Noncommutative Cluster Varieties and Moduli Spaces of Local Systems

Anna Wienhard; Dani Kaufman; Merik Niemeyer; Zachary Greenberg

REVIEW 3 major objections 4 minor 73 references

Noncommutative Cluster Varieties and Moduli Spaces of Local Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-28 · deepseek-v4-flash

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Pith's one-line read This paper introduces Jordan split groups and constructs a noncommutative cluster variety $\mathcal{A}_{R,S}$ whose Jordan points parameterize decorated local systems on any marked surface; for positive structures these points are exactly…

desk verdict An ambitious and largely credible construction of noncommutative cluster varieties whose main input—the existence of the Lie algebras T(R,{J_i})—is asserted, not proved, making Theorem 1.10 conditional. read the letter →

arxiv 2608.27284 v1 pith:LCYDK7UC submitted 2026-08-27 math.RT math.AGmath.CO

Zachary Greenberg , Dani Kaufman , Merik Niemeyer , Anna Wienhard This is my paper · ORCID

classification math.RTmath.AGmath.CO MSC 13F6017B7017C3020G1522E46

keywords noncommutativeclustervarietiesJordansplitgroupsquadraticalgebrasrootsystemgradingdecoratedlocalsystemshigherTeichmüllerspacespositiverepresentationsdoubleBruhatcells

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs, for every reduced root system $R$ and every marked surface $S$, a noncommutative cluster variety $\mathcal{A}_{R,S}$ whose Jordan-algebra points parameterize decorated $G$-local systems on $S$, where $G$ is a newly introduced class of algebraic groups called Jordan split groups of type $R$. This one construction simultaneously extends the cluster varieties used for split groups, the noncommutative cluster algebras built from Jordan algebras, and the polygonal cluster algebras of the authors' earlier work. In the disk case the same points parameterize configurations of decorated flags, and this yields noncommutative cluster structures on the double $R$-Bruhat cells of $G$. When the input Jordan algebras are formally real, the positive Jordan points are shown to be independent of all auxiliary choices and to coincide with the space of decorated positive representations of $\pi_1(S)$. A sympathetic reader would take the paper's aim to be a unified explicit coordinate system for higher Teichmüller spaces and for positivity in a broad new family of real algebraic groups.

What carries the argument

The load-bearing object is the Jordan pinning of a Jordan split group of type $R$: for each simple root one keeps homomorphisms $SL_2(J_i) \to G$ and coroot maps $\check\beta_i:\Gamma(J_i)\to G$, so the structure groups of the quadratic Jordan algebras $J_i$ replace the multiplicative torus. Each $J_i$ is turned into the $A_1$-graded Lie algebra $\mathfrak{sl}_2(J_i)$ by the Tits-Kantor-Koecher construction, and the recognition theorems of Benkart-Moody and Benkart-Zelmanov (as cited in the paper) classify the compatible families $\{J_i\}$ that can be glued into $T(R,\{J_i\})$. On the cluster side, the paper defines seed groups—free groups generated by variables attached to quivers or networks, modulo 'angle' relations—and glues them by noncommutative mutation. The mutation of Jordan points is defined by an angle summation formula that uses the quadratic map $\iota(v)$ to translate sums of vectors in $V$ into products in $\Gamma(J)$; in higher noncommutative rank the same role is played by grounded wiring networks. These two mechanisms together carry the whole argument.

What would settle it

Take the smallest higher-noncommutative-rank case not previously covered, for example a $C_3$-graded group with $J_1=J_2=M_k(\mathbb{C})$ and $J_3=H_k(\mathbb{C})$, pick two triangulations of a punctured surface, compute the same decorated local system's coordinates in both seeds, and check that the coordinate change equals the predicted noncommutative cluster mutation. A mismatch, or a collision of two distinct local systems in the Jordan-point map, would refute Theorem 1.10. A second check is positivity: if a formally real Jordan algebra yields a point that is positive in one seed but not in another, the claimed choice-independence of $\mathcal{A}_{R,S}^{>0}$ fails.

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Extended reading notes

Core claim

The central claim is Theorem 1.10: for a Jordan split group $G$ of type $R$ with associated parabolic $P_\Theta$, the Jordan points of the cluster variety $\mathcal{A}_{R,S}$ parametrize an open and dense set of the space of decorated local systems on $S$, and when $G$ carries a positive structure the set of positive Jordan points $\mathcal{A}_{R,S}^{>0}$ is independent of all choices and corresponds exactly to decorated positive representations of $\pi_1(S)$. Behind the theorem lies a structural discovery about algebraic groups: a semisimple group can be assembled from copies of $SL_2(J)$ for quadratic Jordan algebras $J$, with the structure groups $\Gamma(J)$ playing the role of the torus of a split group. This 'Jordan split' viewpoint lets the authors classify which parabolic subgroups induce a root-system grading (Jordan compatibility), build a root-system calculus of commutation and Weyl-group relations, and then repeat the Fock-Goncharov amalgamation strategy with the torus replaced by Jordan structure groups. The result is a noncommutative cluster variety whose mutations add vectors inside the Jordan algebra while multiplying structure-group elements, and whose positive points are defined by positive cones in the Jordan algebras.

Load-bearing premise

The construction assumes that for every reduced root system $R$ and every compatible family of Jordan algebras $\{J_i\}$, there really is a Lie algebra $T(R,\{J_i\})$ with the prescribed root-system grading and prescribed Jordan factors—the paper invokes the Benkart-Moody and Benkart-Zelmanov recognition theorems to glue the Jordan algebras together, and if those theorems fail for some field or some compatible family, the parametrization of $G$-local systems would lose its main input.

Editorial extensions

If this is right

For any marked surface and any Jordan split group of type $R$, the cluster variety $\mathcal{A}_{R,S}$ gives explicit coordinates on decorated local systems, so holonomies along curves can be written directly in cluster coordinates.
For groups with positive structure, the positive Jordan points $\mathcal{A}_{R,S}^{>0}$ are choice-independent and coincide with decorated positive representations; each cluster seed yields concrete positivity tests expressed as generalized minors.
On a disk, the construction gives noncommutative cluster structures on double $R$-Bruhat cells $P_\Theta u P_\Theta \cap P_\Theta^{\mathrm{opp}} v P_\Theta^{\mathrm{opp}}$, generalizing the Berenstein-Fomin-Zelevinsky cluster algebras on double Bruhat cells.
The cluster modular group $\Gamma_{R,S}$—containing the mapping class group, outer automorphisms, Weyl groups at punctures, and braid groups at boundary components—acts on $\mathcal{A}_{R,S}$ by noncommutative quasi-cluster automorphisms, giving an explicit realization of mapping class group action.
For type $A_p$, $\mathcal{A}_{R,S}$ is birationally equivalent to the moduli space of untwisted rank 1 local systems on the spectral surface, and for $C_p$ there is a symmetric spectral description; this links the new coordinates to spectral-network methods.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the main theorem is correct, the same Jordan-algebra input should produce a noncommutative $X$-cluster variety whose positive points parametrize framed rather than decorated local systems; the paper develops only the $A$-side, so this is a natural extension rather than a claim of the paper.
The dependence on the Jordan split presentation predicts explicit coordinate-change maps between the different cluster structures on the same group arising from different $R$-gradings, for example $SL_6$ as $A_5$, $A_3$, or $A_2$; computing these changes in small examples would test how canonical the new coordinates really are.
Because the positive-structure argument uses only formally real Jordan algebras and their positive cones, the same construction might define higher Teichmüller spaces over other ordered fields, or in $p$-adic settings, by replacing the Euclidean Jordan algebras with suitable positive cones.
The cluster mutation formulas give an algorithm for composing Lusztig parameters under group multiplication; making this algorithm explicit for rank-one $B_p$ and $G_2$ groups would turn the paper's structural result into a computational tool for positive representations.

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Formalized claims in Lean

Claim #1: The central claim is Theorem 1.10: for a Jordan split group $G$ of type $R$ with associated parabolic $P_\Theta$, the Jordan points of the cluster variety $\mathcal{A}_{R,S}$ parametrize an open and dense set of the space of decorated local systems on $S$, and when $G$ carries a positive structure the set of positive Jordan points $\mathcal{A}_{R,S}^{>0}$ is independent of all choices and correspo

/-- @claim 1 The central claim is Theorem 1.10: for a Jordan split group $G$ of type $R$ with associated parabolic $P_\Theta$, the Jordan points of the cluster variety $\mathcal{A}_{R,S}$ parametrize an open and dense set of the space of decorated local systems on $S$, and when $G$ carries a positive structure the set of positive Jordan points $\mathcal{A}_{R,S}^{>0}$ is independent of all choices and correspo -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The headline is that this is a serious, ambitious paper, and the most novel part—the Jordan split framework and the cluster coordinate machine—is largely credible. But the stress-test note lands: Definition 3.37 introduces T(R,{J_i}) by fiat, and that is a genuine gap that makes the main moduli-space theorem conditional.

What is new and good: A_{R,S} genuinely generalizes several cluster varieties (Fock–Goncharov, Goncharov–Shen, Goncharov–Kontsevich, Berenstein–Fomin–Zelevinsky), with the Cp and F4 cases described as new. The Jordan split groups and Jordan pinnings give a clean organizing framework for attaching Jordan algebras to root systems. The classification of Jordan compatible subsets (Theorem 3.28), the Weyl group injections in Section 3.9, and the root system calculus in Section 4 are worked out in detail, with concrete pinnings (SL6, Sp12, Spin(3,4)) that make the claims checkable. These parts look solid. There is no circularity or fitted empirical input, and the self-citations are used appropriately, as building blocks rather than as a substitute for argument.

The weak spot is exactly where the stress-test note points. The recognition theorems quoted from Benkart–Moody and Benkart–Zelmanov classify existing R-graded Lie algebras; they do not prove that every compatible family in Theorem 3.35 actually occurs. Section 3.8 says the Jordan algebras “can be glued” to recover the Tits index, but no bracket construction, universal property, or existence theorem is supplied. Uniqueness up to central extension is also asserted rather than proved. Since Theorem 1.10 parameterizes G-local systems for G whose Lie algebra is T(R,{J_i}), the parameterization has no object if existence fails for some compatible family. The field hypothesis is also sloppy: the paper assumes characteristic not 2 or 3 throughout, while the quoted classifications may require characteristic 0. These are fixable, but they are load-bearing.

Part 3 is harder to verify from the text: it is a long construction that leans on [GKNW26] and prior work, and the F4/exceptional cases are explicitly not fully explored. That is not a fatal flaw for a paper of this scope, but it means Part 3 reads more like a claim-check than a proof-read from the available text.

Net: this paper deserves a serious referee, not a desk reject. The right recommendation is major revision: add a proof or a precise reference for existence and uniqueness of T(R,{J_i}), state the field hypotheses of the classification theorems cleanly, and either prove the F4 cases or mark them conjectural. If the existence gap is closed, this is a substantial contribution. I would bring it to a reading group working on cluster algebras or higher Teichmüller theory, and I would cite it for the root system calculus and the Jordan pinning framework.

Best,

Referee Report

3 major / 4 minor

Summary. The paper constructs noncommutative cluster varieties A_{R,S} for each reduced root system R and marked surface S, with Jordan-split algebraic groups as the main input. Part 1 develops the structure theory of Lie algebras graded by root systems, classifies which parabolic subgroups induce such gradings via the new Jordan-compatibility condition, and introduces Jordan split groups with Jordan pinnings, Jordan weights, a root system calculus, and R-Bruhat decompositions. Part 2 reviews and extends noncommutative cluster formalism in two forms: polygonal cluster algebras for lower noncommutative rank and grounded wiring networks for higher noncommutative rank. Part 3 constructs cluster-like coordinates on configuration spaces of decorated flags, and Part 4 applies these to Gauss decompositions, double R-Bruhat cells, positivity, and decorated local systems. The central claim, Theorem 1.10, is that Jordan points of A_{R,S} parametrize an open dense set of decorated G-local systems, and that for positively structured groups the positive Jordan points correspond exactly to decorated positive representations of the surface group.

Significance. If the main construction is sound, this paper provides a substantial unifying framework: it simultaneously generalizes the cluster varieties of Fock-Goncharov, Goncharov-Shen, Berenstein-Retakh, and Goncharov-Kontsevich, and it gives a cluster parametrization for all groups admitting a Guichard-Wienhard positive structure. The paper contains genuine and largely self-contained contributions in Part 1: the Jordan-compatibility classification of parabolic-induced root gradings, the detailed Serre-relation checks in the root system calculus, the explicit pinnings of Spin(3,4) and Sp_4, and the concrete classification tables for real Lie algebras with A1-gradings are valuable and appear carefully worked out. The proposed framework also yields explicit positivity tests and a uniform account of the mapping class group action via noncommutative quasi-cluster automorphisms. However, the existence of the underlying Lie algebra T(R,{J_i}) for every compatible family is asserted rather than proved, and this gap directly feeds the main parametrization theorem. The paper's significance would be high if this gap is filled; in its current form the central existence input is not yet established.

major comments (3)

[§3.8, Definition 3.37] Definition 3.37 defines T(R,{J_i}) as 'the (adjoint form of) Lie algebra with R-grading and collection {J_i}' and Section 3.8 asserts that the Jordan algebras 'can be glued' to obtain the Tits index, but no bracket construction, universal property, or converse existence theorem is supplied. The recognition theorems in Section 3.3 (Theorems 3.11–3.18) have the direction: if an R-graded Lie algebra exists, then it is a central extension of a model built from Jordan data; they do not state that every compatible tuple in Theorem 3.35 arises from an actual Lie algebra. Since Theorem 1.10 parameterizes local systems for 'G of type R' whose Lie algebra is defined to be T(R,{J_i}), a compatible family for which existence fails would leave the main object of the parameterization undefined. Please provide a construction of T(R,{J_i}) with the promised R-grading and prescribed Jordan factors, or cite and state precisely a converse existence theorem from Benkart-Moody or Benkart-Zelmanov, and also prove the asserted uniqueness up to central extension.
[§3.3, Theorems 3.11–3.18; Theorem 1.5] The paper's blanket hypothesis is that K has characteristic not 2 or 3 (Theorem 1.5), but no field hypotheses are stated for the quoted recognition theorems. If those classifications require characteristic 0, algebraically closed fields, or additional finite-dimensionality assumptions, then the classification of compatible families in Theorem 3.35 and the apparent generality of Theorem 1.10 over arbitrary such fields are not justified. Please state the precise hypotheses under which Theorems 3.11–3.18 apply, and either prove directly that the char-not-2-or-3 assumption suffices or restrict the main statements to the fields for which the quoted classification is valid.
[§8.2, Step 2 and Theorem 12.2] The amalgamation step, which reduces configurations of n flags to configurations of triples, is described by analogy with the split case and is said to be extended in Theorem 12.2, but the stated dependence of that theorem on the new Jordan-split setting is not proved in the material provided before the statement. Since Theorem 1.10 builds the cluster variety A_{R,S} from these configuration spaces of decorated flags, this amalgamation statement is load-bearing for the surface version of the result. Please give a proof of Theorem 12.2 in the Jordan-split generality, or at minimum state explicitly which of its ingredients are proved in the present paper and which are imported from [GS19] and [GKNW26].

minor comments (4)

[Definition 4.42] In the displayed formula for the prefix R-Lusztig map, the last factor is written as x_{i_k}(x_k); it should presumably be x_{i_k}(v_k).
[Lemma 4.35] In the statement of Lemma 4.35 and its proof, the expression 'a1 j' is not defined; the authors should introduce this notation explicitly or replace it with the precise scalar depending on the connection between nodes i and j.
[Example 3.24 and Figure 3] The graphical root notation that lists coefficients over the nodes of a Dynkin diagram is used before it is explained; a one-sentence explanation near Example 3.24 would make this notation immediately readable.
[Section 6.2, Remark 6.7] Remark 6.7 asserts that the coloring rules determine a unique coloring and refers to [GKNW26]; since this uniqueness is used to construct the polygonal quivers, a short proof or a precise quotation of the relevant statement in [GKNW26] would improve self-containedness.

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity: the cluster-coordinate and local-system parametrizations are derived from pinnings and mutation formulas rather than from their conclusions; the main caveat is the stipulated existence of T(R,{Ji}), which is a proof gap rather than a circular reduction.

full rationale

The paper's central chain is self-contained in the relevant sense: A_{R,S} is built from seed groups, angle paths/sums, and wiring-network mutations, and the Jordan points are maps into structure groups of the fixed Jordan algebras. Theorem 1.10 asserts a parametrization of decorated G-local systems by these points, which is a substantive birational statement; the coordinates are not fitted to the local-system moduli, and no fitted parameter is renamed a prediction. The recognition theorems of Benkart-Moody and Benkart-Zelmanov are external classifications, not self-citations, and the paper's own Serre-relation checks carry much of the grading-classification burden. The authors do cite their own prior polygonal cluster algebra framework [GKNW26] and positivity work [GW25, GW26a], but those are used as building blocks or as definitions to be matched, not as the sole justification of the main parametrization theorem. The one genuine concern is Definition 3.37: T(R,{Ji}) is introduced as 'the (adjoint form of) Lie algebra with R-grading and collection {Ji}' without an explicit bracket construction or converse existence theorem; since the quoted recognition theorems classify already-existing R-graded algebras, existence of T(R,{Ji}) for every compatible family is not demonstrated. That is a load-bearing gap in the hypotheses of Theorem 1.10, but it is an unproved input rather than a circular identification of the conclusion with the input. Hence no circular step is exhibited and the circularity score stays low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No parameters are fitted to empirical data; all parameters such as the degree r, capacities, and noncommutative rank are structural and fixed by the root system and Jordan algebra data. The main assumptions are the standard Jordan algebra theory and the external classification of root-graded Lie algebras, plus the paper's own existence claim for T(R,{J_i}).

assumptions (6)

standard math Quadratic Jordan algebra identities and the Tits-Kantor-Koecher construction produce a Lie algebra sl2(J).
Used throughout Part 1; Section 2 defines quadratic Jordan algebras and Section 2.4 defines sl2(J). The paper builds on this standard construction.
standard math Shirshov-Cohn theorem: every Jordan algebra generated by two elements is special.
Invoked in Proposition 3.6 and Proposition 4.12 to reduce computations to an associative matrix algebra.
domain assumption Every finite-dimensional Jordan algebra admits a norm map of degree r, and the structure group Gamma(J) has the stated covering and involution.
Proposition 2.17 and Section 4.1; needed for Jordan pinning, Jordan weights, and positivity.
domain assumption The Benkart-Moody and Benkart-Zelmanov recognition theorems classify R-graded Lie algebras as central extensions of the prescribed T(R,{J_i}) constructions.
Section 3.3, Theorems 3.11 to 3.18; used to justify that every R-graded Lie algebra arises from the listed Jordan data.
ad hoc to paper For any Jordan-compatible family {J_i}, a Lie algebra T(R,{J_i}) and an algebraic group with the promised R-grading exist.
Definition 3.37 and Section 3.8 assume the glued Lie algebra exists; the paper gives examples and cites recognition theorems but does not explicitly construct the bracket identities.
domain assumption Euclidean formally real Jordan algebras provide positive cones matching the Guichard-Wienhard positive structure.
Proposition 2.23 and Section 16; used to identify positive semigroups with previously studied positive structures.

invented entities (2)

Jordan split group of type R independent evidence
purpose: Central algebraic object whose Lie algebra is T(R,{J_i}); the paper develops root system calculus, pinnings, and Bruhat-style decompositions for these groups.
The paper shows many known groups such as SL6(R), Sp6(R), and Spin(3,8) are Jordan split, providing concrete external checkpoints beyond the definitions.
Noncommutative cluster variety A_{R,S} independent evidence
purpose: Gives noncommutative cluster coordinates on flag configurations, double R-Bruhat cells, and moduli spaces of decorated local systems.
Abelianization recovers the Fock-Goncharov and Goncharov-Shen A-cluster varieties, and the Ap case recovers Goncharov-Kontsevich networks with minor modification, giving external benchmarks.

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Pith. "Pith review of Noncommutative Cluster Varieties and Moduli Spaces of Local Systems." pith.science (2026). https://pith.science/paper/LCYDK7UC

@misc{pith2026260827284,
  author       = {Pith},
  title        = {Pith review of: Noncommutative Cluster Varieties and Moduli Spaces of Local Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LCYDK7UC}},
  note         = {Machine review of arXiv:2608.27284}
}

abstract

In this article, we construct noncommutative cluster varieties, $\mathcal{A}_{R,S}$, for each reduced root system $R$ and marked surface $S$ simultaneously generalizing the cluster varieties of Fock-Goncharov, Li, Goncharov-Shen, Berenstein-Retakh, Goncharov-Kontsevich, and our previously introduced polygonal cluster algebras. Additionally, we define a large class of algebraic groups, we call Jordan split groups. Given a reduced root system $R$ and a family of Jordan algebras, the Lie algebra for $G$ is constructed by unifying the Tits-Kantor-Koecher construction for a single Jordan algebra with the construction of a split Lie algebra. The notion of Jordan split groups is closely related to a grading of its Lie algebra by the root system $R$. We show that these gradings are usually induced by a choice of standard parabolic subalgebra $\mathfrak{p}_\Theta$ and we classify $R$-graded pairs $(G,\Theta)$ via a condition depending only on the subset $\Theta\subset \Delta$ of the set of simple roots. Next, we define Jordan algebra points of $\mathcal{A}_{R,S}$ which parameterize $G$-local systems on $S$ with boundary decoration related to cosets $G/U_\Theta$ when $G$ is Jordan split of type $R$. When $S$ is a disk, points of $\mathcal{A}_{R,S}$ parameterize configurations of decorated flags. We use this to give noncommutative cluster structures on the double $R$-Bruhat cells of $G$, generalizing the cluster algebras of Berenstein-Fomin-Zelevinsky. When each Jordan algebra is formally real, we say that $G$ has a positive structure with respect to $\Theta$. This defines a positive semigroup in $G$. For real algebraic groups, the pairs $(G,\Theta)$ which have positive structures are exactly those which admit a positive structure as defined by Guichard-Wienhard and we give algebraic proofs of many of the properties of positive configurations of flags and of positive representations.

Figures

Figures reproduced from arXiv: 2608.27284 by the authors.

**Figure 1.** Relating parabolic subgroups and configurations of 3 decorated flags; using cluster amalgamation to relate an open dense subset of G to configurations of 4 decorated flags. Using cluster amalgamation, we obtain charts on an open dense subset G0 of G as indicated in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** Product on a group using cluster amalgamation [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** by the circled node. A2n−1 : n − 1 n − 1 Bn : Cn : Dn : D∗ 2n : E7 [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

Figures from the paper (58 more)

**Figure 4.** Figure 4: Affine Dynkin diagrams and labeling as in Lemma 3.21. The Dynkin diagrams and labelings are shown in [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

**Figure 5.** Figure 5: Jordan compatible choices of Θ [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]

**Figure 6.** Figure 6: Quiver with weighted nodes. We would draw the same quiver if node 3 had weight 3 instead. In this case the adjacency matrix would be   0 1 0 −1 0 3 0 −1 0   . It will always be clear from the context which case we are dealing with. Definition 5.3. A seed, (Q, a) co…

**Figure 7.** Figure 7: b. a b c d e (a) a b c d f (b) [PITH_FULL_IMAGE:figures/full_fig_p049_7.png]

**Figure 8.** Figure 8: Mutation and zigzag paths. By writing down the cluster variables we encounter along these two paths we get the expression (6) aec + bed = fe2 where f is the new cluster variable obtained by mutation and the equality simply follows by multiplying the exchange relation b…

**Figure 9.** Figure 9: 2-colored quivers and their mutation. Adding color to the quivers is not sufficient to capture the combinatorics of the coordinates which we will discuss in Part 3. For this, we will also need to take the order into account since some of the functions will take values …

**Figure 10.** Figure 10: Decoration of big nodes in an ordered quiver with partition of the arrows. Convention. We draw the noncommutative nodes of an ordered quiver as shown in [PITH_FULL_IMAGE:figures/full_fig_p052_10.png]

**Figure 11.** Figure 11: Examples of decorated polygons. Given a decorated polygon we construct an ordered quiver as follows: (1) For every side, we get a noncommutative node such that the filled half is on the left as given by the orientation. (2) The halves of the nodes facing into the poly…

**Figure 12.** Figure 12: Ordered quivers associated to the decorated polygons in [PITH_FULL_IMAGE:figures/full_fig_p053_12.png]

**Figure 13.** Figure 13: A flip in a decorated tiling. Remark 6.12. The second step is not always possible: There are polygonal quivers for which a mutation of the underlying 2-colored quiver does not lead to a quiver which can be obtained from the prescribed tiling. We do not consider mutati…

**Figure 14.** Figure 14: A polygonal ordered quiver and its mutation, decorated tilings corresponding to the pruned quivers. 1 [PITH_FULL_IMAGE:figures/full_fig_p055_14.png]

**Figure 15.** Figure 15: A quiver for which mutation at node 1 is not admissible and the decorated polygon corresponding to the pruned quiver [PITH_FULL_IMAGE:figures/full_fig_p055_15.png]

**Figure 16.** Figure 16: A closed angle path. To every angle path we associate an expression in the variable from above: Definition 6.16. Let γ be an angle path in a tiling P. The label L(γ) ∈ FQ of γ is computed recursively according to the following rules: • The label of a path following an…

**Figure 17.** Figure 17: Angles used for mutation of J-points. Remark 6.26. Note that in the mutation map, the only elements which are added are in the vector space V (which gets mapped to Γ(J) via ι), and the only elements which are multiplied belong to Γ(J). As such J does not need to be co…

**Figure 18.** Figure 18: Reading off exchange relations from decorated tilings. As before, let the node at which we mutate be k. From the figure a first guess for a mutation formula would be (8) X′ kNk = Ic · στ ε1 [PITH_FULL_IMAGE:figures/full_fig_p059_18.png]

**Figure 19.** Figure 19: Local equivalence moves of plabic networks. We will be interested in control over the boundary edges which cause the network to be nonplanar. Definition 7.2. A triangular grounded wiring network on n strands is a bipartite network embedded in a triangle with a designa…

**Figure 20.** Figure 20: The elementary network J(i). Definition 7.3. Each grounded wiring network P has an associated seed group QP obtained by taking the quotient of the free group generated by the edges of the network by the relation that the counterclockwise product around each vertex of …

**Figure 21.** Figure 21: A leg slide. b1 = (1 + a3a4a1a2)a −1 2 a −1 1 a −1 4 = a −1 2 a −1 1 a −1 4 (1 + a4a1a2a3) b3 = (1 + a1a2a3a4)a −1 4 a −1 3 a −1 2 = a −1 4 a −1 3 a −1 2 (1 + a2a3a4a1) a1 = b3(1 + b4b1b2b3) −1 = (1 + b3b4b1b2) −1 b3 a3 = b1(1 + b2b3b4b1) −1 = (1 + b1b2b3b4) −1 b1 b2 …

**Figure 22.** Figure 22: Square move of a plabic network. are preserved, which ensures the result of a square move is a correctly weighted network. Definition 7.8. The honeycomb networks, Pn, are a family of networks embedded in a triangle following the pattern demonstrated in [PITH_FULL_IMA…

**Figure 23.** Figure 23: The standard GLn-network. In [PITH_FULL_IMAGE:figures/full_fig_p063_23.png]

**Figure 24.** Figure 24: Overview of the general strategy. Our main goal now is to understand how the choices we made effect these coordinates: (1) the choice of triangulation, (2) the choice of edge within each triangle along which to compute an interpolating sequence, (3) the choice of redu…

**Figure 25.** Figure 25: Elementary and opposite elementary configurations for ςi . The space of elementary configurations admits a natural parametrization. Lemma 9.6. Let (A, Al , Ar) ∈ A 3 Θ represent an element of Conf i 3 (AΘ). Then there exist unique g ∈ G, ll , lr ∈ LΘ and a ∈ V × i suc…

**Figure 26.** Figure 26: Left reflection map for elementary configurations. Proposition 9.16. The standard form for the left reflection is r(UΘ, w0UΘll , xi ∗ (a ∗ )w0UΘlr) = (UΘ(l l ) ∗ , yi ∗ (b ∗ )UΘ(l r ) ∗ , w0UΘ) where b = βi(l −1 r )(a −1 ), l l = ςi(l −1 b ll l −1 b ) and l r = lr [P…

**Figure 27.** Figure 27: The elementary quivers E◦(1), E(i), E◦(p − 1), E◦(p), E ¯i . The cluster coordinates are given as follows: (11) ail = Λi(A, Al) air = Λi(A, Ar) aib = Λi(Ar, Al) aj = Λj (A, Al) = Λj (A, Ar) for j ̸= i For i < p, the only noncommutative node can be weaved or switched…

**Figure 28.** Figure 28: Elementary ordered seeds and pruned polygonal tiles with angles. the pruned quiver. These decorations are chosen so that the potential is recovered (up to commutative multiplication) by the angle based near the top vertex. Lemma 10.4. The potential δp(A, Al , Ar) is c…

**Figure 29.** Figure 29: The weighted elementary networks E(i) and E◦(i). Thus, we can use the partial potential to label the networks with a free collection of elements in GL1(R), L1, · · · , Lp+1, Bi , Bi+1, D as in [PITH_FULL_IMAGE:figures/full_fig_p076_29.png]

**Figure 30.** Figure 30: The elementary networks E(i) weighted with the potential. We will also use these networks to describe elementary configurations for groups G with a Cp-grading. In this case, R has an anti-involution ⋆ and G ≃ Sp2p (R, ⋆) by the recognition theorem (Theorem 3.15). Then…

**Figure 31.** Figure 31: The weighted elementary networks Θ type C2. bridges. By construction this path always has ◦ vertices on the right, so all the entries of the L-distance will be the inverse of the weight in the network. 11. Triples of Θ-flags The results of the previous section imply t…

**Figure 32.** Figure 32: Three Move: The quivers E(i, i + 1, i) and E(i + 1, i, i + 1). the additivity of the potential at height i. Essentially, this boils down to the equation [Lus94] xi(a)xi+1(b)xi(c) = xi+1 bc a + c xi(a + c)xi+1 ab a + c . □ We leave the remaining calculations to…

**Figure 33.** Figure 33: Changing orientation of edges in type B2. Example 11.22. We now consider a similar example when G is A2-graded. In this group, the opposition involution acts by reversing the order of components and applying the inverse: l = [A, B, C] 7→ [C −1 , B−1 , A−1 ]). This act…

**Figure 34.** Figure 34: The fan triangulation on Conf× 7 (AΘ). Theorem 12.2. The noncommutative cluster variety AR,n contains seeds for all cluster charts. Moreover the {Ji} points of any cluster chart parametrize an open dense subset of Conf× n (AΘ). Proof. Very nice charts are given by the…

**Figure 35.** Figure 35: Setup for the twisted two move: Configuration of four flags and associated decorated polygon or grounded network, before and after mutation. To finish the proof we apply the appropriate Jordan weights to obtain the mutation formula in the chart E(i¯i). In noncommutati…

**Figure 36.** Figure 36: Setup for realizing flips as cluster mutations. Then the flip is realized by transforming the word r1r2 to r2r1 (where r2 is written in the generators si ). By Lemma 12.4 this is realized by a sequence of cluster mutations, see [PITH_FULL_IMAGE:figures/full_fig_p086_…

**Figure 37.** Figure 37: Realizing a flip by (twisted) two moves for p = 2. We extend the twisted cyclic shift map to n-tuples as follows: Definition 12.5. The twisted cyclic shift map tw : Conf× n (A ) → Conf× n (A ) for n ≥ 3 is given by tw : (A1, A2, A3, . . . , An) 7→ (sGA2, A3, . . . , A…

**Figure 38.** Figure 38: The quiver Qp which is inscribed into the triangle. · · · (a) p even · · · (b) p odd [PITH_FULL_IMAGE:figures/full_fig_p095_38.png]

**Figure 39.** Figure 39: Decorated polygons inscribed into triangles. There is one subtlety in the amalgamation which we must discuss before studying the fourmove or twisted cyclic shift. In [PITH_FULL_IMAGE:figures/full_fig_p095_39.png]

**Figure 40.** Figure 40: Quiver and functions for p = 2. Let u = FD1 (x, v, y, w). Now consider a configuration of flags (A1, A2, A3) = (UΘ, w0UΘ, uw0UΘ) as in Example 13.18. Notice that we set l(A1, A3) = l(A1, A2) = 1 for simplicity. One can check by Proposition 8.8 that these L-distances f…

**Figure 41.** Figure 41: Four move: The ordered quivers [PITH_FULL_IMAGE:figures/full_fig_p098_41.png]

**Figure 42.** Figure 42: Cluster charts in the case p = 2 [PITH_FULL_IMAGE:figures/full_fig_p099_42.png]

**Figure 43.** Figure 43: The seed QD1,1 for type G2. We now consider the mutation sequence, read left to right, S = µ3µ1µ2µ3. Lemma 13.26. The mutation sequence S followed by a switch at 4 changes QD1,1 to QD2,2. Proof. This follows from a very similar calculation to Lemma 13.20. First we cal…

**Figure 44.** Figure 44: Mutation sequence sending QD1,1 to QD2,2 in type G2 The next two mutations commute and we find Y ′ =σ(Y ) −1σ(X′ ) + bcwτ (v)uY −1 = v + u + w , y ′ = N(bcN(v)u + bcuτ (v)u) + b 4 c 2N(u) 2N(v) 2 b 3c 2N(v) 2N(u) = cN(v + u) + bN(u). The final mutation at 2 produces X…

**Figure 45.** Figure 45: Mutation Sequence sending Q1 D1 to Q3 D2 in G2 [PITH_FULL_IMAGE:figures/full_fig_p105_45.png]

**Figure 46.** Figure 46: Networks for A2(Mk(R)) with different systems of Θ-weights. 14.1.2. Three move. Lemma 14.2. The networks given by two seeds which differ by a three move ςiςi+1ςi → ςi+1ςiςi+1 are related by leg slides and a single square move at the single internal face of E(j + 1, j,…

**Figure 47.** Figure 47: Left slid network for A2(Mk(R)). j − 1 1 M1M2 j 1 M1 M2 M−1 1 M3 j + 1 1 1 M3M2 M−1 2 M−1 3 M−1 2 M−1 1 M−1 1 M−1 2 M−1 3 M1 1 N2N3 1 1 N1 N2 N3N −1 1 N1N −1 3 1 1 N2N1 N −1 1 N −1 2 N −1 1 N −1 2 N −1 1 N1N −1 3 N −1 2 [PITH_FULL_IMAGE:figures/full_fig_p108_47.png]

**Figure 48.** Figure 48: The left slid networks E(j + 1, j, j + 1) and E(j, j + 1, j). We then have N1 = b2 = (1 + a4a1a2a3) −1 a4 = (1 + M−1 3 M1) −1M2 = (M3 + M1) −1M3M2 N2 = b1 = (1 + a3a4a1a2)a −1 2 a −1 1 a −1 4 = (1 + M1M−1 3 )M3 = (M3 + M1) N3 = N −1 2 b −1 4 N1 = (M3 + M1) −1 [(1 + M3…

**Figure 49.** Figure 49: A network for Θ type C3 [PITH_FULL_IMAGE:figures/full_fig_p109_49.png]

**Figure 50.** Figure 50: Realizing the four move with square moves. 14.2.3. Rotation and Cp networks. We already proved that rotation by mutation for groups of type Cp is possible by simply rotating the associated chart in for the group of type A2p−1 which contains it. However, we make here a…

**Figure 51.** Figure 51: Relating parabolic subgroups and configurations of 3 decorated flags; using cluster amalgamation to relate an open dense subset of G to configurations of 4 decorated flags. All of this will be made more precise in the subsequent sections but for now it suffices to kn…

**Figure 52.** Figure 52: A flip in the triangulation of the square can be used to compute the opposite Gauss decomposition. Similarly, for two elements g1, g2 ∈ C w0,eC e,w0 , we can compute the Gauss decomposition of g1g2 whenever it exists: Use cluster amalgamation again and consider a conf…

**Figure 53.** Figure 53: By retriangulating this, we may obtain a 4-gon corresponding to g1g2, which allows us to compute the Gauss decomposition of g1g2 [PITH_FULL_IMAGE:figures/full_fig_p113_53.png]

**Figure 54.** Figure 54: Using coordinates to understand that U >0 Θ is a semigroup. Using Section 12.1 we can realize the flip in the triangulation by mutations, meaning that in particular the triple (A1, A2, A4) provides a positive point. Thus u1u2 ∈ U >0 Θ . □ This proof hints already at a…

**Figure 55.** Figure 55: These lift to paths in T ′S using their unit tangent vectors. Let Pi denote parallel transport along the lift of γi to T ′S. In this way we obtain n elements in LA |v1 : (a1, Pn · · · P2a2, Pn · · · P3a3, . . . , Pnan) . D x1 x2 x3 x4 . . . xn C1 C2 C3 Cn y1 y2 y3 yn …

**Figure 56.** Figure 56: Default orientation of sides for cluster chart on Conf3(AΘ). Since the orientation of each edge was fixed the values of the cluster coordinates agree along each edge shared by two triangles after applying this construction to all triangles. Consequently, the charts am…

**Figure 57.** Figure 57: Objects and morphisms in the category GT associated to a triangle. The idea of the proof is now to identify decorated twisted local systems on Sb with certain functors GT → G. We first construct the functor for a decorated twisted G-local system L. Focusing on an edge…

**Figure 58.** Figure 58: Local system on Db4 associated to an element g ∈ G. Example 17.14. Consider the space AG,Db4 . It is (non-canonically) isomorphic to Conf4(AΘ) as described in Example 17.9. Thus we have an embedding G → AG,Db4 , g 7→ (w0UΘ, gw0UΘ, gUΘ, UΘ) induced by the map IΘ : G → …

**Figure 59.** Figure 59: Types of loops in S. of the A -space to the X -space of framed G′ -local systems on S. For full details of the notion of a framed local system see [FG06]. We will only need the fact that a decoration of the twisted local system L determines a decoration of the local s…

**Figure 60.** Figure 60: Triangulation of a genus 2 surface with one boundary component and loop around the boundary. 17.5. LΘ-action at marked points and potentials. Given a decorated local system L we can alter its decoration at a particular marked point by rescaling the decorated flags the…

**Figure 61.** Figure 61: An untwisting of the honeycomb network P4 is given by multiplying all dashed blue edges by −1. (3) Let Σ be a marked surface and ∆ a triangulation of Σ. We get a plabic network on Σ by inscribing Pm into each triangle of ∆, which we denote by P∆,m. Then the space A¯ …

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Berenstein, Arkady and Zelevinsky, Andrei , Title =. Comment. Math. Helv. , ISSN =. 1997 , Language =. doi:10.1007/PL00000363 , Keywords =

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Olivier Guichard and Anna Wienhard , year=. Generalizing Lusztig's total positivity. 2606.10761 , archivePrefix=

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Generalizing Lusztig's total positivity

Olivier Guichard and Anna Wienhard , year=. Generalizing Lusztig's total positivity. ???? , archivePrefix=

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Arkady Berenstein and Sergey Fomin and Andrei Zelevinsky , title =. Duke Mathematical Journal , number =. 2005 , doi =

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