Harder-Narasimhan Filtrations and Zigzag Persistence

We introduce a sheaf-theoretic stability condition for finite acyclic quivers. Our main result establishes that for representations of affine type $\widetilde{\mathbb{A}}$ quivers, there is a precise relationship between the associated Harder-Narasimhan filtration and the barcode of the periodic zigzag persistence module obtained by unwinding the underlying quiver.


Introduction
This paper concerns representations of acyclic quivers of affine type A n . The underlying graph of any such quiver is the n-cycle as drawn below, but one does not obtain a directed cycle after the edges have been oriented: · · · e n−3 x n−3 e n−2 x n−2 Our goal here is to describe remarkable formulas which relate two discrete quantities that are associated to every finite-dimensional representation V of such a quiver. The first one has its roots in geometric invariant theory, and constitutes a numerical reduction of V's Harder-Narasimhan filtration along a special choice of stability condition. The second quantity of interest arises in the algebraic study of persistent homology. To obtain it, one first lifts V to an n-periodic zigzag persistence module over the infinite quiver · · · e n−1 x n−1 e 0 x 0 e 1 x 1 e 2 · · · e n−2 x n−2 e n−1 x n−1 e 0 · · · , and then catalogues the multiplicities of its indecomposable summands. Before outlining the main contributions of our work, we provide brief summaries of both quantities below.
Harder-Narasimhan Filtrations. For the purposes of these introductory remarks, a stability condition on a finite acyclic quiver Q with vertex set Q 0 is a map α : Q 0 → R that assigns a real number α x to each vertex x. The α-slope of a nonzero finitedimensional representation V of Q is the ratio Here V x denotes the vector space assigned by V to each vertex x. We call V semistable if the inequality ϕ α (U) ≤ ϕ α (V) holds for every nonzero subrepresentation U ⊂ V.
Once we fix α, there exists a unique, finite length filtration V • of V: where the successive quotient representations S j := V j /V j−1 are semistable and have strictly decreasing α-slopes. This V • is called the Harder-Narasimhan filtration [16,25] of V along α. Our first invariant, for a specific choice of α to be described later, is the map Q 0 → Z that sends each x in Q 0 to the vector dim(S 1 ) x , . . . , dim(S ) x .
Zigzag Persistence. Gabriel's foundational theorems from [11] establish that the set of indecomposable representations of a type A n quiver can be canonically identified with the collection of subintervals [u, v] ⊂ [0, n − 1] that have integral endpoints. The study of such representations has enjoyed a substantial recent renaissance, induced largely by their appearance in topological data analysis [24,7], where they are called zigzag persistence modules. Thus, by Gabriel's results, every finite-dimensional zigzag persistence module P decomposes uniquely into a direct sum of the form P [u,v] here [u, v] ⊂ [0, n − 1] ranges over a finite set Bar(P) called the barcode of P. For each such interval, I[u, v] is the corresponding indecomposable whose multiplicity, denoted d u,v above, is a strictly positive integer. A similar interval decomposition theorem also holds for infinite zigzag persistence modules [4], where the endpoints of intervals are allowed to attain ±∞ values. Every representation V of a type A n quiver gives rise to an n-periodic infinite zigzag module L V, and the second invariant of interest is the associated multiplicity function Bar(L V) → N >0 . This Paper. We introduce the Euler stability condition : Q 0 → R, which is defined on any finite quiver Q as follows. The underlying graph of Q forms a onedimensional CW complex X; and every representation V of Q functorially induces a cellular sheaf [9] S V on X. From this sheaf, one can build a two-term cochain complex which computes the cohomology of X with S V coefficients. The corresponding Euler characteristic has the form Here deg in (x) equals the number of edges which point to x. Our stability condition is therefore given by (x) := 1 − deg in (x).
Harder-Narasimhan filtrations along enjoy some surprising properties pertaining to barcode decompositions. As a warm-up exercise, we first consider the easiest examples, which are furnished by equioriented quivers of type A n x 0 e 1 / / x 1 e 2 / / · · · e n−1 / / x n−1 .
The representations of such quivers are called (ordinary, discrete) persistence modules. For any such representation V, the Harder-Narasimhan filtration V • along can be used to directly recover the multiplicities of all intervals which have the form [0, j] in Bar(V). Here is a simplified version of Theorem 5.4.
THEOREM (A). Let V be a persistence module and let j 0 < j 1 < . . . < j be the collection of all indices j in {0, . . . , n − 1} satisfying [0, j] ∈ Bar(V). Then V • has length + 1, and for each integer 0 < k ≤ the quotient S k : Thus, the Harder-Narasimhan filtration of V along recovers (the multiplicities of) all those intervals in Bar(V) which have left endpoint 0. We show that one can also extract similar formulas for multiplicities of all the other intervals in Bar(V) -and hence, the entire isomorphism class of V -by availing of Harder-Narasimhan filtrations along of V's restriction to certain truncated subquivers. In future work, we will describe a method for recovering multiplicities of all indecompoables (for representations of a different quiver) by using several different stability conditions at once.
Turning now to the central focus of this paper, we consider a representation V of a type A n quiver Q over an algebraically closed field. The indecomposable summands of V have two possible forms, illustrated below: To the left we have N[u, v], which is obtained by wrapping an interval [u, v] ⊂ Z around Q; here u and v equal u and v modulo n respectively. To the right lies T[λ; w], where all vector spaces have the same dimension w ≥ 1 and all edge-maps are identities except for a Jordan block with diagonal λ = 0 over e 0 . As mentioned above, V lifts to a representation L V of the infinite zigzag quiver UQ obtained by unwinding Q. This lift operation L forms a functor from the category of representations of Q to the full subcategory of representations of UQ spanned by n-periodic objects. Moreover, L V decomposes as a direct sum of the form The first term on the right consists of infinite intervals corresponding to the T[λ; w] summands of V while the second term consists of (infinitely many) n-shifted copies of Here is a condensed description of our main result, Theorem 7.4.
THEOREM (B). Let V be a representation of an acyclic type A n quiver Q over an algebraically closed field. Let V • be the Harder-Narasimhan filtration of V along , and let S j := V j /V j−1 be its successive quotients. If ϕ (S j ) is nonzero, then for each vertex x ∈ Q 0 we have: where the sum is over all N[u, v] appearing in the decomposition of V which have the same -slope as S j . And similarly, if ϕ (S j ) equals 0, then for each vertex x ∈ Q 0 , we have Since L V is periodic, the multiplicities d * ∞ and d * u,v of its indecomposable summands can be computed by restricting to a sufficiently long finite zigzag persistence module via the algorithm of [7] -see Remark 6.4. We hope that the results of this paper will encourage not only the use of tools from geometric invariant theory in the study of persistence modules, but also facilitate an influx of techniques from topological data analysis for efficient computation of Harder-Narasimhan filtrations.
Organisation. Sections 1, 2 and 3 contain notation and preliminary material pertaining to quiver representations, their direct sum decompositions into indecomposable representations, and Harder-Narasimhan filtrations respectively. In Section 4 we introduce the Euler stability condition, and in Section 5 we prove Theorem (A). In Section 6 we describe the infinite zigzag persistence modules arising as lifts of A n quiver representations. Finally, Section 7 contains the proof of Theorem (B).

Related Work.
The recent work of Kinser [21] classifies totally stable conditions on type A quivers, for which every indecomposable is stable, rather than semistable 1 . Using such a stability condition instead of in Section 5 would allow us to recover the entire barcode at once from the HN filtration (rather than only the intervals with left endpoint 0). For more complicated quivers, however, the set of totally stable conditions is always empty. In [17], Hille and de la Peña characterise stable representations of tame quivers when the stability condition is in a neighbourhood of a quantity called the defect. For A n quivers, the defect happens to coincide with the Euler slope, and hence their work yields a different proof of our Proposition 7.3. Unlike their argument, ours does not use any knowledge of tame hereditary algebras (besides the list of indecomposable type A quiver representations). More recently, Apruzzese and Igusa [1] have used a geometric model to determine the maximum finite number of stable indecomposables of type A quivers as the stability condition is varied.
The idea of relating indecomposables of a quiver to those of its universal cover, which is exploited heavily in Section 6, dates back to the work of Riedtmann, [26] Bongartz-Gabriel [3], and Gabriel [12]. An algorithmic perspective on covering theory for representations of strictly alternating A n quivers has been employed in the work of Burghelea and Dey on circle-valued persistence [6], where indecomposables of the form T[λ; w] are called Jordan cells. Similarly, Cheng [8] presents a polynomial time algorithm to compute Harder-Narasimhan filtrations with respect to any stability condition by relating them to the so-called discrepancies of quiver representations. Computing discrepancies requires finding the largest c such that a given space of matrices has a c-shrunk subspace. Although this last problem admits a polynomial-time algorithm [18], we are not aware of any practical implementations.
Finally, the work of Henselman and Ghrist [14] seeks to generalise persistence barcodes for functors to non-abelian categories by viewing (ordinary) persistence modules as lattice homomorphisms from a lattice of intervals with respect to a certain partial order. In contrast, the -slope defines a total preorder on the set of intervals. If the interval modules are all stable (i.e., if the stability condition is totally stable in the sense of [21]), then this preoreder is an order and the HN filtration constitutes a subsaecular series in the language of [14].

Quiver Representation Preliminaries
The study of quiver representations is a vast enterprise spanning algebra and geometry, so we will restrict our focus here on the aspects relevant to this work; comprehensive treatments can be found in [22] and [28].
A quiver Q consists of two sets Q 0 and Q 1 , whose elements are called vertices and edges respectively, equipped with two functions s, t : Q 1 → Q 0 called the source and target map. We typically write e : x → y when indicating that the edge e has source s(e) = x and target t(e) = y. A path of Q is any nonempty finite sequence of edges p = (e 1 , . . . , e k ) so that s(e j ) = t(e j−1 ) holds across all 1 < j ≤ k. The source and target maps extend to any such p by setting s(p) = s(e 1 ) and t(p) = t(e k ), and we call p a cycle of Q whenever the source and target vertex of p are identical. The quiver Q is called acyclic if it does not contain any such cycles.
Let us fix, once and for all, a ground field K so that all vector spaces and linear maps encountered henceforth are understood to be defined over K. A representation V of Q is an assignment of (1) a vector space V x to each vertex x ∈ Q 0 , and (2) a linear map V e : V x → V y to each edge e : x → y in Q 1 .
Unless stated otherwise, we will assume that both Q 0 and Q 1 are finite, and we will only consider finite-dimensional representations of Q -i.e., each V x is required to be finite-dimensional over K. The map dim V : Q 0 → Z sending each x to dim V x is called the dimension vector of V.
A morphism of representations f : V → V is a collection of vertex-indexed linear maps { f x : V x → V x | x ∈ Q 0 } so that for each edge e : x → y in Q 1 the evident diagram of vector spaces commutes: Equivalently, the identity f t(e) • V e = V e • f s(e) holds for every edge e ∈ Q 1 . With this definition of morphisms in place, the representations of Q define an abelian category Rep(Q), see [11,Section 1.2]. Injective and surjective morphisms, kernels, images, quotients, direct sums and the zero object are all defined pointwise in Rep(Q). We say that V is a subrepresentation of another representation V whenever there exists an injective morphism V → V , in which case we simply write V ⊂ V . REMARK 1.1. The quiver Q is a subquiver of another quiver Q = (s , t : Q 1 → Q 0 ) if we have Q 0 ⊂ Q 0 and Q 1 ⊂ Q 1 so that s and t are restrictions of s and t respectively. Given such a subquiver, we note that each representation V of Q automatically induces a representation V of Q by restricting to the available vertices and edges. Namely, V x := V x for all x ∈ Q 0 and V e := V e for all e ∈ Q 1 .

Harder-Narasimhan Filtrations
Harder-Narasimhan filtrations were originally introduced for the purpose of studying moduli spaces of vector bundles over algebraic curves [16]; they have since been employed in a plethora of other contexts [20,5,15], including geometric invariant theory [27,Chapter 4.2].
The Grothendieck group K(A ) of an abelian category A is the abelian group generated by the objects of A subject to the relation that V = U + W whenever there is an exact sequence 0 → U → V → W → 0. By a stability condition on A we mean an abelian group homomorphism Z : K(A ) → (C, +) valued in the (additive) complex numbers, so that every nonzero object U is mapped to the right half plane, i.e. Re Z(U) > 0. Given such a stability condition, the Z-slope of a nonzero object V in A is the real number The importance of semistable objects in the study of moduli spaces stems from the fact that every nonzero object admits a unique filtration for which the successive quotients are semistable and have strictly decreasing slopes. A proof of the following result (for the category of representations of a fixed quiver) can be found in [17, Theorem 2.5]; more generally, see [15,Theorem 4.2]. THEOREM 2.2. Let A be a finite length (i.e., Noetherian and Artinian) abelian category equipped with a stability condition Z : (This unique V • is called the Harder-Narasimhan, or HN, filtration of V along Z.) REMARK 2.3. In particular, it follows from uniqueness that a nonzero object V of A is Z-semistable if and only if the corresponding HN filtration is the trivial one 0 V.
Let Hom(Q 0 , Z) be the abelian group consisting of functions Q 0 → Z with addition defined vertex-wise. When Q is acyclic, the map V → dim V furnishes an isomorphism K(Rep(Q)) Hom(Q 0 , Z) -see for instance [22,Theorem 1.15]. As a result, every stability condition on Rep(Q) amounts to a function β : Q 0 → C sending vertices x to complex numbers β x , subject to the requirement that Re β x > 0. It is customary to assign numbers of the form β x = 1 + i · α x for arbitrary real numbers α x [17,25], which makes no difference to semistability (but is liable to alter HN filtrations). Thus, the stability conditions of interest are precisely the functions α : Q 0 → R, and for such a function the corresponding α-slope of V ∈ Rep(Q) equals Crucially, the uniqueness result of Theorem 2.2 applies only after the stability condition has been fixed -in particular, varying α : Q 0 → R is liable to produce a very different HN filtration of the same representation. If α is identically zero for instance, then all quiver representations are semistable with slope 0, and hence have trivial HN filtrations. Our focus here will be on a specific choice of stability condition arising from cellular sheaf cohomology, which is described in Section 4. Throughout the remainder of this section, however, we fix an arbitrary α and all instances of slopes, stability and HN filtrations encountered here are with respect to this fixed α.
It will be convenient for our purposes to describe the HN filtrations of direct sums in terms of the HN filtrations of constituent factors. To this end, given any representation V in Rep(Q), we consider the HN filtration V • as being indexed over the real line in the following manner. The HN R-filtration of V is the assignment t → V R (t) of quiver representations to real numbers obtained by setting Thus, V R constitutes a functor from the ≥-ordered set of real numbers to the category Rep(Q). There is a natural direct sum operation on such functors, The following result is well-known to experts, but we were unable to find it in the literature and have included a proof for completeness.
PROOF. Writing and m for the lengths of U • and V • respectively, let be the union of slopes attained by the successive quotients of both filtrations. By construction, (U ⊕ V) R is locally constant on R − Φ. Let θ 1 > . . . > θ n be the slopes in Φ arranged in strictly decreasing order, and consider the filtration W • of U ⊕ V given by To see this, note that we have an isomorphism of quotient representations for each k. We will now show that W k /W k−1 is semistable with slope θ k , which -when combined with the uniqueness guarantee of Theorem 2.2 -produces the desired result. By (2), we have a (split) exact sequence of the form The two summands on the right side of (2) are either trivial or semistable with slope θ k by construction. In the nontrivial case, the first assertion of Lemma 2.4 ensures that the slope of W k /W k−1 is also θ k , while the second assertion guarantees semistability.

Indecomposables and Barcodes
A representation V = 0 of a quiver Q is indecomposable if it cannot be written as a direct sum of two nontrivial representations. Since we have assumed that Q is finite and V is finite-dimensional, the Krull-Schmidt theorem [2,23] applies, so there is a unique pair consisting of a finite set Ind(V) of indecomposable representations and a function Ind It follows from Proposition 2.5 that the HN filtrations of all I in Ind(V) determine the HN filtration of V, so in principle it suffices to restrict attention to indecomposable representations. Unfortunately, the task of decomposing a given V ∈ Rep(Q) into indecomposables, and the task of cataloguing all possible indecomposables in Rep(Q) are both remarkably difficult problems for arbitrary Q. Two prominent exceptions are the Dynkin quivers of type A and A, whose indecomposables we describe below.

Quivers of type A.
Gabriel's seminal result [11] established that Rep(Q) is representation finite 2 if and only if the undirected graph obtained from Q by ignoring source and target information is a disjoint union of simply-laced Dynkin diagrams (i.e., of types A, D and E). In particular, we recall that a quiver Q is said to be of type A n for some n ≥ 0 whenever its underlying graph is: Representations of type A n quivers are also called zigzag persistence modules [7]. If s(e i ) = x i−1 and t(e i ) = x i holds for each i, then Q is called equioriented, and its representations are (ordinary) persistence modules [31].
The linear map I[u, v] e j assigned to the edge e j is the identity id K whenever both source and target vector spaces are nontrivial, and is necessarily zero otherwise.
The following result is a direct consequence of Gabriel's theorem [11].
THEOREM 3.2. Let Q be a type A n quiver. Then up to isomorphism, the indecomposables of Rep(Q) are precisely the interval modules It follows directly from the above result that for every zigzag persistence module V there exists a unique finite set Bar(V) containing subintervals of [0, n − 1] and a unique function in Rep(Q). This set Bar(V) is called the barcode of V while [u, v] → d uv is called the multiplicity function. There are practical algorithms which can compute both barcodes and multiplicities for zigzag persistence modules -see [7].
In this paper we will also be interested in representations of type A ∞ quivers; the underlying graph of any such quiver Q has vertices indexed by Z as depicted below:

Affine Quivers of type A.
We say that Q is of type A n if its underlying graph has the form for n > 1. In this section we assume that the ground field K is algebraically closed. With this assumption in place, the indecomposables of Q can be classified into two families, which we describe below (with illustrative examples to follow). (1) For each interval [u, v] ⊂ Z, the representation N[u, v] of Q is defined as follows. Setting := v−u n , the vector space assigned to x j is The linear map over e j is the identity whenever its source and target vector spaces are equidimensional; the exceptional cases occur when j equals either u or v + 1 modulo n. In such cases, if e j has a clockwise orientation, then we have: Similarly, when e j has a counterclockwise orientation, N[u, v] e j is the transpose of the appropriate matrix above. (2) For each field element λ = 0 in K and integer w ≥ 1, let T[λ; w] be the Qrepresentation which assigns to every vertex the vector space K w , to every edge e j for j = 0 the identity map id w , and to e 0 the Jordan block with λ along its diagonal.
We   N[1, 9] of a type A 6 quiver; note that the dimension is larger between vertices x 1 and x 3 , and that unlabelled arrows carry identity maps: Similarly, here is the representation T[2; 3] of the same quiver: A proof of the following result can be found in [10]. THEOREM 3.6. A representation of a type A n quiver is indecomposable if and only if it is isomorphic either to N[u, v] for some interval [u, v] ⊂ Z or to T[λ; w] for some λ = 0 in K and w ≥ 1 in Z.

The Euler Stability Condition
The stability condition that we will use throughout this paper is simple to define. By the discussion following Remark 2.3, we only require a function Q 0 → R. This function is with # denoting cardinality. Readers who are satisfied with this definition may safely proceed to the next section; in this section we will only describe the reasons which have motivated our choice. Consider a CW complex X, write σ ≤ τ to indicate that the cell σ lies in the boundary of the cell τ in X, and denote the poset of cells ordered by this face relation by (X, ≤). A cellular sheaf F on X is a functor from (X, ≤) to the category Vect(K) of K-vector spaces [9]. Thus, F assigns to each cell σ a vector space F σ and to each face relation σ ≤ τ a linear map F στ : F σ → F τ , subject to two axioms: (1) (identity) the map F σσ : F σ → F σ is the identity for each cell σ, and (2) (associativity) across any triple σ ≤ τ ≤ ν of cells, we have When X is one-dimensional, the associativity axiom holds automatically since there are no triples of the form σ < τ < ν. Cellular sheaves on X form an abelian category Shf(X) whose morphisms are given by the natural transformations between functors (X, ≤) → Vect(K).
Let F be a cellular sheaf on a CW complex X, and consider the sequence of vector spaces and linear maps where C j (X; which (as usual) also equals the alternating sum ∑ j≥0 (−1) j · dim C j (X; F) whenever this sum makes sense.
The graph underlying any quiver may be treated as a one-dimensional regular CW complex -cells of dimension 0 and 1 are Q 0 and Q 1 respectively. In this case, the zeroth sheaf cohomology H 0 (X; S V) coincides with the space of sections [29, Section 1] of V.
Our next result shows that the Euler characteristic χ(X; S V) can form the imaginary part of a stability condition on Rep(Q).

PROPOSITION 4.2.
Let Q be a quiver with underlying graph X. The map V → χ(X; S V) is a well-defined group homomorphism K 0 (Rep(Q)) → (R, +).
PROOF. It is not difficult to confirm that the functor S is exact, so every exact sequence 0 → U → V → W → 0 in Rep(Q) produces a short exact sequence of cochain complexes Passing to the long exact sequence on sheaf cohomology, we obtain the desired equality χ(X; S V) = χ(X; S U) + χ(X; S W).
Since there are only two nontrivial cochain groups to consider in (5) when X is the underlying graph of a quiver Q, we have for each V in Rep(Q) the formula by (5) and Def 4.1, Here deg in (x) := # {e ∈ Q 1 | t(e) = x}. We have arrived at the desired stability condition.
is called the Euler slope.

HN Filtrations of Ordinary Persistence Modules
Throughout this section, Q will denote the equioriented quiver of type A n for some n ≥ 1, and V will denote a fixed nonzero object of Rep(Q), i.e., an ordinary persistence module: Our goal here is to link the barcode decomposition of V from (3), recalled below: to its HN filtration with respect to the Euler stability condition (see Theorem 2.2 and Definition 4.3). The first step in this direction is to establish some general results which hold for a wider class of stability conditions.

General Results for Antitone Slopes.
Let denote the lexicographical total ordering on subintervals of [0, n − 1], where [u, v] [u , v ] holds if we have either u < u or both u = u and v ≤ v . Let α : Q 0 → R be a stability condition on Q with associated slope ϕ α . We say that our slope ϕ α is antitone if the inequality PROOF. Fix an interval [u, v] ∈ Bar(V), and consider a nonzero submodule U ⊂ I[u, v]. Using the barcode decomposition of U from Theorem 3.2 followed by Lemma 2.4, we obtain the inequality Thus, we are required prove that It is readily checked that this submodule condition holds if and only if u ≥ u and v = v. In particular, we need u ≥ u for dimension reasons, and v = v follows from the fact that if v exceeds v , then the following diagram of vector spaces fails to commute: Thus, we have [u, v] [u , v ] and the desired conclusion follows because ϕ α is antitone.
For antitone ϕ α 's we can now describe HN filtrations directly in terms of barcodes. PROPOSITION 5.2. Assume that the slope ϕ α is antitone, for each t ∈ R the HN Rfiltration of V evaluated at t is given by where the sum is over all [u, v] ∈ Bar(V) satisfying ϕ α (I[u, v]) ≥ t. The multiplicities d uv are the ones appearing in (3) and the isomorphism can be obtained by restricting the decomposition from (3).
PROOF. We know from Lemma 5.1 that interval modules are semistable for antitone ϕ α . By Remark 2.3, the HN filtration of such a module is the trivial one (0 I[u, v]), which clearly satisfies (8). By Proposition 2.5, the left side of (8) is (also) additive, whence V satisfies (8) as claimed. COROLLARY 5.3. Assume that ϕ α is antitone and that the HN filtration V • of V has length . Then for each j in {1, . . . , }, we have an isomorphism where the sum ranges over all [u, v] , and where d indicates the multiplicity function of V from (3).

Specific Results for Euler Slopes.
We continue to work with a fixed persistence module V as in (6), but now specialise to the Euler stability condition : Q 0 → R from Definition 4.3. It is immediate from this definition that the Euler slope of V is Here is the full version of Theorem (A) from the Introduction. with u = 0, then we have (9) gives

PROOF. Replacing V by an indecomposable module I[u, v] in the formula
Thus, ϕ is antitone and we may apply Corollary 5.3 to establish both assertions.
The preceding result establishes a direct relationship between barcode decompositions and (Euler) HN filtrations of ordinary persistence modules -to obtain the HN filtration from the barcode inductively, one lexicographically orders the intervals in Conversely, it is possible to extract the multiplicity of any interval of the form [0, v] in Bar(V) by using the HN filtration: there is at most one index k for which the There is, however, no separation of intervals [u, v] ∈ J * from each other since all of them have slope 0. Two remedies are available for this unfortunate incompleteness -the first of these comes in the form of slopes which separate all intervals in Bar(V). Any such totally stable slope [21], if used instead of in the proofs above, would allow us to recover the entire barcode of V via successive quotients of the associated HN filtration. The second remedy is described below.
REMARK 5.5. Let us label the vertices and edges of the underlying quiver Q as follows: For each integer k in {0, . . . , n − 1}, define Q ≥k to be the type A n−k subquiver of Q consisting of vertices x j | k ≤ j ≤ n − 1 and all the edges between them. Let V ≥k be the representation of Q ≥k obtained by restricting V to Q k . By the restriction principle [7], the decomposition (3) becomes where the sum is over all intervals [u, v] ∈ Bar(V) satisfying v ≥ k. For a fixed interval [k, v] in Bar(V), the above decomposition induces the relation on multiplicities Here d ≥k 0,v−k denotes the multiplicity of [0, v − k] in Bar(V k ), and the sum is over intervals [u, v] in Bar(V) satisfying u < k. By Theorem 5.4, we have that d ≥k 0,v−k can be recovered from the HN filtration of V ≥k along the Euler stability condition. Thus, we can inductively reconstruct the entire multiplicity function of V from the HN filtrations of V ≥k | k ≥ 0 .

Unwinding Affine Type A Quivers
We say that a quiver is of type A ∞ whenever its underlying graph has the form · · · a −1 Explicitly, the vertex set is identified with the set of integers Z, and there is a unique edge between every adjacent pair of integers. Thus, representations of A ∞ quivers are infinite zigzag persistence modules.
We now fix a quiver Q = (s, t : Q 1 → Q 0 ) of type A n labelled as in (4).
Let ρ : UQ 0 → Z be the bijection (x i , c) → i + cn; for each interval [p, q] ⊂ Z, let UQ p,q ⊂ UQ denote the truncated subquiver spanned by all vertices satisfying ρ(x j , c) ∈ [p, q] and the edges between them. EXAMPLE 6.2. Here is A 6 quiver from Example 3.5 depicted above its unwinding. The labels above and below the vertices of the unwinding are ρ(UQ 0 ) and UQ 0 respectively.
As before, we only consider representations whose assigned vector spaces are all finite-dimensional. Such a representation W ∈ Rep(UQ) is said to be periodic whenever we have: c+1) for all (x j , c) ∈ UQ 0 , and W (e j ,c) = W (e j ,c+1) for all (e j , c) ∈ UQ 1 .
We denote by PRep(UQ) the full subcategory of Rep(UQ) which consists of all such periodic representations -crucially, we do not require that the morphisms in this category are periodic.
Here the sum ranges over intervals [u, v] ⊂ Z lying in the barcode Bar(W), and d u,v is the multiplicity of each such interval as described in (3). We first claim that the multiplicity function is n-periodic -namely, for all [p, q] ⊂ Z, we have d p,q = d p+n,q+n . To verify this claim, let W p,q denote the representation of UQ p,q obtained by restricting W to the available vertices and edges. By (10), we have where the sum is over intervals [u, v]  REMARK 6.4. The argument given above also establishes that the multiplicity function Bar(W) → N >0 of W ∈ PRep(UQ) can be recovered from the irreducible decomposition of a sufficiently large but finite zigzag persistence module. More precisely, let D ∈ Z be chosen so that for any [u, v] Then there is a one-to-one correspondence between the non-zero summands of the decompositions of W and the truncation W 0,D . In fact, D := (dim W 0 + 2)n always suffices: if [u, v] lies in Bar(W), then we have so the translation of [u, v] by nZ whose starting point is in [1, n] There is an evident lift map that sends objects of Rep(Q) to those of Rep(UQ) defined as follows. The lift L V of a representation V of Q assigns V x j to every vertex in UQ 0 of the form (x j , c) and similarly V e j to every edge of the form (e j , c) in UQ 1 . We note that L V is called the infinite cyclic covering in [6]. EXAMPLE 6.5. Continuing Examples 3.5 and 6.2, here are the representation N [1,9] of the A 6 quiver from example 3.5 and its lift L (N [1,9]). As before, unlabelled edges carry identity maps: Given a morphism f : Thus, the morphisms of PRep(UQ) lying in the image of L are always periodic even though -as mentioned above -morphisms in PRep(Q) are not periodic in general.
where the right side is as defined in Proposition 6.3. And moreover, (2) for each field element λ = 0 in K and integer w ≥ 1, we have PROOF. The matrices in the description of N[u, v] from Definition 3.4 are already in barcode form (see [19,Definition 2.2]), so the desired interval decomposition of L N[u, v] can be directly inferred from these matrices. Moreover, since all the matrices in the description of T[λ; w] are isomorphisms, it follows from 3.3 that there exists a change of basis which turns them all into identities.

HN Filtrations of Affine Type A Quivers
Let Q be a quiver of type A n as defined in (4), and fix a representation V = 0 of Q valued in vector spaces over an algebraically closed field K. Our goal here is to relate the HN filtration of V along the Euler stability condition (from Definition 4.3) to the barcode decomposition of the lifted representation L V ∈ Rep(UQ) which has been described in Proposition 6.3.
The first task at hand is to compute the Euler slopes of all indecomposables in Rep(Q) -we recall from Theorem 3.6 that these have form N[u, v] and T[λ; w], both of which have been described in Definition 3.4. There is a single number p u,v ∈ {0, 1, 2} which determines the sign of the Euler slope of each N[u, v], and it admits the following graphical description (with u := u mod n and v := v mod n): Explicitly, p u,v counts whether the edge e u has target u and whether the edge e v +1 has target v . The cases p u,v = 0, 1, 2 are equivalent [13,Sec 11.3] to N[u, v] being called respectively pre-injective, regular and pre-projective in standard texts [22,28]. (1) For each [u, v] ⊂ Z, we have (2) For each nonzero λ ∈ K and integer w ≥ 1, we have PROOF. The second assertion follows from the following elementary calculation: for each λ and w as above, by Definitions 3.4 and 4.3 we have Since ∑ x∈Q 0 deg in (x) equals the cardinality of the edge set, we obtain The numerator on the right side vanishes, as desired, because both cardinalities equal n.
Turning now to N[u, v], by the discussion preceding Definition 4.3 we have Here X denotes the underlying graph of Q, while S : Rep(Q) → Shf(X) is the associated sheaf functor from Definition 4.1. This associated sheaf is locally constant on a decomposition of X into two disjoint intervals X = A B, whose exact nature depends on the value of p u,v . In particular, setting := v−u n , let A ⊂ X be the union of cells on which N[u, v] has dimension + 1, and let B be the complement of A in X (on which N[u, v] necessarily has dimension ). Whether A is closed, clopen or open in X depends on p u,v being 0, 1 or 2. Therefore, A's Euler characteristic is Equivalently, we have χ(A) = 1 − p u,v . Now the numerator in (11) is given by where K +1 A is the constant K +1 -valued sheaf on A, etc. Thus, By additivity of the Euler characteristic, we have χ(B) = −χ(A) since X is homeomorphic to a circle and satisfies χ(X) = 0. Therefore, χ(X; S N[u, v]) equals χ(A), which in turn equals 1 − p u,v . Finally, the denominator in (11) is which, by definition of , simplifies to v − u + 1 as desired.
Next, we seek to establish that all the indecomposable representations in Rep(Q) are -semistable. In light of Lemma 5.1, it suffices to show that the inequality ϕ (I) ≤ ϕ (I ) holds whenever we have an injective morphism I → I between a pair of indecomposables in Rep(Q). There are only three nontrivial cases of I → I to consider, which -based on the slope computations from Proposition 7.1 -we call (++), (0−) and (−−). Explicitly, REMARK 7.2. If we have an injective morphism U → V in Rep(Q), then for every edge e ∈ Q 1 the linear map U e is precisely the restriction of V e . It follows that if V e is injective, then so is U e . We note that all the edges are assigned injective maps in T[λ; w]; but exactly 2 − p u,v edges are assigned a map with a nontrivial kernel by N[u, v]. Therefore, the case (+0) involving N[u, v] → T[λ; w] for p u,v = 0 never arises. And similarly, all the cases involving N[u, v] → N[ū,v] with p u,v < pū ,v are impossible.
The following proposition can be seen as a corollary of [17,Prop 5.2]. We prove it here without using any property of tame hereditary algebras except the decomposition from definition 3.4. PROOF. We treat each of the three cases separately.
Case (++): The injectivity requirement from Remark 7.2 forces both u =ū mod n and v =v mod n, and (translating if necessary) we may as well assume thatū = u andv = v + cn for some c ∈ Z. If c is negative,  (7), no infinite interval can map injectively to a direct sum of finite intervals.
Here is our main result, which is Theorem (B) from the Introduction. PROOF. From Theorem 3.6, we know that there exist integers d u,v > 0 and d λ;w > 0 so that V decomposes as V [u,v] where the first sum is over [u, v] ⊂ Z with u ∈ [0, n − 1] such that N[u, v] ∈ Ind(V) and the second sum is over (λ, w) ∈ (K − {0}) × N >0 such that T[λ; w] ∈ Ind(V). Similarly, by Proposition 6.3 we have integers d * u,v > 0 and d * By Proposition 6.6 and the additivity of the lift functor, we obtain the following relations between the multiplicities of indecomposables in V and in L V: where [u, v] ⊂ Z and (λ, w) ∈ (K − {0}) × N >0 . We use the convention d I = 0 when the indecomposable I is not in Ind(V), and similarly d * I = 0 when I is not in Ind(L V).
Consider a successive quotient S j := V j /V j−1 of the HN filtration of V. By combining (12) with Propositions 7.1 and 7.3, the dimension vector of S j can be decomposed as a sum over indecomposables N[u, v] with the same slope as S j : except when ϕ (S j ) = 0, in which case there is an extra term induced by the summands of the form T[λ; w]: This formula simplifies further since dim T[1;1] equals one at every vertex of Q. We now turn to the three assertions which must be established.
(i) From Remark 6.4 we know that d * can be deduced from the irreducible decomposition of a zigzag persistence module of length O(∆n) with all spaces of dimension at most ∆. This gives the desired worst-case complexity as the algorithm [7] involves performing one Gaussian elimination per map of the zigzag persistence module. Moreover, equations (13) and (14) show that η(V) can be deduced from d * by updating at most ∆ times one of the dimensions dim S j x at some fixed vertex x. The summands in the above sum are null unless either u = u mod n or v + 1 = u mod n. The latter case does not happen as p u ,v = p u,v forces e u mod n and e (v +1) mod n to have different orientations. The former case, u = u mod n, only happens when [u , v ] is a translation by nZ of [u, v] since [u , v ] and [u, v] are required to have the same length.
Finally, we note that although the results obtained above require K to be be algebraically closed, one can always fix bases for the vector spaces of a given V ∈ Rep(Q) and consider a new representation V + with the same matrix description but now over the algebraic closure K of K. The isomorphism class of V + only depends on the isomorphism class of V, and as such, η(V + ) still constitutes an (incomplete) invariant of V. By Theorem 7.4 above, this invariant can be related to the barcode of L V + in the category of UQ-representations valued in vector spaces over K.