Total Stability Functions for Type $\Mathbb {A} $ Quivers

$Total Stability Functions for Type $\Mathbb {A} $ Quivers$

arXiv:2002.12396v2 [math.RT] 4 Nov 2020 (1.2) function slope Nt htti stercpoa ftesoeo h iethro line the of slope the of reciprocal the is this that (Note hsw a dniytesaeo tblt odtoswith conditions stability of space the identify can we Thus iperpeettosa ai,asaiiycniinca condition stability (1.1) a basis, a as representations simple nqaiyo h pc fsaiiyconditions stability of space the on inequality n ah.W ee h edrt Iu n h eeecsthe references the and [Igu] to reader the refer connections. We paths. Harder-N m semi-invariants, and and representations, mathematics of in spaces notions uli other many with connections has ain 0 tations plane.) complex the drawing of convention standard ainof tation 1.1. uhafnto sas nw sa as known also is function a Such Z to attention our restrict 2. Section recollecti in detailed found A is taken. are representations all which ersnain ae1dmninledmrhs ring. endomorphism 1-dimensional have representations : e od n phrases. and words Key 2020 hswr netgtstesto tblt conditions stability of set the investigates work This ersnainof representation A hspprcnen tblt odtosfrqiesa n[ in as quivers for conditions stability concerns paper This K rbe statement. Problem 0 ( ahmtc ujc Classification. Subject Mathematics Q Abstract. fieulte enn h space the defining furthe inequalities We of stable. representations indecomposable all make eesr n ucetfralna tblt odto (a. condition stability linear a for sufficient and necessary o( to of ) Q R V < W < → R Q is > 0 OA TBLT UCIN O TYPE FOR FUNCTIONS STABILITY TOTAL 0 × C ) Z Q ( R uhthat such sal.Ti meitl etit u teto to attention our restricts immediately This -stable. 0 > slnal qiaetto equivalent linearly is o quiver a For 0 ) Z Q oieta ytaking by that Notice . 0 ( ete s hs nqaiist hwta ahfie ftep the of fiber each that show to inequalities these use then We . x Q = ) uvrrpeetto,saiiycniin eta char central condition, stability representation, quiver iersaiiyconditions stability linear scalled is Z w ( Let M · Q µ x a oiieiaiaypr when part imaginary positive has ) Z fDni type Dynkin of ( + Q : Z K ea cci uvradfi nagbaclycoe field closed algebraically an fix and quiver acyclic an be S T - r eta charge central stable 0 · R ( ( x 62,05E10. 16G20, Q Q × 1. ) fttlsaiiycniin,cniee sa pnsubse open an as considered conditions, stability total of ) i ) R YNKINSER RYAN → Introduction if > Q 0 1 Z µ . R o some for Z A R µ , savral,each variable, a as ( n Q 1 W egv e of set a give we , enn hs ie yagophomomorphism group a by given those meaning , 0 Identifying . ) × µ < ( Z Z R ( w rsma lrtos n re sequences green and filtrations, arasimhan no emnlg sdi h introduction the in used terminology of on x > Z mr hwta hs r iia set minimal a are these that show rmore ewitnas written be n uhta vr neopsberepresen- indecomposable every that such ..cnrlcharge) central k.a. = ) 0 ∈ ( ) V R g h rgnand origin the ugh Q R Q o l ozr,poe subrepresen- proper nonzero, all for ) 0 Q w tblt fqie representations quiver of Stability . 0 r 0 c9,Kn4 u9,Bi7.We Bri07]. Rud97, Kin94, Sch91, × · K r , · teaia hsc,sc smod- as such physics, athematical x enfrmr ealaotthese about detail more for rein x ( 0 n R µ ( . M ∈ Q − Z Q > ) ( 0 ( e oa tblt,tp A. type stability, total ge, nqaiiswihare which inequalities 1 W R sanneorepresentation. nonzero a is A ) fDni yesnestable since type Dynkin of ≃ Q > ) 0 QUIVERS 0 Z Such . µ < ) oeto of rojection Z Q Q : 0 0 . K µ Z ytkn lse of classes taking by 0 Z ( ( V Z ( Q x saquadratic a is ) ) fw aethe take we if ) eemnsthe determines → S T C ( Q to t ) k over 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 independent 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. A representation V of Q assigns a ﬁnite-dimensional vector space V (z) to each z ∈ Q0, and to each α ∈ Q1 a choice of linear map V (α): V (tα) → V (hα). A subrepresentation W ⊆ V is a collection of subspaces (W (z) ⊆ V (z))z∈Q0 such that V (α)(W (tα)) ⊆ W (hα) for all α ∈ Q1. The support of a representation V , written Supp V , is the set of vertices z ∈ Q0 such that V (z) 6= 0. Deﬁnitions of standard notions such as morphisms, direct sum, and indecomposability can be found in the references above.

Notation 2.1. For a subset S ⊆ Q0, we let [S] be the representation Q such that

k z ∈ S idk tα, sα ∈ S (2.2) [S](z)= and [S](α)= (0 z 6∈ S (0 otherwise.

For a quiver of type An, it can be seen from repeated use of Gaussian elimination that as S varies over all intervals S = {i,...,j} for 1 ≤ i ≤ j ≤ n, the representations [S] trace out all isomorphism classes of indecomposables for type A quivers (a special case of Gabriel’s theorem [Gab72]). Assume Q is acyclic. We write rep(Q) for the category of representations of Q, and often Q0 identify the Grothendieck group K0(Q) := K0(rep(Q)) with Z by identifying the class of the simple representation [{z}], for z ∈ Q0, with the standard basis vector which is 1 in coordinate z and 0 elsewhere.

Q0 2.2. Stability. A weight on a quiver Q is an element w ∈ R , where we write wz ∈ R for the value in coordinate z ∈ Q0, and we write |w| := z∈Q0 wz. A dimension vector for Q is a weight such that each w is a nonnegative integer. The dimension vector of a representation V of Q is z P (dim V (z))z∈Q0 . Given a weight w and representation V of Q, we write w(V ) := w · dim V , where · is the standard dot product on RQ0 . Remark 2.3. Another notion of stability which is prevalent in quiver literature is the following [Kin94]. A representation V of Q is θ-stable for θ ∈ RQ0 if θ(V )=0 and θ(W ) < 0 for all proper, nonzero subrepresentations W < V . It can be directly seen that a representation which is θ-stable is Z-stable as well for any Z of the form (1.1) with w = θ, but the converse does not hold. However, we can take an arbitrary Z as in (1.1) and define the weight (2.4) θ := r(V ) w − w(V )r, TOTAL STABILITY FOR TYPE A QUIVERS 5 and we have that V is Z-stable if and only if V is θ-stable. The θ-stable representations for fixed θ are the simple objects of the full, abelian subcategory of θ-semistable representations inside the category of all finite-dimensional representations of Q. Thus we can never have all indecomposable representations θ-stable in the above sense if Q has a nonempty arrow set (by Schur’s lemma). 2.3. Initial reductions. The results of this section are valid for all quivers, not just type A. Presumably these lemmas have been observed elsewhere, but we include proofs of everything for completeness. The following lemma can be easily checked (e.g. the proof of [Rei08, Lemma 4.1] generalizes). Lemma 2.5. Let Z be a linear stability function on Q and 0 → A → B → C → 0 a short exact sequence in rep(Q). Then we have the “seesaw property”:

(2.6) µZ(A) <µZ (B) ⇔ µZ (B) <µZ(C) ⇔ µZ(A) <µZ (C), and the same is true when < is replaced by ≤. Furthermore, we have for any c ∈ R that

(2.7) µZ(A), µZ (C) < c ⇒ µZ(B) < c. The next lemma allows us to only consider indecomposable subrepresentations to determine if a representation is stable. Lemma 2.8. Let Q be an arbitrary quiver. A representation V of Q is Z-stable if and only if µZ(W ) <µZ(V ) for all proper nonzero indecomposable subrepresentations W < V .

Proof. The forward implication follows from the deﬁnition. For the converse, assume µZ (W ) < µZ(V ) for all proper indecomposable subrepresentations W < V , and let Y < V be an arbitrary proper nonzero subrepresentation. If every indecomposable summand of Y were to have slope strictly less than Y , this would contradict Lemma 2.5. Therefore, taking W ≤ Y to be an indecomposable summand of Y of maximal slope, we have µZ(Y ) ≤ µZ (W ), and the result follows. Recall that from a quiver Q we obtain its opposite quiver Qop by reversing the orientations of all arrows of Q. Note that a central charge for Q is also one for Qop via the isomorphism op op K0(Q) ≃ K0(Q ) obtained by identifying the simple representations of Q and Q associated to the same vertex. Taking the vector space dual at each vertex, and dual map over each arrow, gives a duality between the categories of representations of Q and Qop. The following lemma gives a helpful connection between stability in these categories. Lemma 2.9. Let Q be an arbitrary quiver and Z a central charge for Q and Qop. Let W < V be a proper nonzero subrepresentation and (V/W )∗ < V ∗ the corresponding dual subrepresentation of op ∗ ∗ Q . Then µZ (W ) <µZ(V ) if and only if µZ((V/W ) ) <µZ (V ). ∗ Proof. Noting that µZ(X) = µZ (X ) for any representation X of Q, this lemma follows from Lemma 2.5. RQ0 The following lemma is used to prove the minimality part of the main theorem. For r ∈ >0, we denote by TSr(Q) the ﬁber of the projection described in Corollary 1.16:

Q0 (2.10) TSr(Q)= w ∈ R | Z(x)= w · x + (r · x)i ∈ T S(Q) . 6 RYAN KINSER

RQ0 Lemma 2.11. Let Q be a connected quiver and r ∈ >0 such that TSr(Q) is nonempty. Then the Q0 only subspace of R which has a translate contained in TSr(Q) is Rr.

Proof. For w ∈ T Sr(Q) and c ∈ R, let µw+cr be the slope function associated to the central charge Z(x) = (w + cr) · x + (r · x)i. It can be directly checked that µw+cr = µw + c as R-valued functions, and thus w ∈ T Sr(Q) if and only if w + cr ∈ T Sr(Q). So if TSr(Q) is nonempty, a translate of the line Rr is contained in TSr(Q). Q0 Now suppose for contradiction that there exists w ∈ T Sr(Q) and η ∈ R , where Rη 6= Rr, but that the aﬃne linear subspace w + Rη ⊂ T Sr(Q). Since Q is connected, Rη 6= Rr implies that there exists α ∈ Q1 such that ηtαrtα 6= ηhαrhα. Let V be the indecomposable representation which is dimension 1 on tα, hα and 0 elsewhere, whose only nonzero map is over α, and let W < V the simple subrepresentation supported at {hα}. Then we compute

ηtα + ηhα ηhα rhαηtα − rtαηhα (2.12) µη(V ) − µη(W )= − = 6= 0. rtα + rhα rhα (rhα + rhα)rhα Since we assumed w + Rη ⊂ T S(Q), for any c ∈ R we have

(2.13) 0 <µw+cη(V ) − µw+cη(W ) = (µw(V ) − µw(W )) + c(µη(V ) − µη(W )), a contradiction since c is arbitrary and the other values on the right hand side are ﬁxed.

3. Proof of the main theorem We begin with an elementary lemma that is used repeatedly throughout the proof of the main theorem.

Lemma 3.1. A stability function Z satisﬁes the sequence of inequalities (1.14) if and only if it satisﬁes

(3.2) µZ ([X0]) >µZ([X1]) > · · · >µZ([Xx(n)]),

Similarly, Z satisﬁes the sequencee of inequalitiese (1.15) if ande only if it satisﬁes

(3.3) µZ ([Y0]) <µZ ([Y1]) < · · · <µZ([Yy(n)]).

Proof. The two ﬁltrations of Q0 ine (1.10) inducee sequences ofe morphisms in rep(Q):

[Xx(n)] ⊂ [Xx(n)−1] ⊂···⊂ [X2] ⊂ [X1] ⊂ [X0] = [Q0] (3.4) [Q ] = [Y ] ։ [Y ] ։ [Y ] ։ . . . ։ [Y ] ։ [Y ]. e0 0 e 1 2 e ey(n)−1 e y(n) Applying Lemma 2.5 ﬁnishes thee proof.e e e e

The following elementary observation is useful in carrying out proof by induction for the main theorem.

Lemma 3.5. Let Q be a type An quiver, and V an indecomposable representation of Q with n ∈ Supp V . Then there exists k such that either V = [Xk] or V = [Yk].

e e TOTAL STABILITY FOR TYPE A QUIVERS 7

Proof of “if and only if” statement of Theorem 1.13. The ⇒ direction of follows immediately from Lemma 3.1 and the sequences of morphisms in (3.4). For the ⇐ direction, we assume that Z is given such that the inequalities (1.14) and (1.15) all hold, and we need to show that Z ∈ T S(Q). By Lemma 2.8, we are reduced to showing µZ(W ) < µZ (V ) for all 0 < W < V with both W, V indecomposable. We observe for later that since X0 = Q0 = Y0, we may concatenate the chains (3.2) and (3.3) to obtain: (3.6) µ ([X ]) < · · · <µ ([X ]) <µ ([Q ]) <µ ([Y ]) < · · · <µ ([Y ]). e Z e x(n) Z 1 Z 0 Z 1 Z y(n) We use induction on the number of vertices of Q, with the statement being vacuously true in e e e e the base case n = 1 (the unique indecomposable is stable with respect to any Z, and there are no inequalities to satisfy). Let Q be a type An quiver and assume the theorem is true for type An−1 quivers. The primary challenge in the induction is that the collection of inequalities (1.14), (1.15) for Q does not simply restrict to the corresponding collection of inequalities for smaller quivers, so we cannot easily apply the induction hypothesis. Consider the arrow n − 1 → n in Q: we can assume n is a sink without loss of generality because Lemma 2.9 gives us the n − 1 ← n case from this by reversing the directions of all inequalities, op noting that the sets Xk and Yk are interchanged when switching between Q and Q . Thus we have x(n)= x(n − 1)+1 and y(n)= y(n − 1). Let Q be the quiver obtained by removing vertex n and the arrow connected to it, and Z¯ ∈ HomZ(K0(Q), C) be the restriction of Z to K0(Q). We furthermore institute superscripts to dis- tinguish between objects associated to Q and Q, whenever necessary. To apply the induction hypothesis to Q, we need to show that Z¯ satisﬁes the sequences of inequalities in (1.14) and (1.15) associated to Q, namely: Q Q Q (3.7) µZ¯([X0 ]) >µZ¯([X1 ]) > · · · >µZ¯([Xx(n−1)]),

Q Q Q (3.8) µZ¯([Y0 ]) <µZ¯([Y1 ]) < · · · <µZ¯([Yy(n−1)]). th Whenever n∈ / S ⊆ Q0, the function µZ([S]) is independent of the n coordinate, and can thus be ¯ Q Q identiﬁed with the function µZ¯([S]) of Z on the space of stability conditions for Q. Since Xk = Xk Q Q ¯ for 0 ≤ k ≤ x(n − 1) and Yk = Yk for 0 ≤ k ≤ y(n − 1) − 1, we know Z satisﬁes all the inequalities in (3.7) and (3.8), except perhaps the far right inequality of (3.8), where we must deal with the Q Q fact that Yy(n−1) = Yy(n−1) \ {n}. Thus to apply the induction hypothesis, it remains to show that the far right inequality of (3.8) holds, which can be written as Q Q (3.9) µZ¯([Yy(n)−1]) <µZ¯([Yy(n)]), Q where we use y(n − 1) = y(n) to simplify the notation here and below. Recalling that Xx(n) = {n} since n is a sink, from (3.6) we can extract Q Q e (3.10) µZ([{n}]) = µZ ([Xx(n)]) <µZ ([Yy(n)−1]). Q Q From the short exact sequence 0 → [{n}] → [Yey(n)−1] → [Yy(ne)−1] → 0 and (3.11) we obtain Q Q (3.11) µZ([Yy(n)−1e]) <µZ([Yy(en)−1]).

e e 8 RYAN KINSER

Q Q Q From the short exact sequence 0 → [Yy(n)−1] → [Yy(n)−1] → [Yy(n)] → 0 and (1.15) we obtain Q Q (3.12) µZ([Yy(n)−1]) <µe Z([Yy(n)−1]).

Q Q Q Finally, from the short exact sequence 0 → [Yy(n)−1] e→ [Yy(n)−1] → [Yy(n)] → 0 and combining (3.12) then (3.11) we obtain e Q Q (3.13) µZ ([Yy(n)−1]) <µZ ([Yy(n)]). Q Q ¯ Since, Yy(n)−1 = Yy(n)−1, restricting the domain to Z gives exactly the inequality (3.9) we set out to show in this paragraph. ¯ Now by the induction hypothesis, Z satisﬁes all inequalities µZ¯(W ) <µZ¯(V ) for V an indecomposable representation of Q and 0 6= W < V . This means Z satisﬁes all such inequalities when n∈ / Supp V . So it remains to consider indecomposable V which are supported at vertex n.

We ﬁrst consider the case when n ∈ Supp W as well, setting out to show µZ(W ) <µZ (V ). Let [n] := [{n}] and W := W/[n] and V := V/[n]. The case where W = [n] follows from the chain (3.6) since its least term is µZ ([n]), and µZ (V ) must appear in this chain by Lemma 3.5. So we may assume now that W 6= 0. By the seesaw property, it is enough to show µZ(W ) <µZ(V/W )= µZ(V /W ), and for this it is enough to show both:

(3.14) (i) µZ(W ) <µZ(V /W ) and (ii) µZ([n]) <µZ (V /W ) by the second statement of Lemma 2.5 applied to the short exact sequence 0 → [n] → W → W → 0. Inequality (i) is immediate from the induction hypothesis since [W ] is a subrepresentation of V . Q For (ii), we have from the far right inequality of (1.14) that µZ ([n]) < µZ(Xx(n)−1). Then noting Q Q Xx(n)−1 = Xx(n)−1, applying the induction hypothesis to the chain of subrepresentations below yields:

Q (3.15) [Xx(n)−1] ≤ W < V ⇒ µZ([n]) <µZ (W ) <µZ (V ).

Since µZ (V ) < µZ (V /W ) by the seesaw property, inequality (ii) is shown, completing the case n ∈ Supp W .

We now consider the case of pairs W < V when n∈ / Supp W but n ∈ Supp V . Fix such Q a V , noting V has the representation [Yy(n)] as a proper, nonzero quotient in order for there to exist nonzero W < V without n in its support (i.e. otherwise V would be uniserial with Q Q socle [n]). Recalling Yy(n) = Yy(n), the chain of inequalities (3.6) and Lemma 3.5 imply that Q µZ(V ) <µZ([Y ]). Now we have a short exact sequence y(n) e Q (3.16) 0 → W0 → V → Yy(n) → 0 Q where W0 = [Supp V \ Yy(n)] is the unique maximal subrepresentation of V not supported at n, so the inequality just shown gives µZ (W0) <µZ(V ) by the seesaw property. The induction hypothesis then implies that µZ (W ) is maximized at W0, as W runs over all subrepresentations of V not supported at n, so we have shown that µZ (W ) <µZ (V ) and the proof of the “if and only if” part of the main theorem is completed. TOTAL STABILITY FOR TYPE A QUIVERS 9

Proof of minimality. To prove minimality, we need to use that each fiber TSr(Q) is nonempty. This is proven independently to this proof in Corollary 1.16 below, so let us assume it for now. RQ0 For a fixed r ∈ >0, we have just proven that the n − 1 inequalities (1.14) and (1.15) cut out the Q0 cone TSr(Q) ⊂ R . If any of them could be omitted, then TSr(Q) could be represented as the intersection of n − 2 or fewer linear half spaces. But then TSr(Q) would contain a translate of a two-dimensional subspace of Rn, contradicting Lemma 2.11. Proof of Corollary 1.16. This proof is due to Hugh Thomas. We begin by setting i i (3.17) xi := rk, yi := rk, xi := xk, yi := yk. k∈XQ k∈Y Q k=1 k=1 Xi Xi X X For Z as in (1.1) for r fixed, we consider the linear functionse of w definede by Q Q Q ∅ fi(w) := µZ([Xi ]) − µZ([Xi+1]), 1 ≤ i ≤ M := max{i : Xi+1 6= } (3.18) Q Q Q ∅ gj(w) := µZ([Yj+1]) − µZ([Yj ]), 1 ≤ j ≤ N := max{j : Yj+1 6= }.

Our main theorem says that Z ∈ T Sr(Q) if and only if these functions are all strictly positive on the weight. We can assume that both M,N ≥ 1, since otherwise the quiver is equioriented and the corollary is immediate [Rei03, Example A]. If TSr(Q) were empty, then by Farkas’ lemma (see for example [BV04, §5.8.3]) there would exist a linear combination with nonnegative coefficients M N (3.19) 0 = aifi(w)+ bjgj(w), ai, bj ∈ R≥0, i j X=1 X=1 where some ai 6= 0 for 1 ≤ i ≤ M or some bj 6= 0 for 1 ≤ j ≤ N. Assume for contradiction that we have such an expression, and take one for which Q has a minimal number of vertices. We will successively consider the coefficients of w1, w2, w3,... and show that (up to a scalar multiple) the vanishing of these coefficients forces ai = xi and bj = yj up to a point, and then yields a contradiction when considering the coefficient of wt when either x(t) or y(t) is maximal (i.e., in Notation 1.6, when we reach a vertex in the furtheste right columne of vertices or furthest up row of vertices). First consider the coefficient of w1. Assume 1 → 2 in Q (without loss of generality by the same application of Lemma 2.9 used in the proof of the main theorem). This variable appears only in a1 b1 f1(w) and g1(w), and the coefficient of w1 in (3.19) is x1 − y1 . Up to a scalar, we are forced to take a1 = r1 = x1 and b1 = y1 = y1. Proceeding inductively up the indices for w, we next consider the coefficient of wt for 1

By induction we already have at−1 = xt−1(= t − 1) and b1 = y1, so a direct substitution into the above expression yields e1 e (3.21) (at − xt−1) − 1 = 0, xt e 10 RYAN KINSER forcing at = xt−1 + xt = xt. However, if x(t) is maximal, then M = t − 1 so for (3.19) to hold we need e e −1 −1 (3.22) at−1 + b1 = 0, xt y1 which is a contradiction since both terms of the left hand side are negative. Continuing up the indices, consider the general situation of t ∈ Q0 such that both k := x(t) > 1 and l := y(t) > 1 and neither is maximal among vertices of Q. The coeﬃcient of wt in (3.19) receives contributions from fk−1(w), fk(w), gl−1(w), gl(w), and (3.19) implies −1 1 1 −1 (3.23) ak−1 + ak + bl−1 + bl = 0. xk xk yl yl

By induction on t we already have ak−1 = xk−1 and bl−1 = yl−1, and either ak = xk (if t − 1 ← t in Q) or bl = yl (if t − 1 → t in Q), so the remaining coeﬃcient is determined in (3.23). In the case that t − 1 ← t in Q, direct substitution intoe the above expressione yields e e 1 (3.24) 1 + (yl−1 − bl) = 0, yl forcing bl = yl−1 + yl = yl. The case that t − 1e → t in Q is similar. At some point we arrive at t ∈ Q0 such that either k or l is maximal, say k (again the other case is similar). Thene the arrowse of Q are oriented like t − 1 → t ←···← n. The coeﬃcient of wt has one fewer term and is by induction equal to −1 1 −1 −1 (3.25) ak−1 + bl−1 + bl = xk−1 − 1 < 0, xk yl yl xk thus nonvanishing. This is the desired contradiction ande the corollary is proven.

We illustrate the main theorem by continuing our running example.

Example 3.26. Continuing Example 1.11, Theorem 1.13 says that the minimal set of inequalities in the variable Z which deﬁne TS(Q) is:

(3.27) µZ ([1]) >µZ ([2]) >µZ([3, 4]) >µZ([5]) >µZ ([6, 7]) >µZ ([8]),

(3.28) µZ ([1, 2, 3]) <µZ ([4, 5, 6]) <µZ ([7, 8]).

Q0 Q0 Taking coordinates w1, . . . , w8, r1, . . . r8 on R × (R>0) , these are explicitly w w w + w w w + w w (3.29) 1 > 2 > 3 4 > 5 > 6 7 > 8 r1 r2 r3 + r4 r5 r6 + r7 r8

w + w + w w + w + w w + w (3.30) 1 2 3 < 4 5 6 < 7 8 , r1 + r2 + r3 r4 + r5 + r6 r7 + r8 and admit a solution in w for any choice of r. TOTAL STABILITY FOR TYPE A QUIVERS 11

References

[AI] P.J. Apruzzese and Kiyoshi Igusa. Stability conditions for affine type A. Algebras and Representation Theory, online Nov 8, 2019, DOI:10.1007/s10468-019-09926-z. [BGMS20] Emily Barnard, Emily Gunawan, Emily Meehan, and Ralf Schiffler. Cambrian combinatorics on quiver representations (type a), 2020. [Bri07] Tom Bridgeland. Stability conditions on triangulated categories. Ann. of Math. (2), 166(2):317–345, 2007. [BV04] Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge University Press, Cambridge, 2004. [DW17] Harm Derksen and Jerzy Weyman. An introduction to quiver representations, volume 184 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2017. [Gab72] Peter Gabriel. Unzerlegbare Darstellungen. I. Manuscripta Math., 6:71–103; correction, ibid. 6 (1972), 309, 1972. [HH] Pengfei Huang and Zhi Hu. Stability and indecomposability of the representations of quivers of An-type. arXiv:1905.11841. [Igu] Kiyoshi Igusa. Linearity of stability conditions. Communications in Algebra, online Jan 8, 2020, DOI: 10.1080/00927872.2019.1705466. [Kin94] A. D. King. Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2), 45(180):515–530, 1994. [Qt] The QPA-team. QPA - quivers, path algebras and representations - a GAP package. Version 1.29; 2018 (https://folk.ntnu.no/oyvinso/QPA/). [Rei03] Markus Reineke. The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli. Invent. Math., 152(2):349–368, 2003. [Rei08] Markus Reineke. Moduli of representations of quivers. In Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., pages 589–637. Eur. Math. Soc., Zürich, 2008. [Rud97] Alexei Rudakov. Stability for an abelian category. J. Algebra, 197(1):231–245, 1997. [Sch91] Aidan Schofield. Semi-invariants of quivers. J. London Math. Soc. (2), 43(3):385–395, 1991. [Sch14] Ralf Schiffler. Quiver representations. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, 2014.

University of Iowa, Department of Mathematics, Iowa City, USA Email address, Ryan Kinser: [email protected]