An Expansion Formula for Decorated Super-Teichmüller Spaces
aa r X i v : . [ m a t h . C O ] F e b An Expansion Formula for Decorated Super-Teichmüller Spaces
Gregg Musiker, Nicholas Ovenhouse, and Sylvester W. Zhang
Abstract.
Motivated by the definition of super Teichmüller spaces, and Penner-Zeitlin’srecent extension of this definition to decorated super Teichmüller space, as examples of superRiemann surfaces, we use the super Ptolemy relations to obtain formulas for super λ -lengthsassociated to arcs in a bordered surface. In the special case of a disk, we are able to givecombinatorial expansion formulas for the super λ -lengths associated to diagonals of a poly-gon in the spirit of Ralf Schiffler’s T -path formulas for type A cluster algebras. We furtherconnect our formulas to the super-friezes of Morier-Genoud, Ovsienko, and Tabachnikov,and obtain partial progress towards defining super cluster algebras of type A n . In particu-lar, following Penner-Zeitlin, we are able to get formulas (up to signs) for the µ -invariantsassociated to triangles in a triangulated polygon, and explain how these provide a steptowards understanding odd variables of a super cluster algebra. Contents
Introduction 1Acknowledgements 21. Decorated Teichmüller Theory 22. Laurent Expression for λ -Lengths 33. Decorated Super-Teichmüller Theory 44. Super T -paths 75. Proof of Theorem 4.9 176. Super-friezes from super λ -lengths and µ -invariants 267. Conclusions and Future Directions 308. Appendix: The Pentagon Relation 33References 38IntroductionCluster algebras were introduced in [FZ02] as certain commutative algebras whose generatorsare defined by a combinatorial recursive procedure called “mutation”. They were originallyconceived to study certain problems in Lie theory and quantum groups, but have since foundmany surprising and deep connections to other areas of mathematics and physics.In [FST08], a class of cluster algebras was defined starting from the data of a surface withboundary, together with a collection of punctures and marked points on the boundary. Itwas shown in [GSV05] and [FG06] that these cluster algebras could be interpreted geomet-rically as functions on decorated Teichmüller spaces. Specifically, the cluster variables arecoordinate functions known as λ -lengths or Penner coordinates [Pen12].
One of the earliest results in the subject is the
Laurent phenomenon , which says that allcluster variables can be expressed as Laurent polynomials in terms of a fixed set of initialcluster variables. Over the years, several explicit formulas have been given for these Laurentexpressions in the case of cluster algebras coming from surfaces, with the Laurent monomialsbeing indexed by various combinatorial objects. Some of these include Schiffler’s “ T -paths”[Sch08], perfect matchings of the snake graphs of Musiker, Schiffler, and Williams [MSW11],and Yurikusa’s “angle matchings” [Yur19].Going beyond the commutative case, Berenstein and Zelevinsky defined quantum cluster al-gebras [BZ05] where cluster variables (of the same cluster) quasi-commute with one another,meaning that exchanging the order of multiplication in a cluster monomial could alter theexpression by yielding a power of q out in front of such a term. More recently, Berensteinand Retakh [BR18] defined in the case of surfaces a (completely) non-commutative model ofcluster variables and obtained non-commutative Laurent expansions analogous to T -paths.Such non-commutative expressions could also be defined as quasi-Plücker coordinates.Recently, Penner and Zeitlin defined the notion of the decorated super-Teichmüller space associated to a bordered marked surface [PZ19]. This work builds off of earlier work onsuper Teichmüller spaces for super Riemann surfaces [CR88]. The coordinates on a super-space are broken into two classes: namely even coordinates and odd coordinates. Evencoordinates are ordinary commutative variables but odd coordinates anti-commute with oneanother. Odd coordinates are also commonly known as Grassmann variables. As in theclassical commutative case, the coordinates correspond to arcs in a fixed triangulation of thesurface. They described a super version of the Ptolemy relation, which is an expression forhow the coordinates change when changing the choice of triangulation.Our main result in this paper is an explicit formula for the super λ -lengths in the case ofmarked disks, generalizing the T -path formulation of Schiffler. Like Schiffler’s formula, theterms in our formula are also indexed by objects which closely resemble the T -paths fromthe classical case. AcknowledgementsThe authors would like to thank the support of the NSF grant DMS-1745638 and the Univer-sity of Minnesota UROP program. We would also like to thank Misha Shapiro and LeonidChekhov for inspiring conversations.1. Decorated Teichmüller TheoryFirst we review some background on decorated Teichmüller spaces. For a detailed reference,see [Pen12]. Let S be a surface with boundary, and let p , . . . , p n be a collection of markedpoints on the boundary, such that each boundary component contains at least one markedpoint. More generally, we can also have a collection of interior marked points (or punctures ),but we will not be concerned with this case in this paper. We also equip the surface witha triangulation, where the arcs terminate at the marked boundary points. The Teichmüllerspace of S , denoted T ( S ) , is the space of (equivalence classes of) hyperbolic metrics on S N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 3 with constant negative curvature, with cusps at the marked points. Because of the cusps,any geodesic between marked points has infinite hyperbolic length.The decorated Teichmüller space of S , written e T ( S ) , is a trivial vector bundle over T ( S ) , withfiber R n> . The fibers represent a choice of a positive real number associated to each markedpoint. At each marked point, we draw a horocycle whose size (or height ) is determined bythe corresponding positive number. Truncating the geodesics using these horocycles, it nowmakes sense to talk about their lengths. If ℓ is the truncated length of one of these geodesicsegments, then the λ -length (or Penner coordinate ) associated to that geodesic arc is definedto be λ := e ℓ/ Fixing a triangulation of the marked surface, the collection of λ -lengths corresponding to thearcs in the triangulation (including segments of the boundary) form a system of coordinatesfor e T ( S ) . Choosing a different triangulation results in a different system of coordinates, butthey are related by simple transformations which are a hyperbolic analogue of Ptolemy’stheorem from classical Euclidean geometry. If two triangulations differ by the flip of a singlearc as in figure 1, then the λ -lengths are related by: ef = ac + bd . a bcd e a bcd f Figure 1.
Ptolemy transformation2. Laurent Expression for λ -LengthsIn this paper, we will only be concerned with the case that the surface S is a disk withmarked points on its boundary (which we will picture as a convex polygon). So we restrictto that case now.Fix a triangulation of an n -gon, and label the vertices through n . Schiffler [Sch08] defineda T -path from i to j to be a sequence α = ( a , . . . , a ℓ ( α ) | t , . . . , t ℓ ( α ) ) such that(T1) a , . . . , a ℓ ( α ) are vertices of the polygon(T2) t k is an arc in the triangulation connecting a k − to a k (T3) no arc is used more than once(T4) ℓ ( α ) is odd(T5) if k is even, then t k crosses the arc connecting vertices i and j GREGG MUSIKER, NICHOLAS OVENHOUSE, AND SYLVESTER W. ZHANG (T6) if k < l and both t k and t l cross the arc from i to j , then the point of intersectionwith t k is closer to i , and the point of intersection with t l is closer to j An example of a T -path in a hexagon is shown in figure 2. The even-numbered edges arecolored blue, and the odd edges red, for emphasis. i j Figure 2. a T -path from i to j .We denote the set of all T -paths from i to j by T ij . Given a T -path α , we define a Laurentmonomial x α (in terms of the λ -lengths of a fixed triangulation) as the product of the λ -lengths used in the T -path, with the even-numbered ones inverted. That is, if the orderedsequence of edges in α is e , e , . . . , e m +1 , then x α := Q mk =0 x e k +1 Q mk =1 x e k Schiffler proved the following theorem relating λ -lengths and T -paths. Theorem 2.1. [Sch08, Theorem 1.2]
Let x ij be the λ -length corresponding to the geodesicarc connecting vertices i and j in a triangulated polygon. Then x ij = X α ∈ T ij x α Remark 2.2.
As a consequence of the definition of T -paths, to compute x ij it is sufficient toconsider only the sub-polygon consisting of the triangles which the geodesic arc connectingvertices i and j crosses. 3. Decorated Super-Teichmüller TheorySuper-Teichmüller spaces have been studied for several years now (see for example [CR88]).Recently, Penner and Zeitlin introcued a decorated version of super-Teichmüller spaces[PZ19]. It is generated by even variables corresponding to the λ -lengths of a triangula-tion, as well as odd variables (called µ -invariants ) corresponding to the triangles. They givea super version of the Ptolemy relation, which reads as follows (see figure 3 for the meaningsof the variables): ef = ( ac + bd ) (cid:18) σθ √ χ χ (cid:19) σ ′ = σ − √ χθ √ χθ ′ = θ + √ χσ √ χ N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 5 where χ = acbd . a bcd e θσ a bcd fθ ′ σ ′ Figure 3. super Ptolemy transformationWe will usually find it convenient to re-write these equations without χ , as follows: ef = ac + bd + √ abcd σθ (1) σ ′ = σ √ bd − θ √ ac √ ac + bd (2) θ ′ = θ √ bd + σ √ ac √ ac + bd (3)An important corollary of these equations is the following:(4) σθ = σ ′ θ ′ which will be used frequently in our proofs.To understand the oriented arrows in Figure 3 and the minus sign in Equation (2), we needthe combinatorial data of a spin structure . In [CR07] and [CR08], an isomorphism wasshown between the set of equivalence classes of spin structures on a surface and the set ofisomorphism classes of Kasteleyn orientations of a fatgraph spine of the surface. Dual toany fatgraph spine is a triangulation of the surface, and so an orientation of the fatgraphcorresponds to an orientation of a triangulation (requiring the dual edge to cross from leftto right). For our purposes a spin structure will be a choice of orientation of the edges of atriangulation, modulo a certain equivalence relation, which we now describe.Fix a triangulation ∆ and an orientation τ of the edges in that triangulation. For any triangle t , consider the transformation which reverses the orientation of the three sides of t . Definean equivalence relation on orientations by declaring that τ ∼ τ ′ if they differ by a sequenceof these transformations. In [PZ19], a spin structure is represented combinatorially by anequivalence class of orientations.In [PZ19], the authors did not consider surfaces with boundary, as we do here. Because theorientation of boundary segments will play no role in our formulas for λ -lengths (but not µ -invariants), we can often ignore the boundary orientations. Note that the hypothesis ofProposition 3.1 on the triangulation is sufficient because of Remark 2.2. Proposition 3.1.
Fix a triangulation of a polygon in which every triangle has at least oneboundary edge. Then there is a unique spin structure after ignoring the boundary edges. In
GREGG MUSIKER, NICHOLAS OVENHOUSE, AND SYLVESTER W. ZHANG particular, this means that, from any representative orientation of a fixed spin structure, onecan obtain all other orientations on the interior diagonals, without leaving this spin structure.Proof.
Because every triangle has at least one boundary edge, one can naturally sequence thetriangles (and internal diagonals) from left-to-right. We may picture the polygon as follows,to emphasize this:We will demonstrate that we may change the orientation of a single edge while remainingin the same equivalence class. Label the triangles from left to right as t , t , . . . and theinternal diagonals as d , d , . . . . We will argue that we can change the orientation of just d k ,by induction on k .If k = 1 , then d is an edge of t , and the other two edges of t are on the boundary. Becausewe ingore boundary orientations, the equivalence relation allows us to reverse the orientationof just d .Now, for arbitrary d k , reverse the orientations around triangle t k . This affects both d k and d k − . But by induction, we may change just d k − while staying in the equivalence class. Thisproves the claim. (cid:3) In Figure 3, the arrows on the edges labelled e and f represent the choice of orientation. Inthe figure, the orientations of the edges around the boundary of the quadrilateral are notindicated. Three of the four edges are unchanged in the super Ptolemy transformation, andonly the orientation of the edge labelled b is changed, see Figure 4. The sign in Equation (2)is determined by the relative positions of the triangular faces with respect to the chosenorientation. ǫ a ǫ b ǫ c ǫ d θσ ǫ a − ǫ b ǫ c ǫ d θ ′ σ ′ − ǫ a − ǫ b ǫ c ǫ d − θσ Figure 4.
Flip effect on spin structures. Here ǫ x denotes the orientation ofan edge x .As illustrated in Figure 4, the super Ptolemy relation is not an involution. Performing aflip twice results in reversing the orientations around the top triangle. This leads us to thefollowing observation that µ -invariants are well-defined only up to sign. Consequently, in themain results of our paper regarding µ -invariants (Theorem 5.4 and Theorem 5.5 ), we mustspecify a mutation sequence to obtain a formula for the µ -invariants. N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 7
Remark 3.2.
The above equivalence relation, i.e. as used in Proposition 3.1, guaranteesthat the result after flipping twice represents the same spin structure, but algebraically it hasthe effect of negating the µ -invariant of that triangle ( θ
7→ − θ in the figure). This means thatthe specific µ -invariants are not a feature of the triangulation and spin structure alone, butalso the choice of representative orientation. Choosing a different orientation corresponds tochanging the sign of some of the µ -invariants. Remark 3.3.
Unlike the µ -invariants, the expressions of λ -lengths in terms of an initialtriangulation (as will be described more fully in Corollary 4.11) are independent of theorientation of the arc as part of a spin structure, and of the flip sequence used to obtain atriangulation containing that arc. This is proven in [PZ19] in the case of surfaces withoutboundaries. The fact that λ -lengths are well-defined can also be seen as a consequence ofthe well-definedness of super-freizes, as we will describe in Sections 6 and 7.2, based on[MGOT15, PZ19]. We provide a more direct proof in Section 8, the Appendix.4. Super T -pathsLet P be an ( n + 3) -gon (a disk with n + 3 marked points on the boundary), and T = { t , . . . , t n +3 } the set of arcs in some triangulation. We will denote by V the set of vertices(marked points). Let a and b be two non-adjacent vertices on the boundary and let ( a, b ) bethe arc that connects a and b .4.1. Fans of a Triangulation and Their Centers.
We call a triangulation a fan if allthe internal diagonals meet at a common vertex. We will define a canonical way to breakany triangulation T of P in to smaller polygons with fan triangulations. For this purpose,certain vertices in P will be distinguished as centers of fans . a = c c b = c c Figure 5. centers of fan segments.Let P be a polygon and T a triangulation. Following Remark 2.2, without loss of generality,we may assume that ( a, b ) crosses all internal diagonals of T . Consider the intersectionof ( a, b ) with triangles in T which do not contain a or b . These intersections create smalltriangles (colored yellow in Figure 5) whose vertices in P we call fan centers . We set a = c and b = c N +1 and as a convention we will name these centers c , · · · , c N such that GREGG MUSIKER, NICHOLAS OVENHOUSE, AND SYLVESTER W. ZHANG (1) For ≤ i ≤ N − , the edge ( c i , c i +1 ) is in T which crosses ( a, b ) ,(2) The intersection ( c i , c i +1 ) ∩ ( a, b ) is closer to a than ( c j , c j +1 ) ∩ ( a, b ) if i < j .Now the edges ( c i , c i +1 ) naturally break the triangulation T into N smaller polygons, each ofwhich comes with an induced fan triangulation. Let F j denote the subgraph of T boundedby c i − , c i and c i +1 , which are called the fan segments of T . We say that c i is the center of F i . See Figure 7 for an illustration, where the fan segments are indicated by different colors.4.2. The Auxiliary Graph.
We shall now define an auxiliary graph associated to ( T, a, b ) ,which will be used to define the super T -paths from a to b .For a triangulation T and a pair of vertices a and b , we define the graph Γ a,bT to be the graphof the triangulation T with some additional vertices and edges.(1) For each face of the triangulation T , we place an internal vertex, which lies on thearc ( a, b ) . We denote the internal vertices V = { θ , · · · , θ n +1 } , such that θ i is closerto a than θ j if and only if i < j .(2) For each face of T , we add an edge σ i := ( θ i , c j ) connecting the internal vertex θ i tothe center of the fan segment which contains θ i . We denote by σ the set of all suchedges.(3) For each θ i and θ j with i < j , we add an edge connecting θ i and θ j . We denote thecollection of these edges as τ = { τ ij : i < j } . For simplicity the τ -edges are drawn tobe overlapping.See figure 6 for example. axc b y c θ θ θ θ σ σ σ σ Figure 6.
The auxiliary graph Γ a,bT . Remark 4.1.
Note that the arc ( a, b ) divides the first and last triangle into two triangles,as opposed to into a triangle and a quadrilateral like the case of every other triangle of T .Consequently, the convention of the yellow coloring as in figure 5 can be extended to thefirst and last triangles of T in multiple ways. Thus we may define the first and last σ -edge ina different way: σ = ( θ , x ) and σ = ( θ , y ) in figure 6, in which case the (not yet defined)super T -paths produces the same weight. We make this choice for sake of consistency ofdoing induction. This means that when P is a quadrilateral, we can view its triangulation T as either a single fan or having 2 fans, and define the auxiliary graph in different ways. N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 9
In figure 7, we give another example of constructing the auxiliary graph. c c c c c c c c c c c c Figure 7. (Left): Yellow shading indicates the fan centers. (Right): Theauxiliary graph where different fans are indicated by different colors.4.3.
Super T -paths. Now we define the super T -paths from a to b to be paths on edges ofthe auxiliary graph Γ a,bT satisfying the certain axioms. Definition 4.2 (super T -paths) . A super T -path t from a to b is a sequence t = ( a , a , · · · , a ℓ ( t ) | t , t , · · · , t ℓ ( t ) ) such that(T1) a = a , a , · · · , a ℓ ( t ) = b ∈ V ∪ V are vertices on Γ a,bT .(T2) For each ≤ i ≤ ℓ ( t ) , t i is an edge in Γ a,bT connecting a i − and a i .(T3) t i = t j if i = j .(T4) ℓ ( t ) is odd.(T5) t i crosses ( a, b ) if i is even. The σ -edges are considered to cross ( a, b ) .(T6) t i ∈ σ only if i is even, t i ∈ τ only if i is odd.(T7) If i < j and both t i and t j cross the arc ( a, b ) , then the intersection t i ∩ ( a, b ) is closerto the vertex a than the intersection t j ∩ ( a, b ) We let T a,b denote the set of super T -paths from a to b . Furthermore, let T a,b be the set ofsuper T -paths from a to b which do not use σ or τ edges. We naturally identify T a,b with T a,b , the set of ordinary T -paths from a to b . We also define T a,b := T a,b − T a,b as the set ofsuper T -paths which do not correspond to ordinary T -paths.An immediate observation is that every super T -path must have an even number of σ -edges.More specifically, they always appear as a sequence of σ -edge, τ -edge, and σ -edge. Here τ -stands for teleportation : instead of following along an edge of the triangulation T , a τ -stepteleports from one internal vertex to another. We call a subsequence of a super T -path of the form ( · · · , θ i , θ j , · · · |· · · , σ i , τ ij , σ j , · · · ) a super step . In other words, a super T -path is aconcatenation of certain ordinary T -paths and super steps. Example 4.3.
Figure 8 illustrates several examples of super T -paths from to . Odd-numbered edges are colored red and even-numbered edges are colored blue.
123 4 5 6 θ θ θ θ x x x x x x x x x t = (1 , , θ , θ , , , θ , θ , , | x , σ , τ , σ , x , σ , τ , σ , x )123 4 5 6 x x x x x x x x x θ θ θ θ t = (1 , , θ , θ , , , , | x , σ , τ , σ , x , x , x ) 123 4 5 6 x x x x x x x x x θ θ θ θ t = (1 , , θ , θ , , | x , σ , τ , σ , x ) Figure 8.
Examples of super T -paths.4.4. Default Orientation and Positive Order.
Fix an arc ( a, b ) , and as mentioned in theprevious section, we assume that this arc crosses all diagonals in the chosen triangulation.We also choose a direction a → b for this arc. Based on these choices, we will define a defaultorientation , which guarantees an ordering of the µ -invariants in which the coefficients inthe λ -length expansion have positive coefficients. Accordingly, we will call this ordering the positive ordering . Notice that only the orientation of interior edges affects our calculation of λ -lengths, therefore the orientation of boundary edges will be omitted.Recall the convention for labelling the vertices of a polygon: given two vertices a and b and achosen direction a → b , we label c = a , c N +1 = b , and the fan centers are labelled c , . . . , c N in such a way that c i is closer to a than c i +1 is. See Figure 9 for an illustration. Definition 4.4 (default orientation) . When the triangulation is a single fan with c beingthe center, every interior edge is oriented away from c . When T is a triangulation with N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 11
N > fans, where c , · · · , c N are the centers, the interior edges within each fan segment areoriented away from its center. The edges where two fans meet each other are oriented as c → c → · · · → c N − → c N See figures 9 and 10. c c c c c c α α α β β γ γ γ δ δ Figure 9.
The default orientation of a generic triangulation where each fansegment is colored differently. The faces are labelled by their µ -invariants. c c c c c c c c c c c c c Figure 10.
More examples of default orientation.
Remark 4.5.
As mentioned above, the definition of default orientation depends on thechoice of direction a → b . In particular, choosing the opposite direction b → a would changethe labelling so that c i becomes c N − i . The effect is that the orientation of the diagonalswithin a fan are unchanged, but the diagonals connecting two fan centers would have thereverse orientation. Definition 4.6 (positive ordering) . For F a single fan triangulation with center c , let θ , · · · , θ k be its faces. The positive ordering is defined to be θ > θ > · · · > θ k , where θ , · · · , θ k are ordered counterclockwise around c .For a triangulation T with fans F , · · · , F N , we order the fans as follows in two different cases(1) If c N − , c N , c N +1 are oriented counterclockwise, then we order the fans as follows ( F > F > · · · > F N − > F N > F N − > · · · > F > F if N is even, F > F > · · · > F N − > F N > F N − > · · · > F > F if N is odd.(2) If c N − , c N , c N +1 are oriented clockwise, then we order the fans as follows ( F > F > · · · > F N − > F N > F N − > · · · > F > F if N is odd, F > F > · · · > F N − > F N > F N − > · · · > F > F if N is even.Then the positive ordering on faces of T is induced by the ordering on fans and the positiveordering within each fan. Remark 4.7.
The positive ordering may also be described inductively, triangle-by-triangle,as follows: Recall from the definition of auxiliary graph that the triangles are labelled θ , θ , . . . , θ n in order from a to b . For each triangle θ k , look at the edge separating θ k and θ k +1 . If the edge is oriented so that θ k is to the right, then we declare that θ k > θ i forall i > k . On the other hand, if θ k is to the left, we declare that θ k < θ i for all i > k .For example, in Figure 9, the positive ordering on the faces is α > α > α > γ > γ > γ > δ > δ > β > β . Expansion Formula.Definition 4.8 (weight) . Let t ∈ T ab be a super T -path which uses edges t , t , . . . in theauxiliary graph Γ Ta,b . We will assign to each edge t i a weight, which will be an element in thesuper algebra R [ x ± , · · · , x ± n +3 | θ , · · · , θ n +1 ] (where θ i ’s are the odd generators) as follows.For the parity of edges t i ∈ σ or τ , we recall axiom (T6) of Definition 4.2. wt( t i ) := x j if t i ∈ T , with λ -length x j , and i is odd x − j if t i ∈ T , with λ -length x j , and i is even q x j x k x l θ s if t i ∈ σ and the face containing t i is as pictured below ( i must be even )1 if t i ∈ τ ( i must be odd ) In keeping with the intuition mentioned after Definition 4.2, teleportation is unweighted.
N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 13 θ s t i x l x k x j Here, θ s is the µ -invariant associated to the face containing t i . Finally, we define the weightof a super T -path to be the product of the weights of its edges wt( t ) = Y t i ∈ t wt( t i ) where the product of µ -invariants is taken under the positive ordering.In what follows, we will use λ ab to denote the λ -length of the arc ( a, b ) , and we will write ijk to denote the µ -invariant associated to the triple of ideal points ( i, j, k ) , subject to Remark3.2.The following theorem is the main result of the current paper, giving an explicit expression forarbitrary super λ -lengths in terms of the λ -lengths and µ -invariants of a fixed triangulation. Theorem 4.9.
Under the default orientation, the λ -length of ( a, b ) is given by λ a,b = X t ∈T a,b wt( t ) When starting with a generic orientation, one can first apply a sequence of equivalencerelations (reversing the arrows around a triangle and negating the µ -invariant) to get tothe default orientation, with some of the µ -invariants having signs changed. This is alwayspossible due to Proposition 3.1. See Example 4.12.Equivalently, we can state the main theorem with respect to an arbitrary choice of orienta-tions (not necessarily the default one) as follows. Corollary 4.10.
With a generic choice of orientation, the λ -length of ( a, b ) is given by λ a,b = X t ∈T a,b ( − inv( t ) wt( t ) , where inv( t ) is the number of edges in the triangulation which cross a τ -edge of t and areoriented opposite the default orientation.Proof. Label the internal diagonals d , d , . . . , d n − in order of their proximity from c to c N , and similarly label the µ -invariants of the triangles θ , θ , . . . , θ n . Suppose d k is the lastdiagonal whose orientation disagrees with the default orientation. Proposition 3.1 describeshow we can find another orientation, representing the same spin structure, where d k is theonly internal diagonal whose orientation is changed. From Remark 3.2, we see that in doingso, θ i must be replaced by − θ i for all i ≤ k .This process may then be repeated for all diagonals whose orientation differs from the defaultone. The end result is as follows: if d i , d i , . . . , d i k are all the internal diagonals whose orientation disagrees with the default orientation (and i < i < · · · < i k ), then the µ -invariants of triangles between d i k − and d i k are negated, those between d i k − and d i k − arenot, those between d i k − and d i k − are negated, those between d i k − and d i k − are not, etc.Any super T -path t contains some number of super-steps. Suppose t contains a super-step ( · · · , θ i , θ j , · · · |· · · , σ i , τ ij , σ j , · · · ) . Let m ij be the number of diagonals between θ i and θ j whose orientation disagrees with the default. If m ij is even, then when we change from thegiven orientation to the default one, either both θ i and θ j are negated, or both stay thesame. In this case, the product θ i θ j is unchanged when passing to the default orientation.On the other hand, if m ij is odd, then one of them is negated, and the other stays the same,in which case the product θ i θ j is negated.The number inv( t ) is simply the sum of these m ij ’s over all super steps in the path t . (cid:3) It is apparent from the super T -path formulation in Theorem 4.9 that these λ -lengths satisfysomething analogous to the Laurent phenomenon exhibited by ordinary cluster algebras.This is summarized in the following corollary. Corollary 4.11.
Let ˜ θ i := wt( σ i ) = q x j x k x l θ i (see Definition 4.8). For any pair of vertices a, b of the polygon, (a) λ ab ∈ R [ x ± , . . . , x ± n +3 | ˜ θ , . . . , ˜ θ n +1 ] . In other words, each term of λ ab is the productof a Laurent monomial in the x i ’s and a monomial in the ˜ θ i ’s. (b) λ ab ∈ R [ x ± , . . . , x ± n +3 | θ , . . . , θ n +1 ] . In other words, each term of λ ab is the productof a Laurent monomial in the square roots of the x i ’s and a monomial in the θ i ’s. Example 4.12.
An example of the expansion formula given in Theorem 4.9 and Corol-lary 4.10 is shown in Figure 11. Continuing from Example 4.3, this figure shows all super T -paths in T , . For example, to obtain the default orientation, we would need to flip thearrow on edge (3 , . We can do this by flipping all arrows of the last triangle while negatingthe µ -invariant to − θ . Keeping the same positive ordering θ > θ > θ > θ , this wouldmake all terms in Figure 11 positive.In this example, the Laurent expansion can also be written in terms of the ˜ θ ’s (defined inCorollary 4.11) as: λ = x x x + x x x x x + x x x x x x x + x x x x x + x x x x x + x x x x ˜ θ ˜ θ + x x x x ˜ θ ˜ θ + x x ˜ θ ˜ θ − x x ˜ θ ˜ θ + x x ˜ θ ˜ θ − x x ˜ θ ˜ θ − x x x x ˜ θ ˜ θ − x x x x ˜ θ ˜ θ − x x x ˜ θ ˜ θ ˜ θ ˜ θ Super T -paths in a single fan triangulation. Let P be an n -gon and T a fantriangulation. Let vertex be the fan center. Lemma 4.13 (Schiffler) . For ≤ i ≤ n − , define: α i − := (2 , , i, i + 1 , , n |· · · ) 2 i n − Then T ,n = { α i : 1 ≤ i ≤ n − } . See Example 4.14. N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 15
123 4 5 6 123 4 5 6 123 4 5 6 123 4 5 6123 4 5 6 123 4 5 6 123 4 5 6 123 4 5 6123 4 5 6 123 4 5 6 123 4 5 6 123 4 5 6123 4 5 6 123 4 5 6 + x x x + x x x x x + x x x x x x x + x x x x x + x x x x x + x √ x x x x x x θ θ + x x √ x x x x x √ x θ θ + x √ x x x √ x x x θ θ − √ x x x x √ x x θ θ + x x √ x x x √ x x θ θ − x √ x x x √ x x x θ θ − x x √ x x x x x √ x θ θ − x √ x x x x x √ x θ θ − √ x x x x x x x x θ θ θ θ
123 4 5 6 x x x x x x x x x θ θ θ θ edge labels
123 4 5 6 spin structure
Figure 11.
Example of Theorem 4.9
Note that in case of i = 2 or i = n − , α i − collapses to a shorter T -path after removingbacktracking. In the special case of n = 3 , there is a unique T -path in T ,n , i.e. α = α n − ,which consists of the single edge (2 , . Example 4.14 (Ordinary T -paths of a Fan) . The following are the ordinary T -paths ofa fan triangulation of an octagon. Notice that each of these T -paths surrounds (and is inbijection with) one of the triangles of T , as illustrated in yellow below. · · · · Lemma 4.15.
Every super T -path in T ,n is of the following form: (2 , , θ i , θ j , , n | x , σ i , τ ij , σ j , x n ) for ≤ i < j ≤ n − . See Example 4.16. We illustrate this lemma with an example before proving it.
Example 4.16.
The following are the (cid:0) (cid:1) = 6 different non-ordinary super T -paths (ele-ments of T , ) of a single-fan hexagon.
123 4 5 6 123 4 5 6 123 4 5 6123 4 5 6 123 4 5 6 123 4 5 6
In this example, λ can be expressed as λ = λ λ λ + λ λ λ λ λ + λ λ λ λ λ + λ λ λ + λ r λ λ λ r λ λ λ λ + λ r λ λ λ r λ λ λ λ + λ r λ λ λ r λ λ λ λ + λ r λ λ λ r λ λ λ λ + λ r λ λ λ r λ λ λ λ + λ r λ λ λ r λ λ λ λ The first four terms are the weights of all ordinary T -paths (as described in Example 4.14),and the remaining six terms correspond to the T -paths pictured above (in the same order). N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 17
It can also be written as λ = λ λ λ + λ λ λ λ λ + λ λ λ λ λ + λ λ λ + λ λ ] ]
134 + λ λ ] ] λ λ ] ]
156 + λ λ ] ]
145 + λ λ ] ]
156 + λ λ ] ] where the g ijk are the weights of the corresponding σ -edges (following the notation ofCorollary 4.11). Proof of Lemma 4.15. A T ,n -path must use one of the σ -edges, therefore the first step inthe super T -path needs to be (2 , . Then after traveling through σ i , what follows must be τ ij which leads the T -path to another internal vertex θ j . The next step is an even step hencehas to be a σ -edge which will take us to vertex : (2 , , θ i , θ j , , · · · | x , σ i , τ ij , σ j , · · · ) Now we are at vertex and have completed even number of steps, therefore the rest of thethis super T -path must be a super T -path from to n — clearly there is only one possibilitywhich is the single edge (1 , n ) . Hence a super T -path in T ,n must have the form (2 , , θ i , θ j , , · · · | x , σ i , τ ij , σ j , · · · ) + ( · · · , , n |· · · , x n ) = (2 , , θ i , θ j , , n | x , σ i , τ ij , σ j , x n ) (cid:3)
5. Proof of Theorem 4.9In this section we prove our main theorem. It turns out that the default orientation guar-antees a positive sign on all terms of the expansion of λ -lengths (Theorem 4.9), and is alsopreserved by induction.Our proof has three parts: we first prove the case of single fan triangulations, and thenprove the case of zig-zag triangulations. Finally we prove Theorem 4.9 in full generality bycombining the two cases mentioned above.Before proving our theorem, we state the following results that will be used in our proofs. Proposition 5.1.
Let
A, β, and Σ be elements in the super algebra A , which for convenience,we assume is written as A = R [ x ± , · · · , x ± n +3 | θ , · · · , θ n +1 ] as in Definition 4.8. Further,we assume that A is an even element with a non-zero body , and that both β and Σ are oddelements of A . Then we have p A + β Σ = √ A + β √ A Σ where the square root of A is taken to be the positive square root. Remark 5.2.
Since A is an even element, its positive square root is well-defined as thechoice such that the body of √ A is the positive square root of the body of A . The body of an element A of super algebra A is the constant term when expanded out in terms of the θ i ’s. Proof.
Squaring the right hand side: (cid:18) √ A + β √ A Σ (cid:19) = A + 2 √ A · β √ A · Σ + (cid:18) β √ A Σ (cid:19) = A + β Σ This clearly equals the square of the left hand side. (cid:3)
Using Proposition 5.1, we can rewrite the super Ptolemy relations in a more symmetricalform.
Proposition 5.3.
The super Ptolemy relations described in figure 3 can be written as follows. θ ′ p ef = θ √ bd + σ √ ac (5) σ ′ p ef = σ √ bd − θ √ ac (6) Proof.
We have θ ′ p ef = θ ′ q ab + cd + √ abcdσθ = θ ′ q ab + cd + √ abcdσ ′ θ ′ = θ ′ √ ab + cd And θ ′ = θ √ bd + σ √ ac √ ac + bdθ ′ √ ac + bd = θ √ bd + σ √ ac Putting these two equations together gives Equation (5): θ ′ p ef = θ √ bd + σ √ ac Equation (6) can be derived in a similar way. (cid:3)
Proof of theorem 4.9 for a single fan.
For sake of readability, in the below, weuse ijk to denote the µ -invariant associated to the triple of ideal points ( i, j, k ) , subject toRemark 3.2, while the λ -length of a pair ( i, j ) will be denoted λ ij . We will also sometimesuse ijk to denote the internal vertex of the auxiliary graph associated to ( i, j, k ) , whentalking about super T -paths.First, we will prove the main theorem in the case of a single fan triangulation. Theorem 5.4.
Consider a single fan triangulation of an n -gon as depicted in Figure 12. Wecontinue our convention of using the default orientation, which for a fan triangulation meansthat arrows on all internal diagonals point away from the fan center. After performing thesuper Ptolemy relations for the flips of arcs (1 , , (1 , , . . . , (1 , k − , we have (a) r λ k λ λ k k = k − X i =1 wt( σ i ) N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 19 n
123 4 5 n − σ σ σ σ n − Figure 12.
Proof of Theorem 5.4(b) λ k = X t ∈T ,k wt( t ) where wt( σ i ) and wt( t ) are defined in Definition 4.8, and as defined in Definition 4.2, T ,k denotes the set of super T -paths from vertex to vertex k .Proof. We will induct on k . We begin with the case of k = 3 where statements (a) and (b)follow immediately since wt( σ ) = q λ λ λ , and the unique super T -path from vertex to vertex is simply the ordinary T -path t = (2 , .In general, for k ≥ , after flipping arcs (1 , , . . . , (1 , k − , the next flip of arc (1 , k − will be inside the following quadrilateral: k − k Note that the µ -invariant , k − , k is in the initial triangulation, but , , k − is not .However, we assume by induction that , , k − is given by part ( a ) .First we prove part (b). The super Ptolemy relation (equation (1)) says that λ k is given by λ ,k − λ k = λ λ k − ,k + λ k λ ,k − + p λ λ ,k − λ k − ,k λ k , , k − , k − , k Except for the special case of k = 4 . After dividing by λ ,k − , we get a formula for λ ,k . On the right hand side, the first termgives λ λ − ,k − λ k − ,k , which is clearly an ordinary T -path. The second term, by induction, is X t ∈T ,k − wt( t ) λ − k − , λ k Taking the ordinary T -paths in T ,k − , and appending the arcs ( k − , and (1 , k ) give therest of the ordinary T -paths in T k . (See Lemma 4.13 and note that appending arc ( k − , as an even step may in fact yield a backtrack that cancels out the final step of an ordinary T -path in T k .) The terms coming from T ,k − , multiplied by λ k λ ,k − , similarly give the super T -paths in T k which do not involve the triangle (1 , k − , k ) (see Lemma 4.15).The last term, using part (a) and induction on , , k − , is p λ λ ,k − λ k − ,k λ k λ ,k − , , k − , k − , k = λ k − X i =1 wt( σ i ) wt( σ k − ) λ k These are the weights of the super T -paths which use the last triangle, namely (1 , k − , k ) (again, see lemma 4.15).Note that the product wt( σ i )wt( σ k − ) is in the positive ordering, since the two µ -invariantsappear in the counter-clockwise order around the fan center.Now, we examine part (a). Looking at the same quadrilateral as above, the super Ptolemyrelation for µ -invariants (equation (5)) says that k p λ ,k − λ k = 1 , k − , k p λ λ k − ,k + 1 , , k − p λ k λ ,k − Dividing by p λ λ ,k − λ k , commuting the λ -lengths past the µ -invariants, we get r λ k λ λ k k = s λ k,k − λ ,k − λ k , k − , k + s λ ,k − λ λ ,k − , , k − The first term on the right hand side is simply wt( σ k − ) . By induction, the second term is P k − i =1 wt( σ i ) . Therefore we have r λ k λ λ k k = wt( σ k − ) + k − X i =1 wt( σ i ) ! = k − X i =1 wt( σ i ) as desired. (cid:3) Proof of theorem 4.9 for a zig-zag triangulation.
Next, we prove our main theo-rem for the case of a zig-zag triangulation.
Theorem 5.5.
Consider a zigzag triangulation T of an n -gon as depicted in Figure 13. Weconsider all the vertices except for and n to be fan centers, so that c i is labelled i + 1 for ≤ i ≤ n − .After flipping the arcs ( n − , n − , ( n − , n − , · · · , ( k + 2 , k + 1) , we have N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 21 (a) s λ kn λ k +1 ,n λ k,k +1 k, k + 1 , n = n − X i = k X t ∈T n,i +1 wt( t ) wt( σ i ) (b) λ k,n = X t ∈T kn wt( t ) 1246 n − n n − θ θ θ θ θ θ n − n − n n − σ σ σ σ σ σ n − Figure 13.
Proof of theorem 5.5.
Left: zig-zag triangulation with the defaultorientation.
Right:
The corresponding auxiliary graph.
Proof.
We assume that vertices n , n − , and n − are oriented as in Figure 13. The casethat they are oriented oppositely is similar.We will induct (backwards) on k . The base case of the induction is the case of a singletriangle, when k = n − . Since the edge ( n − , n ) is already in the triangulation, λ n − ,n isalready one of the generators. Clearly the only T -path in this case is the single edge ( n − , n ) (when zero flips have been performed). This establishes part (b) for the base case. For part(a), the left-hand side is s λ n − ,n λ n − ,n λ n − ,n − n − , n − , n For the right-hand side, there is only a single term in this sum. This is because i only takesthe value n − , and the only T -path from n − to n is the single edge ( n − , n ) . Thus theright-hand side is λ n − ,n wt( σ n − ) = λ n − ,n s λ n − ,n λ n − ,n − λ n − ,n n − , n − , n = s λ n − ,n λ n − ,n λ n − ,n − n − , n − , n This establishes part (a) for the base case. We now assume that ≤ k ≤ n − . After flipping the arcs ( n − , n − , ( n − , n − , · · · , ( k + 3 , k + 2) , we will have one ofthe two following quadrilaterals, depending on the parity of n − k : k + 1 k + 2 nk or k + 2 k + 1 nk Now we flip the edge ( k + 2 , k + 1) while applying the Ptolemy relation equation (5). Inboth cases pictured above, the triangle ( k, k + 1 , n ) will play the role of θ ′ (it will be onthe left, looking in the direction of the arrow after the flip). And because of the oppositeorientations, application of equation (5) in both pictures gives p λ kn λ k +1 ,k +2 k, k + 1 , n = p λ k +1 ,n λ k,k +2 k, k + 1 , k + 2 + p λ k,k +1 λ k +2 ,n k + 1 , k + 2 , n Multiplying both sides by q λ k +1 ,n λ k,k +1 λ k +1 ,k +2 , we get s λ kn λ k +1 ,n λ k,k +1 k, k + 1 , n = λ k +1 ,n s λ k,k +2 λ k,k +1 λ k +1 ,k +2 k, k + 1 , k + 2+ s λ k +1 ,n λ k +2 ,n λ k +1 ,k +2 k + 1 , k + 2 , n First we examine the first term on the right-hand side. Notice that, q λ k,k +2 λ k,k +1 λ k +1 ,k +2 k, k + 1 , k + 2 is the weight of the σ -step σ k , going from vertex k + 1 into the triangle labelled θ k . By in-duction, λ k +1 ,n is the weighted sum of super T -paths from k + 1 to n . Therefore this firstterm is equal to(7) X t ∈T k +1 ,n wt( t ) wt( σ k ) Next, we focus on the second term of the right-hand side: q λ k +1 ,n λ k +2 ,n λ k +1 ,k +2 k + 1 , k + 2 , n . Byinduction, this is equal to(8) s λ k +1 ,n λ k +2 ,n λ k +1 ,k +2 k + 1 , k + 2 , n = n − X i = k +1 X t ∈T n,i +1 wt( t ) wt( σ i ) Adding the terms from equation (7) gives the result for part (a).We should verify that the µ -invariants in each product wt( t ) wt( σ k ) are in the correct positiveordering. We use the same viewpoint as in Remark 4.7. By induction, the µ -invariants ineach term of wt( t ) occur in the positive ordering of the smaller polygon which containstriangles θ k +1 , . . . , θ n − . By construction, the positive ordering of the slightly larger polygon For the case of k = n − , no arcs have yet been flipped. N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 23 (which additionally contains θ k ) is obtained from the smaller by placing θ k either at thebeginning or end of the previous order. But since wt( t ) is an even element, it is central, andso wt( t ) wt( σ k ) = wt( σ k ) wt( t ) , and we may choose whichever one gives the correct positiveordering.Now we will prove part (b). Again after flipping the arcs ( n − , n − , ( n − , n − , · · · , ( k + 3 , k + 2) , we are in one of the two pictures from before. For the picture on the left, anapplication of equation (1), followed by division on both sides by λ k +1 ,k +2 , gives(9) λ kn = λ k,k +2 λ k +1 ,n λ k +1 ,k +2 | {z } part 1 + λ k,k +1 λ k +2 ,n λ k +1 ,k +2 | {z } part 2 + s λ k +1 ,n λ k +2 ,n λ k +1 ,k +2 k + 1 , k + 2 , n | {z } part 3 · s λ k,k +1 λ k,k +2 λ k +1 ,k +2 k, k + 1 , k + 2 | {z } part 4 Note that k + 1 , k + 2 , n lies on the right of the oriented diagonal, and hence plays the role of σ . Analogously, k, k + 1 , k + 2 lies on the left and plays the role of θ . Applying equation (1)to the picture on the right gives the same thing, except that Part(3) and Part(4) appear inthe opposite order.First we will explain that the sum of Part (1) and Part (2) is the weighted sum of all super T -paths from k to n that do not contain a τ -edge which crosses ( k + 1 , k + 2) .Suppose a path t ∈ T kn does not contain a τ edge crossing ( k + 1 , k + 2) (i.e. does not havea super step starting with θ k ). Then the first step of t is either edge ( k, k + 1) or ( k, k + 2) ,and the remainder of t lies in the smaller polygon below the diagonal ( k + 1 , k + 2) . Thereare two cases.The second edge might be ( k + 1 , k + 2) , in which case the remainder of t (after removingthe first two edges) is a super T -path in either T k +1 ,n or T k +2 ,n . This is depicted in the leftof figure 14. Clearly all of these paths occur as terms in Part (1) and Part (2).However, there are more terms in Part (1) and Part (2); namely those in which the denomi-nator λ k +1 ,k +2 cancels a contribution in the numerator. This is the second case, in which thesecond step in t is not the edge ( k + 1 , k + 2) . This is depicted in the right of figure 14. Inthis case, replacing the first edge of t with ( k + 1 , k + 2) gives a super T -path in either T k +1 ,n or T k +2 ,n , and these are the remaining terms in Parts (1) and (2) in which the denominatorcancels.Now we examine Part (3) and Part (4). By induction, Part (3) is given by the formula inpart (a): s λ k +1 ,n λ k +2 ,n λ k +1 ,k +2 k + 1 , k + 2 , n = n − X i = k +1 X t ∈T i +1 ,n wt( t ) wt( σ i ) Part (4) is equal to λ k,k +1 wt( σ k ) , which is the weight of the first two steps of any super T -path t ∈ T kn which does contain a τ -step crossing ( k + 1 , k + 2) . The terms of Part (3) kk + 2 k + 1 1 kk + 2 k + 1 Figure 14.
Left:
Removing the first two edges gives a path in T k +2 ,n . Right:
Replacing the first edge ( k, k + 2) with ( k + 1 , k + 2) gives a path in T k +1 ,n .are all possible ways to complete such a super T -path (after joining them by the appropriate τ -step, which has weight 1).Again, as in part (a), we must check that these expressions are written in the correct positiveordering. As we noted above in the discussion of part (a), the positive ordering of the smallerpolygon (below the diagonal ( k + 1 , k + 2) ) agrees with the positive ordering in the largerpolygon. Therefore any factors appearing the formulas obtained above can be assumed tobe in the correct positive ordering. In Parts (1) and (2), two of the three factors are singleedges in the triangulation, and the terms of the third factor (either λ k +1 ,n or λ k +2 ,n ) appearin positive order by induction. As was discussed in part ( a ) , the terms of Part (3) can bewritten as either wt( t ) wt( σ i ) or wt( σ i ) wt( t ) , whichever is correct. Part (4) only contains asingle µ -invariant. So all that needs to be checked is that Part (3) and Part (4) occur in thecorrect order, depending on whether θ k comes first or last in the positive ordering. But thisis precisely the difference between the left and right pictures above (depending on whethervertex k or n is on the left of the the oriented edge k + 1 → k + 2 ), and Parts (3) and (4)are positioned differently in the two cases. (cid:3) Proof of Theorem 4.9 for generic triangulations.
By a generic triangulation,we mean a polygon in which every triangle has at least one boundary edge. This is thesame hypothesis that appears in Proposition 3.1 because of Remark 2.2. Given a generictriangulation T with n fans, we first apply the flip sequence in Theorem 5.4 on each of thefans, which result in a zig-zag (sub)triangulation T ′ whose vertices are the fan centers of T (including c and c N +1 ). See Figure 15.We will then derive our ultimate formula for λ ab in T via a combination of Theorems 5.4and 5.5. Using Theorem 5.5, we can express λ ab in terms of super T -paths on T ′ . Thisexpression uses certain λ -lengths and µ -invariants that are not in T , so we will substitutetheir expansion in terms of T using Theorem 5.4.We consider the fans of T as subtriangulations, and denote them as F , · · · , F N . We denotethe µ -invariants of T ′ by θ ′ , · · · , θ ′ N and their corresponding σ -edges to be σ ′ , · · · , σ ′ N . Wedenote the µ -invariants of the i -th fan of T by θ i , θ i , · · · (ordered counterclockwise aroundthe fan center), and the corresponding σ -edges σ i , σ i , · · · . See Figure 16. N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 25 a = c c c b = c c c c T = ⇒ a = c c c b = c c c c T ′ Figure 15.
Flipping edges in each fan, to turn a generic triangulation T intoa zig-zag triangulation T ′ . a = c c c b = c c c c σ σ σ σ σ σ σ σ σ σ σ Γ a,bT = ⇒ a = c c c b = c c c c σ ′ σ ′ σ ′ σ ′ σ ′ Γ a,bT ′ Figure 16.
The auxiliary graphs for T and T ′ , with τ -edges ommited.When we substitute super T -path expressions for the fans in T into T ′ , we only need toconsider two cases: (1) substitute a boundary edge ( c i − , c i +1 ) for i n , and (2)substitute a super-step ( · · · , σ ′ i , τ ij , σ ′ j , · · · |· · · ) , because every super T -path is a concatenationof super steps and complete ordinary T -paths. (1) Suppose we were to replace λ c i − ,c i +1 with its super T -path expansion in the i -th fan.In the super T -path of T ′ , the edge ( c i − , c i +1 ) must be an odd step because it doesnot cross the arc ( a, b ) . Therefore when we replace it with a super T -path in F i , theindexing agrees up to parity.Moreover, if an edge crosses the arc ( c i − , c i +1 ) in F i , then it must also cross thearc ( a, b ) in the bigger triangulation T . Therefore the axiom (T5) is satisfied, and itis straightforward to verify that all other axioms will follow.(2) We observe that the left-hand-side of Theorem 5.4 (a) equals wt( σ ′ i ) with respect to T ′ . Using the rest of Theorem 5.4 (a), we get the equality wt( σ ′ i ) = X j wt( σ ji ) , i.e. we have that the weight of σ ′ i in T ′ is equal to the weighted sum of all σ -edgesin the fan F i . This means that a super step ( · · · , σ ′ i , τ ij , σ ′ j , · · · ) will be expanded intothe sum of all super steps from a face of F i to a face of F j . This clearly preserves allaxioms of super T -paths.For the converse, we need to prove that every super T -path in T can be obtained by sucha substitution. First, this is clearly true for ordinary T -paths, or an ordinary sub-path ofa super T -path. Therefore we only need to consider the super steps. Suppose we have asuper-step using two σ -steps σ ∗ and σ • , if σ ∗ and σ • are in the same fan, say F i , then thesuper step is part of the expansion of λ -length of ( c i − , c i +1 ) . If the σ ∗ ∈ F i and σ • ∈ F j arein different fans, then the super step came from the super-step ( · · · , σ ′ i , τ ij , σ ′ j , · · · ) in T ′ .6. Super-friezes from super λ -lengths and µ -invariantsIn this section, we use our formulas for super λ -lengths and µ -invariants to construct arraysthat are variants of the super-friezes appearing in work of Morier-Genoud, Ovsienko, andTabachnikov [MGOT15]. Morier-Genoud et al defined a super-frieze to be an array, whoserows alternate so that one row is all even elements, the next is all odd elements, etc. Considera part of the array, called an “elementary diamond” , of the form B Ξ Ψ
A D
Φ Σ C N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 27
Here, Roman letters are even elements and Greek letters are odd elements. The super-friezerules are: AD − BC = 1 + ΣΞ (10) AD − BC = 1 + ΨΦ (11) A Σ − C Ξ = Φ (12) B Σ − D Ξ = Ψ (13) B Φ − A Ψ = Ξ (14) D Φ − C Ψ = Σ (15)To define a super-frieze, we only need equation (10) or equation (11) and any two of the fourequations (12) to (15). In other words, any two of equations (12) to (15) implies the othertwo, and utilizing these two equations, either of equation (10) or equation (11) implies theother.We will now observe how the λ -lengths and µ -invariants satisfy a modified version of theserelations. Put the λ -lengths in an array so that moving left-to-right along a row rotatesa diagonal of the polygon by shifting indices of both endpoints up by 1, and diagonalsof the array going south-east have a common first endpoint. In between these ordinaryentries, we put a µ -invariant multiplied by the square-root its two adjacent λ -lengths, so that ˜ µ ijk = p λ ij λ jk ijk goes in between λ ij and λ jk . With these conventions, an elementarydiamond looks as follows: b ˜ θ ˜ σ ′ e f ˜ σ ˜ θ ′ d = λ i +1 ,j ˜ µ i,i +1 ,j ˜ µ i +1 ,j,j +1 λ ij λ i +1 ,j +1 ˜ µ i,j,j +1 ˜ µ i,i +1 ,j +1 λ i,j +1 Proposition 6.1.
Every elementary diamond corresponds to a Ptolemy relation of a quadri-lateral, with two boundary edges having λ -length .Proof. Consider the super flip in the following diagram, where a = c = 1 . a bcd e θσ a bcd fθ ′ σ ′ In the super-diamond, we set ˜ θ = θ √ be , ˜ σ = σ √ ed , ˜ θ ′ = θ ′ √ df , and ˜ σ ′ = σ ′ √ bf . Now, thesuper-Ptolemy relation equation (1) is ef = 1 + bd + √ bdσθ Using equation (6), we substitute θ with σ √ bd − σ ′ √ ef : ef = 1 + bd + √ bdσ ( σ √ bd − σ ′ p ef ) Then σ squares to zero, so we have ef = 1 + bd + p bdef σ ′ σ This is exactly equation (11) in terms of these super-frieze entries, i.e. ef = 1 + bd + ˜ σ ′ ˜ σ. The other two Ptolemy relations give rise to the desired super-frieze relations as well. Thesecond Ptolemy relation equation (5) is: θ ′ p ef = θ √ bd + σ Substitute θ ’s with the ˜ θ ’s: ˜ θ ′ √ df p ef = ˜ θ √ be √ bd + ˜ σ √ de ˜ θ ′ r ed = ˜ θ r de + ˜ σ √ de Then multiply by √ de to get ˜ θ ′ e = ˜ θd + ˜ σ The third Ptolemy relation equation (5) is: σ ′ p ef = σ √ bd − θ A similar calculation shows that this is equivalent to ˜ σ ′ e = ˜ σb − ˜ θ These relations on the ‘modified’ µ -invariants are exactly the super-frieze relations. (cid:3) Theorem 6.2.
Every super-frieze pattern comes from a decorated super-Teichmüller spaceof a marked disk. ξ x ξ x . . . x n ξ n +1 n + 3 n + 2 5 43 ξ n +1 ξ ξ ξ x x x n x + + + + ++ Figure 17.
Left:
Diagonal of a super-frieze.
Right:
Initial fan triangula-tion, the + signs records the initial orientation of the spin structure. N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 29
Proof.
Take a diagonal of even and odd variables from a super-frieze, and we declare it tobe the the λ -lengths of a fan triangulation with the default orientation (see figure 17).Using proposition 6.1, the next diagonal to the right corresponds to the triangulation ob-tained by flipping the edges x , x , · · · , x n . Suppose the next diagonal of the frieze is asfollows. ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ ξ θ x y ξ ∗ θ x . . .. . . θ n x n y n ξ n +1 θ n +1 − θ n +1 Meanwhile, applying super-flips on the edges x , · · · , x n gives us the following triangulation . n + 3 n + 2 5 43 θ n θ n +1 θ θ y y y n − y n + + + − ++ This corresponds to the blue-circled entries of the above frieze pattern. To obtain the nextdiagonal, we reverse all arrows on the ‘last’ triangle (the triangle corresponding to θ n +1 ),and negate the µ -invariant. This gives us the following triangulation, which corresponds tothe red-circled entries of the above frieze. Notice that here we apply the clockwise flip sequence, as opposed to the counterclockwise one in Theorem 5.4.The reason of switching the convention here is to have the ‘wrong’ arrow on the edge y n to match up thefrieze relations of the last quiddity row. n + 3 n + 2 5 43 θ n − θ n +1 θ θ y y y n − y n + − − − ++ Now the interior arrows are exactly the ‘same’ as before: all arrows are directed away fromthe (new) fan center. Therefore, inductively, flipping the edges y , · · · , y n and negating thelast triangle will give us the next diagonal.Applying the above operations n times will take us back to the original triangulation, butwith different boundary orientations. In particular, all boundary arrows are reversed, whichis equivalent to reversing the orientation of all triangles and negating all the µ -invariants. − ξ x − ξ x . . . x n − ξ n +1 n + 3 n + 2 5 43 − ξ n +1 − ξ − ξ − ξ x x x n x − − − − −− Therefore the n -th diagonal will have the same even entries and negative odd entries as thefirst diagonal. This explains the glide symmetry of super-frieze patterns. (cid:3)
7. Conclusions and Future Directions7.1.
Expansion Formulas for µ -Invariants. In Theorem 5.4 and Theorem 5.5, we gaveformulas for certain types of µ -invariants. However, these only applied to a subset of suchtriangles which have at least one side being an arc of the triangulation. The proofs dependedon this assumption, and a specific flip sequence, in order to apply the super Ptolemy relations.This begs the following question, subject to the ambiguity of Remark 3.2, and with a specificflip sequence in mind. N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 31
Question 7.1.
What is the correct formula for a µ -invariant of a triangle which has nosides belonging to the triangulation? Looking more broadly, we can consider a study of λ -lenghths and µ -invariants, subject tosuper Ptolemy relations, for other surfaces. Question 7.2.
What is the correct formula for a λ -length (or a µ -invariant) for an arc(resp. a triangle) on an annlus, torus, or other surfaces with boundary? Note that for the special case of a once-punctured torus, both of which with no boundaries,such super structures were studied in [HPZ19]. The cases of a three-punctured sphere andof a once-punctured torus were also investigated in [IPZ18, Appendix B], but differed fromour setup. Therein, they had two odd variables (rather than one) for each triangle.7.2.
Connections to Super Cluster Algebras and Super-friezes.
As we have demon-strated, we have been able to use Penner and Zeitlin’s development of super λ -lengths fordecorated super Teichmüller space to obtain explicit formulas for super λ -lengths on markeddisks which involves a construction of super T -paths. In analogy with the classical case, asin [Sch08] where weighted generating functions of T -paths correspond to cluster variablesin cluster algebras of type A n , we wish to investigate how our formulas for super λ -lengthscould aid in the development of super cluster algebras (of type A n ). Steps towards definingsuper cluster algebras appeared in work of Ovsienko [Ovs15] and separately in the work of Li,Mixco, Ransingh, and Srivastava [LLRS17]. These intial steps were followed up by relatedwork such as [SV19], [OT18], and [OS19].In particular, in [OS19], Ovsienko and Shapiro define a type of super cluster algebra, mo-tivated by super-frieze patterns. In their setup, some of the frozen vertices of the quivercorrespond to odd variables θ , . . . , θ m . There can be paths of length 2 connecting the θ i passsing through the “ordinary” vertices: θ i → x k → θ j . The mutation rules are the samefor all ordinary arrows, and additionally, when mutating at x k ,(1) For θ i → x k → θ j , and for x k → x ℓ , add a 2-path θ i → x ℓ → θ j .(2) Reverse all 2-paths through x k .(3) Cancel oppositely oriented 2-paths through x k .The mutation formula for even variables is given by ˜ µ k ( x k ) = µ k ( x k ) + 1 x k Y θ i → x k → θ j θ i θ j Y x ℓ → x k x ℓ where µ k is the ordinary mutation (ignoring odd variables).Ovsienko and Shapiro noticed that starting with certain initial data, one could construct aquiver with odd vertices such that all entries of the super-frieze can be obtained by mutations.Their choice of initial data consists of all even entries along a NW-SE diagonal, along with the odd entries in the neighboring diagonal. This is pictured below: ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ξ n +1 ∗ θ x ∗ θ x . . .. . . θ m x m ∗ θ m +1 To this set of initial data, they assign the following quiver: x x x · · · x m θ θ θ · · · θ m θ m +1 Remark 7.3.
In Section 6, we described how a super-frieze corresponds to the λ -lengthsand µ -invariants of a triangulated polygon.Following our construction for the case of an initial fan triangulation, we see that the super-frieze that we construct compares with the construction of Ovsienko and Shapiro via a quiverof even and odd variables as follows: The initial data of Ovsienko and Shapiro consists ofthe following: • for even variables, the λ -lengths of all the diagonals of a fan triangulation, • For odd variables, the (modified) µ -invariants √ λ , √ λ λ , √ λ λ , . . . , √ λ n n .Note that unlike our usage of collections of µ -invariants, these µ -invariants correspond totriangles that do not belong to the same triangulation.The results of [OS19] therefore show that all λ -lengths can be expressed in terms of thisinitial data using sequences of mutations.On the other hand, our main theorem (Theorem 4.9) shows that all λ -lengths can be ex-pressed in terms of initial data, where all µ -invariants come from the same initial triangula-tion. From the point of view of cluster algebras, it is more natural to have all initial clustervariables coming from the same triangulation. This leads to the following open question: Question 7.4.
Does there exist some modification of the extended quiver mutation from [OS19] which realizes the super Ptolemy transformations? This would entail (at least) thefollowing:
N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 33 (1) Specifying which 2-paths to include for a given triangulation.(2) Restricting the 2-path mutation to be compatible with this choice.(3) Odd variables must change when mutating at an even vertex.
Section 5 of [SV19] provides yet another conjectural connection to super cluster algebras.In particular, they consider the coordinate ring of the super Grassmannians G r ( R n | m ) of r -planes in ( n + m ) -super-space. In the case of r = 2 , n = 4 or , and m = 1 , these yield acollection of even variables T ab and odd variables θ c where a and b can be identified as verticesof an n -gon, and the θ c ’s can be identified as the possible triangles inside a quadrilateralor pentagon, respectively. This yields super-Plücker relations relating these even and oddvariables to one another. Question 7.5.
Is there an algebraic transformation that relates Shemyakova-Voronov’s T ab ’sand θ c ’s of [SV19] to our λ ab ’s and µ -invariants so that the super-Plücker relations aresatisfied?
8. Appendix: The Pentagon RelationAs we highlighted in Remark 3.3, under the use of super Ptolemy relations, the λ -lengthsassociated to arcs in a polygon are well-defined. To see this, it is sufficient to (1) observethat two consecutive applications of the super Ptolemy relation yields the original arc, and(2) show that the pentagon relation of five alternating applications of the super Ptolemyrelation yields the original two arcs.For (1), we consider the quadrilateral as in figure 3, and flip the arc e . Note that ef = ac + bd + √ abcd σθ. After this flip, if we apply the super Ptolemy relation to the diagonal f in the resultingquadrilateral (and denote the resulting diagonal as g ), we would have f g = bd + ac + √ abcd σ ′ θ ′ . However, using relation Equation (4), we can rewrite the latter as f g = bd + ac + √ abcdσθ, and solving for g , we obtain g = e as desired.The calculations to prove (2) are considerably more involved, and more easily shown usingthe notation of e θ i ’s that appeared in Corollary 4.11.We will examine the sequence of flips illustrated in Figure 18, and apply the super Ptolemyrelations one-by-one.Define the “ modified ” µ -invariants (as in Corollary 4.11) e θ = r eax θ , e θ = r dx x θ , e θ = r cbx θ a b cde x x θ θ θ startingconfiguration a b cde x x θ θ θ flip x a b cde x x θ θ θ flip x a b cde x x θ θ θ flip x a b cde x x θ θ θ flip x a b cde x x θ θ θ flip x Figure 18.
Flip sequence to verify the pentagon relation. Red boundarylabels indicate reversed orientation.
First Flip:
After flipping x , we get x : x = ad + ex x + √ adex x θ θ = ad + ex x + ax e θ e θ and the new θ ’s are θ = √ ad θ − √ ex θ √ x x = 1 √ dex (cid:16) ad e θ − ex e θ (cid:17) and θ = √ ad θ + √ ex θ √ x x = r ax x (cid:16)e θ + e θ (cid:17) Second Flip:
Flip x to get x : x = ac + bx x + √ acbx x θ θ = acx + abd + bex x x + ab (cid:16)e θ e θ + e θ e θ + e θ e θ (cid:17) The new θ ’s are θ = √ ac θ + √ bx θ √ x x = r abx (cid:16)e θ + e θ + e θ (cid:17) N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 35 and θ = √ ac θ − √ bx θ √ x x = 1 √ cx x (cid:16) ac ( e θ + e θ ) − bx e θ (cid:17) Third Flip:
Flip x to get x : x = ce + dx x + √ cdex x θ θ = cex x + dacx + dabd + dbex x x x + dabx ( e θ e θ + e θ e θ + e θ e θ )+ 1 x ( ad e θ − ex e θ ) (cid:16) ac ( e θ + e θ ) − bx e θ (cid:17) = (cid:18) ad + ex x (cid:19) (cid:18) bd + cx x x (cid:19) + (cid:18) dabx + adac + ex acx (cid:19) e θ e θ + (cid:18) dabx − adbx (cid:19) e θ e θ + (cid:18) dab + ex bx (cid:19) e θ e θ = (cid:18) ad + ex x (cid:19) (cid:18) bd + cx x x (cid:19) + (cid:18) dab + acx x (cid:19) e θ e θ + bx e θ e θ = ad + ex + ax x e θ e θ x ! (cid:18) bd + cx x x (cid:19) + bx e θ e θ = bd + cx x + bx e θ e θ with new θ ’s: θ = √ ce θ − √ dx θ √ x x = 1 √ cdx (cid:18) − c ( ad + ex ) x e θ + bd e θ (cid:19) = 1 √ cdx (cid:16) bd e θ − cx e θ (cid:17) θ = √ ce θ + √ dx θ √ x x = 1 x √ ex x (cid:16) a ( ec + dx ) e θ + e ( ac − x x ) e θ − bex e θ (cid:17) = 1 x √ ex x (cid:16) ( ax x − aex b e θ e θ ) e θ − bex ( e θ + e θ ) + eabx e θ e θ e θ (cid:17) = 1 √ ex x (cid:16) ax e θ − be ( e θ + e θ ) (cid:17) Here we used ( ad + ex + ax x e θ e θ ) /x = x Fourth Flip:
Finally, flip x to get x : x = be + ax x + √ abex x θ θ = bex + abd + acx x x + abx x e θ e θ + abx (cid:16) ax e θ − be ( e θ + e θ ) (cid:17) ( e θ + e θ + e θ )= bex + abd + acx x x + abx x e θ e θ + abx (cid:18) ax + bex (cid:19) ( e θ e θ + e θ e θ )= x − abx x ( e θ e θ + e θ e θ + e θ e θ ) + abx x e θ e θ + abx (cid:18) ax + bex (cid:19) ( e θ e θ + e θ e θ ) = x − abx x ( e θ e θ + e θ e θ + e θ e θ ) + abx x e θ e θ + abx (cid:18) x − abx x ( e θ e θ + e θ e θ + e θ e θ ) + abx x e θ e θ (cid:19) ( e θ e θ + e θ e θ )7= x − abx x ( e θ e θ + e θ e θ + e θ e θ ) + abx x e θ e θ + abx x ( e θ e θ + e θ e θ )= x The new θ ’s are: θ = θ √ be + θ √ ax √ x x = r bex x · r abx ( e θ + e θ + e θ ) + r ax x x · r ex x ( ax e θ − be e θ − be e θ )= r aex (cid:18) be + ax x (cid:19) e θ = r aex x e θ = r ax e e θ = θ Here we used x x = ac + bx + abx ( e θ + e θ ) e θ and x x = ce + dx + (cid:16) ac ( ad + ex ) e θ e θ − adbx e θ e θ + ex bx e θ e θ (cid:17) /x . Here we used x = ( acx + abd + bex ) /x x + abx ( e θ e θ + e θ e θ + e θ e θ ) /x Here we used the non-obvious fact that θ θ θ = 0 . We use the equality x = x here. N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 37 θ = θ √ be − θ √ ax √ x x = √ bx √ x x (cid:16) ax e θ − be ( e θ + e θ ) (cid:17) − a √ bx x √ x ( e θ + e θ + e θ )= − r bx x (cid:18) be + ax x (cid:19) ( e θ + e θ ) = − r bx x ( e θ + e θ ) Fifth Flip:
Finally, we flip x to x and get back to the original triangulation (with differentorientation). x = bd + cx x + √ bcdx x θ θ = bd + cx x + √ bcdx x · √ cdx (cid:16) bd e θ − cx e θ (cid:17) − r bx x ( e θ + e θ ) ! = bd + cx x + bx x (cid:18) bd + cx x (cid:19) e θ e θ = (cid:18) bd + cx x (cid:19) (cid:18) bx x e θ e θ (cid:19) = (cid:18) x − bx x x e θ e θ (cid:19) (cid:18) bx x e θ e θ (cid:19) = x + (cid:18) − bx x x + bx x x (cid:19) e θ e θ = x Finally, the new θ ’s are: Similar to above, here we made use of θ θ ( θ + θ ) = 0 We use x x = bd + cx + bx x x e θ e θ here. θ = θ √ bd + θ √ cx √ x x = − (cid:18) bd + cx x (cid:19) r x x d e θ = − (cid:18) x − bx x x e θ e θ (cid:19) r x x d e θ , − r x x d e θ = − θ θ = θ √ bd − θ √ cx √ x x = √ bx √ cx ( bd e θ − cx e θ ) + x √ bcx √ x ( e θ + e θ )= (cid:18) bd + cx x (cid:19) r bcx e θ = (cid:18) x − bx x x e θ e θ (cid:19) r bcx e θ r x bc e θ = θ We see in the end that x = x , x = x . Note also that the edge x , x are oriented oppositeof x , x , and so the λ ij ’s are independent of the orientations. We also see that we get thesame µ -invariants (up to sign), as θ = θ , θ = − θ , and θ = θ .References [BR18] Arkady Berenstein and Vladimir Retakh. Noncommutative marked surfaces. Advances in Math-ematics , 328:1010–1087, 2018.[BZ05] Arkady Berenstein and Andrei Zelevinsky. Quantum cluster algebras.
Advances in Mathematics ,195(2):405–455, 2005.[CR88] Louis Crane and Jeffrey M Rabin. Super riemann surfaces: uniformization and teichmüller the-ory.
Communications in Mathematical Physics , 113(4):601–623, 1988.[CR07] David Cimasoni and Nicolai Reshetikhin. Dimers on surface graphs and spin structures. i.
Com-munications in Mathematical Physics , 275(1):187–208, 2007.[CR08] David Cimasoni and Nicolai Reshetikhin. Dimers on surface graphs and spin structures. ii.
Com-munications in mathematical physics , 281(2):445, 2008.[FG06] Vladimir Fock and Alexander Goncharov. Moduli spaces of local systems and higher teichmüllertheory.
Publications Mathématiques de l’IHÉS , 103:1–211, 2006. Here we use x = x . N EXPANSION FORMULA FOR DECORATED SUPER-TEICHMÜLLER SPACES 39 [FST08] Sergey Fomin, Michael Shapiro, and Dylan Thurston. Cluster algebras and triangulated surfaces.part i: Cluster complexes.
Acta Mathematica , 201(1):83–146, 2008.[FZ02] Sergey Fomin and Andrei Zelevinsky. Cluster algebras i: foundations.
Journal of the AmericanMathematical Society , 15(2):497–529, 2002.[GSV05] Michael Gekhtman, Michael Shapiro, and Alek Vainshtein. Cluster algebras and weil-peterssonforms.
Duke Mathematical Journal , 127(2):291–311, 2005.[HPZ19] Y. Huang, R.C. Penner, and A.M. Zeitlin. Super mcshane identity. arXiv preprintarXiv:1907.09978 , 2019.[IPZ18] Ivan CH Ip, Robert C Penner, and Anton M Zeitlin. N= 2 super-teichmüller theory.
Advancesin Mathematics , 336:409–454, 2018.[LLRS17] James Mixco Li Li, B. Ransingh, and Ashish K. Srivastaval. An introduction to supersymmetriccluster algebras. arXiv preprint arXiv:1708.03851 , 2017.[MGOT15] Sophie Morier-Genoud, Valentin Ovsienko, and Serge Tabachnikov. Introducing supersymmetricfrieze patterns and linear difference operators.
Math. Z. , 281(3-4):1061–1087, 2015. With anappendix by Alexey Ustinov.[MSW11] Gregg Musiker, Ralf Schiffler, and Lauren Williams. Positivity for cluster algebras from surfaces.
Advances in Mathematics , 227(6):2241–2308, 2011.[OS19] Valentin Ovsienko and Michael Shapiro. Cluster algebras with Grassmann variables.
Electron.Res. Announc. Math. Sci. , 26:1–15, 2019.[OT18] Valentin Ovsienko and Serge Tabachnikov. Dual numbers, weighted quivers, and extended Somosand Gale-Robinson sequences.
Algebr. Represent. Theory , 21(5):1119–1132, 2018.[Ovs15] Valetin Ovsienko. A step towards cluster superalgebras. arXiv preprint arXiv:1503.01894 , 2015.[Pen12] Robert C Penner.
Decorated Teichmüller theory , volume 1. European Mathematical Society,2012.[PZ19] RC Penner and Anton M Zeitlin. Decorated super-teichmüller space.
Journal of DifferentialGeometry , 111(3):527–566, 2019.[Sch08] Ralf Schiffler. A cluster expansion formula ( a n case). the electronic journal of combinatorics ,15(R64):1, 2008.[SV19] Ekaterina Shemyakova and Theodore Voronov. On super plücker embedding and cluster algebras. arXiv preprint arXiv:1906.12011 , 2019.[Yur19] Toshiya Yurikusa. Cluster expansion formulas in type a. Algebras and Representation Theory ,22(1):1–19, 2019.
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Email address : [email protected] School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Email address : [email protected] School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Email address ::