TOTAL STABILITY FUNCTIONS FOR TYPE A QUIVERS RYAN KINSER Abstract. For a quiver Q of Dynkin type An, we give a set of n − 1 inequalities which are necessary and sufficient for a linear stability condition (a.k.a. central charge) Z : K0(Q) → C to make all indecomposable representations stable. We furthermore show that these are a minimal set of inequalities defining the space TS(Q) of total stability conditions, considered as an open subset Q0 Q0 of R ×(R>0) . We then use these inequalities to show that each fiber of the projection of TS(Q) R Q0 R RQ1 to ( >0) is linearly equivalent to × >0 . 1. Introduction 1.1. Problem statement. Let Q be an acyclic quiver and fix an algebraically closed field k over which all representations are taken. A detailed recollection of terminology used in the introduction is found in Section 2. This paper concerns stability conditions for quivers as in [Sch91, Kin94, Rud97, Bri07]. We restrict our attention to linear stability conditions, meaning those given by a group homomorphism Z : K0(Q) → C such that Z(M) has positive imaginary part when M is a nonzero representation. Q0 Such a function is also known as a central charge. Identifying K0(Q) ≃ Z by taking classes of simple representations as a basis, a stability condition can be written as Q0 Q0 (1.1) Z(x)= w · x + (r · x)i for some w ∈ R , r ∈ (R>0) . Q0 Q0 Thus we can identify the space of stability conditions with R × (R>0) . Such Z determines the slope function w · x (1.2) µ : K (Q) → R, µ (x)= . Z 0 Z r · x (Note that this is the reciprocal of the slope of the line through the origin and µZ (x) if we take the standard convention of drawing the complex plane.) arXiv:2002.12396v2 [math.RT] 4 Nov 2020 A representation of Q is called Z-stable if µZ (W ) <µZ(V ) for all nonzero, proper subrepresen- tations 0 < W < V . Notice that by taking Z as a variable, each µZ (W ) < µZ(V ) is a quadratic Q0 Q0 inequality on the space of stability conditions R × (R>0) . Stability of quiver representations has connections with many other notions in mathematics and mathematical physics, such as mod- uli spaces of representations, semi-invariants, Harder-Narasimhan filtrations, and green sequences and paths. We refer the reader to [Igu] and the references therein for more detail about these connections. This work investigates the set of stability conditions Z such that every indecomposable represen- tation of Q is Z-stable. This immediately restricts our attention to Q of Dynkin type since stable representations have 1-dimensional endomorphism ring. 2020 Mathematics Subject Classification. 16G20, 05E10. Key words and phrases. quiver representation, stability condition, central charge, total stability, type A. 1 2 RYAN KINSER Definition 1.3. A stability condition Z for a quiver Q is totally stable, or a total stability condition, if every indecomposable representation of Q is Z-stable. The set of total stability conditions for Q is denoted (1.4) TS(Q)= {Z ∈ HomZ(K0(Q), C) | Z is a total stability condition} , Q0 Q0 and identified with an open subset of R × (R>0) . The main results of this paper are about quivers of Dynkin type A. This means that the under- lying undirected graph is of the form (1.5) 1 2 3 · · · n , and we say the quiver is of type An if we want to specify that it has n vertices. For equioriented type A quivers, meaning all arrows point in the same direction, it is easy to describe the set of total stability conditions due to the fact that the all indecomposable representations are uniserial in this case [Rei03, Example A]. For Q of type A and arbitrary orientation, TS(Q) was recently shown to be nonempty in inde- pendent papers of Apruzzese-Igusa [AI] and Huang-Hu [HH], using quite different methods. These papers considered the specific case of standard linear stability conditions as in [Rei03], also called classical slope functions, where r = (1, 1,..., 1) in (1.1). In [AI] it is proven using a geometric model, and that paper also contains more general results about affine type A and maximal green sequences. Another proof was given by a different geometric model in [BGMS20, Thm. 5.3]. The methods of this paper are independent of the above cited papers, and describe all total (linear) stability conditions, not just the standard ones. It would be interesting to interpret our Theorem 1.13 in the geometrical models referenced above. 1.2. Results. The following notation for type A quivers is useful to organize the proof of the main theorem. A running example illustrating the notation starts with Example 1.11. Notation 1.6. Given a type A quiver Q as in (1.5), recursively define functions x,y : Q0 → R by setting x(1) = y(1) = 0, and then for i> 1: x(i +1) = x(i) + 1 and y(i +1) = y(i) if there is an arrow i → i + 1, (1.7) (x(i +1) = x(i) and y(i +1) = y(i) + 1 if there is an arrow i + 1 → i. (Visually, these give us an embedding Q ⊂ R2 by specifying the x,y-coordinates of the vertices and then connecting them with arrows in the simplest way; see (1.12)). This determines two sequences of subsets of Q0, which are pairwise disjoint within each sequence: (1.8) Xk = {z ∈ Q0 | x(z)= k} , Yk = {z ∈ Q0 | y(z)= k} , for k ∈ Z≥1. We furthermore define x(n) y(n) (1.9) Xi := Xk and Yi := Yk k[=i k[=i to get chains of subsets of Q0: e e Xx(n) ⊂ Xx(n)−1 ⊂···⊂ X2 ⊂ X1 ⊂ X0 = Q0 (1.10) Q = Y ⊃ Y ⊃ Y ⊃···⊃ Y ⊃ Y . e 0 e0 1 2 e ye(n)−1 e y(n) e e e e e TOTAL STABILITY FOR TYPE A QUIVERS 3 Example 1.11. The quiver below shows a type A quiver embedded in R2 as in Notation 1.6. 7 8 (1.12) Q = 4 5 6 1 2 3 The corresponding partitions of Q0 come from vertically and horizontally aligned subsets of Q0: X0 = {1}, X1 = {2}, X2 = {3, 4}, X3 = {5}, X4 = {6, 7}, X5 = {8} Y0 = {1, 2, 3}, Y1 = {4, 5, 6}, Y2 = {7, 8}. The chains in (1.10) come from filtering the vertices by x-coordinate and y-coordinate respectively: {8} ⊂ {6, 7, 8} ⊂ {5, 6, 7, 8} ⊂ {3, 4,..., 8} ⊂ {2,..., 8}⊂ Q0 Q0 ⊃ {4, 5, 6, 7, 8} ⊃ {7, 8}. The main result of the paper is below, characterizing membership in TS(Q) by a much smaller set of inequalities than the set resulting a priori from the definition. Here, the notation [S] for S ⊆ Q0 means the indecomposable representation of Q supported on S (see Notation 2.1). Theorem 1.13. Let Q be a quiver of Dynkin type An and recall Notation 1.6. A stability function Z is in TS(Q) if and only if the n − 1 inequalities below hold good: (1.14) µZ ([X0]) >µZ([X1]) > · · · >µZ([Xx(n)]), (1.15) µZ ([Y0]) <µZ ([Y1]) < · · · <µZ([Yy(n)]). Furthermore, the inequalities above are a minimal set of inequalities defining TS(Q) as an open Q0 Q0 set of R × (R>0) . The proof of this theorem is in Section 3. One notices, however, that Theorem 1.13 does not tell us about the solution space to this set of inequalities. With a little more work, we obtain the following corollary, whose proof is in Section 3 as well. A RQ0 RQ0 Corollary 1.16. Let Q be a quiver of Dynkin type . Viewing TS(Q) ⊂ × >0 via (1.1), RQ0 the projection TS(Q) → >0 sending (w, r) 7→ r is surjective with each fiber linearly equivalent to R RQ1 RQ0 × >0. In particular, for any r ∈ >0 there is a total stability function for Q of the form (1.1). A final remark: perhaps unsurprisingly, we expect Theorem 1.13 and Corollary 1.16 to generalize only partially to other Dynkin types. Remark 1.17. Work in progress with Yariana Diaz and Cody Gilbert generalizes Theorem 1.13 to arbitrary Dynkin type using Auslander-Reiten sequences, but without the minimality statement. We have also found that the cone of standard linear stability conditions, that is, the fiber over r = (1, 1,..., 1) in the language of Corollary 1.16, is empty for certain orientations in Dynkin types Dn for all n ≥ 9 and types E7, E8 (cf. [Rei03, Conjecture 7.1]). 4 RYAN KINSER Acknowledgements. The author thanks Øyvind Solberg for discussions about the software QPA [Qt], which was very helpful for completing this paper. The author also thanks Yariana Diaz and Cody Gilbert for discussions on stability of representations of Dynkin quivers and for working together on the QPA and SageMath code which helped finish this work. Special thanks go to Hugh Thomas for the proof of Corollary 1.16, and an anonymous commentor for pointing out that the results in the first version of this article used outdated language. This work was supported by a grant from the Simons Foundation (636534, RK). 2. Background In this section we establish our notation and make some initial reductions for the proof of the main theorem. More detailed background can be found in textbooks such as [Sch14, DW17] and the survey [Rei08]. 2.1. Quiver representations. We write Q0 for the set of vertices of a quiver Q, and Q1 for its α set of arrows, while tα and hα denote the tail and head of an arrow tα −→ hα.