Cluster algebra lab

An arrow i → j is recorded by a positive entry bᵢⱼ. Quivers have no loops, oriented 2-cycles, or arrows between frozen vertices. General skew-symmetrizable matrices are shown as weighted diagrams.

X′ₖ = (∏ᵢ Xᵢ^[bᵢₖ]₊ + ∏ᵢ Xᵢ^{[−bᵢₖ]₊}) / Xₖ,　ŷⱼ = ∏ᵢ Xᵢ^bᵢⱼ.

Lowercase xᵢ denotes an initial variable; Xᵢ denotes a variable in the current seed. There is one ŷ for each mutable vertex, and its product includes frozen rows. Following Theorem 3.5.1 in the notes, Y-mutation results are recomputed exactly from the new cluster and matrix. Numerical substitution is an additional check, not a proof of a general identity.

Finite type is determined from the principal part, excluding frozen rows. Reaching a positive-definite Cartan counterpart establishes finite type; finding |bᵢⱼbⱼᵢ| > 3 after some mutation establishes infinite type. If a bounded search finds neither, the result is undetermined. A finite quiver mutation class is different from finite cluster type.

Fomin–Williams–Zelevinsky, Introduction to Cluster Algebras: Chapters 1–3, Definitions 2.1.2, 2.7.6, 3.1.3, Theorem 3.5.1; Chapters 4–5, Theorems 5.2.8, 5.10.1.

Calculations use exact integer coefficients. If an expression becomes too large, the mutation is stopped and the previous seed is preserved. Up to 24 vertices are supported. Save and Open file use the seed JSON format of this page. The page works offline.

Polygon vertices are labeled 1, …, N clockwise. The N−3 diagonals correspond to mutable vertices and the N sides to frozen vertices. The initial fan triangulation has all diagonals incident to the selected center vertex. Polygons with 3–12 vertices are supported.

Follow the three sides of each triangle clockwise to add arrows between the corresponding quiver vertices, omitting arrows between frozen vertices. Flipping diagonal k applies μₖ at its slot. At every step, B̃ constructed directly from the new triangulation is compared with the mutated B̃.

P_a,cP_b,d = P_a,bP_c,d + P_a,dP_b,c　(a < b < c < d).

Pᵢⱼ is the Plücker coordinate of a chord. Expressions below are exact Laurent expressions in the variables of the initial triangulation. Use Pᵢⱼ / xᵢ to switch the notation for initial variables. The product ŷⱼ = ∏ᵢ Xᵢᵇⁱʲ includes frozen variables. Polygon sides and their variables do not change under flips.

The principal part of the fan triangulation of an N-gon is a path of type A_N−3. Thus all triangulations have the same finite cluster type A_N−3. A triangle has no mutable vertices. Experiments in different tabs are independent. Saved triangulation files contain the number of vertices, fan center, and flip sequence. Reaching a calculation limit leaves the current state unchanged.

Fomin–Williams–Zelevinsky, Introduction to Cluster Algebras, Chapters 1–3: §1.2, relation (1.2.1), Definition 2.2.1, Exercise 2.2.2, Theorem 3.5.1.

Any word on n strands is allowed. Generator i crosses the strands in positions i and i+1 from the bottom. The exterior regions above and below all strands are omitted. A chamber label J is the set of strand labels below it. For a reduced word, P_J is the flag minor with the first |J| rows and columns J.

Bounded chambers correspond to mutable vertices; chambers open to the left or right correspond to frozen vertices. If the word has length L and uses r distinct generators, there are L−r mutable and n−1+r frozen chambers. The empty word is allowed, and the total number L+n−1 of chambers is limited to 24.

The quiver follows the chamber-arrow rules of Definition 2.3.1 in the notes. A braid move applies μₖ at the corresponding chamber slot k. B̃ constructed directly from the new diagram is compared with the mutated B̃. Arrows between frozen vertices are omitted.

For a reduced word, a braid move has exchange relation P_YP_Z = P_AP_C + P_BP_D. For nonreduced words, chamber minors may be dependent, so calculations use a formal seed with an independent variable xᵢ for each initial chamber. Repeated J labels still denote distinct variables; J remains a geometric strand-set label.

Commutation only interchanges unrelated crossings, so the seed is preserved. Braid moves are available at triangular chambers, not at every mutable vertex. Intervening unrelated crossings are commuted automatically.

P_J / xᵢ buttons switch the notation for initial variables of a reduced word. X denotes a current cluster variable, and expressions use the initial variables. The product ŷⱼ = ∏ᵢ Xᵢᵇⁱʲ includes frozen variables. Experiments and saved files in this tab are independent of other tabs.

Finite type is determined from the mutable part of the current quiver. An inconclusive bounded search is marked undetermined. For reduced words of the longest element, the types are A₁ for 3 strands, A₃ for 4, D₆ for 5, and infinite for 6. For other words, the strand count alone does not determine the type.

Fomin–Williams–Zelevinsky, Introduction to Cluster Algebras, Chapters 1–3: §1.3, relation (1.3.1), Definition 2.3.1, Proposition 2.3.2, Theorem 3.5.1; Chapter 6: Example 6.5.6 and Table 6.1.

Two wiring diagrams, each with n strands, are superimposed according to an arbitrary signed word. Negative generators are red crossings and positive generators are blue crossings. Reducedness and the represented permutation are checked separately for each color. Initial strand labels from bottom to top are n,…,1 for red and 1,…,n for blue. The red strand labels below a chamber form I, and the blue labels form J. When both words are reduced, Δ_I,J denotes the minor with row set I and column set J.

Bounded chambers are mutable and unbounded chambers are frozen. If the word has length L and uses r distinct absolute generator values, there are L−r mutable and n+r frozen chambers. The total L+n is limited to 24. The top chamber is included as a frozen variable; for reduced words it is the determinant Δ_[n],[n]. The empty minor at the bottom equals 1.

The quiver follows the chamber-arrow rules of Definition 2.4.1 in the notes. A local move applies μₖ at the corresponding chamber slot k. The matrix B̃ constructed directly from the new diagram is compared with the mutated B̃. Arrows between frozen vertices are omitted.

The three local moves in Figure 1.10 are the red braid, blue braid, and mixed move. When both words are reduced, each move replaces one chamber minor and satisfies Y Z = A C + B D. The mixed move corresponds to the Desnanot–Jacobi (Lewis Carroll) identity. Commutation preserves the seed. Unrelated crossings are commuted automatically when needed for a move.

If either word is nonreduced, the chamber minors may be dependent. Calculations therefore use a formal seed with an independent variable xᵢ for each initial chamber. Repeated (I,J) labels still represent distinct variables; I and J only identify strand sets. Minor notation is disabled in this case.

Δ_I,J / xᵢ buttons switch the notation for initial variables when both words are reduced. X denotes a current variable, and expressions use the initial variables. The product ŷⱼ = ∏ᵢ Xᵢᵇⁱʲ includes frozen variables. Finite type is determined from the mutable part of the actual quiver; an inconclusive bounded search is marked undetermined. Experiments and saved files are independent of other tabs. Conversion to a plabic graph carries over the current seed and the original initial variables.

Fomin–Williams–Zelevinsky, Introduction to Cluster Algebras, Chapters 1–3: §1.4, Figure 1.10, relation (1.4.1), Definition 2.4.1 and Proposition 2.4.3.

Tᵢ starts at boundary vertex i and ends at π_G(i). At a black vertex, take the sharpest right turn; at a white vertex, take the sharpest left turn. Colored paths run to the right of their directed edges. The disk embedding preserves the cyclic order at every vertex.

I_source(f) = {i : f lies to the left of Tᵢ}; I_target(f) = {π_G(i) : f lies to the left of Tᵢ}. Edge labels list the starting indices of the one-way trips that traverse the edge. Rᵢ denotes an internal roundtrip, when present.

Face labels are displayed for reduced leafless graphs, as in Definition 7.12.2. Reducedness is checked using the resonance criterion (Theorem 7.11.5). For a white lollipop all faces lie to the left of its trip; for a black lollipop all lie to its right.

Fomin–Williams–Zelevinsky, Introduction to Cluster Algebras, Chapter 7: Definitions 7.1.8, 7.1.14, 7.11.1 and 7.12.2; Remark 7.12.1; Theorems 7.11.5 and 7.12.4.