combinatorics and triangulations of

Motivated by higher homological algebra, we associate quivers to triangulations of even-dimensional cyclic polytopes and prove two results showing what information about the triangulation is encoded in the quiver. We first show that the cut quivers of Iyama and Oppermann correspond precisely to 2 d -dimensional triangulations without interior ( d + 1)- simplices. This implies that these triangulations form a connected subgraph of the flip graph. Our second result shows how the quiver of a triangulation can be used to identify mutable internal d -simplices. This points towards what a theory of higher-dimensional quiver mutation might look like and gives a new way of understanding flips of triangulations of even-dimensional cyclic polytopes.

For d = 1, these cut quivers are precisely orientations of the A n Dynkin diagram. Hence, this is a higher-dimensional generalisation of the fact that a triangulation of a convex polygon has no internal triangles if and only if its quiver is acyclic, and in this case the quiver is an orientation of the A n Dynkin diagram, and thus provides a neat combinatorial description of a certain class of triangulations of an even-dimensional cyclic polytope. An application of this result is that the set of triangulations of a 2d-dimensional cyclic polytope without internal (d + 1)-simplices forms a connected subgraph of the flip graph of the polytope.
Unlike for the 2-dimensional case, for d > 1 it is not possible to perform a bistellar flip at every internal d-simplex of a 2d-dimensional triangulation, or, equivalently, at every vertex of its quiver. This is an important difference with classical cluster theory, where a key property is that one can mutate a given cluster at every vertex of its quiver. This feature makes higher-dimensional cluster theory much more difficult to work with. Our second result uses the quiver of a triangulation to give a combinatorial criterion for identifying which d-simplices are mutable-that is, admit a bistellar flip. We show how the arrows in the quiver can be partitioned into paths which we call maximal retrograde paths, and prove the following theorem.

2d). A d-arc of T is mutable if and only if it is not in the middle of a maximal retrograde path.
This theorem gives a quiver-theoretic criterion for mutability, and hence points towards what a theory of higher-dimensional quiver mutation [15] could look like. Other extensions of quiver mutation have been of interest in the literature, such as to ice quivers [31]. As an application of this theorem, we give a rule for mutating cut quivers at vertices which are not necessarily sinks or sources. From the perspective of polyhedral geometry, this provides a visual way of understanding mutability for higher-dimensional triangulations, and makes it easier to compute bistellar flips of higher-dimensional triangulations by hand. In the case of polygon triangulations, all retrograde paths are of length one, so that the criterion imposes no restriction and all arcs are mutable.
Our work can also be seen in the context of combinatorial descriptions of algebraic mutation. For example, [29] gives a combinatorial description of the silting mutation from [1], whilst [17] gives a combinatorial procedure for tilting mutation. The algebraic objects corresponding to our triangulations and quivers are cluster-tilting objects in the (d + 2)-angulated cluster categories of type A from [30]. Theorem 0.2 gives a combinatorial way of identifying the mutable summands of these cluster-tilting objects.
The structure of this paper is as follows. In Section 1 we give background to the paper, predominantly on triangulations of even-dimensional cyclic polytopes. In Section 2 we consider triangulations of 2d-dimensional cyclic polytopes without interior (d + 1)-simplices and prove Theorem 0.1. In Section 3 we study mutability of dsimplices in terms of quivers and prove Theorem 0.2. we mean the set of subsets of [m] of size k. For l ∈ [m], using the notation of [28], we use < l to denote the cyclically shifted order on [m] given by l < l l + 1 < l < · · · < l m − 1 < l m < l 1 < l · · · < l l − 1.
For r ⩾ 3, a 1 < · · · < a r is a cyclic ordering if there is an l ∈ [n + 2d + 1] such that a 1 < l · · · < l a r .
Algebraic Combinatorics, Vol. 6 #3 (2023) Throughout this paper, we will often need to change the ordering on [m] to < l for some l. We will usually do this tacitly, rather than explicitly writing < l in place of <. Similarly, we will usually be tacitly using arithmetic modulo n + 2d + 1, where this will be the number of vertices of our cyclic polytope. Hence, we shall write things like j = i − 1, when we mean j ≡ i − 1 (mod n + 2d + 1). Our convention here will also be that our equivalence-class representatives for arithmetic modulo n + 2d + 1 will be [n + 2d + 1] rather than {0, 1, . . . , n + 2d}. This is because we shall label the vertices of our cyclic polytope by [n + 2d + 1].
Finally, given a tuple A ∈ [m] k+1 , we will denote the elements of A by (a 0 , a 1 , . . . , a k ) where a 0 < l a 1 < l · · · < l a k in a cyclically shifted order to be determined by the context. By default this will be the usual order on [m]. Because we wish sometimes to change this order, we are also tacitly identifying tuples which are the same up to cyclically shifted reordering. The same applies to other letters of the alphabet: the upper case letter denotes the tuple; the lower case letter is used for the entries, which are ordered according to their index, which starts from 0. For example, if we write A = (1, 3, 5), then we have a 0 = 1, a 1 = 3, and a 2 = 5. If we changed the cyclic ordering from < 1 to < 3 , then we would have A = (3, 5, 1) with a 0 = 3, a 1 = 5, and a 2 = 1.
A facet of C(n + δ + 1, δ) is a face of codimension one. A circuit of a cyclic polytope C(n + δ + 1, δ) is a pair, (Z + , Z − ), of sets of vertices of C(n + δ + 1, δ) which are inclusion-minimal with respect to the property conv(Z + )∩conv( The circuits and facets of a cyclic polytope are independent of its particular geometric realisation given by the choice of points on the moment curve, by [18,3]. This implies that whether or not a collection of ordered (δ + 1)-tuples of [n + δ + 1] forms a triangulation of C(n + δ + 1, δ) is likewise independent of the particular geometric realisation. Hence, in this paper we primarily consider triangulations combinatorially: a combinatorial triangulation of C(n + δ + 1, δ) is a collection of ordered (δ + 1)-tuples with entries in [n + δ + 1] which gives a geometric triangulation of C(n + δ + 1, δ). Unless otherwise specified, when we write 'triangulation' we mean a combinatorial triangulation. Likewise, if we say that A is a k-simplex, we mean that A is an ordered (k + 1)-tuple with entries in [n + δ + 1]. We write |A| to denote the geometric realisation of A as a geometric simplex which is the convex hull of k + 1 points on the moment curve.
In this paper we are exclusively interested in triangulations of even-dimensional cyclic polytopes. These were given an elegant combinatorial description in [30], which we now explain. This combinatorial description was in turn used to connect triangulations of even-dimensional cyclic polytopes to representation theory of algebras. We will sometimes comment on these connections, but this paper does not require the reader to be familiar with representation theory.
A d-simplex of a triangulation of C(n + 2d + 1, 2d) is internal if it does not lie within a facet of C(n + 2d + 1, 2d). It is clear that a triangulation of a convex polygon is determined by the arcs of the triangulation; similarly, a triangulation of C(n + 2d + 1, 2d) is determined by the internal d-simplices of the triangulation, by a theorem of Dey [8]. For brevity, we refer to an internal d-simplex of a triangulation T of C(n + 2d + 1, 2d) as a d-arc of T . We write d-arcs(T ) for the set of d-arcs of T . We say that a (d + 1)-simplex of T is interior if all of its facets are d-arcs.
In order to recall the description of the triangulations of C(n + 2d + 1, 2d) from [30], and for use later in the paper, we denote the following sets.
and a d + 2 ⩽ a 0 + n + 2d + 1 ; Remark 1.1. Algebraically,Ĩ d n+2d+1 labels the d-cluster-tilting subcategory of the derived category of A d n , whilst ⟲ I d n+2d+1 labels its cluster category [30]. Here A d n is the higher Auslander algebra of type A; see [25]. Triangulations of cyclic polytopes can be mutated by operations known as bistellar flips [7,Section 2.4]. The following theorem can be taken as a definition of this operation in the case of triangulations of C(n + 2d + 1, 2d). 1.3. Cuts and slices. We will associate quivers to triangulations of evendimensional cyclic polytopes and show what properties of the triangulation are encoded in the quiver. Indeed, we now define the quivers which are higher analogues of orientations of the A n Dynkin diagram, following [26]. The main result of the first part of this paper is that these quivers characterise triangulations of C(n + 2d + 1, 2d) with no internal (d + 1)-simplices.
Let Q (d,n) be the quiver with vertices    We say that arrows A → A + F i are arrows of type i. See Figure 2 for pictures of these quivers. A subset C ⊆ Q is called cut if it contains exactly one arrow from each (d + 1)-cycle in Q (d,n) . Given a cut C, we write Q (d,n) C for the quiver with arrows Q (d,n) 1 ∖ C and refer to this as the cut quiver. Examples of cut quivers can be seen in Figure 3. Note that the cut quivers of Q (1,n) are precisely the orientations of the A n Dynkin diagram. Hence, for d > 1, we think of cut quivers of Q (d,n) as higher analogues of orientations of the A n Dynkin diagram.
Iyama and Oppermann [26] show that cut quivers of Q (d,n) are precisely the quivers than can be realised as slices of another family of quivers, denotedQ (d,n) , which we now define. LetQ (d,n) be the quiver with vertices . We also write A + 1, in an analogous way.
where π(a i ) := a i (mod n + 2d + 1) and sort indicates that we should sort the tuple so that it is increasing with respect to the usual order on [n + 2d + 1].
Note that the definition ofĨ d n+2d+1 guarantees that π(A) does not contain any repeated entries.
Following [26,Definition 5.20], we define a slice ofQ (d,n) to be a full subquiver S ofQ (d,n) such that: (1) Any ν d orbit inQ (d,n) contains precisely one vertex which belongs to S.
(2) S is convex, i.e., for any path P inQ (d,n) connecting two vertices in S, all vertices appearing in P belong to S.
Slices are shown in blue in Figure 4 and Figure 5.
Remark 1.5. These combinatorial constructions of course have representationtheoretic interpretations, which are explained in [26]. Cuts and slices correspond to iterated d-APR tilts. The map ν d is the d-th desuspension of the Serre functor on the derived category. The map π corresponds to the projection from the derived category to the (d + 2)-angulated cluster category [30].

Mutation of cuts and slices.
Cuts and slices can be mutated, as was defined in [26]. This corresponds to algebraic mutation and also to bistellar flips.
• Let C be a cut of Q (d,n) and let x be a source of Q which end at x. By [26, Proposition 5.14], we have that µ + x (C) and µ − x (C) are also cuts of Q (d,n) . This process is illustrated in Figure 6.
• Let S be a slice ofQ (d,n) . If x is a source of S, then define a full subquiver µ + x (S) ofQ (d,n) by removing x from S and adding ν −1 d x [26,Definition 5.25]. Dually, if x is a sink of S, define a full subquiver µ − x (S) by removing x and adding ν d x. This process is illustrated in Figure 7. If C S is the cut corresponding to a slice S then C µ + , provided x is a source or sink, respectively. Here we abuse notation by using x to refer both to the relevant vertex of S and to the relevant vertex of Q

Triangulations without interior (d + 1)-simplices
In this section we prove our combinatorial description of triangulations of C(n + 2d + 1, 2d) without interior (d + 1)-simplices and use this description to show that this class of triangulations is connected by bistellar flips. From the perspective of polyhedral geometry, this provides a combinatorial description of the triangulations without interior (d + 1) and shows how they sit within the class of all triangulations. From the perspective of representation theory, this gives a geometric description of iterated d-APR tilts of A d n . 2.1. The quiver of a triangulation. We first define the quiver of a triangulation, which is the higher-dimensional version of the quiver of a polygon triangulation-see, for instance, [   We define the quiver in this way so that it coincides with the Gabriel quiver of the endmorphism algebra of the cluster-tilting object corresponding to the triangulation. The arrows mirror the description of the homomorphisms in the cluster category O A d n from [30]. However, we now prove a simpler description of the quiver, given in Corollary 2.4. In order to obtain this, we first make some observations about cyclic polytopes. Changing the ordering on on [n + 2d + 1] from < 1 to < l induces a (combinatorial) automorphism of the cyclic polytope C(n + 2d + 1, 2d); see [27]. We think of this automorphism as reorienting the cyclic polytope. Generalising [30], given a 2d-simplex T of C(n + 2d + 1, 2d) and an ordering < l of [n + 2d . If the ordering is determined by the context, we simply write e(T ).
Proof. This follows from applying [30,Proposition 2.13] (In fact, by the ordering we have chosen, we must have that i = d.) Therefore A and B are both faces of the 2d-simplex T and they share all but one entry, as desired. □ We call arrows A → A + rE i arrows of type i, analogously to our terminology of arrows of type i in Q (d,n) from Section 1.3. Note that this labelling depends on the choice of an ordering < l .
For an arrow α, we denote the head h(α) and the tail t(α) such that t(α) We note the following property concerning paths in Q(T ) which will be useful later.
Lemma 2.5. Given a triangulation T of C(n + 2d + 1, 2d) and A, B ∈ Q 0 (T ) with Proof. This is clear from Definition 2.1, using induction. □ 2.2. Quiver description. With these preliminaries taken care of, we now move to prove the first main result of this paper, which describes triangulations T without interior (d + 1)-simplices in terms of their quivers Q(T ). Recall that a (d + 1)-simplex of a triangulation T of C(n + 2d + 1, 2d) is interior if all of its facets are d-arcs. Slices ofQ (d,n) give triangulations of C(n + 2d + 1, 2d). We give a direct combinatorial proof of this, although it can also be deduced from the fact that slices correspond to iterated d-APR tilts [26,Theorem 4.15], which implies that projecting to the cluster category will give a triangulation [30,Theorem 6.4]. Recall the map π from Definition 1.4. Proposition 2.6. If S is a slice ofQ (d,n) , then the vertices π(S 0 ) give a triangulation of C(n + 2d + 1, 2d).
Proof. There are as many ν d -orbits as there are elements of ⟲ I d n+2d+1 containing 1, namely n+d−1 d . Suppose that there exist π(A) and π(B) in π(S 0 ) with π(A) π(B). We assume without loss of generality that a 0 < b 0 , noting that π(A) π(B) implies that a i ̸ = b i for all i. We claim that A < B. Suppose for contradiction that b i < a i for some i. We may choose the minimal i such that this is the case. Then we must have a i − a i−1 > n + 2d + 1. Hence a d − a 0 > n + 2d + 1 > n + 2d − 1, which contradicts A ∈Ĩ d n+2d+1 .

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Quiver combinatorics and triangulations of cyclic polytopes Therefore A < A + 1 ⩽ B, which means that here is then a path A A + 1 B, so A + 1 ∈ π(S 0 ) by convexity. But this contradicts the fact that S contains one vertex from every ν d -orbit. Hence π(S 0 ) is a non-intertwining subset of ⟲ I d n+2d+1 of size n+d−1 d , and so gives a triangulation of C(n + 2d + 1, 2d). □ A similar argument also shows the following lemma, which will be useful later.
Lemma 2.7. If S is a convex subquiver ofQ (d,n) such that π(S 0 ) is a triangulation of C(n + 2d + 1, 2d), then S is a slice.
Proof. Suppose that S is a convex subquiver such that π(S 0 ) is a triangulation of C(n + 2d + 1, 2d). Suppose for contradiction that S possesses two vertices A and B which are in the same ν d -orbit. There is then either a path A B or a path B A. Without loss of generality, we suppose the former. But then there is a path A A + 1 B, so we must have A + 1 ∈ S 0 by convexity. This is a contradiction, since π(A) and π(A + 1) are intertwining. □ The main theorem of this section is as follows. This shows how the quivers of triangulations detect the structure of the triangulation, and gives a geometric description of iterated d-APR tilts of A d n .
Theorem 2.8. A triangulation T of C(n + 2d + 1, 2d) has no interior (d + 1)-simplices if and only if its quiver is a cut of Q (d,n) , and this is the case if and only if its quiver has no cycle.
Our strategy for proving this theorem is to prove facts about the quivers Q(T ) for triangulations T of C(n + 2d + 1, 2d) without internal (d + 1)-simplices. We then use these properties to show that one can realise Q(T ) as a slice ofQ (d,n) , which will imply that Q(T ) is a cut of Q (d,n) . The idea of the proofs of both of the following lemmas is that if Q(T ) does not possess a certain property, then we can find an interior (d + 1)-simplex in T . Proof. By Proposition 2.3, every arrow is of the form A → A + rE i for some r > 0 and for each such arrow, we have that (a 0 , a 1 , . . . , a i , a i + r, a i+1 , a i+2 , . . . , a d ) is a face of a 2d-simplex of T . If r > 1, then this is an interior (d + 1)-simplex. □ Lemma 2.10. Suppose that T has no interior (d + 1)-simplices.
. By re-ordering, we may assume that i = d. then  (a 0 , a 1 , . . . , a d ) (x 1 , x 2 , . . . , x d , a d + 1), which is a d-face of T . Hence a j−1 + 2 ⩽ x j < a j + 1. Then (a 0 , a 1 , . . . , a j−1 , x j , x j+1 , . . . , x d , a d + 1) is a (d + 1)-face of T and an interior (d + 1)-simplex of T , a contradiction. □ We now prove some facts about the quiversQ (d,n) , which will be useful in proving the main theorem of this section. These will be used in combination with the previous two lemmas to show that one can realise Q(T ) as a slice if T has no interior (d + 1)simplices. We say that a full subquiver P ofQ (d,n) is switching-closed if whenever Algebraic Combinatorics, Vol. 6 #3 (2023)

, then the sequence of arrows
Lemma 2.11. Let P be a switching-closed full subquiver ofQ (d,n) . If there is a path A B in P , all other paths A B inQ (d,n) must also lie in P .
Proof. Suppose that we have a path A B in P . The length of any such path is d i=0 (b i −a i ). We prove the claim by induction on this quantity. The base case, where the length is 1, follows from the fact that P is a full subquiver ofQ (d,n) .
For the inductive step, we assume that the claim holds for all X and Y with We have that B is the head of up to (d + 1) arrows, namely the ones with tails  By a walk in a quiver Q from A to B we mean a finite sequence of arrows β 1 β 2 . . . β s such that β 1 is incident at A, β s is incident at B, and β i−1 and β i are incident at a common vertex for all i ∈ {2, 3, . . . , s}. In this case, we write A B. That is, a path only consists of forwards arrows, but a walk may contain backwards arrows as well.
Lemma 2.12. Let P be a connected switching-closed full subquiver ofQ (d,n) . If A, B ∈ P 0 are such that there is a path A B inQ (d,n) , then there is a path A B in P .
Proof. Let A, B ∈ P 0 . Suppose that there is a path A B inQ (d,n) . There is certainly a walk W : A B in P , since P is connected. We prove that there is also a path A B in P by induction on the number of backwards arrows in this walk. The base case, in which there are zero backwards arrows in the walk, is immediate.
Hence we suppose for induction that the claim holds for walks with fewer backwards arrows than W . We may assume that the final arrow in W is a backwards one, otherwise we may remove the final arrow and consider instead the walk A B ′ , where A B ′ → B is the original walk W . Therefore we can assume that our walk is of the form A C ← B, where C ∈ P . By the induction hypothesis, we can replace this with a walk of the form A C ← B in P . Moreover, by Lemma 2.11 we have that all paths A C inQ (d,n) lie in P . Since we have a path A B inQ (d,n) , we have A ⩽ B and, moreover, A ⩽ B ⩽ C. There is therefore a path A B C inQ (d,n) . This path is in P , since every path A C is in P , which gives the desired path A B in P . □ These two lemmas imply the following corollary.
Corollary 2.13. Let P be a connected switching-closed full subquiver ofQ (d,n) . Then P is convex inQ (d,n) .
Proof. Suppose that P is a connected switching-closed full subquiver ofQ (d,n) . Let A, B ∈ P 0 be such that there is a path A B inQ (d,n) . By Lemma 2.12, there is a path A B in P . Then, by Lemma 2.11, we have that all paths A B lie in P , and so P is convex. □ This corollary is useful because it is easier to check the property of being connected and switching-closed than the property of being convex. Given a full subquiver P of Algebraic Combinatorics, Vol. 6 #3 (2023) Q (d,n) , we write P for the smallest switching-closed subquiver containing P . The subquiver P is well-defined since the intersection of a set of switching-closed full subquivers is switching-closed, so P may be constructed as the intersection of all switching-closed subquivers containing P . Proof. We first suppose for contradiction that T contains no interior (d + 1)-simplices and that Q(T ) does contain a cycle. By Lemma 2.9, we can realise this cycle as a path P : A B inQ (d,n) , where A is some vertex in the cycle in Q(T ) and π(B) = A with A ̸ = B. We consider this path P as a subquiver ofQ (d,n) . By Lemma 2.11, every path A B inQ (d,n) must lie in P . There is a path A A + 1 B inQ (d,n) . Hence A + 1 is a vertex of P . By Lemma 2.10, if C is a vertex of P , then π(C) is a vertex of Q(T ). Therefore π(A + 1) is a vertex of Q(T ), but this is a contradiction, since π(A + 1) and A are intertwining.
We now suppose that T is a triangulation with an interior (d + 1)-simplex (a 0 , . . . , a d+1 ). Then Q(T ) has a cycle given by concatenating the paths . . . (a 1 , a 2 , . . . , a d , a d+1 )  (a 0 , a 1 , . . . , a d  We are now ready to prove our first main result, noting that the second part of the statement has already been established by Proposition 2.14. Proof of Theorem 2.8. First suppose that T has no interior (d + 1)-simplices. We consider the full subquiver R ofQ (d,n) with vertices We claim that this is disconnected and that each connected component gives Q(T ) by applying π. Let A ∈ Q 0 (T ). If R is connected then it contains a walk W : A  (a 1 , a 2 , . . . , a d , a 0 +n+2d+1). We consider W as a subquiver ofQ (d,n) and consider W . By Lemma 2.12, W contains a path P : A  (a 1 , a 2 , . . . , a d , a 0 + n + 2d + 1). By Lemma 2.10, if B is a vertex of W , then π(B) is a vertex of Q(T ). Hence all vertices of P give vertices of Q(T ), which therefore contains a cycle. But this contradicts Proposition 2.14.
By this argument, R is disconnected and, moreover, the vertices of each connected component are in bijection with the vertices of Q(T ) via π, since Q(T ) is connected. Moreover, the arrows in each connected component of R are the same as the arrows in Q(T ), by Lemma 2.9. Hence, by choosing one of the connected components, we obtain a full subquiver S ofQ (d,n) such that π(S) = Q(T ). We then have that S is a switching-closed connected subquiver ofQ (d,n) , so S is convex by Lemma 2.13. Since π(S 0 ) is a triangulation, it then follows from Lemma 2.7 that S is a slice. Hence Q(T ) is a cut of Q (d,n) by [26,Theorem 5.24].
Algebraic Combinatorics, Vol. 6 #3 (2023) Now suppose that Q(T ) is a cut of Q (d,n) . Then Q(T ) cannot contain any cycles, since cut quivers can be realised as slices, which are full subquivers ofQ (d,n) , which does not contain any cycles. Then we obtain that T contains no interior (d + 1)simplices by Proposition 2.14. □ The flip graph of C(n+2d+1, 2d) is the graph whose vertices are the triangulations of C(n + 2d + 1, 2d), with edges connecting triangulations related by a bistellar flip. Theorem 2.8 implies that the set of triangulations of C(n+2d+1, 2d) without interior (d + 1)-simplices is connected by bistellar flips.

A combinatorial criterion for mutation
Given a triangulation T of C(n + 2d + 1, 2d), we say that a d- It is clear that here A and B must intertwine. For d = 1, where triangulations of C(n + 2d + 1, 2d) are triangulations of convex (n + 3)-gons, all d-arcs are mutable. But this is not true for d > 1. In this section, we prove a criterion for identifying the mutable d-arcs of a triangulation T from its quiver Q(T ). This then leads us to a rule for mutating cut quivers at vertices which are neither sinks nor sources. From the perspective of polyhedral geometry, we again see how properties of the triangulation are encoded in the quiver; from the algebraic perspective, we see what a theory of higher-dimensional quiver mutation might look like. We begin with some motivating observations concerning cuts. We explain how a cut quiver may be decomposed into distinguished cut cycles, and observe that an arc of the triangulation is mutable if and only if it does not occur in the middle of a distinguished cut cycle. Proof. To see this, consider a (d + 1)-cycle with labels F i0 , F i1 , . . . , F i d . Then we must have   3.1. General triangulations. Cut quivers have a very particular form and it is this that allows us to determine the distinguished cut (d + 1)-cycles of the quiver, and then to use these to determine the mutable d-arcs of the triangulation. In general, quivers of triangulations may be much more complicated than cut quivers. Nevertheless, we may generalise Observation 3.2 to arbitrary triangulations of even-dimensional cyclic polytopes using the following notion.
Definition 3.4. Given a triangulation T of C(n + 2d + 1, 2d) and fixing a cyclically shifted order of [n + 2d + 1], a path , we say that A i is in the middle of this retrograde path. We consider paths consisting of a single arrow to be trivially retrograde. We say that a retrograde path is maximal if it is not contained in any longer retrograde paths.
Remark 3.5. The distinguished cut (d + 1)-cycles in a cut quiver are maximal retrograde paths. Arrows in this cut cycle are labelled F l+1 , . . . , F d , F 0 , . . . , F l−1 for some l. The arrows in Q(T ) labelled by F i are of the form A → A + E i , and we always have that a i + 1 < a i+1 . Hence, we obtain that these paths are retrograde. These paths are, furthermore, maximally retrograde, since the next arrow in the path at either end would have to be labelled by F l , but this is precisely the arrow that has been cut out. Proof. If an arrow follows A → A + rE i , then must be of the form A + rE i → A + rE i + sE j with j = i − 1. If an arrow precedes it, then it must be of the form A − sE j → A with j = i + 1. Both such arrows must be unique, by Corollary 2.4. Hence, there is only one way to extend an arrow to a maximal retrograde path. □ Proposition 3.7. The maximal length of a retrograde path in Q(T ) is d.
Proof. Suppose for contradiction that we have a retrograde path in Q(T ) of length (d + 1). By re-ordering, we can represent this in the form a 1 , . . . , a d ) → (a 0 , a 1 , . . . , a d−1 Since this path is retrograde, we have a i < b i < a i+1 for all i ∈ [d]. But this implies that (a 0 , a 1 , . . . , a d ) and (b 0 , b 1 , . . . , b d ) Proof. Let T B be the collection of 2d-simplices of T which have a d-face intertwining with the d-arc B. Let T ∈ T B . We first show that the set of d-faces of T is connected in Q B (T ). Hence, let A, A ′ be two d-faces of T which intertwine with B. Then A and A ′ must have a common vertex, since they are both faces of the same 2d-simplex, so, by re-ordering, we can assume a 0 = a ′ 0 . We know that A and B must be intertwining, so we may also assume that Since a 0 = a ′ 0 and A ′ also intersects B, we also have that where this minimum is taken in the linearly ordered set between b i−1 and b i . Then C ∈ T since it is a d-face of T . Moreover, it is in Q B (T ). There is a path C A in Q B (T ) due to Definition 2.1 since, by construction, (C − 1) A. There is likewise a path C A ′ in Q B (T ). Therefore, A and A ′ are connected to each other in Q B (T ). Hence, any two d-arcs lying in a common 2d-simplex are connected by a walk in Q B (T ).
We now show that the d-arcs in Q B (T ) which lie in different 2d-simplices are connected with each other. Let T, T ′ ∈ T B . Recall from Section 1.2 that we use | − | to denote geometric realisation of simplices, so that |B| is the geometric realisation of B. If one chooses points x ∈ |T | ∩ |B| and x ′ ∈ |T ′ | ∩ |B|, then the line segment xx ′ connecting x and x ′ must lie entirely within |B|, since |B| is convex. If one travels from |T | to |T ′ | along xx ′ , then one runs through a series of 2d-simplices |T | = |T 0 |, |T 1 |, . . . , |T r | = |T ′ | where each pair of 2d-simplices |T l−1 | and |T l | shares a common face |U l | which must also intersect |B|. Then, by the description of the circuits of C(n + 2d + 1, 2d) from Section 1. Remark 3.9. Lemma 3.8 may also be seen quickly using an algebraic argument. [30,Theorem 5.6] implies that Q B (T ) must be the support of an indecomposable module, and so must be connected.
This gives the following useful corollary, which implies that in order to check whether one can mutate a d-arc A to a d-arc B, it suffices only to check whether the d-arcs adjacent to A in the quiver intertwine with B, rather than checking all d-arcs for whether they intertwine with B.
Proof. Suppose that we are in the situation described. We know from Lemma 3.8 that Q B (T ) is connected, and the set-up gives us that it contains at least two vertices, one of which is A. Hence there is a vertex of Q B (T ) which is adjacent to A. □ We can now prove the main theorem of this section.
So A is mutable if and only if this product is non-empty. But the product is nonempty if and only if z i+1 ⩽ b i for all i, which is precisely the condition that none of the paths Z i+1 → A → B i are retrograde. □ Proof. We know from the proof of Theorem 3.11 that any element of may replace A in a bistellar flip. But, we have that the d-arc which can replace A in a bistellar flip must be unique. Hence z i+1 = b i for all i and the unique element of the product must replace A in the bistellar flip. Theorem 3.11 and Corollary 3.12 make it much easier to compute bistellar flips of triangulations of even-dimensional cyclic polytopes by hand. This also makes it easier to compute mutations of cluster-tilting objects of A d n by hand.
Example 3.13. We provide examples of how one may use this criterion to identify the mutable d-arcs of a triangulation. We represent maximal retrograde paths using consecutive arrows of the same colour.
Note that maximal retrograde paths are not always of length d. This is shown by the triangulation of C(10, 6) given in Figure 10. (We use 'A' to denote 10.) The mutable d-arcs of this triangulation are 357A, 1368, 1479.
One can also illustrate Corollary 3.12. Consider the d-arc 1368 in Figure 10. This is mutable by Theorem 3.11, so we can compute what d-arc it is exchanged for. Between 1 and 3 we must have 2 and between 6 and 8 we must have 7. Then, between 3 and 6 we must have 5 since 1368 is adjacent to 1358. Similarly, between 8 and 1 we must have 9, because 1368 is adjacent to 1369. Hence performing a bistellar flip at 1368 exchanges this d-arc for 2579. Observe that the retrograde-path analysis makes it easier to compute the bistellar flips of the triangulation. In the resulting triangulation, shown in Figure 11, none of the retrograde paths are of length d; they are all of length d − 1.
Algebraic Combinatorics, Vol. 6 #3 (2023) 3.2. Mutating cut quivers. There is a rule for mutating cut quivers at sinks and sources [26], as described in Section 1.3.1. In this section, we extend this rule to allow mutation at vertices which are not in the middle of retrograde paths, but which are not necessarily sinks or sources. In the case where the cut quiver is Q(T ) for a triangulation T , we also describe the effect of the mutation on the triangulation T . For the following lemmas, we let C be a cut of Q (d,n) . The purpose of these lemmas is to describe the local structure of a cut quiver around a vertex which is not in the middle of a distinguished cut (d + 1)-cycle. We then use our knowledge of this local structure to describe the effect of mutation at that vertex. Recall the notion of types of arrows from Section 1.3. Proof. If the arrows cut out are of different types, then they form two consecutive arrows of a (d + 1)-cycle, by Lemma 3.1. This (d + 1)-cycle therefore has two arrows cut out. But this is a contradiction, since a cut removes precisely one arrow from each (d + 1)-cycle. □ Lemma 3.15. If a vertex x of Q (d,n) C is neither a source nor a sink of the quiver, nor in the middle of a distinguished cut (d + 1)-cycle, then it is the head of precisely one arrow and the tail of precisely one arrow.
Proof. Suppose that x is neither a source nor a sink of the quiver, nor in the middle of a distinguished cut (d + 1)-cycle. Then, x must be the source of one distinguished cut (d + 1)-cycle and the sink of another distinguished cut (d + 1)-cycle. Moreover, x is the head of at least one arrow α and the tail of at least one arrow β. Since x is not in the middle of a distinguished cut (d + 1)-cycle, the arrows α ′ and β ′ succeeding α and preceding β in their respective distinguished (d + 1)-cycles must be cut out. By Lemma 3.14, α ′ and β ′ have the same type.
Suppose that we have another arrow γ such that x is the head of γ. Then the arrow γ ′ which succeeds it in the distinguished (d + 1)-cycle must be cut out. But then by Lemma 3.14, γ ′ , α ′ , and β ′ all have the same type, so γ = α. A similar argument can be made for an arrow δ with tail x. □ Lemma 3.16. Let T be a triangulation of C(n + 2d + 1, 2d) with A ∈ d-arcs(T ). If a i + 2 < a i+1 for some i, then there is either an arrow (a 0 , . . . , a i , z i+1 , a i+2 , . . . , a d ) → A Algebraic Combinatorics, Vol. 6 #3 (2023)