Generalized double affine Hecke algebras, their representations, and higher Teichmüller theory

Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of 2 -dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we first construct a functor that sends representations of the ˜ D 4 -type GDAHA to representations of the ˜ E 6 -type one for specialised parameters. Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type ˜ D 4 and ˜ E 6 into matrix algebras over quantum cluster X -varieties, thus linking to the theory of higher Teichmüller spaces. For ˜ E 6 , the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations.


Introduction
Double affine Hecke algebras are a class of associative unital algebras linked to root systems, depending on several parameters.They play a fundamental role in the proof of Macdonald conjectures [8,28,31,33] and are deeply related to integrable systems of Calogero-Moser type [13,29] as well as to the Painlevé differential equations [29,26].For any simply laced Dynkin diagram of rank 1 with star-shaped affinization ˜ , the generalized double affine Hecke algebras (GDAHA) were introduced by Etingof, Oblomkov and Rains in [14] as flat deformations of [G l ], the group algebra of the 2-dimensional crystallographic group G l := l ⋉ 2 , for l = 2, 3, 4, 6.In this paper, we focus on l = 2, 3 which correspond to D4 and Ẽ6 .For D4 , the associated GDAHA, denoted here H D 4 (t, q), recovers the Cherednik algebra of type Č C 1 .Namely, H D 4 (t, q) is the family of algebras depending on parameters t 1 , t 2 , t 3 , t 4 , q ∈ × generated by K 1 , K 2 , K 3 , K 4 subject to the relations For Ẽ6 , the GDAHA H E 6 (t, q) is a true generalisation of Č C 1 .It is defined as the family of algebras depending on q, t ( j) i ∈ × , i = 1, 2, 3 and j = 1, 2, by generators J 1 , J 2 , J 3 subject to the relations i t (2) i = 0, i = 1, 2, 3; While the representation theory of H D 4 (t, q) is well understood [24,23], to the best of our knowledge no explicit representations of H E 6 (t, q) are available in the literature.
Our first result gives a way to construct them, at least for a specific choice of parameters: Theorem 1.1.Let Rep H D 4 (t, q) and Rep H E 6 (t, q) be the category of representations of the algebras H D 4 (t, q) and H E 6 (t, q), respectively.There exists a functor where the specialized parameters t are given by t(1) In proving this theorem, we construct the analogue of a basic representation for this specialized GDAHA H E 6 ( t, q), see formulae (43-44-45).The centers of H D 4 (t, 1) and H E 6 (t, 1) are both affine del Pezzo surfaces, obtained by removing a triangle and a nodal 1 respectively from a degree 3 projective del Pezzo: D4 : x 1 x 2 x 3 + x 2  1 + x 2 2 + x 2 3 + a 1 x 1 + a 2 x 2 + a 3 x 1 + a 4 = 0, Ẽ6 : x 1 x 2 x 3 + x 3  1 + x 3 2 + x 2 3 + a 1 x 2 1 + a 2 x 2 2 + a 3 x 1 + a 4 x 2 + a 5 x 3 + a 6 = 0, for the parameters a i ∈ [14].The quantizations of these surfaces are special cases of the generalized Sklyanin-Painlevé algebra introduced in [6].Based on the fact that the D4 del Pezzo can be obtained as a limit from the Ẽ6 one, Chekhov, Rubtsov and the second author of the present paper conjectured that H D 4 (t, q) could be obtained as a limit of H E 6 (t, q).Theorem 1.1 proves this conjecture.In order to tackle the representation theory of the general H E 6 (t, q), we bring the machinery of higher Teichmüller theory and quantum cluster varieties into the GDAHA theory.In [27], the second author constructed an embedding of the D4 -type GDAHA into Mat 2 ( q ).Our second result is an explicit embedding of the Ẽ6 -type GDAHA into 3 × 3 matrices over the coordinate ring of a quantum cluster X -variety: Theorem 1.2.Let X q be the quantum cluster variety generated by 111 with relations encoded by the diamond-shaped quiver of Figure 17.The matrices C, Y , R ∈ SL 3 (X 1 /3 q ) defined by equations (79), where X 1 /3 q is the extension of X q containing cubic roots of the variables {Z C1 , . . ., Z (b) 111 }, satisfy the Hecke relations and the cyclic one where the monomials that appear in the Hecke relations are central elements in the quantum cluster algebra X 1 /3  q .The map defines an embedding of H E 6 (t, q −2 ) into Mat 3 (X Picking any faithful representation of X q , the theorem gives a faithful representation of the Ẽ6 -type GDAHA. Both the quantum torus q and X q appear naturally as coordinate rings of the moduli space of pinnings PSL n (Σ g,s,m ), introduced by Goncharov and Shen [17] as an extension of the moduli space PSL n ( ) (Σ g,s,m ) of framed PSL n -local systems on a Riemann surface Σ g,s,m of genus g with s boundaries and m marked points on the boundaries.Specifically, q corresponds to PSL 2 (Σ 0,4,0 ) and X q to PSL 3 (Σ 0,3,0 ).Our final result introduces a new operation we call quiver seizure: it consists in removing a vertex of a rhombus in a quiver Q having associated cluster algebra X Q .A rhombus in Q is a 4-cycle with vertices labelled Z 1 , Z 2 , Z 3 , Z 4 such that any other arrow of the quiver can be incident at most with Z 1 or Z 3 (left-hand side of Figure 2).Based on the fact that in a rhombus the product Z 2 Z 4 is a central element of X Q , the quiver seizure removes either Z 2 or Z 4 , together with its pair of arrows (right-hand side of Figure 2).After showing that this operation corresponds to performing a quotient in the algebra X Q , we give a constructive proof of the following (see Theorem 5.5 for a more detailed statement) Theorem 1.3.There exists an isomorphism sending the image via q of the Mat 2 ( q )-embedding of the D4 -type GDAHA to a seizure-driven reduction of the Mat 3 (X 1 /3 q )-embedding of Ẽ6 -type one.
The two quiver seizures performing this reduction can be interpreted as the result of two colliding holes in the sense of [4].
The present paper is organised as follows: In Section 2, we define the functor of Theorem 1.1 by introducing a quantum analogue of both Katz's middle convolution and Killing's factorization.In the process, we construct the analogue of the basic representation for H E 6 ( t, q).
In Section 3, we recall the basics of higher Teichmüller theory and the moduli space of pinnings.We give a succinct self-consistent summary of the Fock-Goncharov coordinates of the moduli space of PSL n -local systems and their extension to the moduli space of pinnings due to Goncharov and Shen.We describe the so-called snake calculus, detailing how to compute transport matrices and glue triangles by amalgamations.En passant, we prove that the Fock-Goncharov coordinates define the coordinate ring of the higher bordered cusped Teichmüller space Hom π a (Σ g,s,m ), PSL n ( ) Π m i=1 B i , introduced in [3,4].After giving a recipe to represent fat graph loops by strings of transport matrices, we conclude explaining the quantization of the Fock-Goncharov coordinates and their fractional extensions.In Section 4, after reproducing the Mat 2 ( q )-embedding of the D4 -type GDAHA within the framework of Section 3, we prove Theorem 1.2.
In Section 5, we introduce the quiver seizure operation and prove Theorem 1.3.Finally, in Appendix A we summarise the analytical background behind our constructions while Appendix B lightens the reading by collecting the most involved formulae.

Quantum middle convolution
Katz [21] originally introduced the middle convolution for local systems as a transformation preserving rigidity and irreducibility.In this section, we introduce a noncommutative version of its algebraic analogue as defined in [12], tailored to our GDAHA H D 4 (t, q) case of interest.For convenience, we rescale the generators as so that the Hecke relations can be written as follows: In doing so, the cyclic relation in preserved: For an object (ρ, V ) ∈ Rep H D 4 (t, q) , we denote ρ( K j ) by K j , namely we use the same notation for the generator and its representation K i ∈ End(V ).
Introducing the triple K := ( K 1 , K 2 , K 3 ), the first map we define is where Notice that ( ( K), 3 V ) no longer defines a representation of H D 4 (t, q).The algebra structure in End( 3 V ) is given by the usual matrix multiplication, combined with the algebra operations in H D 4 (t, q).In particular, the ordering is dictated by that of matrix multiplication.The next operation is a quantum quotient to a subspace encoding the Hecke properties of K.
Lemma 2.1.The subspace U ⊂ 3 V , defined as is invariant under the action of N 1 , N 2 and N 3 . Proof.
Analogous computations can be repeated for N 2 and N 3 .
The quantum middle convolution is the restriction of ( K) to the quotient ( 3 V )/U.To construct this quotient, we take advantage of the properties entailed by the Hecke relations.In particular, each operator K i : V → V carries a natural direct sum decomposition of V into eigenspaces: where 1 corresponds to the eigenvalue 1 and 2 to the other eigenvalue t −2 i .Lemma 2.2.The operators e i , defined as are idempotent and project onto the eigenspace 2 : Moreover, denoting ēi := 1 , the following relations hold for i = 1, 2, 3: Proof.It all stems from the Hecke relation Introducing the operator we can finally give the following Definition 2.3.Let VECT 2 be the arrow category of all vector spaces.The quantum middle convolution is the map 3 , where is a functor whose image consists of quantum pseudo-reflections.
Proof.Let us first characterize ( K) explicitly in End(E(V )): since K i acts as the multiplication by t −2 i on e i (V ), it is immediate to obtain the pseudo-reflection formulae Now let (K, V ), (K ′ , V ′ ) be objects in Rep H D 4 (t, q) and φ : V → V ′ be a homomorphism of representations, i.e. for i = 1, 2, 3 the following diagram commutes: Since representations in the same category have the same parameters (q, t), φ also commutes with the rescaled representations.
In order to define the functor on arrows, we first introduce the map as the arrow in VECT 2 making the diagram (17) commute.E.g., Analogously, i φ, the map (16) restricts to E(V ) 3 as defining the functor on arrows.Indeed, the diagram (19) commutes given that (18) restricts as Functoriality is a straightforward consequence of the definitions: for the identity id : V → V , the relation (id) = id is obvious while given two arrows φ : V → V ′ and ψ : Remark 2.5.In the laguage of Katz [21], Definition 2.3 is the quantum algebraic analogue of M (∞, F ).It corresponds to quotient by only the K subspace in [12] (equivalently, to assume λ generic and set it to 1 after the restriction is performed).The quantum construction taking full account of the other subspace is to appear in a different paper: whenever L is nontrivial, the image of the quantum middle convolution must remain an object in Rep H D 4 (t, q) , with no hope of being mapped in a different GDAHA.

Quantum Killing factorization
This section extends to the noncommutative realm a classical result that traces its origin back to Killing [9]: Then, their product is uniquely factorized as for U upper unitriangular and L lower triangular given by Moreover, when Proof.With the ordering in the entries induced by the matrix multiplication one, by direct computation we obtain Multiplying this formula on the left by a suitable upper unitriangular matrix U −1 , we obtain a lower triangular result: To calculate U, we use the fact that U −1 is unipotent: Assuming that a 11 ∈ U( ) and denoting by a −1 11 its multiplicative inverse, and analogous formulae hold for R 2 and R 3 .When a ii ∈ U(R) for i = 1, 2, 3, the triple product R 1 R 2 R 3 can be inverted too.Doing so via the factorisation, since U −1 is known it suffices to invert L: Applying Lemma 2.6 to for i = 1, 2, 3 the factorization (21) takes the form Moreover, defining

The functorial composition
Composing the noncommutative analogues of the Killing factorization and the middle convolution provides a tool to construct representations of the Ẽ6 -type GDAHA: Lemma 2.7.Given an object (ρ, V ) ∈ Rep H D 4 (t, q) , let U and L be the quantum Killing factors of EN 1 EN 2 EN 3 , where (EN 1 , EN 2 , EN 3 ) is the triple of pseudo-reflections (14).Denoting by Π := (EN 1 EN 2 EN 3 ) −1 the inverse triple product, the following relations hold: In particular, the rescaled operators satisfy the Hecke relations together with the cyclic one Proof.By construction, U L Π = 1 and (33) follows immediately.As an upper triangular matrix of operators, U automatically satisfies a Hecke relation with its diagonal entries as parameters-which are forced to be unities by the quantum factorization.Being lower triangular, L satisfies the analogous Hecke relation if and only if its diagonal is made of invertible elements-which is the case for the factorization of (EN 1 , EN 2 , EN 3 ), see (28)(29).
To prove the remaining Hecke relation for Π, we use the basic representation of H D 4 (t, q) [24].This is given by the operators T 0 , T 1 , Z acting on the space of Laurent polynomials ] as follows: These operators satisfy the algebra relations To put these relations in form (7), we set Notice that with this choice, among the new relations we have the cyclic one as (8).Despite the fact that the operators K i act on the infinite dimensional -vector space of Laurent polynomials [z ±1 ], we can give an explicit characterization to their eigenspaces: Lemma 2.8 ([22]).Let Sym denote the space of symmetric Laurent polynomials, and Sym q denote the space of q-symmetric Laurent polynomials, Thanks to Lemma 2.8, we have that allowing to give an explicit restriction of the triple of operators resulting from applying to ( K 1 , K 2 , K 3 ) from (38).The restricted operators act on a generic element in the quotient ) as follows: It is immediate to put these operators in matrix form and read off their Killing factors as explained in Section 2.2.We obtain the following operators: Moreover, we set Π = L −1 U −1 , where L −1 and U −1 are computed as prescribed in Section 2.2: The Hecke relations for L and U can be easily checked directly using formulae (43-44), while U L Π = 1 holds by construction.Verifying the Hecke relation for Π is a heavy computation best performed with Mathematica [11].This concludes the proof of formulae (30) with parameters (38).
We are now ready to conclude the proof of Theorem 1.1, restated here in more detail: Theorem 2.9.The quantum Killing factorization of the quantum middle convolution gives a functor of (faithful) representations with e i defined in Lemma 2.2, and η : with U, L and Π defined in Lemma 2.7 and the parameters t given by (3).
Proof.In Lemma 2.7, we have already proven that q maps objects (ρ, . The faithfulness of (η, E(V )) stems from the same argument used in the proof of Theorem 4.2.Now, let (ρ, V ) and (ρ ′ , V ′ ) be two objects in Rep H D 4 (t, q) and φ : V → V ′ a homomorphism of representations.The map of arrows defined in the proof of Proposition 2.4 carries through the factorization: for i = 1, 2, 3, ⨿ (φ) := ∋ φ gives the commutative diagram Indeed, each Killing factor's entry is a (linear combination of) composition of entries from EN 1 , EN 2 , EN 3 and these suitably commute with φ: as previously observed, φe i = e ′ i φ.To conclude, functoriality holds unaffected: for the identity id : V → V , q (id) = id manifestly while for ψ :

Higher Teichmüller theory
Let Σ g,s,m be a genus g topological surface with s boundary components and m marked points on the boundaries having negative Euler characteristic.In absence of marked points, the Teichmüller space PSL 2 (Σ g,s,0 ), i.e. the moduli space of complex structures on Σ g,s,0 modulo diffeomorphisms isotopic to the identity, is identified with the space of discrete faithful representations π 1 (Σ g,s,0 ) → PSL 2 ( ) modulo conjugation.In the case m ≥ 1, the bordered cusped Teichmüller space PSL 2 ( ) (Σ g,s,m ) was introduced in [3] as where π a (Σ g,s,m ) is the fundamental groupoid of arcs in Σ g,s,m and B 1 , . . ., B m is the choice of a Borel unipotent subgroup in PSL 2 ( ) at each marked point.Notice that an element in PSL 2 ( ) (Σ g,s,m ) uniquely fixes a metric of constant negative curvature on Σ g,s,m , the surface obtained by cutting tubular neighbourhoods of the boundary and adding ideal triangles at the marked points.Both the Teichmüller space and the bordered cusped Teichmüller space admit a higher generalization by replacing PSL 2 with any split semi-simple algebraic group G.In the case of G = PSL n , we denote the higher bordered cusped Teichmüller space by PSL n ( ) (Σ g,s,m ).In Theorem 3.11, we show the coordinate ring of the higher bordered cusped Teichmüller space PSL n ( ) (Σ g,s,m ) can be identified with the moduli space of pinnings PSL n (Σ g,s,m ) introduced by Goncharov and Shen [17].The latter is an extension by additional data of the moduli space PSL n ( ),Σ g,s,m of framed PSL n -local systems, i.e. principal PSL n -bundles with framed flat connections defined by attaching an holonomy-invariant flag to each marked point.

Combinatorial description of the moduli space of pinnings
In this section, we recall the main ingredients of the combinatorial description of PSL n (Σ g,s,m ) due to Goncharov and Shen [17] and its quantisation due to Chekhov and Shapiro [7].We closely follow the latter paper as well as [10]: since notations have been tailored to our needs, for the sake of the reader the exposition is self-consistent.
Throughout this section, we restrict to G = PSL n ( ).In subsection 3.1.1,we describe the moduli space of framed PSL n -local systems for the disk with three marked points 1, 2, 3 on its boundary.We picture such surface Σ 0,1,3 as the equilateral triangle △123 in Figure 10 and assign a clockwise orientation.In subsection 3.1.2,we introduce pinnings on △123 and explain how to glue triangles together to form the moduli space for any Riemann surface Σ g,s,m .

The snake calculus on a triangle
For a given n ∈ >0 , we cover △123 by its unique tessellation of n 2 identical equilateral triangular tiles, arranged between upward and downward.Each vertex of this tessellation is labelled by a triple of non-negative integers (i, j, k) by the minimum number of tiles connecting it to the sides of △123: i for side 23, j for side 31 and k for side 12 (Figure 3).Since i + j + k = n, these triples are called barycentric coordinates.This coordinatization naturally extends to a tile by assigning a triple (a, b, c) to its center: • a + b + c = n − 1 in the upward case, where vertices appear in the form • a + b + c = n − 2 in the downward case, where vertices appear in the form Let us highlight the resulting combinatorics: barycentric coordinates are assigned to vertices of the tessellation and centers of the tiles so that the type of object they label can be detected by just inspecting the (integral) result of their sum.
Since any flat connection on the contractible △123 is trivial, PSL n ( ),△123 is identified with the space of triples of holonomy-invariant complete flags in n .Snake calculus is a way to construct elementary change-of-basis matrices between projective bases of n induced by a choice of flags in generic position.Let us detail the combinatorial features of this construction.
• be the (generic) complete flags in n attached to the vertices of △123.To any center (a, b, c) of a tile in the tessellation of △123, we attach the subspace : a line λ a bc for upward tiles and a plane π abc for downward ones.By construction, a plane π a bc contains the lines λ (a+1)bc , λ a(b+1)c , λ ab(c+1) attached to the three upward tiles adjacent to the downward one it is attached to.Let us visually highlight this correspondence: after labelling each center with its subspace, we stick on each plane a grey upward triangle whose vertices match the three coplanar lines it contains.Figure 4 gives a step-by-step display of the resulting configuration on △123.For the rest of this section, we forget the tessellation focusing on these n 2 grey trianglesand the resulting n−1 2 white downward ones among them-looking at specific paths called snakes that run over their sides.Notice that the upward grey and downward white triangles gives nothing but the n − 1 tessellation of a triangle connecting {λ (n−1)00 , λ 0(n−1)0 , λ 00(n−1) }.Definition 3.3.A snake p is an oriented piece-wise path composed by exactly n − 1 sides of grey triangles, which starts from a tile sharing a vertex with △123 and ends on a tile in contact with the opposite side.
Notice that the length requirement implies no segment can be parallel to the snake's target side of △123.We call p I J , the unique snake running parallel to side I J of △123, a ∂ -snake.Let Greek letters denote a generic triple of barycentric coordinates: e.g., λ i jk is equally denoted by λ α .As shown in Figure 6, each segment of a snake connects two vertices α, β of a grey triangle.The corresponding lines λ α , λ β are coplanar to λ γ , where γ is the remaining vertex of the grey triangle.By coplanarity, a choice of vector v α ∈ λ α uniquely determines v β ∈ λ β by the following orientation rule  Therefore, a snake inductively determines a projective basis of n : chosen the first vector and iteratively applying the rule, the resulting n vectors are defined up to a global scaling factor.Their linear independence is a consequence of the flags being assumed generic.Given any two snakes, a change-of-basis matrix maps the projective basis of one to the one of the other.The snake calculus gives a simple recipe to write down these matrices: since the elementary moves in Figure 7 suffice to decompose any snake transformation, they are constructed out of the elementary building blocks in the following Definition 3.4.For E rs the matrix unit, i.e. (E rs ) i j = δ r i δ s j , the identity matrix, k ∈ {1, . . ., n} and a parameter t ∈ , define the SL n ( ) matrices Figure 7: From left to right, elementary snake moves I,II and III mapping red to blue segments of a sample snake with v 1 ∈ λ n00 .Notice that move I can only be performed on the last segment of a snake, i.e. when no subsequent segments can be affected.In this sense, move II can be thought of as the extension of move I to any other segment.
Let us sketch the origin of this advantageous feature, adapting [10], Appendix A. Move I flips the last segment pivoting its source center across a grey triangle, by rule (47) yielding: Move II flips any two non parallel consecutive segments.Analogously to move I, sweeping the grey triangle yields v α k+1 → v β k+1 = v α k+1 + v α k .However, this drags the second segment in a flip that pivots its target center: we expect the transformed 2-segment portion of the snake to end on a different vector within the same line, i.e. v β k+2 ∝ v α k+2 .Denoting by Z the proportionality constant, Since the elementary blocks L i and H j (t) commute for i ̸ = j, this change-of-basis matrix can be factorized as L k H k+1 (Z) and the move as a whole is well defined.Moreover, the matrix can be made unimodular by replacing H k+1 (Z) with its normalization H k+1 (Z) ∈ SL n : indeed, Finally, move III, inverting a clockwise oriented ∂ -snake, is unravelled tracking reversals of segments: Notice that S −1 = (−1) n−1 S = S T .There are n−1 type II moves, one for each downward white triangle, and the corresponding proportionality constants are the so-called Fock-Goncharov variables.We assume them to be positive: this restriction is known to provide a parametrization of the moduli space, in that they describe a connected component of PSL n ( >0 ),△123 known as the higher Teichmüller space of △123.Topologically, notice that Fock-Goncharov variables are in bijection with inner vertices of the tessellation of △123: there is exactly one such vertex inside any white triangle.We thus denote them Z i jk by the barycentric coordinates of the unique corresponding vertex, i, j, k ∈ >0 .Taking advantage of this calculus, the general formula for the change-of-basis matrix corresponding to the ∂ -snake map p 12 → p 31 reads Example 3.5.For n = 2, there are no inner vertices and formula (54) simplifies to For n = 3, there is just a single Fock-Goncharov variable Z 111 : (56)

Transport matrices and amalgamation
In order to glue triangles together, we need to attach additional variables to the sides of △123.This is formally done by extending PSL n ,△123 to the moduli space of pinnings PSL n ,△123 , in which each oriented side of the triangle comes equipped with a 1-dimensional subspace of n in generic position to the corresponding pair of flags.
A pinning on side I J is given by the triple A choice of Λ equals a choice of projective basis {v α 1 , . . .v α n } in n , a vector for each vertex along I J from the corresponding line, via the condition n i=1 v α i ∈ Λ.Therefore, each oriented side I J comes with two projective bases, one from the pinning and the other from the corresponding ∂ -snake p I J , and the unimodular change-of-basis matrix between them takes the form n−1 i=1 H i (t i ).These n − 1 proportionality constants are thought of as additional Fock-Goncharov variables Z i jk , labelled by the vertices on the interior of I J. Adding these extra variables from all three sides to the ones birthed by type II moves, we get a total of 3(n − 1) Fock-Goncharov variables (Figure 9).As a whole, they parametrize PSL n ( >0 ),△123 and are in bijection with the tessellation's vertices except 1, 2, 3. We finally define the transport matrices T i ∈ SL n ( >0 ), i = 1, 2, 3, in Figure 10.They correspond to the special change-of-basis matrices between the pinning-induced projective bases associated to the oriented sides of △123.For example, T 1 maps the pinning of side 12 first to the snake p 12 , then maps the snake p 12 to the snake p 31 , and finally maps the snake p 31 to the pinning of side 31.Definition 3.6.The transport matrices T 1 , T 2 , T 3 are the following n × n matrices Together with their inverses, T 1 , T 2 , T 3 suffice to map between any two sides.Notice that the permutation map σ acts on matrices T (Z i jk ) depending on Fock-Goncharov variables Z i jk as σT (Z i jk ) := T (Z ki j ), so that we have T 2 = σT 1 and T 3 = σ 2 T 1 .We introduce the following shorthand notation: Remark 3.7.These diagonal factors modifying the change-of-basis matrix can be visualized as passing from the side's pinning to the inner ∂ -snake and vice versa: H 12 in for the oriented side 12 the path crosses to enter the triangle (pinning-to-snake), H 31  out for the oriented side of exit (snake-to-pinning).
Example 3.8.Explicitly, for n = 2 For n = 3, Notice that, in both cases, no Fock-Goncharov variables from side 23 appear, in accordance with the crossing of △123 associated with T 1 .Cyclicly permute the indices once and twice to get the expressions for T 2 and T 3 .
As anticipated, pinnings allow to amalgamate variables of two adjacent triangles, creating the set of parameters describing the moduli space PSL n ,□ of the quadrangle obtained by gluing the pair along the common side.The amalgamation procedure, as a translation of the topological gluing of triangles to the parametrization, orderly identifies the two (n − 1)-tuples of vertices on the interior of the sides to be glued, assigning to each resulting vertex a Fock-Goncharov amalgamated variable via the product of the parent ones: if α 2 ∼ α 1 results in the vertex α, Z α := Z α 2 Z α 1 .This operation allows to parameterize PSL n , , for any (suitable) triangulated surface , by amalgamation of the moduli spaces of pinnings assigned to the individual triangles.

Loop representation via transport matrices
Using the machinery of Section 3.1, we here explain how to assign sequences of transport matrices to loops on a fat graph Γ .In order to explain this transport matrix factorization of fat graph paths, we assume a clockwise labelling of vertices from the set {1, 2, 3} is chosen for each triangle coming from the fat graph Γ g,s,n of the surface Σ g,s,m .One should picture a path as transporting an oriented side along the triangulation by the action of transport matrices.In order to consistently compose two transport matrices-that we defined as maps between the clockwise oriented sides of a triangle-a reversal of the transported side must be performed.This is done by inserting an S block between the matrices.In terms of Fock-Goncharov variables, this transport composition rule enforces the amalgamation, performed in the language of transport matrices by letting the diagonal pinning factors multiply each other.Notice that the side reversal is exactly the operation needed to absorb the leftmost S factor of a transport matrix and let the H blocks generate the amalgamated variable.Example 3.9.For n = 2, the composition in Figure 11 is the amalgamation , for Z α the amalgamated variable.For n = 3, the mechanism reads with Z α , Z β as the two amalgamated variables.Remark 3.10.When dealing with loops, the amalgamation at the base-point cannot be captured by the mere factorization over transport matrices: the unavoidable choice of a starting point prevents the composition between the first and last transport matrices from happening.Nevertheless, this issue is easily fixed by a global conjugation.E.g., the path in Figure 11 can be closed into a loop conjugating its factorization S T Notice that in this last factorization none of the variables forming Z ′′ remains.
Summing up, once the sequence of transport matrices associated to the directed crossings of triangles is read off, each loop matrix is assembled by transport compositions and finalized by a global conjugation.

Higher bordered cusped Teichmüller space
In this section, following the work by the second author and her collaborators [3,4,5], we recall the definition of the higher bordered cusped Teichmüller space and show that it can be identified with the moduli space of pinnings PSL m (Σ g,s,m ).Given a Riemann surface Σ g,s,m of genus g with s > 0 boundaries and m ≥ 0 marked points on the boundary, denote by Σ g,s,m the surface resulting by removing tubular domains around each boundary of Σ g,s,m and adding an ideal triangle at each marked point.The Riemann surface Σ g,s,m is called hyperbolic if the hyperbolic area of Σ g,s,m , Area of Σ g,s,m = 4g − 4 + 2s + m π, is strictly positive.Given a hyperbolic Riemann surface Σ g,s,m , denote by p 1 , . . ., p m the marked points on the boundary.The fundamental groupoid of arcs π a (Σ g,s,m ) is the set of all directed paths γ i j : [0, 1] → Σ g,s,m such that γ i j (0) = p i and γ i j (1) = p j modulo homotopy.The groupoid structure is dictated by the usual path-composition rules.The higher bordered cusped Teichmüller space is defined as where B 1 , . . ., B m is the choice of an unipotent Borel element in PSL m ( ) for every marked point.The unquotiented space Hom π a (Σ g,s,m ), PSL m ( ) is endowed with the Fock-Rosly pre-Poisson bracket, which is compatible with the cluster variety Poisson structure [7] .
Theorem 3.11.The higher bordered cusped Teichmüller space is isomorphic to the moduli space of pinnings The Fock-Goncharov coordinates define the coordinate ring on PSL n (Σ g,s,m ).
Proof.For a hyperbolic Riemann surface Σ g,s,m , Σ g,s,m can be triangulated by 4g − 4 + 2s + n ideal triangles.Each arc γ i j ∈ π a uniquely intersects two sides of any triangle it crosses.As explained in Section 3.1, a transport matrix T is associated to any such oriented crossing, with T −1 corresponding to the opposite direction.Transport matrices (Figure 10) fulfill the role of building blocks for associating a matrix M i j ∈ SL m ( ) to each arc γ i j ∈ π a (Σ g,s,m ): as in Section 3.2, they are suitably composed resulting in a given matrix M i j for any path γ i j .The entries of each M i j are expressed in Fock-Goncharov variables.Therefore, to prove that the Fock-Goncharov variables provide the coordinate ring of PSL n (Σ g,s,m ) we only need to prove that Let us compute the left-hand side first.As explained above, Σ g,s,m can be triangulated by F = 4g − 4 + 2s + m ideal triangles.Each of these triangles in endowed with (n − 1)(n − 2)/2 inner variables and n − 1 pinnings for each side, resulting in a total of (n + 4)(n − 1)/2 Fock-Goncharov variables.However, most of the sides of these triangles are amalgamated: indeed, the only non-amalgamated sides are the ones dual to an open edge in the cusped fat graph corresponding to a marked point.Therefore, we have 3F −m = 2(6g −6+3s+m) amalgamated sides that absorbs (6g − 6 + 3s + m)(n − 1) variables, leading to We now compute the dimension of PSL n (Σ g,s,m ).The fundamental groupoid of arcs is generated by 2g + s − 2 + m arcs.For each of these, we associate a matrix in PSL n and, taking into account the quotient by the unipotent Borel subgroups, we obtain The two numbers coincide.

Quantization
The moduli space of pinnings PSL n ,△123 is quantized by promoting the Fock-Goncharov variables to generators of a quantum cluster algebra, with relations encoded by a quiver constructed from the tessellation of △123.Provided the removal of 1, 2, 3, the quiver's vertices coincide with the tessellation's ones and arrows are defined by consistently extending the clockwise orientation of △123 to the tiles: upward ones are clockwise and downward ones counterclockwise.Arrows from the sides of △123 are dashed.The resulting quiver is displayed in Figure 12.The set of vertices of the quiver is in bijection with quantum Fock-Goncharov variables Z α and arrows rule their commutation relations: for a fixed parameter q ∈ × , (63) The resulting noncommutative algebra of Laurent polynomials reads where # counts the number of arrows from α to β ( 1 /2 for a dashed one).It naturally carries the Weyl quantum ordering: for any monomial Z α 1 • • • Z α n , we denote it by double bullets A handy way to master rule (65) is to imagine the weight w i j = −w ji measuring a flow carried by the arrows: −1 for an outgoing arrow α i → α j (outflow) and +1 for an incoming arrow α i ← α j (inflow), with dashed arrows corresponding to half flows.Notice that inside the double bullets the order does not matter, e.g., • .This is the very reason this definition provides a well-defined ordering for quantum variables.Due to the normalized H k block, we need an extension of X q containing n-th roots of Fock-Goncharov variables: The n 2 denominator is found factorizing each Z as n i=1 Z 1 /n , while the quantum ordering formula remains valid.Transport matrices are quantized following a straightforward recipe: Definition 3.12.For the diagonal matrix (Q) ii := q 2−i− n+1 n 2 , the triplet of quantum transport matrices is given by where the Weyl quantum ordering acts linearly on each entry.
The matrix Q is uniquely defined by enforcing the quantum groupoid relation [7] T Within the framework of Section 3.2, this translates to the topological consistency visualized by Figure 13.In the classical case, T 1 T 2 T 3 = 1 follows automatically from (57).Notice that the quantum correction introduced with the matrix Q causes the entries of the quantum transport matrices not to be Weyl-ordered monomials.Moreover, interpreting this correction as the quantization S → S q := QS, we quantize the transport composition rule (Figure 11) by prescribing to insert a S q block instead.
Finally, the quantum extension of the amalgamation procedure is straightforward: under the simplest rule prescribing commutation for quantum Fock-Goncharov variables coming from different triangles, a quantum amalgamated variable reads Figure 13: Interpretation of the quantum groupoid relation for paths: being topologically equivalent, blue and red must be assigned the same matrix, i.e. (T For the special self-gluing case where two sides of the same triangle are identified, only the first equality holds and the amalgamated variable must be taken as the quantum ordering.

GDAHAs from higher Teichmüller theory
For a special pair of fat graphs, we prove that the matrix algebras resulting from the loop representation provide embeddings of GDAHAs.The n = 2 case recovers a known representation, serving as both a quantum showcase of the machinery developed in Section 3.2 and an appetizer for the more involved n = 3 one.
Our proofs are supported by the NCAlgebra extension for Mathematica [20].This package allows to perform noncommutative multiplications and simplify symbolic expressions by repeated substitution of a prescribed set of relations.All Mathematica-aided computations can be found in [11].
In the following two sections, the notation drops the q superscript for better readability, namely T i and S stay for the respective quantum matrices.

The matrix algebra for H D 4
For n = 2, the expected recovery of classical Teichmüller theory manifests by choosing the fat graph of the four punctured Riemann sphere Σ 0,4 , see Appendix A for the analytic rationale underlying this choice.Indeed, the matrix algebra resulting from the loop factorization delivers the same representation of the GDAHA H D 4 (t, q) that was found in the classical Teichmüller framework [27].
The transport matrix (60), computed in Example 3.8, needs to be quantized and multiplied by the quantum correction Q = diag(q 1 /4 , q −3 /4 ): Quantum T 2 and T 3 follow the same recipe.The loop representation can be read off from Figure 14: denoting by O the matrix corresponding to the ochre loop, B the matrix of the blue loop, G the one of the green loop and P that of the pink one, we have S, (68) where T (a) i stays for the quantum transport matrix T i in the Fock-Goncharov variables Z (a)  α of the triangle (a).The q and q factors have been introduced in (68) to set the product of each pair of Hecke parameters to the unit.The base-point amalgamation is achieved conjugating formulae (68) by the diagonal matrix For a matrix M representing a path in a fat graph, we denote by M := C M C −1 the one conjugated by C in (69).
The final matrices O, B, G, P depend only on the amalgamated variables whose algebra relations are encoded by the quiver in Figure 15.On the one hand, being isolated vertices in this quiver, the variables Z O1 , Z B1 , Z G1 are central elements and for convenience we treat them as complex parameters.On the other hand, despite the transport matrices involve square roots of Fock-Goncharov variables, namely q ), no fractional Z O2 , Z B2 or Z G2 appear in the whole quadruple (O, B, G, P).Therefore, the matrix entries can be thought of as elements in the quantum torus As anticipated, the next theorem recovers the very same Mat 2 ( q )-embedding of the D4 -type GDAHA found by the second author in [27].
volve fractional powers of all variables and we must resort to the cubic root extension of the amalgamated cluster algebra.
Theorem 4.2.Let X q be the quantum cluster variety with 111 and q-commutations encoded by Figure 17.The SL 3 (X ) with (80) satisfy the relations The map Proof.Proving that Y , C and R satisfy relations (81) is a direct computation, which can be reproduced in the Mathematica companion [11].The Hecke parameters are central being products of pairs of variables having arrows with opposite directions.As an example, take For each bracketed pair, arrows cancel out: e.g., the q-factors due to arrows Z C2 → Z 111 .The fact that the map is an embedding can be proved by choosing a faithful representation of X 1 /3 q , namely a vector space V and an algebra homomorphism ρ : X 1 /3 q → End(V ).The resulting map ρ : H E 6 (t, q) → Mat 3 (End(V )) gives a representation of H E 6 (t, q) on 3 V .Now, the rank 1 GDAHA of type E 6 is prime.Indeed, for generic values of parameters, it is Morita equivalent to its spherical subalgebra, whose associated graded algebra is a twisted homogeneous coordinate ring of an irreducible curve, and therefore is a domain (see Theorems 6.5 and 6.10 in [14]) 1 .This proves that ρ is injective thus so is our map.

Cluster seizure
In this section, after applying the functor q to the matrix triple (K 1 , K 2 , K 3 ) = (O, B, G) from Theorem 4.1, we prove Theorem 1.3.As prescribed in Section 2.1, we start by rescaling the triple: Since our input to q is given by 2 × 2 matrices, the operation will produce 6 × 6 matrices N 1 , N 2 , N 3 .
In order to perform concretely the quotient in Proposition 2.4, we need an explicit characterization of the eigenspaces V (1) 2 , V (3) 2 we have to resctict to.Selecting a representation of q on a vector space V, we fit the framework of genuine representations on vector spaces developed in Section 2: indeed, this allows to view the matrices K i as elements in End(V ⊕ V), namely V := V ⊕ V. Now, solving for eigenspaces is more conveniently carried out by reading formulae (82) as arrows in Hom Mod-q (⊕ 2 q , ⊕ 2 q ), where Modq denotes the category of right q -modules, namely having the rescaled matrices act in the usual way on columns in Mat 2×1 ( q ).Indeed, computing eigenspaces in V ⊕ V amounts to solving q -linear equations.
Remark 5.1.The quantum torus is known to be a Noetherian domain whose ring of fractions 1 We thank P. Etingof for clarifying this argument to us.
We choose a more polished (R 1 , R 2 , R 3 ) by performing a global diagonal conjugation, which manifestly preserves the pseudo-reflection structure of the whole triple: As detailed below in m( Ẑi ) the monomial with Z i removed, the assignment extends to a quantum cluster algebra isomorphism X Q 〈m(Z i ) − 1〉 ∼ − → X Q\Z i .We are now ready to prove our final Theorem 1.3, namely show that the triple (U, L, Π) can be found within (C, Y , R) from Theorem 1.2.Theorem 5.5.Let X q,I := X q I be the quotient by the ideal and denote by (C I , Y I , R I ) the restriction of the triple (79) to X q,I .Then, via the entry-wise action of the following maps: the algebra isomorphism reversing the parameter of the quantum torus (71), i.e.
the algebra isomorphism ι : and the quantum cluster mutation 111 . (95) Proof.We start by noticing that X q,I is a well-defined quantum quotient: both Z C1 Z C3 and 111 Z C2 Z Y 3 are central in the algebra X q .Moreover, both these monomials in X q can be recognized as seizures for the quiver in Figure 17: at vertex Z C3 by setting m(Z C3 ) = Z C1 Z C3 for the rhombus {Z 111 , Z C2 }.By the seizure's properties, the q-commutations for X q,I are encoded by the reduced quiver in which we have erased the vertices Z Y 3 and Z C3 together with their incident arrows.This is displayed in two equivalent shapes in Figure 18.Therefore, the reduced triple (C I , Y I , R I ) is obtained via the identifications , so that entries are Laurent polynomials in the six variables 111 } generating X q,I .It turns out that C I is free from fractional powers and thus a genuine element in SL 3 (X q,I ): (96) By the very definition of I, all its diagonal elements are turned into unities matching diag(U).To push the match further, we need to take advantage of the cluster nature of the X -space.Indeed, to connect the cluster algebra X q,I to the quantum torus q of the triple (U, L, Π), we need the quantum mutation (95).In quiver terms, mutating at vertex α translates as a 3-step recipe [16]: 1.For each oriented two-arrow path i → α → j, add a new arrow i → j 2. Flip all arrows incident with α 3. Remove all pairwise disjoint 2-cycles, i.e. closed paths with shape i ⇄ j Therefore, mutating at vertex Z (b) 111 , we turn the reduced quiver in Figure 18 from star-shaped to box-shaped as in Figure 19.As expected, this mutated quiver encodes the q-commutations in X ′ q,I ′ .On the corresponding cluster algebras, µ acts as a quantum analogue of a pullback sending X ′ q,I ′ to X q,I .The gain in using µ is made manifest by the algebra isomorphism (94).Indeed, ι reveals that the mutated quiver is equivalent to the quantum torus one in the right hand side of Figure 15, provided all arrows are reversed.This is visually displayed in Figure 20.The cluster algebra counterpart of this arrow reversal is the τ map (93).Now that quantum algebras agree, by direct computation the entry-wise action of the composition µ ι τ on the triple (U, L, Π) is proven to match the reduced one (C I , Y I , R I ).Before we detail these computations, let us illustrate the phenomena allowing them to run successfully.
On the one hand, only Z O1 , Z B1 , Z G1 make a fractional appearance in (U, L, Π) and their image under µ ι does not involve the formal inverse of 1 + qZ Therefore, no fractional power of a formal inverse appears.On the other hand, 1+qZ does appear through µ ι(Z −1 G2 ) but its algebra relations are easily figured out: indeed, for a formal power series f (x), As a result, despite resorting to the fraction field for the mutation to act, the entry-wise action of µ delivers genuine elements in SL 3 (X 1 /3 q,I ): using (98), each formal inverse simplifies.Further theoretical evidence is given by the fact that, in the restricted quiver, Z (b) 111 is a 4-valent node with alternating incoming and outgoing arrows: as proved in [32], mutations at these special vertices preserve the transport matrix calculus.We conclude the proof detailing the computations behind the correspondence U → C I : with ) corresponds to the outer ribbon while the inner one, passing via the centers of the triangles, triggers the appearance of both Z (t)  111 and Z (b)  111 in m(Z Y 3 ).

A. Analytical theory
In this appendix we briefly discuss the analytic counterpart of the functor q .As proved in [27], the classical limit of the representation of H D 4 (t, q) on SL 2 ( q ) given in Theorem 4.1 produces the generators of the monodromy group of a 2 × 2 Fuchsian system with 4 poles: where each matrix A k ∈ sl 2 ( ) has spectrum {± θ k 2 } for constants θ k ̸ ∈ , and for a constant θ ∞ ̸ ∈ .Fixing the fundamental solution at infinity and a basis in π 1 of non intersecting loops circling each singularity once, the monodromy group of this system is given by The classical limits are obtained by the correspondence as q → 1 and the Fock-Goncharov quantum cluster variety is replaced by the classical cluster Poisson variety.
In the following, we prove a lemma stating that taking the classical limit of the matrices in Proposition 5.3 leads to the generalized monodromy data (namely Stokes and monodromy matrices) of a 3 × 3 linear system with a simple pole at 0 and a double pole at ∞: where U = diag(u 1 , u 2 , u 3 ) and V is any constant matrix with diagonal part and eigenvalues µ 1 = 1 2 (Tr(Θ) + θ ∞ ), µ 2 = 1 2 (Tr(Θ) − θ ∞ ), µ 3 = 0.
Remark A.1.According to the theory developed in [3], this scenario corresponds to a SL 3connection on Σ 0,2,2 .This surface is precisely represented by the fat graph in Figure 16 with two open edges, see also Remark 5.6.
Let us briefly describe the generalized monodromy data of the linear system (102): fixing both the fundamental matrix on a sector at infinity and an appropriate branch cut, the monodromy data are collected in the following subset of SL 3 ( ) 3 : (M 0 , S 1 , S 2 ) ∈ SL 3 ( ) × B where B (1) + is the Borel subgroup of upper unitriangular matrices and B − is the Borel subgroup of lower triangular matrices.The two linear systems (99) and (102) are dual in the sense of Harnad [19].This duality was first produced by Dubrovin in the case θ i = 0 for i = 1, 2, 3.He showed that, given a loop γ avoiding branch cuts in the λ-plane, the inverse Laplace transform converges in a non-empty sector as z → ∞ and maps the system (102) to a 3 × 3 Fuchsian system of the form where E k is the matrix with zero entries everywhere except a 1 in position kk.In fact, the system (105) can be reduced to (99) by a simple quotient.Subsequently, this result was generalized to any values of θ i by the second author of the present paper in [25].Following this, Filipuk and Haraoka [15] showed that the link between the 3×3 Fuchsian system (105) and the 2×2 one (99) could be recast in terms of the so-called additive middle convolution.
We summarise this chain of results in the following diagram: It is important to point out that the Harnad duality maps the isomonodromic deformations equations of system (99) to the ones of system (102).On the level of monodromy data, Dettweiler and Reiter showed that the monodromy matrices of two Fuchsian systems related by addittive middle convolution transform by the Katz middle convolution, also called multiplicative middle convolution [12].In our case, the monodromy matrices obtained via multiplicative middle convolution are given by three pseudo-reflections R 1 , R 2 , R 3 and R 4 = (R 1 R 2 R 3 ) −1 .By constructing S 1 ∈ B (1) + and S 2 ∈ B − as Killing factors of the triple (R 1 , R 2 , R 3 ) and setting M 0 = R −1 4 , one obtains exactly an element in the set (103) [2].However, to prove that these are indeed the monodromy data of the system (102) corresponding to the fundamental solution coming out of the inverse Laplace transform, one needs to be careful with the loops of integration, the basis of loops in π 1 and the choice of sectors.This computation was performed by Dubrovin, following [1], in the case when θ i = 0 for i = 1, 2, 3 and extended to all possible cases in [18].This discussion can be visualized by expanding (106) to the following diagram, in which the commutativity of the left square is due to Dettweiler and Reiter while that of the right square is due to Dubrovin and Guzzetti: (107) The blue map, defined as the classical analogue c : (M 1 , M 2 , M 3 , M ∞ ) → (S 1 , S 2 , M 0 ) of the functor q , delivers the main result of this appendix: Lemma A.2.The classical limit as q → 1 of the matrices U, L, Π appearing in Proposition 5.3 gives the monodromy data of the linear system (102): Proof.This is a straightforward consequence of the fact that, by construction of q and c ,

Figure 1 :
Figure 1: Affine Dynkin diagrams: D4 on the left and Ẽ6 on the right.Each leg contributes a generator and its length determines the order of the corresponding Hecke relation.These two cases are special in that all n legs have same length, which is itself equal to n.

Figure 3 :
Figure 3: n = 7 tessellation of △123 and barycentric coordinates (1, 4, 2) for the pink vertex, with colors highlighting the tile-counting ruling the coordinatization.The total n 2 tiles are all similar to △123 and arranged between n+1 2 upward and n 2 downward ones.

Figure 4 :
Figure 4: For n = 3, from left to right: tessellation of △123, barycentric coordinates for vertices of the tessellation and centers of the tiles, configuration of subspaces with the grey triangles (and one white triangle enclosed by them).

Figure 6 :
Figure 6: Segments of two oppositely oriented snakes.The vertices of the grey triangle correspond to 3 coplanar lines λ α , λ β , λ γ and v γ = v β ± v α depending on whether the segment is oriented clockwise or counterclockwise with respect to its grey triangle.

Figure 10 :
Figure 10: The triple of transport matrices on the oriented triangle △123.T 1 corresponds to the map of oriented sides 12 → 31, T 2 to 23 → 12 and T 3 to 31 → 23.

Figure 11 :
Figure 11: The blue path transports the oriented side indicated by the thick black arrow.When constructing its representation, the transport composition rule prescribes the insertion of the S block between the transport matrices.The orientation is explicitly indicated on those sides interacting via the composition: the 31 oriented side of the right (r) triangle is reversed by S to match the 23 one of the left (l) triangle.The leftmost S block performs a final reversal: without it, the path would flip the side it is just allowed to transport.
amalgamated variable due to the closure,

Figure 14 :
Figure 14: Fat graph's triangulation of Σ 0,4 and relevant loops for n = 2, with the transported edge displayed by the thick black arrow.The four triangles are labelled as follows: (c) for the central one, (r) for the rightmost one, (l) for the leftmost one and (d) for the down most one.For each triangle, the 1 indicates the choice of labelling and thus dictates its triple of transport matrices as in Figure 10.

Figure 15 :
Figure 15: On the left, amalgamated pairs are highlighted in red, the shaded one triggered by the global conjugation.No variables from the original four triangles remain.On the right, the resulting quiver of amalgamated variables.The variables Z O1 , Z B1 , Z G1 together with Z O2 Z B2 Z G2 generate the subalgebra of Casimir elements.
111 are respectively absorbed by the ones due toZ Y 3 ← Z (b) 111 and Z Y 3 → Z (t) (t) 111 , Z C1 , Z (b) 111 , Z C3 } and at vertex Z Y 3 by setting m(Z Y 3 ) = Z

Figure 18 :
Figure 18: The reduced quiver in two equivalent shapes.On the left, the diamond obtained erasing the vertices Z Y 3 and Z C3 directly in Figure 17.On the right, a rearranged star allowing for a better visualization of the mutation's action in Figure 19.

Figure 19 :
Figure 19: Reduced quiver, before and after quantum cluster mutation.

Figure 20 :
Figure 20: The quiver counterpart of the isomorphism ι.Highlighted are the 3-cycles identified by the map, while we color-coded each isolated vertex on the right with the corresponding pair of vertices on the left: e.g., Z B1 ∝ Z ′ (t) 111 Z ′ (b)−1 111 .

Figure 21 :
Figure 21: On the left, the n = 3 fat graph with the cyan loop represented by C (the red segments are identified).On the right, the only Fock-Goncharov variables involved by this loop.The shaded closed ribbons highlight the way the central monomials of the two seizures in Theorem 5.5 are formed as cycles, with the amalgamations bridging the gaps between the quivers of the two triangles.E.g., the monomialm(Z C3 ) = Z C1 Z C3 = (Z (t)021 Z (b) 201 )(Z (b) 210 Z (t)120 ) corresponds to the outer ribbon while the inner one, passing via the centers of the triangles, triggers the appearance of both Z(t)  111 and Z(b)  111 in m(Z Y 3 ).