is an analytic coordinate patch on W . If Q doesnot intersect some member W of U , then Q is entirely contained in the singularpoints of V , which contradicts the hypothesis that the nonsingular points of V form a dense subset of V . Thus, Q intersects some W ∈ U , and hence there is aconnected, nonempty open subset R ⊂ Q ∩ W .Now, there are real polynomials f and g such that V = { x : f ( x ) = 0 } and U = { x : g ( x ) = 0 } since V and U are real algebraic varieties. Thus, f and g must coincide on R .Because W admits an analytic coordinate system φ : Q → W for Q some nonemptyopen subset of a Euclidean space, we have that f ◦ φ − g ◦ φ = 0on φ − ( R ) as a multivariate analytic function. We claim that f ◦ φ − g ◦ φ = 0 on allof φ − ( W ). This is simply a consequence of the Identity Principle for single-variableanalytic functions. That is, suppose h : C k → C is analytic and that h ( x ) = 0 forall x ∈ R k with (cid:107) x (cid:107) < r . Then by fixing x , . . . , x k with x + · · · + x k < r , wehave that h ( x ) = h ( x , x , . . . , x k )is a one-dimensional analytic function that vanishes on a sequence of points havingan accumulation point in C . Consequently, h ( x ) = 0 for all x ∈ C , and hence h ( x , . . . , x k ) = h ( x , . . . , x k ) . Now, since h ( x , . . . , x k ) still vanishes on a ball contained in R k − , we may useinduction to see that h = 0 on all of C k .We then have that f = g on all of W . If W (cid:48) ∈ U intersects W nontrivially, thenthe same reasoning as above shows that f = g on W (cid:48) . Now, let A denote the unionof all W (cid:48) ∈ U such that f = g on W (cid:48) and let B denote the union of all open sets W (cid:48) ∈ U such that f (cid:54) = g on W (cid:48) . Then A and B are both open, and if A and B intersect nontrivially, there is a nontrivial intersection between some W (cid:48) and W (cid:48)(cid:48) in U such that f = g on W (cid:48) and f (cid:54) = g on W (cid:48)(cid:48) . This is a contradiction by our aboveobservation, so we see that the set of nonsingular points of V coincides with thedisjoint union of open A and B , which, by connectedness of V and nonemptynessof A , implies that B is empty and hence f = g on all nonsingular points of V .Since U is closed, the above reasoning immediately implies that V ⊂ U . (cid:3) Overall, if the nonsingular points of a real algebraic variety form a dense con-nected subset, then the nonempty Zariski-open subsets of that variety are generic,as desired. It should be noted that the above proposition employs the topologicaldefinition of connectivity, but Corollaries 6.10 and 6.11 say that our set of nonsin-gular points form a path-connected set. However, it is clear that the nonsingularpoints form an analytic manifold, and hence these two definitions of connectivitycoincide. 6. Connectivity of the subset of NOD frames in F F N,d In this section, we refine the results of the previous section to show that, giventwo NOD frames F, G ∈ F F N,d , there is a continuous path connecting F and G in F F N,d that does not pass through the OD frames. ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 23 To begin with, we need some lemmas that reveal the structure of the OD frames.Our first lemma essentially allows us to know that a frame is NOD if we can extracta NOD basis from the frame. Lemma 6.1 (Proposition 4.2 in [23]) . A frame F is OD if and only if every basiscontained in F is OD. This next lemma tells us that the eigensteps of an OD frame must be on theboundary of Λ N,d . Thus, when we use Lemma 3.3 to lift paths through the interiorof Λ N,d , we know that the path never crosses an OD frame. Lemma 6.2. Suppose F ∈ F F N,d is OD then Λ( F ) ∈ ∂ Λ N,d .Proof. Since F is OD, there is an index k > f k is orthogonal to f i foreach i < k . Thus, the nonzero values of λ k ( F ) consists of a 1 and all of the nonzerovalues of λ k − ( F ). Since the largest nonzero value of λ k − is at least 1, this meansthat λ k − ( F ) = λ k ;1 ( F ) and hence Λ( F ) is on the boundary of Λ N,d . (cid:3) Our final lemma for this section tells us that if we reorder any NOD frame so thatthe first d vectors of the frame form a NOD basis, then there are no OD frames thatmap to the same eigensteps. This means that when the V n ’s are being continuouslydiagonalized in Lemma 3.3, the path avoids the OD frames. Lemma 6.3. Let F = { f n } Nn =1 ∈ F F N,d . (i) If the first d vectors in F , { f n } dn =1 , form a NOD basis, then Λ( F ) is notin the image of the OD frames under the eigensteps map. (ii) If F is NOD, then there is a permutation σ such that { f σ ( n ) } dn =1 is a NODbasis and hence F (cid:48) = { f σ ( n ) } Nn =1 has eigensteps which are not in the imageof the OD frames under the eigensteps map.Proof. First, we prove part (i). For the eigensteps of an OD basis { g n } Nn =1 , therewill be a k > λ k ( { g n } Nn =1 ) consist of the nonzerovalues of λ k − ( { g n } dn =1 ) together with 1. Consequently, there are no OD frameswith eigensteps Λ( { f n } Nn =1 ).To prove part (ii), we first identify a NOD basis from F , B = { b n } dn =1 . Let b = f and set n = 1. Inductively, we now choose b k = f n k for d ≥ k > f n k is not in span ⊥ { b n } k − n =1 or span { b n } k − n =1 , and n k is not in { n , . . . , n k − } . Weclaim that there is always such an f n k . Suppose there is not. Then all f n such that n (cid:54)∈ { n , . . . , n k − } are either in span ⊥ { b n } k − n =1 or span { b n } k − n =1 , and since neither ofthese spaces will be empty, this implies that F is necessarily OD, which contradictsour hypothesis. As such, we may take σ to be any permutation sending k to n k forall k = 1 , . . . , d . (cid:3) Now that we have these lemmas, it is fairly straightforward to prove the refinedconnectivity result for all of the cases where N (cid:54) = 2 d . Theorem 6.4. Suppose N ≥ d + 2 ≥ and N (cid:54) = 2 d . Then the NOD members of F F N,d form a path-connected set.Proof. The structure of this proof is similar to that of Theorem 1.2. We proceedwith the exact induction structure, but we now take care to construct paths thatdo not pass through OD frames. Additionally, the application of Proposition 2.4in Theorem 1.2 is replaced with the application of Proposition 2.5, and the case N ≥ d + 1 only needs the connectivity result in Theorem 1.2. The first observation we must make is that if there are no OD frames witheigensteps λ or µ , then the path connecting F and G with eigensteps λ and µ provided in Theorem 1.2 never passes through any OD frames. The argumentsupporting this statement occurs in two steps. First, two frames F and G in F F N,d with N > d are connected to frames H and H (cid:48) with eigensteps ν that ensurethe first d + 1 members of H and H (cid:48) each form a a member of F R d +1 ,d . Sincethere are no OD frames with eigensteps ν (Lemma 6.1 gives this since the first d vectors of a member of F R d +1 ,d form a NOD basis), the path provided by Lemma3.3 does not pass through eigensteps that are in the image of the OD frames underthe eigensteps map. Finally, note that each frame in the path from H to H (cid:48) constructed in Theorem 1.2 always contains d vectors of a a member of F R d +1 ,d ,which is necessarily a nonorthdecomposable basis and hence Lemma 6.1 ensuresthat the full frame is not OD.The only remaining issue to deal with is that either F or G may have eigen-steps in the image of the OD frames under the eigensteps map, so the continuousdiagonalization of the V n ’s might cross an OD frame. Let us suppose that thereis an OD frame with the same eigensteps as F . Since F is NOD Lemma 6.3 pro-vides us with a permutation σ such that F (cid:48) = { f σ ( n ) } Nn =1 has that Λ( F (cid:48) ) is notin the image of the OD frames under the eigensteps map. Therefore we may usethe reasoning in the preceding paragraph to connect F (cid:48) to a frame H such that { h n } d +1 n =1 is a member of F R d +1 ,d , and in such a way that no frame along this path isOD. Let (cid:96) ( t ) = { f (cid:48) n ( t ) } Nn =1 denote this path and note that { f (cid:48) σ − ( n ) ( t ) } Nn =1 is alsoa path through NOD frames which takes our original F to a frame which containsa member of F F d +1 ,d . Let H denote this frame.Now, we use Lemma 4.2 to rearrange H so that the first d + 1 vectors in H forma member of F F d +1 ,d . Note that the intermediate frames in the path produced byLemma 4.2 will always contain at least d vectors from a member of F F d +1 ,d , andhence all these intervening frames along the path are NOD.Finally, we finish the proof by observing that our paths produced in Theorem 1.2to connect frames containing a member of F F d +1 ,d as the first d + 1 vectors alwaysconsist of frames which have at least d members of a frame from F F d +1 ,d . Thus,the final paths connecting F and G also avoid the OD frames, and the proof iscomplete. (cid:3) Using Lemma 6.5, it is not difficult to show that F C d,d is also path connected. Lemma 6.5 (Lemma 2.2 in [8]) . Two generic orthonormal bases are full spark. Theorem 6.6. The NOD members of F C d,d are path-connected.Proof. As in the proof of Theorem 6.4 if both F and G have eigensteps which arenot in the image of the OD frames, then the paths constructed in Section 4 will notpass through OD frames. Without loss of generality, we can now assume that thereis an OD frame that has the same eigensteps as F . Since F is not OD, Lemma6.3 provides us with a permutation σ such that F (cid:48) = { f σ ( n ) } Nn =1 has that Λ( F (cid:48) ) isnot in the image of the OD frames under the eigensteps map and note that thispermutation satisfies σ (1) = 1. By Lemma 6.5, we can choose two orthonormalbases, say { e i } di =1 and { u i } di =1 such that { e i } di =1 ∪ { u i } di =1 is full spark. Reorderthese vectors into a frame H = { h n } dn =1 so that h = e , h = u , and h σ − (2) = u .Observe that { h n } dn =1 is a NOD basis so Λ( H ) is not in the image of the OD frames ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 25 by Lemma 6.3. Now we can connect F (cid:48) to H so that the path does not go throughany OD frames. Next apply σ − to this path to get a path from F to σ − ( H ) whichdoes not pass through any OD frame. Note that Λ( σ − ( H )) is not in the image ofthe OD frames since h σ (1) = e and h σ (2) = u and H is full spark so { h σ ( n ) } dn =1 is a NOD basis. We have shown that any NOD frame in F C d,d can be connected toa frame whose eigensteps are not in the image of the OD frames. It follows thatwe can connect any two NOD frames in this set without passing through an ODframe. (cid:3) The above result does not apply to the real case when N = 2 d . In the process ofconnecting our F (cid:48) to H , in the real case we may only ensure that we can connect F (cid:48) to a frame sharing the same eigensteps as H . Additionally, the path constructedin Theorem 1.2 for the case N = 2 d requires that we align frame vectors directlyon top of each other. Once this alignment occurs, we have reached an OD frame.This makes the proof for F R d,d much more involved.In order to show that we can connect two NOD frames in F R d,d without passingthrough an OD frame, we shall still use the unions of two orthonormal bases asthe central nexus of our paths. As in Theorem 1.2, once we get to a single unionof two orthonormal bases which constitutes a NOD frame (a nontrivial task), wejust need to show that we can permute the vectors via continuous paths whileremaining NOD. The next lemma demonstrates that continuous permutations canbe performed for a particular NOD frame consisting of the union of two orthonormalbases. Lemma 6.7. Suppose d ≥ , N = 2 d , and fix F ∈ F R d − ,d − (in the case d = 3 ,this is vacuous). Then the following are true: (i) There is a ξ ∈ {− , } d − such that U = √ √ ,d − √ − √ ,d − √ d − ξ d − , d − , (cid:113) d − d − F , V = √ √ ,d − − √ d − ξ √ − √ ,d − d − , d − , (cid:113) d − d − F are both positively oriented orthonormal bases and F ∗ = ( u v u v v · · · v d u · · · u d ) is not OD, where u i and v i are the i th columns of U and V respectively. (ii) There is a continuous path through the NOD members of F R d,d which con-nects F ∗ to G ∗ = ( v u u v v · · · v d u · · · u d ) . Proof. First, we let ξ be a member of F R d − , which correponds to the Naimarkcomplement of F . Now, ξ is unique up to a global sign factor, so we choose thesign so that (cid:32) √ d − ξ (cid:113) d − d − F (cid:33) is a negatively oriented orthonormal basis. Thus the resulting U and V are posi-tively oriented orthonormal bases. Now, note that u has a nonzero inner productwith each member of v i and hence F ∗ is not OD. We now describe the continous path connecting F ∗ to G ∗ . First, we note thatthe frame operator of the collection F ∗ = { u , v , u , v } is diag(2 , , , , . . . , F ∗ = { u i } di =3 ∪ { v i } di =3 is diag(0 , , , , . . . , F ∗ and F ∗ in theplane spanned by the standard orthonormal vectors e and e , and the resultingframe is still always in F R d,d and is NOD. Our first action is to continuously rotate F ∗ in the span of e and e to arrive at the frame √ √ √ √ ,d − ,d − √ − √ √ d − ξ − √ d − ξ √ − √ 22 1 √ d − ξ √ d − ξ d − , d − , d − , d − , (cid:113) d − d − F (cid:113) d − d − F . The reason for doing this is to avoid any OD frames during a motion that will onlyinvolve the first four vectors. At the end of the motion involving only the first fourvectors, we shall undo this rotation.We now restrict our attention to the frame √ √ √ √ √ − √ √ − √ . By a continuous rotation of π/ e and e , we maymove to the frame √ √ √ √ − √ √ √ − √ . This has permuted the columns by a 4-cycle. We now produce a 3-cycle on the lastthree columns. First, we may continuously rotate the orthonormal pair consistingof the second and fourth columns by Lemma 4.1 until we arrive at the frame √ √ 00 0 0 1 √ − √ . Now, the third and fourth columns are orthonormal and we use Lemma 4.1 againto continuously rotate to √ √ − √ − √ . One final application of Lemma 4.1 on the second and third columns yields √ √ √ √ √ − √ √ − √ . Clearly, this path lifts to a continuous path taking the collection { u , v , u , v } to the collection { v , u , u , v } such that all intervening frames have unit-normmembers and frame operator diag(2 , , , , . . . , ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 27 In this paragraph, we justify why this path never meets an OD frame. First, wenote that the projection of F ∗ into the span of { e , e } contains an orthogonal pairand the inner product of any vector in the rotation of F ∗ with any vector in F ∗ isequal to the inner products of the vectors if they are first projected into the spanof { e , e } . Now, simply rotating in the span of { e , e } by some Q , it is easy tosee that QF ∗ still contains an orthogonal pair and thus the rotated vectors from F ∗ shall have nonzero inner product with at least one vector from this pair. Usingthis fact and the fact that QF ∗ always has a connected correlation network, weshall have that the entire frame has a connected correlation network and is henceNOD. Furthermore, note that v persists as a vector throughout the remainingoperations. Since v has nonzero correlation with all the rotations of the vectorsin F ∗ and e , v together with the rotations of the vectors in F ∗ has a connectedcorrelation network and forms a frame for R d . Consequently, the full frame is neverOD throughout the continuous motions described above.Finally, we undo the starting rotation on F ∗ to arrive at G ∗ . (cid:3) Theorem 6.8. The set of NOD frames in F R d,d is path-connected for d ≥ .Proof. The proof occurs in three parts. First, we show that any NOD frame con-nects to a frame in F R d,d containing a member of F R d,d − in a ( d − R d , which is necesarily NOD. In the next step, we show that thereis a relatively simple motion that takes any frame in F R d,d containing a memberof F R d,d − in a ( d − F ∈ F R d,d is NOD. Let µ denote the eigensteps of a frame in F R d,d such that the first d vectors of the frame form a member of F R d,d − in a ( d − R d . Using the same permutation argument as in Theorem1.2, we may essentially employ Lemma 3.3 to connect F to a frame containing amember of F R d,d − in ( d − R d without crossing an ODframe.Now, let { u , . . . , u d } denote the frame vectors forming the member of F R d,d − ina ( d − { v , . . . , v d } denote the remaining vectorsin the frame. By continuous rotation, we may assume (without loss of generality)that span { u , . . . u d } = span { e , . . . e d } . Thus, the frame operator of { v , . . . , v d } is diag (cid:18) , d − d − , . . . , d − d − (cid:19) , and thus there is a ξ ∈ { , − } d and an H ∈ F R d,d − such that the coordinaterepresentation of { v , . . . , v d } is given by (cid:113) d ξ (cid:113) d − d H . Let us identify { u , . . . , u d } with H (cid:48) ∈ F R d,d − , and let ζ ∈ {− , } d be a Naimarkcomplement of H (cid:48) . Thus, the coordinate representation of { u , . . . , u d } is given by (cid:18) · ζH (cid:48) (cid:19) . when viewed as a member of F R d +1 ,d in span { e , . . . , e d } . By continuous rotation,we may assume that ζ ξ i d + d − d (cid:104) h (cid:48) , h i (cid:105) (cid:54) = 0where h (cid:48) is the first vector of H (cid:48) and h i is i th vector of H for all i = 1 , . . . , d . Thisis because h (cid:48) may be rotated to any point on the unit sphere in ( d − 1) dimensions,and the intersection of the sphere with the set-theoretic complement of any finitenumber of hyperplanes is always dense in the sphere. We now use the followingpath: V ( t ) = (cid:113) − td ξ (cid:113) d − td H and U ( t ) = (cid:113) td ξ (cid:113) d − td H (cid:48) . By construction, V (0) = { v , . . . , v d } and U (0) = { u , . . . , u d } , and V (1) and U (1)are both orthonormal bases. Moreover, the frame operators of V ( t ) and U ( t ) arediag (cid:18) − t, d − td − , . . . , d − td − (cid:19) and diag (cid:18) t, d − td − , . . . , d − td − (cid:19) so the union of V ( t ) and U ( t ) always forms a member of F R d,d . By construction,we also have that V ( t ) is an equiangular set with nonzero mutual inner productsuntil t = 1. Thus, V ( t ) is NOD for t ∈ [0 , 1) and hence the union of V ( t ) and U ( t )is NOD for t ∈ [0 , h (cid:48) , h (cid:48) (1) has nonzero inner product withall of the vectors in V (1), and hence we also get that the union of V (1) and U (1)is also NOD. This completes the second step of our proof.Our final goal is to demonstrate that the set of all NOD unions of two orthonor-mal bases is connected. This task is divided into two parts. First, we show that if { i k } dk =1 and { j k } dk =1 form a partition of [2 d ] and ( a, b ) ∈ {− , } and if G ( { i k } dk =1 , { j k } dk =1 , a, b )is the set of all NOD frames G = { g i } di =1 with { g i k } dk =1 and { g j k } dk =1 both orthonor-mal bases having orientations a and b respectively, then G ( { i k } dk =1 , { j k } dk =1 , a, b )is path-connected. The final step is to then use Lemma 6.7 to permute the framevectors, thus arriving at a NOD frame for which the first d vectors are the standardorthonormal basis and the last d vectors form another orthonormal basis.Let G ∈ G ( { i k } dk =1 , { j k } dk =1 , a, b ). For convenience, we first permute so that { i k } dk =1 = { k } dk =1 and { j k } dk =1 = { k } dk = d +1 and { g i } di =1 and { g i } di = d +1 are bothpositively oriented orthonormal bases. If we construct a path for this configuration,then we may invert the index permutation over the entire path to get a path through G ( { i k } dk =1 , { j k } dk =1 , a, b ). Additionally, by continuous rotation, we may assumethat { g i } di =1 is the standard orthonormal basis. Now, consider g d +1 . There is acontinuous rotation U ( t ) and an ε > U ( t ) g d +1 ) j (cid:54) = 0 for all j = 1 , . . . , d ,and g ji (cid:54) = 0 implies ( U ( t ) g i ) j (cid:54) = 0 for all i = d + 1 , . . . , d and all t ∈ (0 , ε ).Thus, without loss of generality, we may assume that g j,d +1 are all nonzero. Wenow show that we may assume that all of g j,d +1 are strictly positive. Suppose g j,d +1 is negative, and choose another coordinate index i (cid:54) = j . Without moving g d +1 , rotate g d +2 so that its nonzero projection onto the span of e i and e j is notorthogonal or parallel to the projection of g d +1 onto the span of e i and e j . Thisis possible because d > g d +1 are nonzero, and since g d +1 stays fixed and has all nonzero entries, the full frame remains NOD while g d +2 rotates. At this point, we continuously rotate { g i } di = d +1 in the span of e and ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 29 e until the i and j entries of g d +1 are strictly positive. Since the j th entry wasnegative, the intermediate value theorem tells us that this entry becomes zero atsome point during this rotation. Thus, e i or e j may become orthogonal to g d +1 atsome point. However, our positioning of g d +2 ensures that g d +2 will have nonzeroinner product with both e i and e j at these points. Thus, the entire frame remainsNOD during this procedure. Once all of the entries of g d +1 are all strictly positive,we continuously rotate g d +1 to the vector √ d d while keeping all of the entries of g d +1 strictly positive and hence the full frame remains NOD during this procedure.We are now done if we have path-connectivity of the set of frames such that g i = e i for i = 1 , . . . , d , g d +1 = √ d d , and such that { g i } di = d +1 is a positively orientedorthonormal basis. All of these frames are NOD, and the path-connectivity followsfrom the connectivity of SO ( d − d andlast d vectors form two positively oriented orthonormal bases, and then connectthis to a frame with the standard orthonormal basis as the first d vectors and theconstant vectors as the ( d + 1)th vector without passing through the OD frames.Since this set of frames is path-connected and contains no OD frames, and we cancontinuously connect any NOD unions of orthonormal bases by a path throughNOD unions of orthonormal bases. This completes the proof. (cid:3) Example 6.9. Here we give an example for the motion in F R , . In Figure 3 wesee the motion from a frame consisting of a member of F R , in the x - y plane andthe subframe with frame operator diag(1 , , F R , up and the subframe vectorsdown to get the second frame, which is a union of two orthonormal bases. (a) (b) Figure 3. Subfigure (a) illustrates the starting point: a unionof a member of F R , for a two-dimensional subspace of R and asubframe with frame operator diag(1 , , F R , towards the top pole of the sphere and pushing the other vectorsaway. For the swapping phase of the motion, we first align the orthonormal basis whichcomplements the subframe with frame operator diag(1 , , (cid:13) and (cid:52) . (a) (b) Figure 4. Subfigure (a) indicates the starting position. Subfigure(b) shows how the orthonormal pair spins so that the remainingmotions never pass through an OD frame.These last corollaries summarize all of the results of this section. Corollary 6.10. If N and d satisfy N ≥ d ≥ 1, then the NOD frames in F C N,d form a path-connected set. Corollary 6.11. If N and d satisfy N ≥ d + 2 > d = 2 and N ≥ 5, then theNOD frames in F R N,d form a path-connected set.7. Irreducibility of F F N,d and consequences Combining Corollaries 6.10 and 6.11 and Lemma 5.10, Theorem 1.4 follows. Inthis section, we explore an interesting consequence of this irreducibility. In par-ticular, we turn our attention to the demonstration of Theorem 1.6. By Proposi-tion 5.11, finishing our demonstration that a generic frame in F F N,d is full spark justrequires that there exists a full spark frame in F F N,d . Theorem 7.1 (Theorems 4 and 5 in [20]) . For every d and every N ≥ d , thereexists a full spark FUNTF { f n } Nn =1 ⊆ R d .Proof of Theorem 1.6. By Theorem 7.1, the set of full spark frames in F F N,d isnonempty, and hence the real generic property holds by Proposition 5.11 since thenonsingular points of F F N,d form a connected dense subset (Corollaries 6.10 and 6.11,the discussion preceding Lemma 5.10, and Lemma 5.4). (cid:3) ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 31 (a) (b)(c) (d) Figure 5. Subfigure (a) shows the spinning of the vectors labeled( (cid:52) (cid:3)♦ (cid:13) ) in Figure 4(b) to the ordering ( (cid:13)(cid:52) (cid:3)♦ ). The remainingfigures illustrate how to perform a cycle of ( (cid:52) (cid:3) ♦ ) to end at theordering ( (cid:13) (cid:3) ♦ (cid:52) ). In Subfigures (b) through (d), we spin ( (cid:52) (cid:3) ),( ♦ (cid:3) ), and finally ( (cid:3) (cid:52) ). 8. Discussion Proposition 5.11 is often identified with the additional property that V ∩ U iseither a null set or has full measure in V . This additional property presupposes theexistence of a uniform measure on F F N,d . If the algebraic variety happens to also be amanifold, we can be sure that this uniform distribution exists (see [25], for example).On the other hand, in private communications with Christopher Manon of GeorgeMason University, it has been suggested that the theory of Duistermaat-Heckmanmeasures [13] allows us to induce a uniform distribution on F F N,d by (1) drawing apoint uniformly from the the polytope of eigensteps, (2) uniformly drawing fromthe possible U and the V n in Theorem 7 of [9], and (3) reconstructing the framefrom the iteration from the data provided in (1) and (2), as indicated by Theorem7 of [9]. Because this procedure is still under active research, we only definitivelysay that the full spark frames have full measure in the uniform measure of F F N,d when N and d > Another interesting question is whether these results can be extended to theinfinite-dimensional setting. Since the eigensteps construction is our primary tool,the first step of this process would involve a generalization of this construction.One could also study whether similar results hold for sets of frames with differentframe operators and different norms of the frame vectors. Acknowledgements We would like to thank Bernhard Bodmann, Gitta Kutyniok, and Tim Roemerfor organizing the American Institute of Mathematics workshop “Frame Theoryintersects Geometry” where this work began. We thank the American Instituteof Mathematics for their great generosity. We also would like to thank Eva-MariaFeichtner and Emily King for organizing the workshop “Frames and Algebraic &Combinatorial Geometry,” which has brought more light to the application of al-gebraic geometry in frame theory. J. Cahill was supported by NSF Grant No.ATD-1321779. D. G. Mixon was supported by an AFOSR Young InvestigatorResearch Program award, NSF Grant No. DMS-1321779, and AFOSR Grant No.F4FGA05076J002. N. Strawn was supported by NSF Grant No. DMS-10-45153.The views expressed in this article are those of the authors and do not reflect theofficial policy or position of the United States Air Force, Department of Defense,or the U.S. Government. References 1. B. Alexeev, J. Cahill, D. G. Mixon, Full spark frames, J. Fourier Anal. Appl. 18 (2012)1167–1194.2. J. J. Benedetto, M. Fickus, Finite normalized tight frames, Adv. Comput. Math. 18 (2003)357–385.3. J. J. Benedetto, A. M. Powell, O. Yılmaz, Sigma-Delta (Σ∆) quantization and finite frames,IEEE Trans. Inf. Theory 52 (2006) 1990–2005.4. J. J. Benedetto, D. F. Walnut, Gabor frames for L and related spaces, In: Wavelets: Math-ematics and Applications, 1994, pp. 97–162.5. J. Bochnak, M. Coste, M. F. Roy, Real Algebraic Geometry. Vol. 36. Springer Science &Business Media, 2013.6. J. Cahill, Flags, Frames, and Bergman Spaces, Master’s Thesis, San Francisco State Univer-sity, 2009.7. J. Cahill, P. G. Casazza, A. Heinecke, A notion of redundancy for infinite frames, Proc. Sampl.Theory Appl. (2011).8. J. Cahill, P. G. Casazza, J. Peterson, L. Woodland, Phase retrieval by projections, Availableonline: arXiv:1305.62269. J. Cahill, M. Fickus, D. G. Mixon, M. J. Poteet, N. Strawn, Constructing finite frames of agiven spectrum and set of lengths, Appl. Comput. Harmon. Anal. 35 (2013) 52–73.10. J. Cahill, P. G. Casazza, The Paulsen problem in operator theory, Available online:arXiv:1102.234411. P. G. Casazza, G. Kutyniok, Finite frames: Theory and applications, Springer, 2013.12. R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc.72 (1952) 341–366.13. J. J. Duistermaat, G. J. Heckman. On the variation in the cohomology of the symplectic formof the reduced phase space. Inventiones Mathematicae 69.2 (1982): 259-268.14. K. Dykema, N. Strawn, Manifold structure of spaces of spherical tight frames, Int. J. PureAppl. Math. 28 (2006) 217–256.15. I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of waveletframes, Appl. Comput. Harmon. Anal. 14 (2003) 1–46. ONNECTIVITY AND IRREDUCIBILITY OF FUNTF SPACES 33 16. J. Giol, L. V. Kovalev, D. Larson, N. Nguyen, J. E. Tener, Projections and idempotentswith fixed diagonal and the homotopy problem for unit tight frames, Available online:arXiv:0906.013917. T. Haga, C. Pegel, Polytopes of eigensteps of finite equal norm tight frames, arXiv preprintarXiv:1507.04197 (2015).18. D. Han, D. R. Larson, Frames, bases, and group representations, Mem. Amer. Math. Soc.174, 2000.19. W. Lenhart and S. Whitesides, Reconfiguring closed polygonal chains in Euclidean d-space.Discrete & Computational Geometry 13, no. 1 (1995): 123-140.20. M. P¨uschel, J. Kovaˇcevi´c, Real, tight frames with maximal robustness to erasures, Proc. DataCompr. Conf. (2005) 63–72.21. N. Strawn, Geometry and constructions of finite frames, Master’s Thesis, Texas A&M Uni-versity, 2007.22. N. Strawn, Finite frame varieties: nonsingular points, tangent spaces, and explicit local pa-rameterizations, J. Fourier Anal. Appl. 17 (2011) 821–853.23. N. Strawn, Optimization over finite frame varieties and structured dictionary design, Appl.Comput. Harmon. Anal. 32 (2012) 413–434.24. T. Strohmer, R. W. Heath, Grassmannian frames with applications to coding and communi-cation, Appl. Comput. Harmon. Anal. 14 (2003) 257–275.25. D. W. Stroock, Essentials of integration theory for analysis, Springer, 2011. Department of Mathematical Sciences, New Mexico State University, Las Cruces,New Mexico, 88003 E-mail address : [email protected] Department of Mathematics and Statistics, Air Force Institute of Technology,Wright-Patterson AFB, Ohio, 45433 E-mail address : [email protected] Department of Mathematics and Statistics, Georgetown University, Washington,District of Columbia, 20007 E-mail address ::