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Introduction
Cluster coordinates and skein theory give two descriptions of functions on moduli spaces of surface local systems. Cluster coordinates describe these functions in overlapping algebraic tori, whereas skein theory represents them by diagrams and local relations. For a quantum cluster algebra, the Laurent phenomenon gives an inclusion in the intersection of all quantum seed tori, its upper cluster algebra. Identifying this intersection with a diagram algebra requires both a construction of cluster variables by diagrams and a converse statement expressing every universally Laurent element by diagrams. This paper establishes that identification for the projected stated S L n SL_n S L n
Fomin and Zelevinsky introduced cluster algebras to study canonical bases and total positivity [9 ] . Berenstein, Fomin, and Zelevinsky introduced upper cluster algebras and proved equality with cluster algebras under acyclicity hypotheses [BFZ]. Berenstein and Zelevinsky subsequently developed quantum seeds, compatible skew forms, quantum mutation, and the quantum Laurent phenomenon [BZ]. These constructions distinguish the algebra generated by cluster variables from the intersection of Laurent coordinate algebras; equality is a further assertion about the seed.
Fock and Goncharov constructed coordinate systems on moduli spaces of decorated and framed local systems, extending the coordinates of Teichmüller theory to higher rank [7 ] . For S L n SL_n S L n n n n n > 2 n > 2 n > 2 S L 2 SL_2 S L 2 [8 ] . These geometric coordinate systems supply the seeds considered here.
Bonahon and Wong constructed the S L 2 SL_2 S L 2 [3 ] . Muller introduced a skein algebra whose diagrams may end at marked boundary points and proved that, after inverting boundary elements, it lies between the quantum cluster algebra and the quantum upper cluster algebra. For a triangulable surface with at least two marked points in each connected component, he proved equality of all three algebras [21 ] . This result is the S L 2 SL_2 S L 2
Stated skein algebras place endpoints on open boundary intervals and label them by states. The states make cutting along ideal arcs compatible with finite sums over the newly created endpoints. Lê established triangular decomposition for the stated S L 2 SL_2 S L 2 [18 ] ; Higgins developed the corresponding stated S L 3 SL_3 S L 3 [13 ] . Lê and Sikora defined stated S L n SL_n S L n n n n [19 ] . Their ordinary and transmuted quantum matrix factors will provide the finite filtration used in our proof.
Lê and Yu constructed higher-rank quantum traces in both X \mathcal{X} X A \mathcal{A} A [20 ] . Their full quantum trace is injective on the unpunctured surfaces considered here. They also defined the reduced skein algebra by killing bad corner arcs and constructed its quantum trace, whose image is the projected algebra. Initial A \mathcal{A} A
Several results address the cluster side of this comparison. Ishibashi and Yuasa constructed quantum cluster structures related to s l 3 \mathfrak{sl}_3 sl 3 [16 ] . In the classical setting, Ishibashi, Oya, and Shen proved A = U \mathcal{A} = \mathcal{U} A = U G G G [15 ] . Their use of pinnings and Wilson lines is also suited to the classical full-skein comparison. Goncharov and Shen’s work on local-system coordinates and Donaldson–Thomas transformations supplies the coordinate conventions and the mutation sequence used below [12 , 11 ] .
Huang and Wang studied the projected stated algebra inside the A \mathcal{A} A [14 ] . Their Theorems 1.2 and 1.3 establish its inclusions in the quantum cluster and upper cluster algebras, respectively, under the stated surface hypotheses. In Conjecture 1.5 they asked whether the projected algebra, the quantum cluster algebra, and the quantum upper cluster algebra coincide. Conjecture 1.6 specifies generators of the upper algebra over the Laurent algebra of frozen variables: the exchangeable variables appearing in their elementary-arc mutation formulas. Both conjectures concern triangulable surfaces without interior punctures and with at least two boundary marked points in each connected component. Their coefficient ring is an arbitrary commutative domain with the indicated invertible quantum parameter.
Cao, Huang, and Wang subsequently proved the projected and full, boundary-localized equalities for polygons, and constructed rotation-invariant bases in that setting. They also established the full-skein equality for n = 2 n = 2 n = 2 [4 ] . These results settle substantial special cases and give the full-seed matrices used in our comparison. Kim and Wang proved naturality of the higher-rank quantum traces under changes of triangulation [17 ] .
Their network transformations, together with the complete flip word of [26 , 7 ] , allow local formulas to be compared across adapted triangulations.
We prove both Huang–Wang conjectures in their original range. The marked-point requirement is imposed on each connected component of the surface; an individual boundary circle may have a single marked point. All frozen variables in the projected cluster algebra are inverted.
Theorem 1.1. Let n ≥ 2 n \ge2 n ≥ 2 Σ \Sigma Σ R R R v ∈ R × v \in R^{\times} v ∈ R × ω = v 2 \omega= v^{2} ω = v 2
S ~ ω ( Σ ) = A ω ( Σ ) = U ω ( Σ ) . \widetilde{\mathcal{S}}_{\omega}(\Sigma)=\mathcal{A}_{\omega}(\Sigma)=\mathcal{U}_{\omega}(\Sigma). S ω ( Σ ) = A ω ( Σ ) = U ω ( Σ ) . Over the Laurent algebra of frozen variables, this algebra is generated by the exchangeable variables specified in Huang–Wang Conjecture 1.6.
Here S ~ ω ( Σ ) \widetilde{\mathcal{S}}_{\omega}(\Sigma) S ω ( Σ ) A ω ( Σ ) \mathcal{A}_{\omega}(\Sigma) A ω ( Σ ) U ω ( Σ ) \mathcal{U}_{\omega}(\Sigma) U ω ( Σ ) § 3 defines these objects and fixes the normalizations. § 9 identifies the generating family with the one in the original conjecture, including the extremal arc entry recovered by a reverse-arc cofactor identity.
Proof strategy. The first part of the proof expresses elementary arc coefficients by mutation words on triangles and quadrilaterals. Cutting and gluing then give the inclusion of the projected skein algebra in the cluster algebra. The local calculations retain all endpoint cases and all occurrences of a boundary mark. They also produce the frozen units needed for the passage to the original generating family.
For the reverse inclusion, we use the full stated skein algebra. Attaching a triangle to each boundary interval embeds its initial torus into a larger projected torus. In the classical fiber, a collar correspondence with decorated local systems identifies the frozen localization of the full algebra with the full upper algebra. Primality of the individual corner minors removes the frozen poles. A fixed saturated arc system then gives finite free filtered pieces. We prove that the initial mutable webs retain their degrees at the classical specialization, using reduced arc and tripod representatives and a nonvanishing calculation on products of determinant cones. This strictness makes each bounded-degree upper lattice finite, so Nakayama’s lemma lifts the classical equality to the generic quantum parameter.
Finally, injectivity of the full trace in prime fibers gives integral descent. Positive quantum theta functions [5 ] , the surface Donaldson–Thomas transformation [GSI6], and the pointed-basis theorem of [25 , 24 ] give a basis of the projected upper algebra over arbitrary coefficient rings. Specializing finite skein preimages of this basis proves the asserted equality for every coefficient domain and parameter in the conjectures. § 4 –§ 9 carry out these steps; Appendix § 10 records the complete local exchange rows.
Skein algebras and quantum seeds
Surface and seed conventions
A marked bordered surface is obtained from a compact oriented surface by deleting a finite set of boundary points meeting every boundary circle. A boundary interval is a connected component of the remaining boundary. Ideal arcs join deleted points; boundary arcs have endpoints on boundary intervals. An ideal triangulation cuts the surface into triangles with their vertices deleted. We exclude the monogon and bigon, as in [14 ] . The ends of a based ideal arc, and the occurrences of a marked point in different triangle corners, remain distinct incidences.
For a triangle whose vertices are clockwise ordered, its small vertices have coordinates
( a , b , c ) ∈ Z ≥ 0 3 , a + b + c = n , ( a , b , c ) ∉ { ( n , 0 , 0 ) , ( 0 , n , 0 ) , ( 0 , 0 , n ) } . (a,b,c) \in\mathbb{Z}_{\geq0}^{3}, \qquad a+b+c=n, \qquad(a,b,c)\notin\{(n,0,0),(0,n,0),(0,0,n)\}. ( a , b , c ) ∈ Z ≥ 0 3 , a + b + c = n , ( a , b , c ) ∈ / {( n , 0 , 0 ) , ( 0 , n , 0 ) , ( 0 , 0 , n )} . Draw arrows with increments ( − 1 , 1 , 0 ) , ( 0 , − 1 , 1 ) , ( 1 , 0 , − 1 ) (-1,1,0),(0,-1,1),(1,0,-1) ( − 1 , 1 , 0 ) , ( 0 , − 1 , 1 ) , ( 1 , 0 , − 1 ) 1 / 2 1/2 1/2 V λ V_{\lambda} V λ
Q λ ( u , w ) = wt ( u ⟶ w ) − wt ( w ⟶ u ) . Q_{\lambda}(u,w)=\operatorname{wt}(u\longrightarrow w)-\operatorname{wt}(w\longrightarrow u). Q λ ( u , w ) = wt ( u ⟶ w ) − wt ( w ⟶ u ) . The mutable rows and columns are integral. Define H λ ( u , w ) = Q λ ( u , w ) H_{\lambda}(u,w)=Q_{\lambda}(u,w) H λ ( u , w ) = Q λ ( u , w ) u , w u,w u , w H λ ( u , u ) = 1 H_{\lambda}(u,u)=1 H λ ( u , u ) = 1 − 1 -1 − 1 w w w u u u Q \mathbb{Q} Q
K λ = n H λ − 1 , P λ = 2 K λ Q λ K λ T . K_{\lambda}=nH_{\lambda}^{-1}, \qquad P_{\lambda}=2K_{\lambda}Q_{\lambda}K_{\lambda}^{T}. K λ = n H λ − 1 , P λ = 2 K λ Q λ K λ T . The matrix K λ K_{\lambda} K λ P λ P_{\lambda} P λ P λ Q λ ( − , k ) = − 2 n 2 e k T P_{\lambda}Q_{\lambda}(-,k)=-2n^{2}e_{k}^{T} P λ Q λ ( − , k ) = − 2 n 2 e k T k k k [20 Lemma 11.9 and Theorem 11.7] . Here e k e_{k} e k k k k Q ( − , k ) Q(-,k) Q ( − , k ) T T T
For a finite seed index set V t V_{t} V t P t P_{t} P t R R R { M t ( a ) : a ∈ Z V t } \{M_{t}(a):a\in\mathbb{Z}^{V_{t}}\} { M t ( a ) : a ∈ Z V t }
M t ( a ) M t ( b ) = v a P t b T M t ( a + b ) . M_{t}(a)M_{t}(b)=v^{aP_{t}b^{T}}M_{t}(a+b). M t ( a ) M t ( b ) = v a P t b T M t ( a + b ) . We denote this torus by T t , R T_{t,R} T t , R x j = M t ( e j ) x_{j}=M_{t}(e_{j}) x j = M t ( e j ) V t V_{t} V t
M t ( a ) = v − ∑ i < j a i a j P t ( i , j ) ∏ j x j a j . M_{t}(a)=v^{-\sum_{i<j}a_{i}a_{j}P_{t}(i,j)}\prod_{j}x_{j}^{a_{j}}. M t ( a ) = v − ∑ i < j a i a j P t ( i , j ) j ∏ x j a j . Square brackets around a product of coordinates of one seed mean this Weyl normalization. More generally, if X Y = v 2 d Y X XY=v^{2d}YX X Y = v 2 d Y X [ X Y ] = v − d X Y [XY]=v^{-d}XY [ X Y ] = v − d X Y q k = Q t ( k , ∗ ) q_{k}=Q_{t}(k,*) q k = Q t ( k , ∗ ) q k = q k + − q k − q_{k}=q_{k}^{+}-q_{k}^{-} q k = q k + − q k − k k k x k x_{k} x k
x k ′ = M t ( − e k + q k + ) + M t ( − e k + q k − ) x_{k}'=M_{t}(-e_{k}+q_{k}^{+})+M_{t}(-e_{k}+q_{k}^{-}) x k ′ = M t ( − e k + q k + ) + M t ( − e k + q k − ) and fixes the other coordinates. Matrix mutation changes the sign of row and column k k k i , j ≠ k i,j\ne k i , j = k Q t ( i , j ) Q_{t}(i,j) Q t ( i , j )
Q t ( i , j ) + [ Q t ( i , k ) ] + [ Q t ( k , j ) ] + − [ − Q t ( i , k ) ] + [ − Q t ( k , j ) ] + , [ r ] + = max ( r , 0 ) . Q_{t}(i,j)+[Q_{t}(i,k)]_{+}[Q_{t}(k,j)]_{+}-[-Q_{t}(i,k)]_{+}[-Q_{t}(k,j)]_{+}, \qquad[r]_{+}=\max(r,0). Q t ( i , j ) + [ Q t ( i , k ) ] + [ Q t ( k , j ) ] + − [ − Q t ( i , k ) ] + [ − Q t ( k , j ) ] + , [ r ] + = max ( r , 0 ) . Let C C C k k k − e k + q k + -e_{k}+q_{k}^{+} − e k + q k + C P t C T CP_{t}C^{T} C P t C T t ∼ λ t\sim\lambda t ∼ λ t t t Frac ( R ) \operatorname{Frac}(R) Frac ( R ) R R R
The based X-torus has monomials M X ( b ) M_{\mathcal{X}}(b) M X ( b ) b ∈ Z V λ b\in\mathbb{Z}^{V_{\lambda}} b ∈ Z V λ M X ( b ) M X ( c ) = v 2 b Q c T M X ( b + c ) M_{\mathcal{X}}(b)M_{\mathcal{X}}(c)=v^{2bQc^{T}}M_{\mathcal{X}}(b+c) M X ( b ) M X ( c ) = v 2 b Q c T M X ( b + c ) n Z V τ n\mathbb{Z}^{V_{\tau}} n Z V τ M A ( a ) M_{\mathcal{A}}(a) M A ( a ) (7) identifies the A-torus with the balanced X-subtorus.
For integers a , b a,b a , b [ a , b ] [a,b] [ a , b ] a a a b b b a > b a>b a > b I , J I,J I , J X I , J X_{I,J} X I , J det X I , J \det X_{I,J} det X I , J q q q q = q L Y = v − 2 n 2 q=q_{\mathrm{LY}}=v^{-2n^2} q = q LY = v − 2 n 2
det q X I , J = ∑ σ ∈ S d ( − q ) ℓ ( σ ) X i 1 , j σ ( 1 ) ⋯ X i d , j σ ( d ) , d = ∣ I ∣ = ∣ J ∣ , \det\nolimits_q X_{I,J}=\sum_{\sigma\in\mathfrak{S}_d}(-q)^{\ell(\sigma)}X_{i_1,j_{\sigma(1)}}\cdots X_{i_d,j_{\sigma(d)}},\qquad d=\lvert I\rvert=\lvert J\rvert, det q X I , J = σ ∈ S d ∑ ( − q ) ℓ ( σ ) X i 1 , j σ ( 1 ) ⋯ X i d , j σ ( d ) , d = ∣ I ∣ = ∣ J ∣ , where ℓ ( σ ) \ell(\sigma) ℓ ( σ ) S d \mathfrak{S}_d S d { 1 , … , d } \{1,\ldots,d\} { 1 , … , d } M J I ( c ) M^I_J(\mathsf{c}) M J I ( c ) I I I J J J i ˉ = n + 1 − i \bar{i}=n+1-i i ˉ = n + 1 − i 1 C \mathbf{1}_C 1 C C C C δ a b \delta_{ab} δ ab a = b a=b a = b N = Z ≥ 0 \mathbb{N}=\mathbb{Z}_{\geq0} N = Z ≥ 0
Three skein algebras
Fix an integer n ≥ 2 n\geq2 n ≥ 2 Σ \Sigma Σ λ \lambda λ n n n Σ × ( − 1 , 1 ) \Sigma\times(-1,1) Σ × ( − 1 , 1 ) n n n { 1 , … , n } \{1,\ldots,n\} { 1 , … , n } R R R [19 Section 4] ; multiplication stacks the first web above the second. We use precisely the normalization of [20 Section 4.2, (64)–(71)] , with the parameter conversion in (4) .
A corner arc cuts off one deleted boundary point. Choose an orientation-preserving planar picture of its corner sector with the marked point at the top and the surface below it. The positional indices i , j i,j i , j [20 Section 7.1, Figure 7] . A bad corner arc has i < j i<j i < j I b a d I_{\mathrm{bad}} I bad tr λ , R A \operatorname{tr}^{A}_{\lambda,R} tr λ , R A R R R [20 Theorem 13.1] , computed by cutting a web into triangles, summing over matched states on the cuts, and applying the triangle trace and the balanced A / X A/X A / X T λ , R \mathcal{T}_{\lambda,R} T λ , R
B R ( Σ ) = S R ( Σ ) , S R r e d ( Σ ) = S R ( Σ ) / I b a d , S R ( Σ ) = im ( tr λ , R A : S R r e d ( Σ ) → T λ , R ) . (1) \mathcal{B}_R(\Sigma)=\mathcal{S}_R(\Sigma),\qquad\mathcal{S}^{\mathrm{red}}_R(\Sigma)=\mathcal{S}_R(\Sigma)/I_{\mathrm{bad}},\qquad\mathcal{S}_R(\Sigma)=\operatorname{im}\left(\operatorname{tr}^{A}_{\lambda,R}:\mathcal{S}^{\mathrm{red}}_R(\Sigma)\to\mathcal{T}_{\lambda,R}\right). \tag*{(1)} B R ( Σ ) = S R ( Σ ) , S R red ( Σ ) = S R ( Σ ) / I bad , S R ( Σ ) = im ( tr λ , R A : S R red ( Σ ) → T λ , R ) . (1) Here B R \mathcal{B}_R B R S R \mathcal{S}_R S R S f ω \mathcal{S}^{f\omega} S f ω
Denote the quantum cluster algebra and upper cluster algebra of the projected seed by
A R ( Σ ) , U R ( Σ ) = ⋂ t ∼ λ T t , R . (2) \mathcal{A}_R(\Sigma),\qquad\mathcal{U}_R(\Sigma)=\bigcap_{t\sim\lambda}\mathcal{T}_{t,R}. \tag*{(2)} A R ( Σ ) , U R ( Σ ) = t ∼ λ ⋂ T t , R . (2) All frozen variables are inverted. Thus B f r , R = R [ x f ± 1 : f frozen ] \mathbb{B}_{\mathrm{fr},R}=R[x_f^{\pm1}:f\text{ frozen}] B fr , R = R [ x f ± 1 : f frozen ]
The full seed of [20 , 4 ] is constructed by attaching a triangle to each boundary interval. In an attached triangle, label the original interface e 1 e_1 e 1 e 2 e_2 e 2 e 3 e_3 e 3 e 2 e_2 e 2 M t M_t M t H H H e 3 e_3 e 3 (38) specifies its torus inside the enlarged projected torus, and the mutable exchange columns are the columns transported by that shear. For this seed and all its mutations, let V t V_t V t M t M_t M t
T t , R H + = ⨁ a ∈ Z V t , a h ≥ 0 ( h ∈ H ) R M t ( a ) , U f u l l , R + = ⋂ t T t , R H + , U f u l l , R f r = ⋂ t T t , R . (3) T_{t,R}^{H^+}=\bigoplus_{\substack{a\in\mathbb{Z}V_t,\ a_h\geq0\\(h\in H)}} R M_t(a),\qquad U_{\mathrm{full},R}^{+}=\bigcap_t T_{t,R}^{H^+},\qquad U_{\mathrm{full},R}^{\mathrm{fr}}=\bigcap_t T_{t,R}. \tag*{(3)} T t , R H + = a ∈ Z V t , a h ≥ 0 ( h ∈ H ) ⨁ R M t ( a ) , U full , R + = t ⋂ T t , R H + , U full , R fr = t ⋂ T t , R . (3) The full frozen set H H H H H H H H H
Integral parameters
For universal formulas, v v v
A = Z [ v ± 1 ] , v = ω 1 / 2 = q ^ L Y − 1 , q L Y = v − 2 n 2 . (4) A=\mathbb{Z}[v^{\pm1}],\qquad v=\omega^{1/2}=\widehat{q}_{\mathrm{LY}}^{-1},\qquad q_{\mathrm{LY}}=v^{-2n^2}. \tag*{(4)} A = Z [ v ± 1 ] , v = ω 1/2 = q LY − 1 , q LY = v − 2 n 2 . (4) The convention q = q ^ 2 n 2 q=\widehat{q}^{2n^2} q = q 2 n 2 [20 Section 2.2] . We use HW’s matrix conventions. For the barred reduced matrices of LY, the dictionary is
Q H W = − Q ‾ L Y / 2 , P H W = − P ‾ L Y , K H W = K ‾ L Y , H H W = H ‾ L Y . (5) Q_{\mathrm{HW}}=-\overline{Q}_{\mathrm{LY}}/2,\qquad P_{\mathrm{HW}}=-\overline{P}_{\mathrm{LY}},\qquad K_{\mathrm{HW}}=\overline{K}_{\mathrm{LY}},\qquad H_{\mathrm{HW}}=\overline{H}_{\mathrm{LY}}. \tag*{(5)} Q HW = − Q LY /2 , P HW = − P LY , K HW = K LY , H HW = H LY . (5) In particular, H H H H = − Q ‾ L Y / 2 H=-\overline{Q}_{\mathrm{LY}}/2 H = − Q LY /2 − 1 -1 − 1 u → v u\to v u → v − 1 -1 − 1 v → u v\to u v → u H H H H K = n I HK=nI H K = n I H = Q H W H=Q_{\mathrm{HW}} H = Q HW q ^ L Y P ‾ L Y \widehat{q}_{\mathrm{LY}}^{\overline{P}_{\mathrm{LY}}} q LY P LY v P H W v^{P_{\mathrm{HW}}} v P HW v 2 Q H W = ω Q H W v^{2Q_{\mathrm{HW}}}=\omega^{Q_{\mathrm{HW}}} v 2 Q HW = ω Q HW [20 Section 11.7, Lemma 11.9] and [14 Remarks 2.4 and 2.7] . HW PDF equations (51)–(53) are HTML equations (57)–(59); we use the PDF numbering.
In BZ conventions we take q B Z 1 / 2 = v q_{\mathrm{BZ}}^{1/2}=v q BZ 1/2 = v Λ = P \Lambda=P Λ = P 2 n 2 I 2n^2I 2 n 2 I
The filtered lifting argument uses only
R 0 = C [ v ± 1 ] ( v − 1 ) , p = v − 1 , K 0 = C ( v ) . (6) R_0=\mathbb{C}[v^{\pm1}]_{(v-1)},\qquad p=v-1,\qquad K_0=\mathbb{C}(v). \tag*{(6)} R 0 = C [ v ± 1 ] ( v − 1 ) , p = v − 1 , K 0 = C ( v ) . (6) The denominators of normalized exterior projectors are units in this discrete valuation ring. Integral descent and upper-algebra base change will give the result over arbitrary coefficient domains, including positive characteristic and roots of unity.
Conventions for maps and specializations
The bar map on the universal based torus sends v v v v − 1 v^{-1} v − 1 M t ( a ) M_t(a) M t ( a ) [20 Proposition 4.11] . A based map sends each based monomial to a based monomial with coefficient one. For a lattice matrix E E E M ( a ) ↦ M ( a E ) M(a)\mapsto M(aE) M ( a ) ↦ M ( a E ) f f f f f f
When the coefficient ring is clear, subscripts on algebras and traces are omitted. A subscript 0 0 0 v = 1 v=1 v = 1 C \mathbb{C} C K 0 K_0 K 0 C ( v ) \mathbb{C}(v) C ( v ) § 5.3 .
Local formulas and projected skein elements
Cutting and mutation
Proposition 3.1. The canonical cutting map is an integral based embedding of the initial A A A A A A
Proof. The A / X A/X A / X By [20 Theorem 11.7, Lemma 11.9(c), Proposition 11.10] , the integral matrices K λ K_\lambda K λ H λ H_\lambda H λ
H λ K λ = K λ H λ = n I , ψ λ ( M A ( a ) ) = M X ( a K λ ) . (7) H_\lambda K_\lambda= K_\lambda H_\lambda= nI,\qquad\psi_\lambda(M_A(a)) = M_X(aK_\lambda). \tag*{(7)} H λ K λ = K λ H λ = n I , ψ λ ( M A ( a )) = M X ( a K λ ) . (7) The balanced X X X b H λ / n bH_\lambda/n b H λ / n M X ( b ) ↦ M A ( b H λ / n ) M_X(b) \mapsto M_A(bH_\lambda/n) M X ( b ) ↦ M A ( b H λ / n )
Cut Σ \Sigma Σ D D D X X X X X X S X ( M X ( b ) ) = M X , c u t ( b D ) S_X(M_X(b)) = M_{X,\mathrm{cut}}(bD) S X ( M X ( b )) = M X , cut ( b D ) [20 Theorem 12.1] is compatible with X X X
Φ = ψ c u t − 1 S X ψ Σ , E = K Σ D H c u t / n (8) \Phi= \psi_{\mathrm{cut}}^{-1}S_X\psi_\Sigma,\qquad E = K_\Sigma D H_{\mathrm{cut}}/n \tag*{(8)} Φ = ψ cut − 1 S X ψ Σ , E = K Σ D H cut / n (8) is an integral, based, injective map of A A A
Uncut columns. For an uncut small vertex w w w
D H c u t ( ∗ , w ) = H Σ ( ∗ , w ) , E ( ∗ , w ) = e w T . (9) D H_{\mathrm{cut}}(*,w) = H_\Sigma(*,w),\qquad E(*,w) = e_w^T. \tag*{(9)} D H cut ( ∗ , w ) = H Σ ( ∗ , w ) , E ( ∗ , w ) = e w T . (9) Indeed, by the definition of H H H w w w (7) gives the second identity. This calculation retains corner occurrences separately and thus permits based loops and repeated occurrences of a marked point.
For an uncut mutable vertex i i i H ( i , ∗ ) = Q ( i , ∗ ) H(i,*) = Q(i,*) H ( i , ∗ ) = Q ( i , ∗ )
Q Σ ( i , ∗ ) E = Q c u t ( i , ∗ ) . (10) Q_\Sigma(i,*)E = Q_{\mathrm{cut}}(i,*). \tag*{(10)} Q Σ ( i , ∗ ) E = Q cut ( i , ∗ ) . (10) The exchange submatrices on uncut vertices also coincide. These identities concern uncut columns; cut columns can have additional contributions when an edge is a based loop.
Common frozen factors along mutation words. Freeze the cut directions and mutate only in uncut directions. We use row vectors. Let q i q_i q i r i r_i r i E E E q i E = r i q_iE = r_i q i E = r i
Mutate at k k k q k = q k + − q k − q_k = q_k^+ - q_k^- q k = q k + − q k − r k = r k + − r k − r_k = r_k^+ - r_k^- r k = r k + − r k −
C = q k − E − r k − = q k + E − r k + . C = q_k^-E - r_k^- = q_k^+E - r_k^+. C = q k − E − r k − = q k + E − r k + . The vector C C C
d = e k − E k + C , (11) d=e_k-E_k+C, \tag*{(11)} d = e k − E k + C , (11) which has the same support restriction. Compatibility gives ( r k + − r k − ) P t a r g e t d T = 0 (r_k^+-r_k^-)P_{\mathrm{target}}d^T=0 ( r k + − r k − ) P target d T = 0 M ( d ) M(d) M ( d )
Φ ( x k ′ ) = [ x k , t a r g e t ′ M ( d ) ] . (12) \Phi(x'_k)=[x'_{k,\mathrm{target}}M(d)]. \tag*{(12)} Φ ( x k ′ ) = [ x k , target ′ M ( d )] . (12) Only row k k k
E k ′ = 2 e k − E k + C . (13) E'_k=2e_k-E_k+C. \tag*{(13)} E k ′ = 2 e k − E k + C . (13) Its uncut columns remain identity columns. For i ≠ k i\ne k i = k h = q i ( k ) = r i ( k ) h=q_i(k)=r_i(k) h = q i ( k ) = r i ( k ) k k k
q i ′ = q i + [ h ] + q k + − [ − h ] + q k − − 2 h e k . q'_i=q_i+[h]_+q_k^+-[-h]_+q_k^--2he_k. q i ′ = q i + [ h ] + q k + − [ − h ] + q k − − 2 h e k . Since q i ′ ( k ) = − h q'_i(k)=-h q i ′ ( k ) = − h
q i ′ E ′ = r i + [ h ] + ( r k + + C ) − [ − h ] + ( r k − + C ) − 2 h E k − h ( 2 e k − 2 E k + C ) = r i + [ h ] + r k + − [ − h ] + r k − − 2 h e k = r i ′ . (14) \begin{aligned}
q'_iE'=r_i+[h]_+(r_k^++C)-[-h]_+(r_k^-+C)-2hE_k-h(2e_k-2E_k+C) \\
&=r_i+[h]_+r_k^+-[-h]_+r_k^--2he_k=r'_i. \tag*{(14)}
\end{aligned} q i ′ E ′ = r i + [ h ] + ( r k + + C ) − [ − h ] + ( r k − + C ) − 2 h E k − h ( 2 e k − 2 E k + C ) = r i + [ h ] + r k + − [ − h ] + r k − − 2 h e k = r i ′ . (14) The identity for i = k i=k i = k H , K H,K H , K
Triangulation naturality. The A-seeds of all adapted triangulations define the same cluster algebra. We obtain this from the complete flip argument of [17 v2] , the balanced tori of [20 ] , and the following finite A/X mutation identities.
In a mutable direction put X = A k X=A_k X = A k Y = A Q ( k , ∗ ) Y=A^{Q(k,*)} Y = A Q ( k , ∗ ) q = v 2 n 2 q=v^{2n^2} q = v 2 n 2 Y X = q − 2 X Y YX=q^{-2}XY Y X = q − 2 X Y f f f Y Y Y f ( Y ) X = X f ( q − 2 Y ) f(Y)X=Xf(q^{-2}Y) f ( Y ) X = X f ( q − 2 Y ) f ( Y ) X − 1 = X − 1 f ( q 2 Y ) f(Y)X^{-1}=X^{-1}f(q^2Y) f ( Y ) X − 1 = X − 1 f ( q 2 Y ) T = X ( 1 + q − 1 Y ) − 1 T=X(1+q^{-1}Y)^{-1} T = X ( 1 + q − 1 Y ) − 1 T − 1 = X − 1 ( 1 + q Y ) T^{-1}=X^{-1}(1+qY) T − 1 = X − 1 ( 1 + q Y ) X X X X − 1 X^{-1} X − 1 r = s r=s r = s s = 0 s=0 s = 0
T s = X s ∏ r = 0 s − 1 ( 1 + q − ( 2 r + 1 ) Y ) − 1 , T − s = X − s ∏ r = 0 s − 1 ( 1 + q 2 r + 1 Y ) , s ≥ 0. (15) T^s=X^s\prod_{r=0}^{s-1}(1+q^{-(2r+1)}Y)^{-1},\qquad
T^{-s}=X^{-s}\prod_{r=0}^{s-1}(1+q^{2r+1}Y),\qquad s\ge0. \tag*{(15)} T s = X s r = 0 ∏ s − 1 ( 1 + q − ( 2 r + 1 ) Y ) − 1 , T − s = X − s r = 0 ∏ s − 1 ( 1 + q 2 r + 1 Y ) , s ≥ 0. (15) The left mutation matrix C C C k k k C k k = − 1 C_{kk}=-1 C k k = − 1 C i k = [ Q ( i , k ) ] + C_{ik}=[Q(i,k)]_+ C ik = [ Q ( i , k ) ] + i ≠ k i\ne k i = k C 2 = I C^2=I C 2 = I H ′ = C H C T H'=CHC^T H ′ = C H C T (15) .
For a directed path p p p ℓ p \ell_p ℓ p p i p_i p i κ = ∑ i ℓ p i \kappa=\sum_i\ell_{p_i} κ = ∑ i ℓ p i Θ \Theta Θ [17 Lemma 4.5] with its printed normalization sign corrected to M X ( − κ ) M_X(-\kappa) M X ( − κ ) [20 Theorem 10.5] . The exponent there is n ℓ p − κ n\ell_p-\kappa n ℓ p − κ Θ ( M X ( κ ′ ) ) = M X ( κ ) \Theta(M_X(\kappa'))=M_X(\kappa) Θ ( M X ( κ ′ )) = M X ( κ ) v a v^a v a v − a v^{-a} v − a a = 0 a=0 a = 0
We apply [26 Proposition 4.2] along the standard complete flip word. Layer s s s s ( n − s ) s(n-s) s ( n − s ) a + b + c + d = n − 2 a+b+c+d=n-2 a + b + c + d = n − 2 ( a , b , c , d ) + e 2 + e 4 (a,b,c,d)+e_2+e_4 ( a , b , c , d ) + e 2 + e 4 ( a , b , c , d ) + e 1 + e 3 (a,b,c,d)+e_1+e_3 ( a , b , c , d ) + e 1 + e 3 e 1 + e 2 e_1+e_2 e 1 + e 2 e 2 + e 3 e_2+e_3 e 2 + e 3 e 3 + e 4 e_3+e_4 e 3 + e 4 e 4 + e 1 e_4+e_1 e 4 + e 1 [7 Section 10.3] gives the four-valent condition and the next-layer update. X-cutting reduces a general surface to the ordinary quadrilateral. This yields naturality at the universal and generic parameters; subsequent specializations concern finite Laurent identities.
Boundary identifications preserve this complete flip. Write the octahedral coordinates as x = ( a , b , c , d ) ∈ N 4 x=(a,b,c,d)\in\mathbb{N}^4 x = ( a , b , c , d ) ∈ N 4 c = d = 0 c=d=0 c = d = 0 a = b = 0 a=b=0 a = b = 0 x = 0 x=0 x = 0 n = 2 n=2 n = 2
The terminal H H H S = H + H T S=H+H^{T} S = H + H T H = Q + S / 2 H=Q+S/2 H = Q + S /2 S S S C = I + u e k T C=I+ue_k^{T} C = I + u e k T e k T S = S e k = 0 e_k^{T}S=Se_k=0 e k T S = S e k = 0 C S C T = S CSC^{T}=S C S C T = S H H H Q Q Q Q Q Q S S S H H H
Elementary arcs
Proposition 3.2. For every allowed coefficient domain, S R ( Σ ) ⊆ A R ( Σ ) S_{\mathrm{R}}(\Sigma)\subseteq A_{\mathrm{R}}(\Sigma) S R ( Σ ) ⊆ A R ( Σ )
Proof. We first calculate the mutation words on ordinary polygons. Section 3.2 then glues their neutral-support expressions to a general surface.
Quantum lifting for coefficient-one arcs. Suppose a mutation word visits each pivot at most once. At each visit the denominator is the unchanged initial variable, while all other variables enter the exchange monomials with nonnegative exponents. Recursive expansion uses addition, multiplication, and inverses of initial monomials. Every Laurent coefficient therefore lies in N [ v ± 1 ] \mathbb{N}[v^{\pm1}] N [ v ± 1 ]
By LY10.5, either orientation of a simple triangle corner arc is a finite sum of based X-monomials with coefficient one. Distinct paths have distinct left-region vectors: closing them by the same boundary path expresses their difference as the directed boundary of bounded faces, each containing an undeleted small vertex. Equal indicators force the face-chain difference, and hence the path difference, to vanish. The based A / X A/X A / X
Suppose an expression obtained from such a word, normalized by an incident frozen monomial, equals the arc at v = 1 v=1 v = 1 v a v^a v a a = 0 a=0 a = 0
The two triangle orientations. In the first triangle write
A p q = g n − p , q , p − q , 0 ≤ q ≤ p ≤ n . A_{pq}=g_{n-p,q,p-q}, \qquad0\leq q\leq p\leq n. A pq = g n − p , q , p − q , 0 ≤ q ≤ p ≤ n . Here g a , b , c g_{a,b,c} g a , b , c [20 Section 13.2] , whose trace is the coordinate at ( a , b , c ) (a,b,c) ( a , b , c ) F q p F_{qp} F q p ( q , n − p , p − q ) (q,n-p,p-q) ( q , n − p , p − q ) 0 ≤ q ≤ p ≤ n 0\leq q\leq p\leq n 0 ≤ q ≤ p ≤ n y s = A s s = F s s y_s=A_{ss}=F_{ss} y s = A ss = F ss y 0 = y n = 1 y_0=y_n=1 y 0 = y n = 1 C = ( C i j ) C=(C_{ij}) C = ( C ij ) C ~ = ( C ~ i j ) \widetilde{C}=(\widetilde{C}_{ij}) C = ( C ij ) C C C C ˉ \bar{C} C ˉ § 3.2 . Their entries above the diagonal vanish in the reduced algebra. Classical formulas use their v = 1 v = 1 v = 1
LY(179) and complementary minors give the classical coordinates
A p q = A p 0 A q q det C ˉ [ p − q + 1 , p ] , [ 1 , q ] , A p 0 = ( C 11 ⋯ C p p ) − 1 , (16) A_{pq}=A_{p0}A_{qq}\det\bar{C}_{[p-q+1,p],[1,q]},\qquad A_{p0}=(C_{11}\cdots C_{pp})^{-1}, \tag*{(16)} A pq = A p 0 A q q det C ˉ [ p − q + 1 , p ] , [ 1 , q ] , A p 0 = ( C 11 ⋯ C pp ) − 1 , (16) F q p = F p p det C ˉ [ n − p + q + 1 , n ] , [ q + 1 , p ] . (17) F_{qp}=F_{pp}\det\bar{C}_{[n-p+q+1,n],[q+1,p]}. \tag*{(17)} F q p = F pp det C ˉ [ n − p + q + 1 , n ] , [ q + 1 , p ] . (17) To check (16) at v = 1 v=1 v = 1 E a b = ( − 1 ) n − a δ n + 1 − a , b E_{ab}=(-1)^{n-a}\delta_{n+1-a,b} E ab = ( − 1 ) n − a δ n + 1 − a , b R R R C C C § 3.2 . Its start–end matrix is R T = E C − 1 E − 1 R^{T}=EC^{-1}E^{-1} R T = E C − 1 E − 1 [20 (179)] , with ( i , j , k ) = ( n − p , q , p − q ) (i,j,k)=(n-p,q,p-q) ( i , j , k ) = ( n − p , q , p − q ) [ n − p + 1 , n − q ] [n-p+1,n-q] [ n − p + 1 , n − q ] [ n − p + q + 1 , n ] [n-p+q+1,n] [ n − p + q + 1 , n ] R T R^{T} R T
det ( R T ) [ n − p + 1 , n − q ] , [ n − p + q + 1 , n ] = ( − 1 ) ( p − q ) q det ( C − 1 ) [ q + 1 , p ] , [ 1 , p − q ] . \det(R^{T})_{[n-p+1,n-q],[n-p+q+1,n]}=(-1)^{(p-q)q}\det(C^{-1})_{[q+1,p],[1,p-q]}. det ( R T ) [ n − p + 1 , n − q ] , [ n − p + q + 1 , n ] = ( − 1 ) ( p − q ) q det ( C − 1 ) [ q + 1 , p ] , [ 1 , p − q ] . Since C C C p × p p\times p p × p C − 1 C^{-1} C − 1 C C C
det ( C − 1 ) [ q + 1 , p ] , [ 1 , p − q ] = ( − 1 ) ( p − q ) ( p + 1 ) det C [ p − q + 1 , p ] , [ 1 , q ] det C [ 1 , p ] , [ 1 , p ] . \det(C^{-1})_{[q+1,p],[1,p-q]}=(-1)^{(p-q)(p+1)}\frac{\det C_{[p-q+1,p],[1,q]}}{\det C_{[1,p],[1,p]}}. det ( C − 1 ) [ q + 1 , p ] , [ 1 , p − q ] = ( − 1 ) ( p − q ) ( p + 1 ) det C [ 1 , p ] , [ 1 , p ] det C [ p − q + 1 , p ] , [ 1 , q ] . The combined sign is ( − 1 ) p ( p + 1 ) − q ( q + 1 ) = 1 (-1)^{p(p+1)-q(q+1)}=1 ( − 1 ) p ( p + 1 ) − q ( q + 1 ) = 1 A p 0 − 1 A_{p0}^{-1} A p 0 − 1 A q q A_{qq} A q q [20 (179)] proves (16) . Empty minors give the same calculation at q = 0 q=0 q = 0 q = p q=p q = p C , C ˉ C,\bar{C} C , C ˉ (17) is the transpose of the corresponding LY minor.
Fix i i i p = i − 1 , i − 2 , … , 2 p=i-1,i-2,\ldots,2 p = i − 1 , i − 2 , … , 2 q = 1 , … , p − 1 q=1,\ldots,p-1 q = 1 , … , p − 1 § 10 .§ 2 lists the complete exchange rows. Their classical recurrence is
A p q Z p q = A p − 1 , q Z p + 1 , q + A p , q + 1 Z p , q − 1 , (18) A_{pq}Z_{pq}=A_{p-1,q}Z_{p+1,q}+A_{p,q+1}Z_{p,q-1}, \tag*{(18)} A pq Z pq = A p − 1 , q Z p + 1 , q + A p , q + 1 Z p , q − 1 , (18) The boundary values are Z i , q = A i − 1 , 0 A i , q + 1 Z_{i,q}=A_{i-1,0}A_{i,q+1} Z i , q = A i − 1 , 0 A i , q + 1 Z p , 0 = A i 1 A p − 1 , 0 Z_{p,0}=A_{i1}A_{p-1,0} Z p , 0 = A i 1 A p − 1 , 0
Z p q = A p − 1 , 0 A i 0 A q + 1 , q + 1 det C [ p − q , p − 1 ] ∪ { i } , [ 1 , q + 1 ] . (19) Z_{pq}=A_{p-1,0}A_{i0}A_{q+1,q+1}\det C_{[p-q,p-1]\cup\{i\},[1,q+1]}. \tag*{(19)} Z pq = A p − 1 , 0 A i 0 A q + 1 , q + 1 det C [ p − q , p − 1 ] ∪ { i } , [ 1 , q + 1 ] . (19) The boundary values follow from (16) . To check the recurrence, use increasing row lists and write Δ I , J = det C I , J \Delta_{I,J}=\det C_{I,J} Δ I , J = det C I , J S = [ p − q + 1 , p − 1 ] S=[p-q+1,p-1] S = [ p − q + 1 , p − 1 ] a = p − q a=p-q a = p − q b = p b=p b = p c = i c=i c = i
Δ S ∪ { b } , [ 1 , q ] Δ S ∪ { a , c } , [ 1 , q + 1 ] = Δ S ∪ { a } , [ 1 , q ] Δ S ∪ { b , c } , [ 1 , q + 1 ] + Δ S ∪ { c } , [ 1 , q ] Δ S ∪ { a , b } , [ 1 , q + 1 ] . \Delta_{S\cup\{b\},[1,q]}\Delta_{S\cup\{a,c\},[1,q+1]}
=\Delta_{S\cup\{a\},[1,q]}\Delta_{S\cup\{b,c\},[1,q+1]}+\Delta_{S\cup\{c\},[1,q]}\Delta_{S\cup\{a,b\},[1,q+1]}. Δ S ∪ { b } , [ 1 , q ] Δ S ∪ { a , c } , [ 1 , q + 1 ] = Δ S ∪ { a } , [ 1 , q ] Δ S ∪ { b , c } , [ 1 , q + 1 ] + Δ S ∪ { c } , [ 1 , q ] Δ S ∪ { a , b } , [ 1 , q + 1 ] . It follows by expanding the last column, or by the Plücker relation for these nested column sets. Substituting (16) and (19) into (18) , all three terms have the common factor
A p 0 A p − 1 , 0 A i 0 A q q A q + 1 , q + 1 . A_{p0}A_{p-1,0}A_{i0}A_{qq}A_{q+1,q+1}. A p 0 A p − 1 , 0 A i 0 A q q A q + 1 , q + 1 . Canceling it leaves exactly the displayed identity. This verifies every interior recurrence, including q = 1 q=1 q = 1 S S S p = k , q = k − 1 p=k,q=k-1 p = k , q = k − 1 Z k , k − 1 = A i 0 A k k C i k Z_{k,k-1}=A_{i0}A_{kk}C_{ik} Z k , k − 1 = A i 0 A k k C ik § 4.2 lifts the identity to the quantum setting.
For i ≥ 2 i\ge2 i ≥ 2 U 1 = A i 1 , U k = Z k , k − 1 U_1=A_{i1},U_k=Z_{k,k-1} U 1 = A i 1 , U k = Z k , k − 1 ( 1 < k < i ) (1<k<i) ( 1 < k < i ) U i = A i − 1 , 0 U_i=A_{i-1,0} U i = A i − 1 , 0
C i k = { [ U k A i 0 − 1 A k k − 1 ] , k < i , [ U i A i 0 − 1 ] , k = i . (20) C_{ik} =
\begin{cases}
[U_k A_{i0}^{-1} A_{kk}^{-1}], & k < i,\\
[U_i A_{i0}^{-1}], & k = i.
\end{cases} \tag*{(20)} C ik = { [ U k A i 0 − 1 A k k − 1 ] , [ U i A i 0 − 1 ] , k < i , k = i . (20) For i = 1 i=1 i = 1 C 11 = [ A 10 − 1 ] C_{11}=[A_{10}^{-1}] C 11 = [ A 10 − 1 ] U 1 = U i U_1=U_i U 1 = U i
For the reverse orientation fix j < n j<n j < n F k , k + t F_{k,k+t} F k , k + t k = j − 1 , … , 1 k=j-1,\ldots,1 k = j − 1 , … , 1 t = n − j , … , 1 t=n-j,\ldots,1 t = n − j , … , 1 W k , t W_{k,t} W k , t § 10 .§ 3 supplies the complete rows. The recurrence and its solution are
F k , k + t W k , t = W k , t + 1 F k , k + t − 1 + W k + 1 , t F k − 1 , k + t − 1 , (21) F_{k,k+t}W_{k,t}=W_{k,t+1}F_{k,k+t-1}+W_{k+1,t}F_{k-1,k+t-1}, \tag*{(21)} F k , k + t W k , t = W k , t + 1 F k , k + t − 1 + W k + 1 , t F k − 1 , k + t − 1 , (21) W k , n − j + 1 = F j n F k − 1 , k + n − j , W j , t = F j − 1 , n F j , j + t − 1 , W k , t = F k + t − 1 , k + t − 1 F j n det C ˉ { j } ∪ [ n − t + 2 , n ] , [ k , k + t − 1 ] . (22) \begin{aligned}
W_{k,n-j+1}&=F_{jn}F_{k-1,k+n-j},\qquad W_{j,t}=F_{j-1,n}F_{j,j+t-1},\\
W_{k,t}&=F_{k+t-1,k+t-1}F_{jn}\det\bar{C}_{\{j\}\cup[n-t+2,n],[k,k+t-1]}. \tag*{(22)}
\end{aligned} W k , n − j + 1 W k , t = F j n F k − 1 , k + n − j , W j , t = F j − 1 , n F j , j + t − 1 , = F k + t − 1 , k + t − 1 F j n det C ˉ { j } ∪ [ n − t + 2 , n ] , [ k , k + t − 1 ] . (22) For the reverse recurrence take the ordered row list R = { j } ∪ [ n − t + 1 , n ] R=\{j\}\cup[n-t+1,n] R = { j } ∪ [ n − t + 1 , n ] J = [ k , k + t ] J=[k,k+t] J = [ k , k + t ] a = j a=j a = j b = n − t + 1 b=n-t+1 b = n − t + 1 c = k c=k c = k d = k + t d=k+t d = k + t a < b a<b a < b t ≤ n − j t\le n-j t ≤ n − j Δ I , L = det C ˉ I , L \Delta_{I,L}=\det\bar{C}_{I,L} Δ I , L = det C ˉ I , L
Δ R ∖ { a } , J ∖ { c } Δ R ∖ { b } , J ∖ { d } = Δ R , J Δ R ∖ { a , b } , J ∖ { c , d } + Δ R ∖ { a } , J ∖ { d } Δ R ∖ { b } , J ∖ { c } . \Delta_{R\setminus\{a\},J\setminus\{c\}}\Delta_{R\setminus\{b\},J\setminus\{d\}}
=
\Delta_{R,J}\Delta_{R\setminus\{a,b\},J\setminus\{c,d\}}
+\Delta_{R\setminus\{a\},J\setminus\{d\}}\Delta_{R\setminus\{b\},J\setminus\{c\}}. Δ R ∖ { a } , J ∖ { c } Δ R ∖ { b } , J ∖ { d } = Δ R , J Δ R ∖ { a , b } , J ∖ { c , d } + Δ R ∖ { a } , J ∖ { d } Δ R ∖ { b } , J ∖ { c } . After (17) and (22) are substituted, the common scalar factor in (21) is F k + t , k + t F k + t − 1 , k + t − 1 F j n F_{k+t,k+t}F_{k+t-1,k+t-1}F_{jn} F k + t , k + t F k + t − 1 , k + t − 1 F j n t = 1 t=1 t = 1 k = j k=j k = j t = n − j + 1 t=n-j+1 t = n − j + 1 (17) . At t = 1 t=1 t = 1 W k , 1 = F k k F j n C ˉ j k W_{k,1}=F_{kk}F_{jn}\bar{C}_{jk} W k , 1 = F k k F j n C ˉ j k V k = W k , 1 V_k=W_{k,1} V k = W k , 1 ( k < j ) (k<j) ( k < j ) V j = F j − 1 , n V_j=F_{j-1,n} V j = F j − 1 , n j = n j=n j = n
V k = F k − 1 , k , 1 ≤ k ≤ n . (23) V_k=F_{k-1,k},\qquad1\le k\le n. \tag*{(23)} V k = F k − 1 , k , 1 ≤ k ≤ n . (23) Both cases are expressed by
C ˉ j k = { [ V k F k k − 1 F j n − 1 ] , k < j , [ V j F j n − 1 ] , k = j . (24) \bar{C}_{jk}=
\begin{cases}
[V_kF_{kk}^{-1}F_{jn}^{-1}], & k<j,\\
[V_jF_{jn}^{-1}], & k=j.
\end{cases} \tag*{(24)} C ˉ j k = { [ V k F k k − 1 F j n − 1 ] , [ V j F j n − 1 ] , k < j , k = j . (24) These finite identities invert only the incident sides. The initial frozen exponents on the unused third side are nonnegative, and bad triangle states vanish.
The quadrilateral seam. Let D i j D_{ij} D ij i i i j j j ∑ k C i k ⊗ C ˉ j k \sum_k C_{ik}\otimes\bar{C}_{jk} ∑ k C ik ⊗ C ˉ j k 1 ≤ i , j ≤ n 1\le i,j\le n 1 ≤ i , j ≤ n m = min ( i , j ) ≥ 2 m=\min(i,j)\ge2 m = min ( i , j ) ≥ 2
p = i − 1 , … , m , q = 1 , … , m − 1 ; p = m − 1 , … , 2 , q = 1 , … , p − 1 , p=i-1,\ldots,m,\ q=1,\ldots,m-1;\qquad p=m-1,\ldots,2,\ q=1,\ldots,p-1, p = i − 1 , … , m , q = 1 , … , m − 1 ; p = m − 1 , … , 2 , q = 1 , … , p − 1 , then the reverse rectangle word, and finally mutate y 1 , … , y m − 1 y_1,\ldots,y_{m-1} y 1 , … , y m − 1 q ≥ m ≥ 2 q\ge m\ge2 q ≥ m ≥ 2 q q q § 10 .§ 2 is the already visited predecessor Z p , q − 1 Z_{p,q-1} Z p , q − 1 A i − 1 , 0 A_{i-1,0} A i − 1 , 0 y s y_s y s s < m s<m s < m
After the two words the complete seam exchange column is the following; under the conventions of Appendix § 10 .§ 5 it is the negative of the seam row:
Q ( − , y s ) = U s + V s − U s + 1 − V s + 1 − 1 s > 1 y s − 1 + ( 1 − 1 i = s + 1 − 1 j = s + 1 ) y s + 1 . (25) Q(-,y_s)=U_s+V_s-U_{s+1}-V_{s+1}-1_{s>1}y_{s-1}+(1-1_{i=s+1}-1_{j=s+1})y_{s+1}. \tag*{(25)} Q ( − , y s ) = U s + V s − U s + 1 − V s + 1 − 1 s > 1 y s − 1 + ( 1 − 1 i = s + 1 − 1 j = s + 1 ) y s + 1 . (25) The letters here denote coordinate incidence vectors: for example, U s U_s U s U s U_s U s n n n § 10 .§ 4 derives this column from the complete triangle rows. Each seam mutation removes the preceding U , V U,V U , V X X X
X = [ U 1 V 1 y 1 ] + ∑ k = 2 m − 1 [ U k V k y k − 1 y k ] + [ y m 1 i = j U m V m y m − 1 ] . (26) X=\left[\frac{U_1V_1}{y_1}\right]+\sum_{k=2}^{m-1}\left[\frac{U_kV_k}{y_{k-1}y_k}\right]+\left[\frac{y_m^{1_{i=j}}U_mV_m}{y_{m-1}}\right]. \tag*{(26)} X = [ y 1 U 1 V 1 ] + k = 2 ∑ m − 1 [ y k − 1 y k U k V k ] + [ y m − 1 y m 1 i = j U m V m ] . (26) Each quantum coefficient equals one. The preceding changed seam coordinate occurs linearly at each step, so each summand is produced once with a parameter power. The V k V_k V k i = j < n i=j<n i = j < n y i y_i y i i = j = n i=j=n i = j = n
Cutting and normalization in a quadrilateral. The four physical corners of an ordinary quadrilateral are distinct. Appendix § 10 .§ 5 verifies H E 0 = D H c u t HE_0=DH_{\mathrm{cut}} H E 0 = D H cut
S ( A p q ) = [ A p q U ⊗ F p p V ] , S ( F q p ) = [ A q q U ⊗ F q p V ] . (27) S(A_{pq})=[A^U_{pq}\otimes F^V_{pp}],\qquad S(F_{qp})=[A^U_{qq}\otimes F^V_{qp}]. \tag*{(27)} S ( A pq ) = [ A pq U ⊗ F pp V ] , S ( F q p ) = [ A q q U ⊗ F q p V ] . (27) These formulas concern the ordinary polygon. The triangle exchange rows are homogeneous. Starting from the initial degrees deg A p q = e p \deg A_{pq}=e_p deg A pq = e p deg F q p = e q \deg F_{qp}=e_q deg F q p = e q
S ( U k ) = [ U k ⊗ F k − 1 , k − 1 F i i ] , k < i , S ( U i ) = [ U i ⊗ F i − 1 , i − 1 ] , S ( V k ) = [ A k − 1 , k − 1 A j j ⊗ V k ] , k < j , S ( V j ) = [ A j − 1 , j − 1 ⊗ V j ] . (28) \begin{aligned}
S(U_k)&=[U_k\otimes F_{k-1,k-1}F_{ii}],\qquad k<i,\\
S(U_i)&=[U_i\otimes F_{i-1,i-1}],\\
S(V_k)&=[A_{k-1,k-1}A_{jj}\otimes V_k],\qquad k<j,\\
S(V_j)&=[A_{j-1,j-1}\otimes V_j]. \tag*{(28)}
\end{aligned} S ( U k ) S ( U i ) S ( V k ) S ( V j ) = [ U k ⊗ F k − 1 , k − 1 F ii ] , k < i , = [ U i ⊗ F i − 1 , i − 1 ] , = [ A k − 1 , k − 1 A j j ⊗ V k ] , k < j , = [ A j − 1 , j − 1 ⊗ V j ] . (28) The two original endpoint-side factors satisfy S ( [ A i 0 F j n ] ) = [ A i 0 A j j ⊗ F i i F j n ] S([A_{i0}F_{jn}])=[A_{i0}A_{jj}\otimes F_{ii}F_{jn}] S ([ A i 0 F j n ]) = [ A i 0 A j j ⊗ F ii F j n ] (26) identifies each normalized summand with C i k ⊗ C ‾ j k C_{ik}\otimes\overline{C}_{jk} C ik ⊗ C j k y m y_m y m
S ( [ X A i 0 − 1 F j n − 1 ] ) = ∑ k = 1 min ( i , j ) C i k ⊗ C ‾ j k = S ( D i j ) . S([XA_{i0}^{-1}F_{jn}^{-1}])=\sum_{k=1}^{\min(i,j)}C_{ik}\otimes\overline{C}_{jk}=S(D_{ij}). S ([ X A i 0 − 1 F j n − 1 ]) = k = 1 ∑ m i n ( i , j ) C ik ⊗ C j k = S ( D ij ) . The right-hand side is the skein state sum. Injectivity of initial cutting from § 4.1 gives
D i j = [ X A i 0 − 1 F j n − 1 ] , m ≥ 2. (29) D_{ij}=[XA_{i0}^{-1}F_{jn}^{-1}],\qquad m\ge2. \tag*{(29)} D ij = [ X A i 0 − 1 F j n − 1 ] , m ≥ 2. (29) The boundary cases m = 1 m=1 m = 1
D i 1 = [ A i 1 A i 0 − 1 F 1 n − 1 ] , D 1 j = [ V 1 A 10 − 1 F j n − 1 ] , j > 1. (30) D_{i1}=[A_{i1}A_{i0}^{-1}F_{1n}^{-1}],\qquad D_{1j}=[V_1A_{10}^{-1}F_{jn}^{-1}],\qquad j>1. \tag*{(30)} D i 1 = [ A i 1 A i 0 − 1 F 1 n − 1 ] , D 1 j = [ V 1 A 10 − 1 F j n − 1 ] , j > 1. (30) In particular, D 1 n = [ F 01 A 10 − 1 ] D_{1n}=[F_{01}A_{10}^{-1}] D 1 n = [ F 01 A 10 − 1 ] F 12 / A 10 F_{12}/A_{10} F 12 / A 10
Gluing on neutral support. An essential arc has endpoints on two distinct open boundary intervals. A narrow neighborhood of the arc and its endpoint sides is an ordinary triangle for a corner arc and a quadrilateral strip otherwise. Its interior is embedded. The unused lateral sides can be two copies of one global ideal edge, and marked corners can repeat. Retain a lateral side that is a physical boundary side; if it degenerates to a corner, use the triangle case. The remaining sides and strip diagonal extend to a triangulation. Cutting the lateral sides gives an ordinary patch P P P R R R
Let J J J P P P a a a r = a K P r=aK_P r = a K P r r r r g = r J r_g=rJ r g = r J
H Σ ( u , ∗ ) = H P ( u , ∗ ) J . H_\Sigma(u,*)=H_P(u,*)J. H Σ ( u , ∗ ) = H P ( u , ∗ ) J . Indeed, a nonseam u u u Q Q Q e u Q P J = e p ( u ) Q Σ e_uQ_PJ=e_{p(u)}Q_\Sigma e u Q P J = e p ( u ) Q Σ H − Q H-Q H − Q r r r
a J = 1 n r g H Σ , a J E = 1 n r g D H c u t = a ⊕ 0. (31) aJ=\frac{1}{n}r_gH_\Sigma,\qquad aJE=\frac{1}{n}r_gDH_{\mathrm{cut}}=a\oplus0. \tag*{(31)} a J = n 1 r g H Σ , a J E = n 1 r g D H cut = a ⊕ 0. (31) This identity holds on the neutral sublattice and proves that J J J
The local arc has no endpoints on a cut side. Boundary X-homogeneity [20 Proposition 10.4, Corollary 12.2] makes every X-exponent vanish on the cut sides. Thus each term of its finite A-expansion h P = ∑ a c a M P ( a ) h_P=\sum_a c_aM_P(a) h P = ∑ a c a M P ( a ) (31) , and ∑ a c a M Σ ( a J ) \sum_a c_aM_\Sigma(aJ) ∑ a c a M Σ ( a J ) Φ \Phi Φ h P ⊗ 1 h_P\otimes1 h P ⊗ 1
On the other hand, § 4.1 applied to the global mutation word shows that the normalized cluster expression c c c
Φ ( c ) = [ h P M c u t ( d ) ] (32) \Phi(c)=[h_PM_{\mathrm{cut}}(d)] \tag*{(32)} Φ ( c ) = [ h P M cut ( d )] (32) where d d d d = 0 d=0 d = 0
At v = 1 v=1 v = 1 J J J i = j = n i=j=n i = j = n U , V U,V U , V
By (31) , the two sides of (32) at v = 1 v=1 v = 1 h P , 1 ⊗ 1 h_{P,1}\otimes1 h P , 1 ⊗ 1 h P , 1 x d h_{P,1}x^d h P , 1 x d h P , 1 h_{P,1} h P , 1 d d d d = 0 d=0 d = 0
The final denominators involve only actual endpoint boundary variables. Each numerator is a global cluster variable, or a frozen variable already in the coefficient algebra.
From arcs to S A ⊆ A A S_A\subseteq A_A S A ⊆ A A
The full stated skein algebra is generated by the finitely many state-labeled arcs of a saturated system [20 Theorem 6.2(3)] . If a generator has both endpoints on an open interval c c c c ′ c' c ′ c ′ c' c ′ c ′ c' c ′ c c c c ′ c' c ′
The coefficient is ϵ i − 1 \epsilon_i^{-1} ϵ i − 1 ϵ i \epsilon_i ϵ i ϵ i \epsilon_i ϵ i [ n − 2 ] ! [n-2]! [ n − 2 ]!
By § 4.2 every such essential arc state belongs to A A \mathcal{A}_A A A § 4.1 identifies these embedded cluster algebras, and generation passes to the projected quotient. Hence
S A ( Σ ) ⊆ A A ( Σ ) ⊆ U A ( Σ ) . (33) \mathcal{S}_A(\Sigma) \subseteq\mathcal{A}_A(\Sigma) \subseteq\mathcal{U}_A(\Sigma). \tag*{(33)} S A ( Σ ) ⊆ A A ( Σ ) ⊆ U A ( Σ ) . (33) The second inclusion is the quantum Laurent phenomenon [BZ]. The local identities are finite Laurent identities over the universal ring, with denominators given by allowed boundary monomials. The same inclusions therefore hold after every allowed specialization.
Boundary units
Proposition 3.3. Every projected frozen coordinate has its inverse in S R ( Σ ) S_R(\Sigma) S R ( Σ )
Proof. When the incident boundary intervals are distinct, [20 Lemma 7.6] applies. If a compact boundary circle has one mark, its corner endpoints lie on the same open interval; we treat this case directly.
Let F = O q ( S L n ) F=\mathcal{O}_q(SL_n) F = O q ( S L n ) F ~ \widetilde{F} F κ : F → F ~ \kappa:F\to\widetilde{F} κ : F → F [20 Section 5.2] . Let I − I^- I − I ~ − \widetilde{I}^- I − u i j u_{ij} u ij i < j i<j i < j [20 Proposition 5.5] ,
κ ( I − ) = I ~ − . (34) \kappa(I^-)=\widetilde{I}^-. \tag*{(34)} κ ( I − ) = I − . (34) Take x = u i i x=u_{ii} x = u ii y = u ← i ˉ i ˉ y=\overleftarrow{u}_{\bar{i}\bar{i}} y = u i ˉ i ˉ x y = y x = 1 xy=yx=1 x y = y x = 1 κ ( x y ) \kappa(xy) κ ( x y ) κ ( x ) κ ( y ) \kappa(x)\kappa(y) κ ( x ) κ ( y ) x x x u ~ i , j ′ \widetilde{u}_{i,j'} u i , j ′ j ′ > i j'>i j ′ > i y x yx y x x x x u ~ k ′ , i \widetilde{u}_{k',i} u k ′ , i k ′ < i k'<i k ′ < i
1 = λ x ~ y ~ , 1 = μ y ~ x ~ , λ , μ ∈ A × . (35) 1=\lambda\widetilde{x}\widetilde{y},\qquad1=\mu\widetilde{y}\widetilde{x},\qquad\lambda,\mu\in A^\times. \tag*{(35)} 1 = λ x y , 1 = μ y x , λ , μ ∈ A × . (35) A left and a right inverse in a unital ring coincide. Hence every diagonal entry is a unit. This uses the linear formula for κ \kappa κ
For an ordered subset K K K ∏ a ∈ K u a a \prod_{a\in K}u_{aa} ∏ a ∈ K u aa I − I^- I − (34) and [LY(110)] to this diagrammatic minor. Every strict word correction is again bad, giving
κ ‾ ( Δ K , K ) = λ K ∏ a ∈ K u ~ a a , λ K ∈ A × . (36) \overline{\kappa}(\Delta_{K,K})=\lambda_K\prod_{a\in K}\widetilde{u}_{aa},\qquad\lambda_K\in A^\times. \tag*{(36)} κ ( Δ K , K ) = λ K a ∈ K ∏ u aa , λ K ∈ A × . (36) Thus these diagrammatic principal minors are units over the original coefficient ring. The projected boundary seed web is such a minor. The k = 0 k=0 k = 0 [20 Lemma 10.3] identifies the side vertex i + j = n i+j=n i + j = n Δ [ n − i + 1 , n ] , [ n − j + 1 , n ] ( C ( v 2 ) ) \Delta_{[n-i+1,n],[n-j+1,n]}(C(v_2)) Δ [ n − i + 1 , n ] , [ n − j + 1 , n ] ( C ( v 2 )) [20 Section 13.2] follows the finite link interval of a boundary mark. Combining the successive triangle sectors is a ribbon isotopy in the corner collar, preserving parallel strands and ports. Even if the two link endpoints lie on the same open boundary interval, they remain distinct physical ports. The resulting diagram is the minor of κ \kappa κ
The skein map from the local monogon collar sends its bad ideal into the global bad ideal. These local units therefore give global projected units, and
B f r , A ⊆ S A , (37) \mathbb{B}_{\mathrm{fr},A} \subseteq S_A, \tag*{(37)} B fr , A ⊆ S A , (37) All these identities specialize to the allowed coefficient rings.
The full algebra and its classical fiber
The full seed and mixed upper algebra
Proposition 4.1. The full trace identifies B R 0 ( Σ ) B_{R_0}(\Sigma) B R 0 ( Σ ) U f u l l , R 0 + U^{+}_{\mathrm{full},R_0} U full , R 0 +
Proof. Coordinates of the attached triangles. Attach a triangle to every boundary interval of Σ \Sigma Σ Σ ∗ \Sigma^* Σ ∗ e 1 e_1 e 1 e 2 e_2 e 2 e 3 e_3 e 3 H H H e 3 e_3 e 3 W W W e 2 e_2 e 2
Formula [LY(240)] and [4 Lemmas 4.3–4.4] give a based monomial embedding L L L Σ ∗ \Sigma^* Σ ∗ V ‾ λ \overline{V}_\lambda V λ V λ ′ = V Σ ∗ ∖ W V'_\lambda=V_{\Sigma^*}\setminus W V λ ′ = V Σ ∗ ∖ W p ( i , j , k ) = ( 0 , n − k , k ) ∈ e 2 p(i,j,k)=(0,n-k,k)\in e_2 p ( i , j , k ) = ( 0 , n − k , k ) ∈ e 2 a v a_v a v A v A_v A v
a v ⟼ { A v , v ∈ V ‾ λ , [ A v A p ( v ) − 1 ] , v ∈ V λ ′ ∖ V ‾ λ , y w = A w ( w ∈ W ) . (38) a_v\longmapsto
\begin{cases}
A_v, & v\in\overline{V}_\lambda,\\
[A_vA_{p(v)}^{-1}], & v\in V'_\lambda\setminus\overline{V}_\lambda,
\end{cases}
\qquad y_w=A_w\quad(w\in W). \tag*{(38)} a v ⟼ { A v , [ A v A p ( v ) − 1 ] , v ∈ V λ , v ∈ V λ ′ ∖ V λ , y w = A w ( w ∈ W ) . (38) The coordinate at W W W A w A_w A w A w / A p ( w ) = 1 A_w/A_{p(w)}=1 A w / A p ( w ) = 1
The full and enlarged projected seeds have the same mutable set. Adjoin W W W y w y_w y w C ^ \widehat{C} C e v e_v e v e v − e p ( v ) e_v-e_{p(v)} e v − e p ( v ) e w e_w e w Q ^ ( w , k ) = 0 \widehat{Q}(w,k)=0 Q ( w , k ) = 0 w ∈ W w\in W w ∈ W k k k
P ^ = C ^ P Σ ∗ C ^ T , \widehat{P}=\widehat{C}P_{\Sigma^*}\widehat{C}^{T}, P = C P Σ ∗ C T , including its frozen block. The integral matrix identity CHW4.1 gives its restriction to the full sublattice and the mutable-column correspondence. We verify the latter directly below.
Give the attached triangle clockwise vertices ( n , 0 , 0 ) (n,0,0) ( n , 0 , 0 ) ( 0 , n , 0 ) (0,n,0) ( 0 , n , 0 ) ( 0 , 0 , n ) (0,0,n) ( 0 , 0 , n ) e 1 = { k = 0 } e_1=\{k=0\} e 1 = { k = 0 } e 2 = { i = 0 } e_2=\{i=0\} e 2 = { i = 0 } e 3 = { j = 0 } e_3=\{j=0\} e 3 = { j = 0 } Q ( z , y ) = wt ( z → y ) − wt ( y → z ) Q(z,y)=\operatorname{wt}(z\to y)-\operatorname{wt}(y\to z) Q ( z , y ) = wt ( z → y ) − wt ( y → z ) ( − 1 , 1 , 0 ) (-1,1,0) ( − 1 , 1 , 0 ) ( 0 , − 1 , 1 ) (0,-1,1) ( 0 , − 1 , 1 ) ( 1 , 0 , − 1 ) (1,0,-1) ( 1 , 0 , − 1 )
R r = { ( i , n − r − i , r ) : 0 ≤ i ≤ n − r } , 1 ≤ r < n . R_r=\{(i,n-r-i,r):0\le i\le n-r\},\qquad1\le r<n. R r = {( i , n − r − i , r ) : 0 ≤ i ≤ n − r } , 1 ≤ r < n . whose first point w r = ( 0 , n − r , r ) w_r=(0,n-r,r) w r = ( 0 , n − r , r ) y = ( a , b , c ) y=(a,b,c) y = ( a , b , c ) a , b ≥ 1 a,b\ge1 a , b ≥ 1 c ≥ 0 c\ge0 c ≥ 0 a + b + c = n a+b+c=n a + b + c = n ∑ z ∈ R r Q ( z , y ) \sum_{z\in R_r}Q(z,y) ∑ z ∈ R r Q ( z , y )
Table 2.
Every pair lies in the allowed coordinate region and avoids the deleted ideal corners. All relevant small edges have weight one: even if a neighbor lies on e 2 e_2 e 2 e 3 e_3 e 3 y y y + 1 , − 1 +1,-1 + 1 , − 1 c = 0 c=0 c = 0 r = 1 r=1 r = 1 c = 1 c=1 c = 1 r = 0 r=0 r = 0 r = n − 1 r=n-1 r = n − 1 c = n − 2 c=n-2 c = n − 2 a = b = 1 a=b=1 a = b = 1
Every row R r R_r R r r > 0 r>0 r > 0 e 1 e_1 e 1 y y y e 1 e_1 e 1 e 2 , e 3 e_2,e_3 e 2 , e 3 y y y
∑ z ∈ R r Q Σ ∗ ( z , y ) = 0. \sum_{z\in R_r} Q_{\Sigma^*}(z,y)=0. z ∈ R r ∑ Q Σ ∗ ( z , y ) = 0. Here c = 0 c=0 c = 0 1 ≤ r < n 1\le r<n 1 ≤ r < n R 0 R_0 R 0
Let C C C C ^ \widehat{C} C W W W
C ^ T Q f u l l ( : , M u t ) = Q Σ ∗ ( : , M u t ) . \widehat{C}^{T}Q_{\mathrm{full}}(:,\mathrm{Mut})=Q_{\Sigma^*}(:,\mathrm{Mut}). C T Q full ( : , Mut ) = Q Σ ∗ ( : , Mut ) . Restoring the e w e_w e w C ^ \widehat{C} C
The exchange and common-factor induction of § 4.1 shows that, at every pair of corresponding seeds, L L L y w ± 1 y_w^{\pm1} y w ± 1
The finite trace square for the collar embedding followed by reduction sends B R 0 B_{R_0} B R 0 S R 0 ( Σ ∗ ) S_{R_0}(\Sigma^*) S R 0 ( Σ ∗ ) (33) . This supplies the map and coordinate identities needed below.
Descent from the enlarged torus. We use the following lattice fact. If M ↪ N M\hookrightarrow N M ↪ N T M ⊂ T N T_M\subset T_N T M ⊂ T N
T N ∩ Frac ( T M ) = T M . (39) T_N\cap\operatorname{Frac}(T_M)=T_M. \tag*{(39)} T N ∩ Frac ( T M ) = T M . (39) Grade T N T_N T N N / M N/M N / M z z z d ∈ T M d\in T_M d ∈ T M z d ∈ T M zd\in T_M z d ∈ T M c c c z c d = 0 z_cd=0 z c d = 0 z c = 0 z_c=0 z c = 0 M M M
Apply (39) in each corresponding seed to obtain
B R 0 ⊆ U f u l l , R 0 f r . (40) B_{R_0}\subseteq U^{\mathrm{fr}}_{\mathrm{full},R_0}. \tag*{(40)} B R 0 ⊆ U full , R 0 fr . (40) Nonnegative full frozen exponents. Represent a full skein element by a finite web. In a thin collar, its portions in the attached triangles can be chosen as simple corner tails from e 1 e_1 e 1 e 2 e_2 e 2 e 3 e_3 e 3
By § 4.2 , each nonzero triangle tail is a cluster coordinate obtained without repeated pivots, divided by an incident e 1 , e 2 e_1,e_2 e 1 , e 2 e 3 e_3 e 3 e 3 e_3 e 3 § 4.1 preserves every e 3 e_3 e 3 e 3 e_3 e 3
Formula (38) changes only the e 2 e_2 e 2 h ∈ H h\in H h ∈ H e 3 e_3 e 3
B R 0 ⊆ T f u l l , 0 , R 0 H + . (41) B_{R_0} \subseteq T_{\mathrm{full},0,R_0}^{H+}. \tag*{(41)} B R 0 ⊆ T full , 0 , R 0 H + . (41) Mutation invariance of the minimum frozen exponent. In a fully Laurent torus, fix a frozen variable f f f w f w_f w f f f f w f ( 0 ) = + ∞ w_f(0)=+\infty w f ( 0 ) = + ∞
Mutate at k ≠ f k\ne f k = f f f f [ q f k ] + [q_{fk}]_+ [ q f k ] + [ − q f k ] + [-q_{fk}]_+ [ − q f k ] + q f k ≠ 0 q_{fk}\ne0 q f k = 0 P P P
x ‾ k ′ = x k − 1 P , \overline{x}'_k=x_k^{-1}P, x k ′ = x k − 1 P , where P P P f , x k f,x_k f , x k ( x k − 1 P ) a (x_k^{-1}P)^a ( x k − 1 P ) a x k x_k x k a a a x k x_k x k P P P f f f
In each seed, mixed membership is consequently equivalent to w h ≥ 0 w_h\ge0 w h ≥ 0 (40) –(41) , this gives
B R 0 ⊆ U f u l l , R 0 + . (42) B_{R_0} \subseteq U_{\mathrm{full},R_0}^{+}. \tag*{(42)} B R 0 ⊆ U full , R 0 + . (42) The same argument applies over K 0 K_0 K 0 v = 1 v=1 v = 1
Filtration and frozen divisors
Choose a saturated oriented boundary arc system E = { a 1 , … , a r } \mathcal{E}=\{a_1,\ldots,a_r\} E = { a 1 , … , a r } [19 Theorem 8.8] . Fix the order and orientation of these arcs throughout the degree argument. Give each of the n 2 n^2 n 2 a e a_e a e D ≥ 0 D\ge0 D ≥ 0 F D B R 0 F_DB_{R_0} F D B R 0 D D D F D = 0 F_D=0 F D = 0 D < 0 D<0 D < 0 F 0 F_0 F 0 ⨁ D ≥ 0 F D / F D − 1 \bigoplus_{D\ge0}F_D/F_{D-1} ⨁ D ≥ 0 F D / F D − 1
Proposition 4.2. The word filtration associated with a fixed saturated arc system has finite free layers and a domain as its associated graded algebra. In the classical fiber, each full frozen variable generates a prime ideal, and its divisibility order equals its minimum Laurent exponent.
Proof. The saturated word filtration. By [20 Proposition 3.3, Lemma 3.5] , the ordinary matrix-factor basis is indexed by
Γ = Mat n ( N ) / Z I n \Gamma=\operatorname{Mat}_n(\mathbb{N})/\mathbb{Z}I_n Γ = Mat n ( N ) / Z I n Here I n I_n I n I n I_n I n m ^ \widehat m m min i m ^ i i = 0 \min_i\widehat m_{ii}=0 min i m ii = 0 ℓ ( m ) = ∑ i j m ^ i j \ell(m)=\sum_{ij}\widehat m_{ij} ℓ ( m ) = ∑ ij m ij [20 Theorem 5.2] . For m ∈ Γ m \in\Gamma m ∈ Γ b e ( m ) b_e(m) b e ( m ) a e a_e a e
Let G e G_e G e a e a_e a e span R 0 ( G e ) span R 0 ( G f ) = span R 0 ( G f ) span R 0 ( G e ) \operatorname{span}_{R_0}(G_e)\operatorname{span}_{R_0}(G_f)=\operatorname{span}_{R_0}(G_f)\operatorname{span}_{R_0}(G_e) span R 0 ( G e ) span R 0 ( G f ) = span R 0 ( G f ) span R 0 ( G e )
F D B R 0 = span R 0 { b 1 ( m 1 ) ⋯ b r ( m r ) : ∑ e ℓ ( m e ) ≤ D } . (43) F_D B_{R_0}=\operatorname{span}_{R_0}\{b_1(m_1)\cdots b_r(m_r):\sum_e \ell(m_e)\le D\}. \tag*{(43)} F D B R 0 = span R 0 { b 1 ( m 1 ) ⋯ b r ( m r ) : e ∑ ℓ ( m e ) ≤ D } . (43) This is finite free, and the successive quotient has the basis elements of length exactly D D D ℓ \ell ℓ Γ \Gamma Γ
At v = 1 v=1 v = 1
B 0 ≅ O ( S L n ) ⊗ r , gr F B 0 ≅ ⨂ e = 1 r C [ M n ] / ( det X e ) . (44) B_0\cong\mathcal{O}(SL_n)^{\otimes r},\qquad\operatorname{gr}_F B_0\cong\bigotimes_{e=1}^{r}\mathbb{C}[M_n]/(\det X_e). \tag*{(44)} B 0 ≅ O ( S L n ) ⊗ r , gr F B 0 ≅ e = 1 ⨂ r C [ M n ] / ( det X e ) . (44) For one factor the highest homogeneous part of det − 1 \det-1 det − 1 det \det det det \det det
Each quantum graded layer is finite free and hence p p p p p p
gr F B R 0 , gr F B K 0 are domains. (45) \operatorname{gr}_F B_{R_0},\quad\operatorname{gr}_F B_{K_0}\quad\text{are domains.} \tag*{(45)} gr F B R 0 , gr F B K 0 are domains. (45) It follows that δ K ( b ) = min { D : b ∈ F D B K 0 } \delta_K(b)=\min\{D:b\in F_D B_{K_0}\} δ K ( b ) = min { D : b ∈ F D B K 0 } δ 0 ( b ) = min { D : b ∈ F D B 0 } \delta_0(b)=\min\{D:b\in F_D B_0\} δ 0 ( b ) = min { D : b ∈ F D B 0 } − ∞ -\infty − ∞ δ ( a + b ) ≤ max { δ ( a ) , δ ( b ) } \delta(a+b)\le\max\{\delta(a),\delta(b)\} δ ( a + b ) ≤ max { δ ( a ) , δ ( b )}
To extend degree to fractions, let P P P T T T P ⊆ B ⊆ T P\subseteq B\subseteq T P ⊆ B ⊆ T P P P V 0 V_0 V 0
dim P ≤ N = ( N + ∣ V 0 ∣ ∣ V 0 ∣ ) , dim P ≤ N + d dim P ≤ N ⟶ 1 ( N ⟶ ∞ ) \dim P_{\le N}=\binom{N+|V_0|}{|V_0|},\qquad\frac{\dim P_{\le N+d}}{\dim P_{\le N}}\longrightarrow1\quad(N\longrightarrow\infty) dim P ≤ N = ( ∣ V 0 ∣ N + ∣ V 0 ∣ ) , dim P ≤ N dim P ≤ N + d ⟶ 1 ( N ⟶ ∞ ) for fixed d ≥ 0 d\ge0 d ≥ 0 a , b a,b a , b d d d N N N 2 dim P ≤ N > dim P ≤ N + d 2\dim P_{\le N}>\dim P_{\le N+d} 2 dim P ≤ N > dim P ≤ N + d ( r , s ) ↦ r a − s b (r,s)\mapsto ra-sb ( r , s ) ↦ r a − s b R 0 R_0 R 0 a , b ∈ B a,b\in B a , b ∈ B d d d d a , d b ∈ P da,db\in P d a , d b ∈ P P P P r , s r,s r , s r a = s d b ra=sdb r a = s d b ( r d ) a = ( s d ) b (rd)a=(sd)b ( r d ) a = ( s d ) b B B B B B B B B B
δ K ( a b − 1 ) = δ K ( a ) − δ K ( b ) (46) \delta_K(ab^{-1})=\delta_K(a)-\delta_K(b) \tag*{(46)} δ K ( a b − 1 ) = δ K ( a ) − δ K ( b ) (46) is well defined on the common division ring. Indeed, if a b − 1 = c d − 1 ab^{-1}=cd^{-1} a b − 1 = c d − 1 r , s ∈ B r,s\in B r , s ∈ B b r = d s br=ds b r = d s a r = c s ar=cs a r = cs
δ K ( a ) + δ K ( r ) = δ K ( c ) + δ K ( s ) , δ K ( b ) + δ K ( r ) = δ K ( d ) + δ K ( s ) . \delta_K(a) + \delta_K(r) = \delta_K(c) + \delta_K(s), \qquad\delta_K(b) + \delta_K(r) = \delta_K(d) + \delta_K(s). δ K ( a ) + δ K ( r ) = δ K ( c ) + δ K ( s ) , δ K ( b ) + δ K ( r ) = δ K ( d ) + δ K ( s ) . Subtracting proves that the two degree differences agree. To check additivity on fractions, choose c t = b u ≠ 0 ct = bu \ne0 c t = b u = 0 b − 1 c = u t − 1 b^{-1}c = ut^{-1} b − 1 c = u t − 1
( a b − 1 ) ( c d − 1 ) = a u ( d t ) − 1 , δ K ( u ) − δ K ( t ) = δ K ( c ) − δ K ( b ) . (ab^{-1})(cd^{-1}) = au(dt)^{-1}, \qquad\delta_K(u) - \delta_K(t) = \delta_K(c) - \delta_K(b). ( a b − 1 ) ( c d − 1 ) = a u ( d t ) − 1 , δ K ( u ) − δ K ( t ) = δ K ( c ) − δ K ( b ) . This gives the required sum of degrees. A common right denominator similarly reduces subadditivity for a sum of fractions to subadditivity in B B B
Geometric integrality of the extreme-minor quotient. We prove the required primality in the classical fiber. Over any field k k k
d = det X [ 1 , i ] , [ n − i + 1 , n ] , 1 ≤ i < n . (47) d = \det X_{[1,i],[n-i+1,n]}, \qquad1 \le i < n. \tag*{(47)} d = det X [ 1 , i ] , [ n − i + 1 , n ] , 1 ≤ i < n . (47) The ring k [ M n ] / ( d ) k[M_n]/(d) k [ M n ] / ( d ) d d d i × i i \times i i × i i = 1 i = 1 i = 1 i > 1 i > 1 i > 1 i − 1 i - 1 i − 1 i × ( i − 1 ) i \times(i - 1) i × ( i − 1 ) ( i − 1 ) × i (i - 1) \times i ( i − 1 ) × i i − 1 i - 1 i − 1
The determinant of X X X d = 0 d = 0 d = 0 det X = 1 \det X = 1 det X = 1 Y = Spec ( ( k [ M n ] / ( d ) ) [ ( det X ) − 1 ] ) Y = \operatorname{Spec}((k[M_n]/(d))[(\det X)^{-1}]) Y = Spec (( k [ M n ] / ( d )) [( det X ) − 1 ]) t t t D i + 1 ( t ) D_{i+1}(t) D i + 1 ( t ) i + 1 i+1 i + 1 t t t
\begin{aligned}
Y &\longrightarrow Z \times\mathbb{G}_m, & X &\longmapsto\left(D_{i+1}((\det X)^{-1})X,\det X\right),\\
Z \times\mathbb{G}_m &\longrightarrow Y, & (Z_0,t) &\longmapsto D_{i+1}(t)Z_0,
\end{aligned}
\qquad Z = \{X \in SL_n : d(X) = 0\}. \tag*{(48)}\end{aligned}
Scaling this row preserves d d d k [ Y ] ≃ k [ Z ] [ t ± 1 ] k[Y] \simeq k[Z][t^{\pm1}] k [ Y ] ≃ k [ Z ] [ t ± 1 ] k [ Z ] k[Z] k [ Z ]
k [ S L n ] / ( d ) is geometrically integral. (49) k[SL_n]/(d) \text{ is geometrically integral.} \tag*{(49)} k [ S L n ] / ( d ) is geometrically integral. (49) Primality of each full frozen variable. Each full frozen variable f h f_h f h v = 1 v = 1 v = 1 S L n SL_n S L n κ ( d ) \kappa(d) κ ( d )
For this individual corner choose a saturated system containing it. Cutting off the corner disk containing its original mark leaves another mark of the component, so the system can be completed [19 Theorem 8.8] . The construction also applies to a singleton compact boundary circle, whose endpoints are distinct physical ports.
In the classical decomposition for this auxiliary system, the chosen corner is an independent factor. § 5.2 therefore gives
B 0 / ( f h ) ≃ ( C [ S L n ] / ( d ) ) ⊗ C [ S L n ] ⊗ ( r − 1 ) , (50) B_0/(f_h) \simeq\left(\mathbb{C}[SL_n]/(d)\right) \otimes\mathbb{C}[SL_n]^{\otimes(r-1)}, \tag*{(50)} B 0 / ( f h ) ≃ ( C [ S L n ] / ( d ) ) ⊗ C [ S L n ] ⊗ ( r − 1 ) , (50) which is a domain by geometric integrality. Thus each ( f h ) (f_h) ( f h ) B 0 B_0 B 0 § 5.2 .
Prime order and the minimum Laurent exponent. Combine the nonnegative-support inclusion of § 5.1 with the primality from § 5.2 . In the initial classical torus,
P 0 = C [ x v : v ∈ V ] ⊆ B 0 ⊆ T 0 H + , B 0 / ( f ) is a domain , f = x h . (51) P_0=\mathbb{C}[x_v:v\in V]\subseteq B_0\subseteq T_0^{H^+},\qquad B_0/(f)\text{ is a domain},\qquad f=x_h. \tag*{(51)} P 0 = C [ x v : v ∈ V ] ⊆ B 0 ⊆ T 0 H + , B 0 / ( f ) is a domain , f = x h . (51) For every nonzero b ∈ B 0 b\in B_0 b ∈ B 0 r ≥ 0 r\geq0 r ≥ 0
b ∈ f r B 0 ⟺ w f ( b ) ≥ r . (52) b\in f^rB_0\Longleftrightarrow w_f(b)\geq r. \tag*{(52)} b ∈ f r B 0 ⟺ w f ( b ) ≥ r . (52) The forward implication follows from B 0 ⊆ T 0 H + B_0\subseteq T_0^{H^+} B 0 ⊆ T 0 H + m m m f f f b b b m b = f r P mb=f^rP mb = f r P P P P P ∈ P 0 ⊆ B 0 P\in P_0\subseteq B_0 P ∈ P 0 ⊆ B 0 x j x_j x j f B 0 fB_0 f B 0 x j / f ∈ B 0 x_j/f\in B_0 x j / f ∈ B 0 f f f (51) . Primality yields m ∉ ( f ) m\notin(f) m ∈ / ( f ) m b ∈ ( f ) mb\in(f) mb ∈ ( f ) b = f b 1 b=fb_1 b = f b 1 f f f (52) . Equivalently, after inverting the other initial variables, both ends of (51) become C [ f , x j ± 1 : j ≠ h ] \mathbb{C}[f,x_j^{\pm1}:j\neq h] C [ f , x j ± 1 : j = h ]
Writing v f ( b ) = max { r ≥ 0 : b ∈ f r B 0 } v_f(b)=\max\{r\geq0:b\in f^rB_0\} v f ( b ) = max { r ≥ 0 : b ∈ f r B 0 } v f ( b ) = w f ( b ) v_f(b)=w_f(b) v f ( b ) = w f ( b ) v f i ( f j ) = δ i j v_{f_i}(f_j)=\delta_{ij} v f i ( f j ) = δ ij [10 Example 4.1.1] , the ring C [ y , z , X , ( y + z ) / X ] \mathbb{C}[y,z,X,(y+z)/X] C [ y , z , X , ( y + z ) / X ] X X X y + z y+z y + z X X X X X X (51) , since ( y + z ) / X (y+z)/X ( y + z ) / X X X X
Classical coordinates and the collar correspondence
Proposition 4.3. In the classical fiber, B 0 = U f u l l , 0 + B_0=U_{\mathrm{full},0}^+ B 0 = U full , 0 +
Proof. Work over C \mathbb{C} C C \mathbb{C} C
Fully pinned twisted local systems. Write B + B^+ B + B − B^- B − S L n SL_n S L n U + U^+ U + S L n / U + SL_n/U^+ S L n / U + S L n / B + SL_n/B^+ S L n / B + ( U + , B − ) (U^+,B^-) ( U + , B − ) U + ∩ B − U^+\cap B^- U + ∩ B −
The fixed saturated arcs give a free fundamental groupoid graph with boundary intervals as objects. The tangent bundle of an oriented surface with nonempty boundary is trivializable. Require the tangent-fiber loop Ω \Omega Ω s I s_I s I s = ( − 1 ) n − 1 s=(-1)^{n-1} s = ( − 1 ) n − 1 r r r S L n SL_n S L n P e P_e P e e e e T γ T_\gamma T γ γ \gamma γ U + ∩ B − = 1 U^+\cap B^-=1 U + ∩ B − = 1
The ordinary map in [27 Section 3.5] from the surface S O ( 2 ) SO(2) S O ( 2 ) S O ( 3 ) SO(3) S O ( 3 ) Z → Z / 2 \mathbb{Z}\to\mathbb{Z}/2 Z → Z /2 Ω 2 \Omega^2 Ω 2 S L n SL_n S L n Ω ↦ s I \Omega\mapsto sI Ω ↦ s I s 2 = 1 s^2=1 s 2 = 1
Section 4.2 gives B 0 = O ( S L n ) ⊗ r B_0=\mathcal{O}(SL_n)^{\otimes r} B 0 = O ( S L n ) ⊗ r [27 Theorem 3.15] . Put
J a b = δ a , b ˉ , E a b = ( − 1 ) n − a δ a , b ˉ . (53) J_{ab}=\delta_{a,\bar b},\qquad E_{ab}=(-1)^{n-a}\delta_{a,\bar b}. \tag*{(53)} J ab = δ a , b ˉ , E ab = ( − 1 ) n − a δ a , b ˉ . (53) so that det E = 1 \det E=1 det E = 1 E 2 = s I E^2=sI E 2 = s I E − 1 = s E E^{-1}=sE E − 1 = s E ρ W \rho_W ρ W
M = J ρ W T E J , ρ W = ( J M J E − 1 ) T . (54) M=J\rho_W^{T}EJ,\qquad\rho_W=(JMJE^{-1})^{T}. \tag*{(54)} M = J ρ W T E J , ρ W = ( J M J E − 1 ) T . (54) This is a coordinate automorphism of each S L n SL_n S L n
We fix its relation to the IOS lift. IOS Definition 3.2 starts at the negative object E − E^- E − E + E^+ E + π \pi π E − E^- E − β − \beta_- β − E + E^+ E + E − E^- E − π \pi π E + E^+ E + β = β + \beta=\beta_+ β = β + R π ( x , v ) = ( x , − v ) R_\pi(x,v)=(x,-v) R π ( x , v ) = ( x , − v ) R π ( β − ) = β + R_\pi(\beta_-)=\beta_+ R π ( β − ) = β +
Let T T T J Σ J_\Sigma J Σ π / 2 \pi/2 π /2
j + ( v ) = ( T , v , J Σ v ) , j − ( v ) = ( T , − v , − J Σ v ) = j + ( R π v ) . (55) j_+(v)=(T,v,J_\Sigma v),\qquad j_-(v)=(T,-v,-J_\Sigma v)=j_+(R_\pi v). \tag*{(55)} j + ( v ) = ( T , v , J Σ v ) , j − ( v ) = ( T , − v , − J Σ v ) = j + ( R π v ) . (55) Under j + j_+ j + t e t_e t e ( T , t e , J Σ t e ) (T,t_e,J_\Sigma t_e) ( T , t e , J Σ t e ) π / 2 \pi/2 π /2 2 π 2\pi 2 π ρ W = s ρ β \rho_W=s\rho_\beta ρ W = s ρ β σ ( g ) = J g − T J \sigma(g)=Jg^{-T}J σ ( g ) = J g − T J (54) gives
M E − 1 = J ρ β T J = σ ( ρ β ) − 1 . (56) ME^{-1}=J\rho_\beta^{T}J=\sigma(\rho_\beta)^{-1}. \tag*{(56)} M E − 1 = J ρ β T J = σ ( ρ β ) − 1 . (56) This fixes the transpose, dual, inverse, and central sign s s s
A lift half-turn must also be distinguished from a change of canonical pinning. For a boundary interval denoted by E E E E − E^- E − E + E^+ E + A − = U + A_-=U^+ A − = U + A + = E − 1 h E U + A_+=E^{-1}h_EU^+ A + = E − 1 h E U + h E h_E h E E − 1 E^{-1} E − 1 (53) . If r E : E − → E + r_E:E^-\to E^+ r E : E − → E + [15 Section 3.3, Example 3.20] gives the gauge
P E + = T r E ( P E − ) E − 1 h E . (57) P_{E^+}=T_{r_E}(P_{E^-})E^{-1}h_E. \tag*{(57)} P E + = T r E ( P E − ) E − 1 h E . (57) The transformation R π R_\pi R π E − 1 h E E^{-1}h_E E − 1 h E
At an original mark let e − , e + e_-,e_+ e − , e + γ p \gamma_p γ p (56) gives the corner formula
C p = P e − − 1 T γ p ( P e + ) E . (58) C_p=P_{e_-}^{-1}T_{\gamma_p}(P_{e_+})E. \tag*{(58)} C p = P e − − 1 T γ p ( P e + ) E . (58) After transporting both frames to the same fiber this can be abbreviated as C p = P − − 1 P + E C_p=P^{-1}_{-}P_{+}E C p = P − − 1 P + E e − = e + e_-=e_+ e − = e + γ p \gamma_p γ p
Raw volumes and normalized triangle coordinates. In a reduced triangle the bad relations make the corner matrices C v C_v C v P ′ = P C v E − 1 P' = PC_vE^{-1} P ′ = P C v E − 1 P ′ B + = P B − P'B^+ = PB^- P ′ B + = P B − [15 Remark 3.10] , giving the same decorated-representation functor.
Take clockwise vertices v 1 , v 2 , v 3 v_1,v_2,v_3 v 1 , v 2 , v 3 E + E^+ E + e 3 e_3 e 3 v 3 v_3 v 3 C 1 = C v 1 C_1=C_{v_1} C 1 = C v 1 C 2 = C v 2 C_2=C_{v_2} C 2 = C v 2
P 3 = I , P 1 = C 1 E − 1 , P 2 = C 1 E − 1 C 2 E − 1 . (59) P_3=I,\qquad P_1=C_1E^{-1},\qquad P_2=C_1E^{-1}C_2E^{-1}. \tag*{(59)} P 3 = I , P 1 = C 1 E − 1 , P 2 = C 1 E − 1 C 2 E − 1 . (59) The IOS counterclockwise order is ( v 3 , v 2 , v 1 ) (v_3,v_2,v_1) ( v 3 , v 2 , v 1 ) v 1 , v 2 v_1,v_2 v 1 , v 2 ( U + , s P 2 U + , s P 1 U + ) (U^+,sP_2U^+,sP_1U^+) ( U + , s P 2 U + , s P 1 U + ) D = diag ( C 1 ) D=\operatorname{diag}(C_1) D = diag ( C 1 ) H = J diag ( C 2 ) J H=J\operatorname{diag}(C_2)J H = J diag ( C 2 ) J C 1 , C 2 C_1,C_2 C 1 , C 2 s P 1 U + = C 1 E U + = D E U + sP_1U^+=C_1EU^+=DEU^+ s P 1 U + = C 1 E U + = D E U + s P 2 U + = C 1 ( E − 1 C 2 E ) U + = C 1 H U + sP_2U^+=C_1(E^{-1}C_2E)U^+=C_1HU^+ s P 2 U + = C 1 ( E − 1 C 2 E ) U + = C 1 H U +
( A 1 , A 2 , A 3 ) = ( U + , C 1 H U + , D E U + ) . (60) (A_1,A_2,A_3)=(U^+,C_1HU^+,DEU^+). \tag*{(60)} ( A 1 , A 2 , A 3 ) = ( U + , C 1 H U + , D E U + ) . (60) For a matrix X X X X [ 1 , d ] X_{[1,d]} X [ 1 , d ] d d d i , j , k ≥ 0 i,j,k\ge0 i , j , k ≥ 0 i + j + k = n i+j+k=n i + j + k = n (16) and exterior expansion give
g i j k = det [ I [ 1 , k ] ∣ ( C 1 H ) [ 1 , j ] ∣ ( D E ) [ 1 , i ] ] . (61) g_{ijk}=\det\left[I_{[1,k]}\mid(C_1H)_{[1,j]}\mid(DE)_{[1,i]}\right]. \tag*{(61)} g ij k = det [ I [ 1 , k ] ∣ ( C 1 H ) [ 1 , j ] ∣ ( D E ) [ 1 , i ] ] . (61) Here Λ i E ( e 1 ∧ ⋯ ∧ e i ) = e n − i + 1 ∧ ⋯ ∧ e n \Lambda^iE(e_1\wedge\cdots\wedge e_i)=e_{n-i+1}\wedge\cdots\wedge e_n Λ i E ( e 1 ∧ ⋯ ∧ e i ) = e n − i + 1 ∧ ⋯ ∧ e n ( − 1 ) ( i 2 ) (-1)^{\binom{i}{2}} ( − 1 ) ( 2 i ) (61) has no additional sign.
For 0 ≤ a ≤ n 0\le a\le n 0 ≤ a ≤ n ω a \omega_a ω a diag ( t 1 , … , t n ) ↦ t 1 ⋯ t a \operatorname{diag}(t_1,\ldots,t_n)\mapsto t_1\cdots t_a diag ( t 1 , … , t n ) ↦ t 1 ⋯ t a ω 0 = ω n = 0 \omega_0=\omega_n=0 ω 0 = ω n = 0
Δ ω a , ω b , ω c = ( − 1 ) b c δ a b c , a + b + c = n . (62) \Delta_{\omega_a,\omega_b,\omega_c}=(-1)^{bc}\delta_{abc},\qquad a+b+c=n. \tag*{(62)} Δ ω a , ω b , ω c = ( − 1 ) b c δ ab c , a + b + c = n . (62) where δ \delta δ ( A , U + , E U + ) (A,U^+,EU^+) ( A , U + , E U + ) [ b + 1 , a + b ] [b+1,a+b] [ b + 1 , a + b ] b b b ( − 1 ) a b (-1)^{ab} ( − 1 ) ab ( − 1 ) ( n − 1 ) b (-1)^{(n-1)b} ( − 1 ) ( n − 1 ) b ( − 1 ) b c (-1)^{bc} ( − 1 ) b c [15 Remark 4.14] :
x i j k : = Δ ω i , ω k , ω j ( A 3 , A 1 , A 2 ) . x_{ijk}:=\Delta_{\omega_i,\omega_k,\omega_j}(A_3,A_1,A_2). x ij k := Δ ω i , ω k , ω j ( A 3 , A 1 , A 2 ) . Moving the last i i i
x i j k = ( − 1 ) i j + i k + j k g i j k . (63) x_{ijk}=(-1)^{ij+ik+jk}g_{ijk}. \tag*{(63)} x ij k = ( − 1 ) ij + ik + j k g ij k . (63) [12 Definitions 8.5–8.6, Theorem 8.7, Example 9.21] allow this distinguished vertex and standard reduced word in each face, followed by amalgamation. These are standard triangle seeds. The edge sign is ( − 1 ) i ( n − i ) (-1)^{i(n-i)} ( − 1 ) i ( n − i )
A global Cartan action realizes this symmetric sign. Choose z n = ( − 1 ) ( n 2 ) z^n=(-1)^{\binom{n}{2}} z n = ( − 1 ) ( 2 n )
h = diag ( z , − z , z , − z , … ) ∈ S L n , χ a ( h ) = z a ( − 1 ) ( a 2 ) . h=\operatorname{diag}(z,-z,z,-z,\ldots)\in SL_n,\qquad\chi_a(h)=z^a(-1)^{\binom{a}{2}}. h = diag ( z , − z , z , − z , … ) ∈ S L n , χ a ( h ) = z a ( − 1 ) ( 2 a ) . The determinant of h h h z n ( − 1 ) ( n 2 ) = 1 z^n(-1)^{\binom{n}{2}}=1 z n ( − 1 ) ( 2 n ) = 1 i + j + k = n i+j+k=n i + j + k = n ( n 2 ) = ( i 2 ) + ( j 2 ) + ( k 2 ) + i j + i k + j k \binom{n}{2}=\binom{i}{2}+\binom{j}{2}+\binom{k}{2}+ij+ik+jk ( 2 n ) = ( 2 i ) + ( 2 j ) + ( 2 k ) + ij + ik + j k
χ i ( h ) χ j ( h ) χ k ( h ) = ( − 1 ) i j + i k + j k . (64) \chi_i(h)\chi_j(h)\chi_k(h)=(-1)^{ij+ik+jk}. \tag*{(64)} χ i ( h ) χ j ( h ) χ k ( h ) = ( − 1 ) ij + ik + j k . (64) Apply the same h h h (64) . By the proof of [15 Proposition 6.2] , each mutable column has boundary weight zero; equivalently, its exchange monomials have equal Cartan weight. This property persists under mutation. The global action therefore rescales coordinates by scalars in every seed and preserves every Laurent ring. The normalized IOS and raw LY upper algebras are the same subalgebra of the common function field.
On a general surface, the elongated Y turns left along each mark’s finite link interval. Corner triangularity multiplies an extremal exterior vector by its diagonal weight. These are exactly the factors of the unique surviving state in LY13.3. Thus the global raw seed is still the transported-flag volume (61) , and the single global h h h
The inverse collar construction. The original surface here carries a full pinned system: its C p C_p C p S L n SL_n S L n § 5.3 was a reduced-triangle condition. We first restrict the full system to its F-open locus, then compare it with generic decorations on the enlarged surface Σ ∗ \Sigma^* Σ ∗
Let N N N Σ ∗ \Sigma^* Σ ∗ r → p r\to p r → p ( p , r , q e ) (p,r,q_e) ( p , r , q e ) e 2 e_2 e 2 r → q e r\to q_e r → q e e 3 e_3 e 3 q e → p q_e\to p q e → p r r r q e q_e q e
Let H S L n H_{SL_n} H S L n χ k ( diag ( t 1 , … , t n ) ) = t 1 ⋯ t k \chi_k(\operatorname{diag}(t_1,\ldots,t_n))=t_1\cdots t_k χ k ( diag ( t 1 , … , t n )) = t 1 ⋯ t k χ 0 = χ n = 1 \chi_0=\chi_n=1 χ 0 = χ n = 1 P e P_e P e
P e E − 1 U + . P_eE^{-1}U^+. P e E − 1 U + . Every decoration of this underlying flag is uniquely P e E − 1 U + ⋅ h e P_eE^{-1}U^+\cdot h_e P e E − 1 U + ⋅ h e h e ∈ H S L n h_e\in H_{SL_n} h e ∈ H S L n G / U + → G / B + G/U^+\to G/B^+ G / U + → G / B +
Put
d p , i = det C p [ 1 , … , i ; n − i + 1 , … , n ] , F = { d p , i : p , 1 ≤ i < n } . (65) d_{p,i}=\det C_p[1,\ldots,i;n-i+1,\ldots,n],\qquad F=\{d_{p,i}:p,1\le i<n\}. \tag*{(65)} d p , i = det C p [ 1 , … , i ; n − i + 1 , … , n ] , F = { d p , i : p , 1 ≤ i < n } . (65) By (58) , in the common frame of the preceding pinning, the underlying flags at the bridge endpoints are ( B − , C p B − ) (B^-,C_pB^-) ( B − , C p B − ) d p , i d_{p,i} d p , i (65) , and a square matrix over a test ring is invertible precisely when its determinant is a unit.
Starting with an F F F h e h_e h e F F F Σ ∗ \Sigma^* Σ ∗ e 2 e_2 e 2 h e h_e h e C \mathbb C C
A S L n , Σ ∗ × ≃ Spec B 0 [ F − 1 ] × H S L n N . (66) \mathcal{A}^{\times}_{SL_n,\Sigma^*}\simeq\operatorname{Spec} B_0[F^{-1}]\times H^N_{SL_n}. \tag*{(66)} A S L n , Σ ∗ × ≃ Spec B 0 [ F − 1 ] × H S L n N . (66) The two transported ports of a singleton circle and the T γ p T_{\gamma_p} T γ p (58) are retained. Even if p = r p=r p = r (66) in that case.
We identify the initial coordinates of this functor with (38) . In the attached triangle of clockwise order ( p , r , q e ) (p,r,q_e) ( p , r , q e ) v 3 = q e v_3=q_e v 3 = q e P e = I P_e=I P e = I A r = U + A_r=U^+ A r = U + A q e = E − 1 h e U + A_{q_e}=E^{-1}h_eU^+ A q e = E − 1 h e U + p p p C p = P e − 1 P + E C_p=P_e^{-1}P_+E C p = P e − 1 P + E
( A p , A r , A q e ) = ( C p E U + , U + , E − 1 h e U + ) , s ( P e − 1 P + ) U + = C p E U + . (67) (A_p,A_r,A_{q_e})=(C_pEU^+,U^+,E^{-1}h_eU^+),\qquad s(P_e^{-1}P_+)U^+=C_pEU^+. \tag*{(67)} ( A p , A r , A q e ) = ( C p E U + , U + , E − 1 h e U + ) , s ( P e − 1 P + ) U + = C p E U + . (67) For i + j + k = n i+j+k=n i + j + k = n
g i j k = det [ ( E − 1 h e ) [ 1 , k ] ∣ I [ 1 , j ] ∣ ( C p E ) [ 1 , i ] ] = det ( C p ) [ j + 1 , j + i ] , [ n − i + 1 , n ] χ k ( h e ) . (68) \begin{aligned}
g_{ijk}=\det\left[(E^{-1}h_e)_{[1,k]}\mid I_{[1,j]}\mid(C_pE)_{[1,i]}\right] \\
&=\det(C_p)_{[j+1,j+i],[n-i+1,n]}\,\chi_k(h_e). \tag*{(68)}
\end{aligned} g ij k = det [ ( E − 1 h e ) [ 1 , k ] ∣ I [ 1 , j ] ∣ ( C p E ) [ 1 , i ] ] = det ( C p ) [ j + 1 , j + i ] , [ n − i + 1 , n ] χ k ( h e ) . (68) The second equality has no additional s i s^i s i E e a = ( − 1 ) a − 1 e n + 1 − a E e_a=(-1)^{a-1}e_{n+1-a} E e a = ( − 1 ) a − 1 e n + 1 − a E − 1 = s E E^{-1}=sE E − 1 = s E k k k k k k s k ( − 1 ) k ( n − k ) = ( − 1 ) k ( 2 n − k − 1 ) = 1 s^k(-1)^{k(n-k)}=(-1)^{k(2n-k-1)}=1 s k ( − 1 ) k ( n − k ) = ( − 1 ) k ( 2 n − k − 1 ) = 1 (68) for every rank.
For i = 0 i=0 i = 0 w k = ( 0 , n − k , k ) w_k=(0,n-k,k) w k = ( 0 , n − k , k )
y w k = g 0 , n − k , k = χ k ( h e ) . (69) y_{w_k}=g_{0,n-k,k}=\chi_k(h_e). \tag*{(69)} y w k = g 0 , n − k , k = χ k ( h e ) . (69) Dividing (68) by this y p ( v ) y_{p(v)} y p ( v ) det ( C p ) [ j + 1 , j + i ] , [ n − i + 1 , n ] \det(C_p)_{[j+1,j+i],[n-i+1,n]} det ( C p ) [ j + 1 , j + i ] , [ n − i + 1 , n ] (38) . In particular, j = 0 j=0 j = 0 d p , i d_{p,i} d p , i [4 v1, Section 2.6] , M J I M^I_J M J I I I I J J J s k = ( − 1 ) k ( n − k ) s^k=(-1)^{k(n-k)} s k = ( − 1 ) k ( n − k ) (63) .
The χ 1 , … , χ n − 1 \chi_1,\ldots,\chi_{n-1} χ 1 , … , χ n − 1 X ∗ ( H S L n ) X^*(H_{SL_n}) X ∗ ( H S L n ) t 1 = χ 1 t_1=\chi_1 t 1 = χ 1 t a = χ a / χ a − 1 t_a=\chi_a/\chi_{a-1} t a = χ a / χ a − 1 t n = χ n − 1 − 1 t_n=\chi_{n-1}^{-1} t n = χ n − 1 − 1 (66) identifies the coordinate rings as
O ( A S L n , Σ ∗ × ) = B 0 [ F − 1 ] [ y w ± 1 : w ∈ W ] . (70) \mathcal{O}\left(\mathcal{A}^{\times}_{SL_n,\Sigma^*}\right)=B_0[F^{-1}][y_w^{\pm1}:w\in W]. \tag*{(70)} O ( A S L n , Σ ∗ × ) = B 0 [ F − 1 ] [ y w ± 1 : w ∈ W ] . (70) Classical upper equality and removal of frozen poles. On the independent saturated matrix generators, the map in (70) is (54) –(56) , hence is the classical skein collar map. Its initial-coordinate formula is (38) . The augmented C ~ \widetilde C C y y y
T Σ ∗ , t , 0 = T f u l l , t , 0 [ y w ± 1 : w ∈ W ] . (71) \mathcal{T}_{\Sigma^*,t,0}=\mathcal{T}_{\mathrm{full},t,0}[y_w^{\pm1}:w\in W]. \tag*{(71)} T Σ ∗ , t , 0 = T full , t , 0 [ y w ± 1 : w ∈ W ] . (71) All algebras here are commutative. An element of the initial enlarged torus has a unique finite expansion ∑ α ∈ Z W u α y α \sum_{\alpha\in\mathbb{Z}^W}u_\alpha y^\alpha ∑ α ∈ Z W u α y α y w y_w y w (71) , this element belongs to the enlarged torus at t t t u α u_\alpha u α T f u l l , t , 0 \mathcal{T}_{\mathrm{full},t,0} T full , t , 0 t t t
U 0 ( Σ ∗ ) = U f u l l , 0 f r [ y w ± 1 ] . (72) \mathcal{U}_0(\Sigma^*)=\mathcal{U}^{\mathrm{fr}}_{\mathrm{full},0}[y_w^{\pm1}]. \tag*{(72)} U 0 ( Σ ∗ ) = U full , 0 fr [ y w ± 1 ] . (72) By [15 Theorem 4.3] and the seed comparison in § 5.3 , the left side is the ring in (70) . Comparing y 0 y^0 y 0
B 0 [ F − 1 ] = U f u l l , 0 f r . (73) B_0[F^{-1}] = U^{\mathrm{fr}}_{\mathrm{full},0}. \tag*{(73)} B 0 [ F − 1 ] = U full , 0 fr . (73) For u ∈ U f u l l , 0 + u \in U^+_{\mathrm{full},0} u ∈ U full , 0 + (73) gives a presentation u = ( f r ) − 1 a u=(f^{\mathbf r})^{-1}a u = ( f r ) − 1 a a ∈ B 0 a \in B_0 a ∈ B 0 w f i ( u ) ≥ 0 w_{f_i}(u) \ge0 w f i ( u ) ≥ 0 (52) makes a a a f i r i f_i^{r_i} f i r i w f i ( f j ) = δ i j w_{f_i}(f_j)=\delta_{ij} w f i ( f j ) = δ ij u ∈ B 0 u \in B_0 u ∈ B 0 (42) , so
B 0 = U f u l l , 0 + . (74) B_0 = U^+_{\mathrm{full},0}. \tag*{(74)} B 0 = U full , 0 + . (74) Degrees of the initial full seed webs
Proposition 5.1. For every initial full mutable web g g g δ K ( g ) = δ 0 ( g 0 ) \delta_K(g)=\delta_0(g_0) δ K ( g ) = δ 0 ( g 0 )
Proof. Both degrees use the saturated oriented arc system fixed in § 5.2 .
Embedded arcs and tripods. [20 Section 11.3] constructs embedded elongated weighted Y-graphs on surfaces without interior punctures, and [20 Section 13.2] gives the initial seed webs. Boundary small vertices give colored arcs; interior small vertices give colored tripods. Full variables from attached triangles are the corner consecutive minors M [ i ˉ , n ] [ j + 1 , j + i ] ( c ) M^{[j+1,j+i]}_{[\bar{i},n]}(c) M [ i ˉ , n ] [ j + 1 , j + i ] ( c ) Σ \Sigma Σ 1 , … , n − 1 1,\ldots,n-1 1 , … , n − 1 [20 Lemma 13.3] , the initial trace is a nonzero monomial, so each g 0 g_0 g 0
The physical ports are distinct even when they correspond to the same compact marked point or open boundary interval. An embedded arc or tripod has a disk as regular neighborhood. The following argument concerns these webs.
To handle repeated compact corners, blow up each boundary mark p p p 0 < r < ε 0<r<\varepsilon 0 < r < ε 0 ≤ θ ≤ π 0\le\theta\le\pi 0 ≤ θ ≤ π θ = 0 \theta=0 θ = 0 θ 1 < ⋯ < θ s \theta_1<\cdots<\theta_s θ 1 < ⋯ < θ s 0 < r s < ⋯ < r 1 < ε 0<r_s<\cdots<r_1<\varepsilon 0 < r s < ⋯ < r 1 < ε h h h θ h \theta_h θ h r h r_h r h a < b a<b a < b a a a θ b \theta_b θ b b b b r b < r a r_b<r_a r b < r a a a a
These nested tracks preserve the source cyclic order and produce distinct ordered terminal ports. A singleton boundary circle can identify both compact edge endpoints with p p p
Proper walls and a median representative. By [19 Theorem 8.8] , the complementary regions of the fixed saturated system E \mathcal{E} E a e a_e a e c e c_e c e a e a_e a e
Cutting along the walls gives disk cells, each containing exactly one original open boundary interval. Indeed, the spokes divide each ideal monogon into sectors containing segments of original boundary intervals. Pairings across arc-sides join their half-sides in the linear order along the interval. Since the boundary component is an open interval, the sector chain cannot close and forms a disk. The dual ribbon graph G G G a e a_e a e G G G p p p γ p \gamma p γ p
Choose one metric for all seed webs. If the compact surface has negative Euler characteristic, take a compact hyperbolic core with geodesic boundary. Its universal cover Ω ⊂ H 2 \Omega\subset\mathbb{H}^2 Ω ⊂ H 2 ε > 0 \varepsilon> 0 ε > 0 Ω ε = { x : d ( x , Ω ) ≤ ε } \Omega_\varepsilon= \{x : d(x,\Omega) \le\varepsilon\} Ω ε = { x : d ( x , Ω ) ≤ ε }
For an annulus use the ε \varepsilon ε [6 Theorem 3.1] successively to the finite disjoint arc family, obtaining a simultaneous proper-arc isotopy from the walls to the chords, and extend it through the ideal collars. Pull the metric back by this isotopy. The original walls are then these chords, with the original a e a_e a e
Let Y Y Y D D D
The boundary of D D D D D D D D D γ D \gamma D γ D [6 Lemma 1.6] .
The lifted wall cells are intersections of Ω ε \Omega_\varepsilon Ω ε D D D D D D D D D D D D D D D
This median tripod is ribbon-isotopic to Y Y Y [6 Theorem 3.1] they are ambient-isotopic. Slide the trivalent vertex along the fixed path to the chosen source, then cut the surface along this path. Cyclic order determines the same side and cut component for the third arms. The fundamental group of that component injects into that of the surface by van Kampen for a proper-arc cut. The third arms are therefore relatively homotopic in the cut component. Apply the same theorem there, fixing the first two arms. Gluing gives an isotopy preserving the three leaves, boundary order, and ribbon framing. The chord and proper-arc argument likewise gives a reduced representative of a colored arc.
The cost in the next subsection is computed on the median representative. The isotopy identifies this representative with the original skein element.
The determinant-cone lower bound. Write the fixed oriented-edge variables of § 5.2 as independent X e X_e X e a e : start → end a_e:\text{ start}\to\text{ end} a e : start → end § 5.3 gives
X e = M e E − 1 = σ ( ρ e ) − 1 , σ ( g ) = J g − T J . X_e=M_eE^{-1}=\sigma(\rho_e)^{-1},\qquad\sigma(g)=J_g^{-T}J. X e = M e E − 1 = σ ( ρ e ) − 1 , σ ( g ) = J g − T J . Here M e M_e M e a e a_e a e ρ e \rho_e ρ e (56) . The map ρ e \rho_e ρ e σ ( ρ e ) \sigma(\rho_e) σ ( ρ e ) X e X_e X e a a a b b b ρ a b = ρ b ρ a \rho_{ab}=\rho_b\rho_a ρ ab = ρ b ρ a
X a b = σ ( ρ b ρ a ) − 1 = X a X b . X_{ab}=\sigma(\rho_b\rho_a)^{-1}=X_aX_b. X ab = σ ( ρ b ρ a ) − 1 = X a X b . The fixed endpoint basis change E − 1 E^{-1} E − 1 s = ( − 1 ) n − 1 s=(-1)^{n-1} s = ( − 1 ) n − 1 M e ↦ X e M_e\mapsto X_e M e ↦ X e Λ i X e \Lambda^iX_e Λ i X e Λ i X e − 1 \Lambda^iX_e^{-1} Λ i X e − 1 i i i n − i n-i n − i
A forward traversal of color i i i F i ( X ) = Λ i X F_i(X)=\Lambda^iX F i ( X ) = Λ i X i i i Λ i X \Lambda^iX Λ i X i i i i i i X X X
H i ( X ) = det ( X ) Λ i ( X − 1 ) , (75) H_i(X)=\det(X)\Lambda^i(X^{-1}), \tag*{(75)} H i ( X ) = det ( X ) Λ i ( X − 1 ) , (75) For increasing size-i i i I , J ⊆ { 1 , … , n } I,J\subseteq\{1,\ldots,n\} I , J ⊆ { 1 , … , n }
( H i ( X ) ) I , J = ( − 1 ) ∑ a ∈ I a + ∑ b ∈ J b det X J c , I c . (H_i(X))_{I,J}=(-1)^{\sum_{a\in I}a+\sum_{b\in J}b}\det X_{J^c,I^c}. ( H i ( X ) ) I , J = ( − 1 ) ∑ a ∈ I a + ∑ b ∈ J b det X J c , I c . where complements are increasing lists in { 1 , … , n } \{1,\ldots,n\} { 1 , … , n } n − i n-i n − i H i H_i H i Λ i ( X − 1 ) \Lambda^i(X^{-1}) Λ i ( X − 1 ) det X = 1 \det X=1 det X = 1
C ( g ) = ∑ forward traversals i + ∑ reverse traversals ( n − i ) . (76) C(g)=\sum_{\text{forward traversals}}i+\sum_{\text{reverse traversals}}(n-i). \tag*{(76)} C ( g ) = forward traversals ∑ i + reverse traversals ∑ ( n − i ) . (76) Tensor contraction along the reduced arms gives a homogeneous polynomial representative P g P_g P g g 0 g_0 g 0 C ( g ) C(g) C ( g ) C ( g ) C(g) C ( g ) P g P_g P g
Let η \eta η L e , R e ∈ S L n L_e,R_e\in SL_n L e , R e ∈ S L n
X e = L e diag ( η 1 , … , η n − 1 , 0 ) R e . (77) X_e=L_e\operatorname{diag}(\eta^1,\ldots,\eta^{n-1},0)R_e. \tag*{(77)} X e = L e diag ( η 1 , … , η n − 1 , 0 ) R e . (77) For an increasing subset I = { i 1 < ⋯ < i d } I=\{i_1<\cdots<i_d\} I = { i 1 < ⋯ < i d } ∣ I ∣ = e i 1 ∧ ⋯ ∧ e i d |I|=e_{i_1}\wedge\cdots\wedge e_{i_d} ∣ I ∣ = e i 1 ∧ ⋯ ∧ e i d ⟨ I ∣ \langle I| ⟨ I ∣ i i i [ 1 , i ] [1,i] [ 1 , i ] H i H_i H i η \eta η
F i ( X e ) = η i ( i + 1 ) / 2 ( Λ i L e ∣ [ 1 , i ] ⟩ ) ( ⟨ [ 1 , i ] ∣ Λ i R e ) + ⋯ (78) F_i(X_e)=\eta^{i(i+1)/2}(\Lambda^iL_e|[1,i]\rangle)(\langle[1,i]|\Lambda^iR_e)+\cdots \tag*{(78)} F i ( X e ) = η i ( i + 1 ) /2 ( Λ i L e ∣ [ 1 , i ]⟩) (⟨[ 1 , i ] ∣ Λ i R e ) + ⋯ (78) H i ( X e ) = η ( n − i ) ( n − i + 1 ) / 2 ( Λ i R e − 1 ∣ [ n − i + 1 , n ] ⟩ ) ( ⟨ [ n − i + 1 , n ] ∣ Λ i L e − 1 ) + ⋯ (79) H_i(X_e)=\eta^{(n-i)(n-i+1)/2}(\Lambda^iR_e^{-1}|[n-i+1,n]\rangle)(\langle[n-i+1,n]|\Lambda^iL_e^{-1})+\cdots \tag*{(79)} H i ( X e ) = η ( n − i ) ( n − i + 1 ) /2 ( Λ i R e − 1 ∣ [ n − i + 1 , n ]⟩) (⟨[ n − i + 1 , n ] ∣ Λ i L e − 1 ) + ⋯ (79) The only structurally forced zero between consecutive lowest factors is an immediate reversal on the same edge: its pairing is ⟨ [ 1 , i ] ∣ [ n − i + 1 , n ] ⟩ = 0 \langle[1,i]|[n-i+1,n]\rangle=0 ⟨[ 1 , i ] ∣ [ n − i + 1 , n ]⟩ = 0 0 < i < n 0<i<n 0 < i < n R e L e R_eL_e R e L e
The three first halfedges at a median are distinct, so the central volume uses independent edge-end flags. The two ends of a single loop arise from R e R_e R e L e − 1 L_e^{-1} L e − 1
The parameter scheme of all L e , R e L_e,R_e L e , R e η \eta η P g P_g P g P g P_g P g (44) ,
δ 0 ( g 0 ) = C ( g ) . (80) \delta_0(g_0)=C(g). \tag*{(80)} δ 0 ( g 0 ) = C ( g ) . (80) The quantum upper bound. Work over R 0 = C [ v ± 1 ] ( v − 1 ) R_0=\mathbb{C}[v^{\pm1}]_{(v-1)} R 0 = C [ v ± 1 ] ( v − 1 ) l l l [19 Section 7.2, equation (82)] gives a finite state sum with the same l l l k = 0 k=0 k = 0 [19 ] identifies its matrix coefficients with these counit values.
For the exterior projectors put q = q L Y = v − 2 n q=q_{\mathrm{LY}}=v^{-2n} q = q LY = v − 2 n β s \beta_s β s
T s = q 1 / n β s = v − 2 n β s . T_s=q^{1/n}\beta_s=v^{-2n}\beta_s. T s = q 1/ n β s = v − 2 n β s . The defining relations [LY(64), LS(43)] give T s − T s − 1 = ( q − q − 1 ) 1 T_s-T_s^{-1}=(q-q^{-1})1 T s − T s − 1 = ( q − q − 1 ) 1 ( T s − q ) ( T s + q − 1 ) = 0 (T_s-q)(T_s+q^{-1})=0 ( T s − q ) ( T s + q − 1 ) = 0 [19 (9)] : the equal-state block is q q q a < b a<b a < b ( e a ⊗ e b , e b ⊗ e a ) (e_a\otimes e_b,e_b\otimes e_a) ( e a ⊗ e b , e b ⊗ e a )
( q − q − 1 1 1 0 ) \begin{pmatrix}
q-q^{-1} & 1\\
1 & 0
\end{pmatrix} ( q − q − 1 1 1 0 ) It satisfies the quadratic relation, with ( 1 , − q ) (1,-q) ( 1 , − q ) − q − 1 -q^{-1} − q − 1 n n n
Let T σ T_\sigma T σ σ ∈ S i \sigma\in\mathfrak{S}_i σ ∈ S i
A i = ∑ σ ∈ S i ( − q ) − ℓ ( σ ) T σ , d i = ∑ σ ∈ S i q − 2 ℓ ( σ ) , P i = d i − 1 A i . (81) A_i=\sum_{\sigma\in\mathfrak{S}_i}(-q)^{-\ell(\sigma)}T_\sigma,\qquad
d_i=\sum_{\sigma\in\mathfrak{S}_i}q^{-2\ell(\sigma)},\qquad
P_i=d_i^{-1}A_i. \tag*{(81)} A i = σ ∈ S i ∑ ( − q ) − ℓ ( σ ) T σ , d i = σ ∈ S i ∑ q − 2 ℓ ( σ ) , P i = d i − 1 A i . (81) Fix an adjacent transposition s s s { σ , s σ } \{\sigma,s\sigma\} { σ , s σ } ℓ ( s σ ) = ℓ ( σ ) + 1 \ell(s\sigma)=\ell(\sigma)+1 ℓ ( s σ ) = ℓ ( σ ) + 1 A i A_i A i ( − q ) − ℓ ( σ ) ( T σ − q − 1 T s σ ) (-q)^{-\ell(\sigma)}(T_\sigma-q^{-1}T_{s\sigma}) ( − q ) − ℓ ( σ ) ( T σ − q − 1 T s σ ) T s T σ = T s σ T_sT_\sigma=T_{s\sigma} T s T σ = T s σ T s T s σ = T σ + ( q − q − 1 ) T s σ T_sT_{s\sigma}=T_\sigma+(q-q^{-1})T_{s\sigma} T s T s σ = T σ + ( q − q − 1 ) T s σ
T s ( T σ − q − 1 T s σ ) = − q − 1 T σ + q − 2 T s σ = − q − 1 ( T σ − q − 1 T s σ ) . T_s(T_\sigma-q^{-1}T_{s\sigma})=-q^{-1}T_\sigma+q^{-2}T_{s\sigma}=-q^{-1}(T_\sigma-q^{-1}T_{s\sigma}). T s ( T σ − q − 1 T s σ ) = − q − 1 T σ + q − 2 T s σ = − q − 1 ( T σ − q − 1 T s σ ) . Summing proves T s A i = − q − 1 A i T_sA_i=-q^{-1}A_i T s A i = − q − 1 A i A i 2 = ∑ σ ( − q ) − ℓ ( σ ) ( − q − 1 ) ℓ ( σ ) A i = d i A i A_i^2=\sum_{\sigma}(-q)^{-\ell(\sigma)}(-q^{-1})^{\ell(\sigma)}A_i=d_iA_i A i 2 = ∑ σ ( − q ) − ℓ ( σ ) ( − q − 1 ) ℓ ( σ ) A i = d i A i i i i d i ( 1 ) = i ! ≠ 0 d_i(1)=i!\ne0 d i ( 1 ) = i ! = 0 d i ∈ R 0 × d_i\in R_0^\times d i ∈ R 0 × P i 2 = P i P_i^2=P_i P i 2 = P i
Perform the following operations in order.
For a tripod, use the D-relation in the boundary-adjacent source cell to compare its external matrix coefficients and insert P i P_i P i P i 2 = P i P_i^2=P_i P i 2 = P i k = 0 k=0 k = 0
Resolve every vertex cell by the D-relation. Removing its actual boundary interval leaves an ordered interval. Connect the external ports to the true boundary by noncrossing fans in reverse order, following the fixed halfedge order of a e a_e a e
Replace a reverse color block using i ↦ n − i i\mapsto n-i i ↦ n − i a e a_e a e
This last operation can be realized on the fixed generating arc. For a wall c e c_e c e a e a_e a e a e a_e a e E \mathcal{E} E
For each passage, its first fan leg, strip, and second fan leg now form an embedded ribbon rectangle along the same a e a_e a e a e a_e a e a e a_e a e
In an ordinary bigon, the matrix coefficients of W i = im P i W_i=\operatorname{im}P_i W i = im P i i i i i i i
W i ⊗ W n − i ⟶ W n = R 0 . (82) W_i\otimes W_{n-i}\longrightarrow W_n=R_0. \tag*{(82)} W i ⊗ W n − i ⟶ W n = R 0 . (82) In the subset basis, only complementary subsets pair nontrivially, with value ± q a \pm q^a ± q a R 0 R_0 R 0 det q = 1 \det_q=1 det q = 1 W i ∗ ≃ W n − i W_i^*\simeq W_{n-i} W i ∗ ≃ W n − i R 0 R_0 R 0 ( n − i ) (n-i) ( n − i )
Transport these ordinary-bigon identities linearly back to the tubular neighborhoods. The result uses parallel copies of the fixed generating arcs. The endpoint fans already follow the halfedge order on the original intervals, so these are the generators a e a_e a e § 5.2 . If their heights differ from the chosen product order, [20 Lemma 4.8] changes states and boundary heights without changing the number of copies of an arc. Crossings, source coefficients, projectors, and state permutations contribute finite coefficients in R 0 R_0 R 0
A forward traversal consequently uses at most i i i n − i n-i n − i
g ∈ F C ( g ) B R 0 . (83) g \in FC_{(g)}BR_0. \tag*{(83)} g ∈ F C ( g ) B R 0 . (83) The free successive layers imply that reduction cannot increase degree. Equations (80) –(83) give C ( g ) = δ 0 ( g 0 ) ≤ δ K ( g ) ≤ C ( g ) C(g)=\delta_0(g_0)\leq\delta_K(g)\leq C(g) C ( g ) = δ 0 ( g 0 ) ≤ δ K ( g ) ≤ C ( g )
Quantum lifting and integral descent
Finite filtered lifting
Proposition 6.1. One has B R 0 = U f u l l , R 0 + B_{R_0}=U^+_{\mathrm{full},R_0} B R 0 = U full , R 0 + B K 0 = U f u l l , K 0 + B_{K_0}=U^+_{\mathrm{full},K_0} B K 0 = U full , K 0 +
Proof. The common parameter valuation. For 0 ≠ c ∈ C ( v ) 0\ne c\in\mathbb{C}(v) 0 = c ∈ C ( v ) ord p ( c ) = r \operatorname{ord}_p(c)=r ord p ( c ) = r c = p r g ( v ) / h ( v ) c=p^r g(v)/h(v) c = p r g ( v ) / h ( v ) g , h ∈ C [ v ] g,h\in\mathbb{C}[v] g , h ∈ C [ v ] g ( 1 ) h ( 1 ) ≠ 0 g(1)h(1)\ne0 g ( 1 ) h ( 1 ) = 0 T K 0 T_{K_0} T K 0
ν p ( ∑ a c a M ( a ) ) = min c a ≠ 0 ord p ( c a ) . (84) \nu_p\left(\sum_a c_aM(a)\right)=\min_{c_a\ne0}\operatorname{ord}_p(c_a). \tag*{(84)} ν p ( a ∑ c a M ( a ) ) = c a = 0 min ord p ( c a ) . (84) Set ν p ( 0 ) = + ∞ \nu_p(0)=+\infty ν p ( 0 ) = + ∞ ν p \nu_p ν p p r a 0 , p s b 0 p^r a_0,p^s b_0 p r a 0 , p s b 0 a 0 , b 0 a_0,b_0 a 0 , b 0 R 0 R_0 R 0 r − s r-s r − s a ‾ 0 / b ‾ 0 \overline{a}_0/\overline{b}_0 a 0 / b 0
At v = 1 v=1 v = 1 ν p \nu_p ν p
Consequently U R 0 + U^+_{R_0} U R 0 + p p p T R 0 H + T^{H+}_{R_0} T R 0 H + p u ∈ U R 0 + pu\in U^+_{R_0} p u ∈ U R 0 + u ∈ T R 0 H + u\in T^{H+}_{R_0} u ∈ T R 0 H + u u u u ∈ U R 0 + u\in U^+_{R_0} u ∈ U R 0 + u u u (74) ,
U R 0 + ‾ ⊆ U 0 + = B 0 . (85) \overline{U^+_{R_0}}\subseteq U^+_0=B_0. \tag*{(85)} U R 0 + ⊆ U 0 + = B 0 . (85) Injectivity of LY13.1 at v = 1 v=1 v = 1 B 0 B_0 B 0
Finite rank of bounded-degree upper lattices. For an integer D D D
E D = { 0 } ∪ { u ∈ U R 0 + : δ K ( u ) ≤ D } . (86) E_D=\{0\}\cup\{u\in U^+_{R_0}:\delta_K(u)\leq D\}. \tag*{(86)} E D = { 0 } ∪ { u ∈ U R 0 + : δ K ( u ) ≤ D } . (86) Subadditivity of δ K \delta_K δ K E D E_D E D R 0 R_0 R 0 p p p E D E_D E D p p p
E D / p E D ↪ T R 0 H + / p T R 0 H + . (87) E_D/pE_D\hookrightarrow T^{H+}_{R_0}/pT^{H+}_{R_0}. \tag*{(87)} E D / p E D ↪ T R 0 H + / p T R 0 H + . (87) Take u ∈ E D u\in E_D u ∈ E D u ‾ ≠ 0 \overline{u}\ne0 u = 0 b b b u u u a = b u a=bu a = b u B R 0 B_{R_0} B R 0
δ 0 ( b ˉ ) = δ K ( b ) . \delta_0(\bar{b})=\delta_K(b). δ 0 ( b ˉ ) = δ K ( b ) . We have b ˉ ≠ 0 \bar{b}\ne0 b ˉ = 0 a ˉ = b ˉ u ˉ ≠ 0 \bar{a}=\bar{b}\bar{u}\ne0 a ˉ = b ˉ u ˉ = 0 (85) , u ˉ ∈ B 0 \bar{u}\in B_0 u ˉ ∈ B 0
δ 0 ( u ˉ ) = δ 0 ( a ˉ ) − δ 0 ( b ˉ ) ≤ δ K ( a ) − δ K ( b ) = δ K ( u ) ≤ D . (88) \delta_0(\bar{u})=\delta_0(\bar{a})-\delta_0(\bar{b})
\le\delta_K(a)-\delta_K(b)=\delta_K(u)\le D. \tag*{(88)} δ 0 ( u ˉ ) = δ 0 ( a ˉ ) − δ 0 ( b ˉ ) ≤ δ K ( a ) − δ K ( b ) = δ K ( u ) ≤ D . (88) To recover an integral filtered bound from the generic one, use the fixed basis slices in (43) : F D B R 0 = B R 0 ∩ F D B K 0 F_D B_{R_0}=B_{R_0}\cap F_D B_{K_0} F D B R 0 = B R 0 ∩ F D B K 0 B K 0 B_{K_0} B K 0 δ K ( a ) = d \delta_K(a)=d δ K ( a ) = d a ∈ F d B R 0 a\in F_dB_{R_0} a ∈ F d B R 0 δ 0 ( a ˉ ) ≤ d \delta_0(\bar{a})\le d δ 0 ( a ˉ ) ≤ d a a a b b b (87) lies in the finite-dimensional space F D B 0 F_D B_0 F D B 0
Enumerate the initial mixed-torus monomials as e 1 , e 2 , … e_1,e_2,\ldots e 1 , e 2 , … L m = ⨁ j = 1 m R 0 e j L_m=\bigoplus_{j=1}^m R_0e_j L m = ⨁ j = 1 m R 0 e j L 0 = 0 L_0=0 L 0 = 0 pr m : L m → R 0 \operatorname{pr}_m:L_m\to R_0 pr m : L m → R 0
0 ⟶ E D ∩ L m − 1 ⟶ E D ∩ L m → pr m pr m ( E D ∩ L m ) ⟶ 0. 0\longrightarrow E_D\cap L_{m-1}\longrightarrow E_D\cap L_m
\xrightarrow{\operatorname{pr}_m}\operatorname{pr}_m(E_D\cap L_m)\longrightarrow0. 0 ⟶ E D ∩ L m − 1 ⟶ E D ∩ L m pr m pr m ( E D ∩ L m ) ⟶ 0. The image is an ideal of the DVR R 0 R_0 R 0 m m m m m m E D ∩ L m E_D\cap L_m E D ∩ L m E D E_D E D
Reduction of this direct-sum basis modulo p p p E D / p E D E_D/pE_D E D / p E D (87) –(88) , that vector space embeds in F D B 0 F_D B_0 F D B 0 dim C F D B 0 \dim_{\mathbb{C}}F_D B_0 dim C F D B 0 E D E_D E D
Nakayama and generic equality. By (42) , F D B R 0 ⊆ E D F_D B_{R_0}\subseteq E_D F D B R 0 ⊆ E D F D B 0 F_D B_0 F D B 0 (88) places the residue image of E D E_D E D u ∈ E D u\in E_D u ∈ E D b ∈ F D B R 0 b\in F_D B_{R_0} b ∈ F D B R 0 u − b = p w u-b=pw u − b = pw u − b ∈ E D u-b\in E_D u − b ∈ E D E D E_D E D p p p w ∈ E D w\in E_D w ∈ E D
E D = F D B R 0 + p E D . (89) E_D=F_D B_{R_0}+pE_D. \tag*{(89)} E D = F D B R 0 + p E D . (89) The quotient E D / F D B R 0 E_D/F_D B_{R_0} E D / F D B R 0 p p p z 1 , … , z l z_1,\ldots,z_l z 1 , … , z l z i = p ∑ j a i j z j z_i=p\sum_j a_{ij}z_j z i = p ∑ j a ij z j a i j ∈ R 0 a_{ij}\in R_0 a ij ∈ R 0 ( δ i j − p a i j ) (\delta_{ij}-pa_{ij}) ( δ ij − p a ij ) z i z_i z i p p p R 0 R_0 R 0 D D D
E D = F D B R 0 , B R 0 = U f u l l , R 0 + . (90) E_D=F_D B_{R_0},\qquad B_{R_0}=U^+_{\mathrm{full},R_0}. \tag*{(90)} E D = F D B R 0 , B R 0 = U full , R 0 + . (90) For D < 0 D<0 D < 0 F D = 0 F_D=0 F D = 0 E D = 0 E_D=0 E D = 0
For u ∈ U f u l l , K 0 + u\in U^+_{\mathrm{full},K_0} u ∈ U full , K 0 + ν p \nu_p ν p m ≥ 0 m\ge0 m ≥ 0 p m u p^m u p m u R 0 R_0 R 0 p m u ∈ U R 0 + p^m u\in U^+_{R_0} p m u ∈ U R 0 + (90) gives
B K 0 = U f u l l , K 0 + . (91) B_{K_0}=U^+_{\mathrm{full},K_0}. \tag*{(91)} B K 0 = U full , K 0 + . (91) For the fully frozen-inverted version, the equality B K 0 = U + B_{K_0}=U^+ B K 0 = U + h h h h B K 0 = B K 0 h hB_{K_0}=B_{K_0}h h B K 0 = B K 0 h w h w_h w h f N f^N f N u ∈ U f r u\in U^{\mathrm{fr}} u ∈ U fr
U + [ H − 1 ] = U f r , B K 0 [ H − 1 ] = U f u l l , K 0 f r . (92) U^{+}[H^{-1}] = U^{\mathrm{fr}}, \qquad B_{K_0}[H^{-1}] = U^{\mathrm{fr}}_{\mathrm{full},K_0}. \tag*{(92)} U + [ H − 1 ] = U fr , B K 0 [ H − 1 ] = U full , K 0 fr . (92) Integral descent and the projected algebra
Proposition 6.2. Over A = Z [ v ± 1 ] A = \mathbb{Z}[v^{\pm1}] A = Z [ v ± 1 ]
Proof. Descent from C ( v ) \mathbb{C}(v) C ( v ) Q ( v ) \mathbb{Q}(v) Q ( v ) Put K = Q ( v ) K = \mathbb{Q}(v) K = Q ( v ) L = C ( v ) L = \mathbb{C}(v) L = C ( v )
B L = B K ⊗ K L ⊆ T K ⊗ K L = T L . B_L = B_K \otimes_K L \subseteq T_K \otimes_K L = T_L. B L = B K ⊗ K L ⊆ T K ⊗ K L = T L . For u ∈ U f u l l , K + u \in U^{+}_{\mathrm{full},K} u ∈ U full , K + L L L (91) the element belongs to B L B_L B L 0 → B K → T K → T K / B K → 0 0 \to B_K \to T_K \to T_K/B_K \to0 0 → B K → T K → T K / B K → 0 u u u
U f u l l , K + ⊆ B K . (93) U^{+}_{\mathrm{full},K} \subseteq B_K. \tag*{(93)} U full , K + ⊆ B K . (93) Full-trace saturation at scalar primes. Continue with A = Z [ v ± 1 ] A = \mathbb{Z}[v^{\pm1}] A = Z [ v ± 1 ] π ∈ A \pi\in A π ∈ A A / ( π ) A/(\pi) A / ( π ) v v v [20 Theorem 13.1(a)] .
The skein presentation is the free module on finite diagrams modulo Laurent-coefficient relations, so it commutes with coefficient reduction. The trace uses finite state sums, and the balanced A / X A/X A / X b ∈ B A b \in B_A b ∈ B A π T A \pi T_A π T A B A / π B A B_A/\pi B_A B A / π B A A / ( π ) A/(\pi) A / ( π ) b ∈ π B A b \in\pi B_A b ∈ π B A
B A ∩ π T A = π B A . (94) B_A \cap\pi T_A = \pi B_A. \tag*{(94)} B A ∩ π T A = π B A . (94) For u ∈ U f u l l , A + u \in U^{+}_{\mathrm{full},A} u ∈ U full , A + (93) provides nonzero d ∈ A d \in A d ∈ A d u ∈ B A du \in B_A d u ∈ B A d d d A A A π ∣ d \pi\mid d π ∣ d d u ∈ B A ∩ π T A du \in B_A \cap\pi T_A d u ∈ B A ∩ π T A (94) , write d u = π b 1 du = \pi b_1 d u = π b 1 π \pi π A A A T A T_A T A ( d / π ) u = b 1 ∈ B A (d/\pi)u = b_1 \in B_A ( d / π ) u = b 1 ∈ B A
U f u l l , A + ⊆ B A . (95) U^{+}_{\mathrm{full},A} \subseteq B_A. \tag*{(95)} U full , A + ⊆ B A . (95) The finite-neighbor criterion. For a compatible quantum seed over A A A
U f r = T 0 ∩ ⋂ k ∈ Mut T μ k 0 . (96) U^{\mathrm{fr}} = T_0 \cap\bigcap_{k \in\operatorname{Mut}} T_{\mu_k0}. \tag*{(96)} U fr = T 0 ∩ k ∈ Mut ⋂ T μ k 0 . (96) Compatibility gives full column rank of the extended exchange matrix, the condition in the BZ coprimality argument. The P , 2 n 2 P, 2n^2 P , 2 n 2 § 3.3 makes the theorem applicable.
By the common frozen valuations of § 5.1 , mixed membership is equivalent to (96) together with nonnegative initial H H H
U + = T 0 H + ∩ ⋂ k ∈ Mut T μ k 0 H + . (97) U^{+} = T_0^{H+} \cap\bigcap_{k \in\operatorname{Mut}} T_{\mu_k0}^{H+}. \tag*{(97)} U + = T 0 H + ∩ k ∈ Mut ⋂ T μ k 0 H + . (97) The valuation comparison is established over the fraction field. Requiring coefficients in A A A (97) also holds integrally. General specializations will be treated separately in Section 7.
Clearing denominators before taking the quotient. Partition the full initial indices as
I ⊔ F ⊔ E m u t ⊔ H . (98) I \sqcup F \sqcup E_{\mathrm{mut}} \sqcup H. \tag*{(98)} I ⊔ F ⊔ E mut ⊔ H . (98) Here I I I F F F E m u t E_{\mathrm{mut}} E mut H H H
Formula [LY(208)] and the CHW initial construction give two facts. The full P P P I ∪ F I \cup F I ∪ F i ∈ I i \in I i ∈ I E m u t ∪ H E_{\mathrm{mut}} \cup H E mut ∪ H I ∪ F I \cup F I ∪ F
Take u ∈ U A ( Σ ) u \in U_A(\Sigma) u ∈ U A ( Σ ) P P P I ∪ F I \cup F I ∪ F F F F m m m
b = m u (99) b = mu \tag*{(99)} b = m u (99) has nonnegative initial F F F (97) :
In the initial mixed torus, the H H H
At a neighbor in i ∈ I i \in I i ∈ I u u u m m m b b b
At a neighbor in e ∈ E m u t e \in E_{\mathrm{mut}} e ∈ E mut u , m u,m u , m
At a neighbor in f ∈ F f \in F f ∈ F b b b x f x_f x f x f x_f x f x f − 1 x_f^{-1} x f − 1 H H H
Equations (97) and (95) give b ∈ B A b \in B_A b ∈ B A (99) may still contain mutable inverses. Choose a positive I I I c c c
c b = P ( x v : v ∈ I ∪ F ) (100) cb = P(x_v : v \in I \cup F) \tag*{(100)} c b = P ( x v : v ∈ I ∪ F ) (100) is a finite linear combination of positive monomials. The three elements c , b , P c,b,P c , b , P B A B_A B A (100) is an identity in that algebra.
Now apply the algebra map π : B A → S A ⊆ T p r o j , A \pi: B_A \to S_A \subseteq T_{\mathrm{proj},A} π : B A → S A ⊆ T proj , A I ∪ F I \cup F I ∪ F [20 Section 13.2, Lemma 13.3] . Hence
π ( c ) π ( b ) = P p r o j = π ( c ) m p r o j u . \pi(c)\pi(b) = P_{\mathrm{proj}} = \pi(c)m_{\mathrm{proj}}u. π ( c ) π ( b ) = P proj = π ( c ) m proj u . In the projected torus, π ( c ) \pi(c) π ( c ) π ( b ) = m p r o j u \pi(b) = m_{\mathrm{proj}}u π ( b ) = m proj u m p r o j m_{\mathrm{proj}} m proj S A S_A S A
U A ( Σ ) ⊆ S A ( Σ ) . (101) U_A(\Sigma) \subseteq S_A(\Sigma). \tag*{(101)} U A ( Σ ) ⊆ S A ( Σ ) . (101) Together with (33) , this gives
S A = A A = U A . (102) S_A = A_A = U_A. \tag*{(102)} S A = A A = U A . (102) The two denominator clearings keep the argument inside the algebra on which the quotient is defined: full frozen variables H H H π \pi π
Coefficient specialization
Proposition 7.1. For every coefficient domain R R R v R v_R v R S R ( Σ ) = A R ( Σ ) = U R ( Σ ) S_R(\Sigma)=A_R(\Sigma)=U_R(\Sigma) S R ( Σ ) = A R ( Σ ) = U R ( Σ )
Proof. To pass from (102) to nonflat coefficient changes, let M = Z V A M=\mathbb{Z}^{V_A} M = Z V A I I I Z I \mathbb{Z}^{I} Z I
Conventions and reddening. For the actual projected seed P , Q P,Q P , Q
B D M = − Q , Λ D M = − P / ( 2 n 2 ) , t = v − 2 n 2 . (103) B_{\mathrm{DM}}=-Q,\qquad\Lambda_{\mathrm{DM}}=-P/(2n^2),\qquad t=v^{-2n^2}. \tag*{(103)} B DM = − Q , Λ DM = − P / ( 2 n 2 ) , t = v − 2 n 2 . (103) Then Z [ t ± 1 / ( 2 n 2 ) ] = A \mathbb{Z}[t^{\pm1/(2n^2)}]=A Z [ t ± 1/ ( 2 n 2 ) ] = A t Λ D M ( a , b ) = v P ( a , b ) t^{\Lambda_{\mathrm{DM}}(a,b)}=v^{P(a,b)} t Λ DM ( a , b ) = v P ( a , b ) B 1 ( e i ) B_1(e_i) B 1 ( e i ) Q ( − , i ) Q(-,i) Q ( − , i ) P Q ( − , i ) = − 2 n 2 e i T PQ(-,i)=-2n^2e_i^T P Q ( − , i ) = − 2 n 2 e i T + Q N I +Q\mathbb{N}^{I} + Q N I B 1 ( Z I ) B_1(\mathbb{Z}^{I}) B 1 ( Z I )
More explicitly, DM mutation changes the basis to e k ′ = − e k e'_k=-e_k e k ′ = − e k e r ′ = e r + [ − Q ( r , k ) ] + e k e'_r=e_r+[-Q(r,k)]_+e_k e r ′ = e r + [ − Q ( r , k ) ] + e k r ≠ k r\ne k r = k T k ( m ) = m + [ m k ] + Q ( − , k ) T_k(m)=m+[m_k]_+Q(-,k) T k ( m ) = m + [ m k ] + Q ( − , k ) Q ( r , k ) [ m k ] + + [ − Q ( r , k ) ] + m k Q(r,k)[m_k]_+ + [-Q(r,k)]_+m_k Q ( r , k ) [ m k ] + + [ − Q ( r , k ) ] + m k r r r m k ≥ 0 m_k\ge0 m k ≥ 0 m k ≤ 0 m_k\le0 m k ≤ 0
m k ′ = − m k , m r ′ = m r + [ Q ( r , k ) ] + [ m k ] + − [ − Q ( r , k ) ] + [ − m k ] + . (104) m'_k=-m_k,\qquad m'_r=m_r+[Q(r,k)]_+[m_k]_+-[-Q(r,k)]_+[-m_k]_+. \tag*{(104)} m k ′ = − m k , m r ′ = m r + [ Q ( r , k ) ] + [ m k ] + − [ − Q ( r , k ) ] + [ − m k ] + . (104) These are the coordinates of [24 Section 3.3] . Thus both the torus product and compatible pointedness use the same conventions.
[11 Theorems 1.3 and 1.12] give a cluster DT transformation, and hence reddening, for admissible surface principal quivers. Here g g g μ \mu μ μ \mu μ g + μ ≥ 3 g+\mu\ge3 g + μ ≥ 3 g ≥ 1 g\ge1 g ≥ 1 μ ≥ 2 \mu\ge2 μ ≥ 2 g = 0 g=0 g = 0 μ = 2 \mu=2 μ = 2 g = 0 g=0 g = 0 μ ≥ 3 \mu\ge3 μ ≥ 3
By GS16(260), deleting boundary frozen vertices from the A-quiver gives the X-quiver, hence the required principal mutable quiver. For the c/g-vector comparison, use principal coefficients, obtained by adjoining an identity frozen block to the principal exchange matrix. Let C i C_i C i C i = e i C_i=e_i C i = e i sgn ( C i ) \operatorname{sgn}(C_i) sgn ( C i ) C C C G G G ε = − Q p r i n = Q p r i n T \varepsilon=-Q_{\mathrm{prin}}=Q_{\mathrm{prin}}^T ε = − Q prin = Q prin T i ≠ k i\ne k i = k
C i ′ = C i + [ sgn ( C k ) ε i k ] + C k , C k ′ = − C k . (105) C'_i=C_i+[\operatorname{sgn}(C_k)\varepsilon_{ik}]_+C_k,\qquad C'_k=-C_k. \tag*{(105)} C i ′ = C i + [ sgn ( C k ) ε ik ] + C k , C k ′ = − C k . (105) After transposing the row matrix, ε i k = Q k i \varepsilon_{ik}=Q_{ki} ε ik = Q k i Q Q Q G T C = I G^TC=I G T C = I [25 Theorem 2.2.6] makes the terminal principal g-vectors a negative basis permutation. Extended labels add only integral frozen coordinates, and the principal g-vectors are independent of the added frozen rows. GS Theorem 3.10 supplies the corresponding result for the opposite quiver. A seed is injective-reachable when a finite mutation sequence gives mutable variables whose principal g g g n = 2 n=2 n = 2
Finiteness of all theta functions. A Laurent series is m m m m m m m + Q N I m+\mathrm{Q}\mathbb{N}^{I} m + Q N I Q N I = { ∑ i ∈ I c i Q ( − , i ) : c i ≥ 0 } \mathrm{Q}\mathbb{N}^{I}=\{\sum_{i\in I}c_iQ(-,i):c_i\geq0\} Q N I = { ∑ i ∈ I c i Q ( − , i ) : c i ≥ 0 } (104) in every seed under consideration. For this lattice, [5 Theorems 1.1–1.2, Corollaries 4.13 and A.4] construct formal ϑ m \vartheta_m ϑ m m + Q N I m+\mathrm{Q}\mathbb{N}^{I} m + Q N I ϑ p ϑ q \vartheta_p\vartheta_q ϑ p ϑ q ϑ p + q \vartheta_{p+q} ϑ p + q
Let Θ ⊆ M \Theta\subseteq M Θ ⊆ M p , q ∈ Θ p,q\in\Theta p , q ∈ Θ t Λ D M ( p , q ) ϑ p + q t^{\Lambda_{\mathrm{DM}}(p,q)}\vartheta_{p+q} t Λ DM ( p , q ) ϑ p + q ϑ p + q \vartheta_{p+q} ϑ p + q p + q ∈ Θ p+q\in\Theta p + q ∈ Θ
The initial variables give e i ∈ Θ e_i\in\Theta e i ∈ Θ g g g − e σ ( i ) + c i -e_{\sigma(i)}+c_i − e σ ( i ) + c i c i c_i c i − c i -c_i − c i − e i ∈ Θ -e_i\in\Theta − e i ∈ Θ
Θ = M . (106) \Theta=M. \tag*{(106)} Θ = M . (106) Every formal theta function is therefore a finite element of U A U_A U A
The pointed basis theorem over arbitrary rings. We use Section 3 of [24 2512.05228v2, 2026-01-29] . Full rank in its Section 3.3 is an integral-lattice condition, satisfied here by compatibility. Theorem 3.17 states that, for an injective-reachable full-rank seed, an upper-algebra family indexed by all lattice labels and compatibly pointed along a reaching sequence is a basis over the coefficient ring.
The restriction involving the possible zero divisor 2 in Assumption 3.1 concerns isolated extended mutable columns. Our extended matrix has full column rank over the integers, hence no zero mutable column. Lemma 3.15 applies: the exchange products are monic regular polynomials. Through [25 v4, Proposition 4.2.1, Theorem 4.3.1, Lemmas 3.1.2 and 3.4.12] , Theorem 3.17 obtains finite spanning from monic leading terms, a truncated nilpotent-ideal Nakayama argument, and finite intervals between dominance bounds. The nilpotence of that ideal makes the argument valid over the coefficient ring in Theorem 3.17.
By (106) , our theta family satisfies this theorem. For a commutative unital ring R R R v R ∈ R × v_R\in R^\times v R ∈ R ×
U A ⊗ A R ≅ U R , { ϑ m , R : m ∈ M } is an R -basis . (107) U_A\otimes_A R\cong U_R,\qquad\{\vartheta_{m,R}:m\in M\}\text{ is an }R\text{-basis}. \tag*{(107)} U A ⊗ A R ≅ U R , { ϑ m , R : m ∈ M } is an R -basis . (107) All frozen variables are inverted here. We apply (107) to the coefficient domains allowed in the conjectures.
Specializing finite skein preimages. By (102) , every universal ϑ m \vartheta_m ϑ m A A A ϑ m , R \vartheta_{m,R} ϑ m , R S R S_R S R U R U_R U R (107) , so u R ⊆ S R u_R \subseteq S_R u R ⊆ S R
The finite local formulas of § 4.2 and the quantum Laurent identities specialize as well, giving S R ⊆ A R ⊆ U R S_R \subseteq A_R \subseteq U_R S R ⊆ A R ⊆ U R
S R ( Σ ) = A R ( Σ ) = U R ( Σ ) (108) S_R(\Sigma)=A_R(\Sigma)=U_R(\Sigma) \tag*{(108)} S R ( Σ ) = A R ( Σ ) = U R ( Σ ) (108) for every allowed coefficient domain and parameter.
For disconnected surfaces, apply the argument directly to the block seed of a finite disjoint union. Take the union of the saturated systems, the sum of word lengths, and the product of determinant cones. The full classical pinned and collar functors are componentwise products; the y y y
The Huang–Wang generating family
Proposition 8.1. The algebra generated over the frozen Laurent coefficient algebra by the variables in Huang–Wang Conjecture 1.6 equals S R ( Σ ) S_R(\Sigma) S R ( Σ )
Proof. The coefficient algebra in the conjecture is B f r , R \mathbb{B}_{\mathrm{fr},R} B fr , R (30) at ( i , j ) = ( 1 , n ) (i,j)=(1,n) ( i , j ) = ( 1 , n ) F 12 F_{12} F 12 F 01 F_{01} F 01
Ordered quantum cofactors of the reverse arc. In the ordinary full bigon, index u i j u_{ij} u ij u ⃗ \vec{u} u S S S
S ( u i j ) = ( − q ) i − j det q ( u j i ) = ( − q ) i − j u ⃗ j ˉ , i ˉ , (109) S(u_{ij})=(-q)^{i-j}\det\nolimits_q(u^{ji})=(-q)^{i-j}\vec{u}_{\bar{j},\bar{i}}, \tag*{(109)} S ( u ij ) = ( − q ) i − j det q ( u j i ) = ( − q ) i − j u j ˉ , i ˉ , (109) where q = q L Y = v − 2 n 2 q=q_{\mathrm{LY}}=v^{-2n^2} q = q LY = v − 2 n 2 u j i u^{ji} u j i j j j i i i [20 Proposition 4.4] to obtain
u j ˉ , i ˉ = det q ( u ⃗ j i ) . (110) u_{\bar{j},\bar{i}}=\det\nolimits_q(\vec{u}^{ji}). \tag*{(110)} u j ˉ , i ˉ = det q ( u j i ) . (110) For an oriented essential arc γ \gamma γ [20 Section 4.12] induces an algebra homomorphism. Apply it to (110) in the full surface skein algebra and then project.
Let U i j U_{ij} U ij γ \gamma γ W a b W_{ab} W ab W a b = U ⃗ b a W_{ab}=\vec{U}_{ba} W ab = U ba i = 1 , j = n i=1,j=n i = 1 , j = n (110) , the row-ordered determinant becomes the following column-ordered polynomial in W W W
U 1 n = ∑ σ ∈ S n − 1 ( − q ) ℓ ( σ ) W σ ( 1 ) + 1 , 1 W σ ( 2 ) + 1 , 2 ⋯ W σ ( n − 1 ) + 1 , n − 1 . (111) U_{1n}=\sum_{\sigma\in S_{n-1}}(-q)^{\ell(\sigma)}W_{\sigma(1)+1,1}W_{\sigma(2)+1,2}\cdots W_{\sigma(n-1)+1,n-1}. \tag*{(111)} U 1 n = σ ∈ S n − 1 ∑ ( − q ) ℓ ( σ ) W σ ( 1 ) + 1 , 1 W σ ( 2 ) + 1 , 2 ⋯ W σ ( n − 1 ) + 1 , n − 1 . (111) Each first index on the right is at least two and each second index is at most n − 1 n-1 n − 1 ( 1 , n ) (1,n) ( 1 , n ) n = 2 n=2 n = 2 U 12 = W 21 U_{12}=W_{21} U 12 = W 21
The original generating family and S R S_R S R Let G R G_R G R B f r , R \mathbb{B}_{fr,R} B f r , R
The corrections of § 4.2 retain the original mutation words for all m ≥ 2 m \ge2 m ≥ 2 D 1 n D_{1n} D 1 n G R G_R G R D 1 n D_{1n} D 1 n (111) . Each reverse-oriented entry on its right has an unchanged valid formula and belongs to G R G_R G R § 4.2 gives S R ⊆ G R S_R \subseteq G_R S R ⊆ G R
Conversely, multiply the correct arc formula of each unchanged numerator by its actual boundary factors to place that numerator in S R S_R S R F 12 F_{12} F 12 [20 Lemma 13.3] and [14 Theorem 2.11(a)] . If frozen, it is already in the coefficient algebra. § 4.4 supplies all frozen units. Hence
G R = S R . (112) G_R = S_R. \tag*{(112)} G R = S R . (112) Together with (108) , this proves Conjecture 1.6 for its original generating family.
Proof of Theorem 1.1 . Proposition 7.1 gives the equality of the projected skein, cluster, and upper cluster algebras over every allowed coefficient domain. Proposition 8.1 identifies their generators over the frozen Laurent algebra with the family in the original Conjecture 1.6. The surface and parameter assumptions in both propositions are those of the theorem.
Local mutation rows and quadrilateral cutting
We give the complete exchange-row calculations used in § 4.2 , for arbitrary integral indices in the stated ranges.
Forward complete rows
Lemma A.1. The forward mutation word of § 4.2 has exactly the four exchange-neighbor lists in the table below. Deleting pivots with second index at least m m m
Proof. Fix i i i p = i − 1 , … , 2 p=i-1,\ldots,2 p = i − 1 , … , 2 q = 1 , … , p − 1 q=1,\ldots,p-1 q = 1 , … , p − 1 Z p q Z_{pq} Z pq
Table 3.
Corner labels equal one in products and zero as incidence vectors. When boundary labels coincide, their row contributions are added or canceled.
Describe mutation at k k k I k I_k I k O k O_k O k S k S_k S k − 1 -1 − 1 k k k
Q n e w = S k ( Q o l d + I k ∧ O k ) S k , ( I ∧ O ) u v = I u O v − O u I v , (113) Q_{\mathrm{new}}=S_k(Q_{\mathrm{old}}+I_k\wedge O_k)S_k,\qquad(I\wedge O)_{uv}=I_uO_v-O_uI_v, \tag*{(113)} Q new = S k ( Q old + I k ∧ O k ) S k , ( I ∧ O ) uv = I u O v − O u I v , (113) with conventions I u = [ Q ( u , k ) ] + I_u=[Q(u,k)]_+ I u = [ Q ( u , k ) ] + O u = [ Q ( k , u ) ] + O_u=[Q(k,u)]_+ O u = [ Q ( k , u ) ] +
If previous pivots have each been visited once, let S ( r ) S^{(r)} S ( r ) − 1 -1 − 1
S ( r ) Q ( r ) S ( r ) = Q ( 0 ) + ∑ ℓ < r ( S ( ℓ ) I ℓ ) ∧ ( S ( ℓ ) O ℓ ) . (114) S^{(r)}Q^{(r)}S^{(r)}=Q^{(0)}+\sum_{\ell<r}(S^{(\ell)}I_\ell)\wedge(S^{(\ell)}O_\ell). \tag*{(114)} S ( r ) Q ( r ) S ( r ) = Q ( 0 ) + ℓ < r ∑ ( S ( ℓ ) I ℓ ) ∧ ( S ( ℓ ) O ℓ ) . (114) In a previously verified complete row, every interior neighbor visited later is an axial grid neighbor; exceptional terms lie on unvisited boundary indices. Hence the row at ( p , q ) (p,q) ( p , q ) ( p + 1 , q ) (p+1,q) ( p + 1 , q ) ( p , q − 1 ) (p,q-1) ( p , q − 1 )
Substituting the top boundary values
Z i , q = A i − 1 , 0 A i , q + 1 , Z p , 0 = A i 1 A p − 1 , 0 Z_{i,q}=A_{i-1,0}A_{i,q+1},\qquad Z_{p,0}=A_{i1}A_{p-1,0} Z i , q = A i − 1 , 0 A i , q + 1 , Z p , 0 = A i 1 A p − 1 , 0 in (18) recovers the extra top-row and first-column factors in the table.
Under truncation at q ≥ m ≥ 2 q\ge m\ge2 q ≥ m ≥ 2 A i − 1 , 0 A_{i-1,0} A i − 1 , 0 A i − 2 , q A_{i-2,q} A i − 2 , q A i , q + 1 A_{i,q+1} A i , q + 1 Z p , q − 1 Z_{p,q-1} Z p , q − 1 A p , q + 1 A_{p,q+1} A p , q + 1 q q q A i − 1 , 0 A_{i-1,0} A i − 1 , 0 Z p + 1 , q Z_{p+1,q} Z p + 1 , q A p − 1 , q A_{p-1,q} A p − 1 , q Z p , q − 1 Z_{p,q-1} Z p , q − 1 A p , q + 1 A_{p,q+1} A p , q + 1 q q q Z p , q − 1 Z_{p,q-1} Z p , q − 1 m m m y s y_s y s s < m s<m s < m
Reverse complete rows
Lemma A.2. The reverse mutation word of Section 3.2 has the four exchange-neighbor lists below and yields recurrence (21) .
Proof. Fix j < n j<n j < n k = j − 1 , … , 1 k=j-1,\ldots,1 k = j − 1 , … , 1 t = n − j , … , 1 t=n-j,\ldots,1 t = n − j , … , 1 F k , k + t F_{k,k+t} F k , k + t W k + 1 , t W_{k+1,t} W k + 1 , t W k , t + 1 W_{k,t+1} W k , t + 1
Table 1.
In (A.2), the earlier axial pivots contributing to this row are ( k + 1 , t ) (k+1,t) ( k + 1 , t ) k < j − 1 k<j-1 k < j − 1 ( k , t + 1 ) (k,t+1) ( k , t + 1 ) t < n − j t<n-j t < n − j
W j , t = F j − 1 , n F j , j + t − 1 , W k , n − j + 1 = F j , n F k − 1 , k + n − j , W_{j,t}=F_{j-1,n}F_{j,j+t-1},\qquad W_{k,n-j+1}=F_{j,n}F_{k-1,k+n-j}, W j , t = F j − 1 , n F j , j + t − 1 , W k , n − j + 1 = F j , n F k − 1 , k + n − j , @page 38/43 starts_mid=1 ends_mid=1 Describe mutation at k k k I k I_k I k O k O_k O k S k S_k S k − 1 -1 − 1 k k k
Q n e w = S k ( Q o l d + I k ∧ O k ) S k , ( I ∧ O ) u v = I u O v − O u I v , (115) Q_{\mathrm{new}}=S_k(Q_{\mathrm{old}}+I_k\wedge O_k)S_k,\qquad(I\wedge O)_{uv}=I_uO_v-O_uI_v, \tag*{(115)} Q new = S k ( Q old + I k ∧ O k ) S k , ( I ∧ O ) uv = I u O v − O u I v , (115) with conventions I u = [ Q ( u , k ) ] + I_u=[Q(u,k)]_+ I u = [ Q ( u , k ) ] + O u = [ Q ( k , u ) ] + O_u=[Q(k,u)]_+ O u = [ Q ( k , u ) ] +
If previous pivots have each been visited once, let S ( r ) S^{(r)} S ( r ) − 1 -1 − 1
S ( r ) Q ( r ) S ( r ) = Q ( 0 ) + ∑ ℓ < r ( S ( ℓ ) I ℓ ) ∧ ( S ( ℓ ) O ℓ ) . (116) S^{(r)}Q^{(r)}S^{(r)}=Q^{(0)}+\sum_{\ell<r}(S^{(\ell)}I_\ell)\wedge(S^{(\ell)}O_\ell). \tag*{(116)} S ( r ) Q ( r ) S ( r ) = Q ( 0 ) + ℓ < r ∑ ( S ( ℓ ) I ℓ ) ∧ ( S ( ℓ ) O ℓ ) . (116) In a previously verified complete row, every interior neighbor visited later is an axial grid neighbor; exceptional terms lie on unvisited boundary indices. Hence the row at ( p , q ) (p,q) ( p , q ) ( p + 1 , q ) (p+1,q) ( p + 1 , q ) ( p , q − 1 ) (p,q-1) ( p , q − 1 )
Substituting the top boundary values
Z i , q = A i − 1 , 0 A i , q + 1 , Z p , 0 = A i 1 A p − 1 , 0 Z_{i,q}=A_{i-1,0}A_{i,q+1},\qquad Z_{p,0}=A_{i1}A_{p-1,0} Z i , q = A i − 1 , 0 A i , q + 1 , Z p , 0 = A i 1 A p − 1 , 0 in (18) recovers the extra top-row and first-column factors in the table.
Under truncation at q ≥ m ≥ 2 q\ge m\ge2 q ≥ m ≥ 2 A i − 1 , 0 A_{i-1,0} A i − 1 , 0 A i − 2 , q A_{i-2,q} A i − 2 , q A i , q + 1 A_{i,q+1} A i , q + 1 Z p , q − 1 Z_{p,q-1} Z p , q − 1 A p , q + 1 A_{p,q+1} A p , q + 1 q q q A i − 1 , 0 A_{i-1,0} A i − 1 , 0 Z p + 1 , q Z_{p+1,q} Z p + 1 , q A p − 1 , q A_{p-1,q} A p − 1 , q Z p , q − 1 Z_{p,q-1} Z p , q − 1 A p , q + 1 A_{p,q+1} A p , q + 1 q q q Z p , q − 1 Z_{p,q-1} Z p , q − 1 m m m y s y_s y s s < m s<m s < m
Reverse complete rows
Lemma A.2. The reverse mutation word of Section 3.2 has the four exchange-neighbor lists below and yields recurrence (21) .
Proof. Fix j < n j<n j < n k = j − 1 , … , 1 k=j-1,\ldots,1 k = j − 1 , … , 1 t = n − j , … , 1 t=n-j,\ldots,1 t = n − j , … , 1 F k , k + t F_{k,k+t} F k , k + t W k + 1 , t W_{k+1,t} W k + 1 , t W k , t + 1 W_{k,t+1} W k , t + 1
Table 1 (PDF p. 38).
In (A.2), the earlier axial pivots contributing to this row are ( k + 1 , t ) (k+1,t) ( k + 1 , t ) k < j − 1 k<j-1 k < j − 1 ( k , t + 1 ) (k,t+1) ( k , t + 1 ) t < n − j t<n-j t < n − j
W j , t = F j − 1 , n F j , j + t − 1 , W k , n − j + 1 = F j , n F k − 1 , k + n − j , (117) W_{j,t}=F_{j-1,n}F_{j,j+t-1},\qquad W_{k,n-j+1}=F_{j,n}F_{k-1,k+n-j}, \tag*{(117)} W j , t = F j − 1 , n F j , j + t − 1 , W k , n − j + 1 = F j , n F k − 1 , k + n − j , (117) the first-side product in every row is W k + 1 , t F k − 1 , k + t − 1 W_{k+1,t}F_{k-1,k+t-1} W k + 1 , t F k − 1 , k + t − 1 W k , t + 1 F k , k + t − 1 W_{k,t+1}F_{k,k+t-1} W k , t + 1 F k , k + t − 1 (21) , including both boundary rows. If j = 1 j=1 j = 1 n − j = 1 n-j=1 n − j = 1 t = n − j t=n-j t = n − j § 10 .§ 2 .
Seam columns and endpoint coincidences
Lemma A.3. After the two triangle sweeps, the seam column is (A.4). Successive seam mutations give the expansion (26) , including the single and double terminal cases.
Proof. Record exchange adjacency by the actual matrix column Q ( − , y s ) Q(-,y_s) Q ( − , y s ) Q ( y s , − ) Q(y_s,-) Q ( y s , − ) a = n − s , b = s a=n-s,b=s a = n − s , b = s
Q ( − , y s ) = + A s + 1 , s − A s , s − 1 + F s − 1 , s − F s , s + 1 . (118) Q(-,y_s)=+A_{s+1,s}-A_{s,s-1}+F_{s-1,s}-F_{s,s+1}. \tag*{(118)} Q ( − , y s ) = + A s + 1 , s − A s , s − 1 + F s − 1 , s − F s , s + 1 . (118) The four terms of (A.9) are the row terms A s , s − 1 , − A s + 1 , s , F s , s + 1 , − F s − 1 , s A_{s,s-1},-A_{s+1,s},F_{s,s+1},-F_{s-1,s} A s , s − 1 , − A s + 1 , s , F s , s + 1 , − F s − 1 , s n , s n,s n , s
We compute the rows chronologically through the two triangle sweeps. In the forward table, the only pivots adjacent to y s y_s y s A s + 1 , s A_{s+1,s} A s + 1 , s s + 1 < i s+1<i s + 1 < i A s , s − 1 A_{s,s-1} A s , s − 1 s > 1 s>1 s > 1 h = Q ( y s , k ) = − 1 h=Q(y_s,k)=-1 h = Q ( y s , k ) = − 1 row ( y s ) ′ = row ( y s ) − row ( k ) − + 2 e k \operatorname{row}(y_s)'=\operatorname{row}(y_s)-\operatorname{row}(k)^-+2e_k row ( y s ) ′ = row ( y s ) − row ( k ) − + 2 e k s > 1 s>1 s > 1 Z s + 1 , s − 1 + y s + 1 Z_{s+1,s-1}+y_{s+1} Z s + 1 , s − 1 + y s + 1 s = 1 s=1 s = 1 A 10 + U 1 + y 2 A_{10}+U_1+y_2 A 10 + U 1 + y 2
At the second pivot, h = + 1 h=+1 h = + 1 row ( y s ) ′ = row ( y s ) + row ( k ) + − 2 e k \operatorname{row}(y_s)'=\operatorname{row}(y_s)+\operatorname{row}(k)^+-2e_k row ( y s ) ′ = row ( y s ) + row ( k ) + − 2 e k s < i − 1 s<i-1 s < i − 1 Z s + 1 , s − 1 + y s − 1 Z_{s+1,s-1}+y_{s-1} Z s + 1 , s − 1 + y s − 1 Z Z Z s = i − 1 s=i-1 s = i − 1 U i + y s − 1 + A i , s U_i+y_{s-1}+A_{i,s} U i + y s − 1 + A i , s s = 1 s=1 s = 1 i = 2 i=2 i = 2
− U s + U s + 1 + 1 s > 1 y s − 1 − ( 1 − 1 i = s + 1 ) y s + 1 . -U_s+U_{s+1}+\mathbf{1}_{s>1}y_{s-1}-(1-\mathbf{1}_{i=s+1})y_{s+1}. − U s + U s + 1 + 1 s > 1 y s − 1 − ( 1 − 1 i = s + 1 ) y s + 1 . For j < n j<n j < n F s , s + 1 F_{s,s+1} F s , s + 1 k = s , t = 1 k=s,t=1 k = s , t = 1 h = + 1 h=+1 h = + 1 V s + 1 + F s − 1 , s V_{s+1}+F_{s-1,s} V s + 1 + F s − 1 , s s + 1 < j s+1<j s + 1 < j V j + y j + F s − 1 , s V_j+y_j+F_{s-1,s} V j + y j + F s − 1 , s s = j − 1 s=j-1 s = j − 1 F s − 1 , s F_{s-1,s} F s − 1 , s § 10 .§ 3 gives the same expressions at n − j = 1 n-j=1 n − j = 1 j = n j=n j = n V s = F s − 1 , s V_s=F_{s-1,s} V s = F s − 1 , s y n y_n y n
− V s + V s + 1 + 1 j = s + 1 y s + 1 . -V_s+V_{s+1}+\mathbf{1}_{j=s+1}y_{s+1}. − V s + V s + 1 + 1 j = s + 1 y s + 1 . Add these row contributions and negate to obtain the complete exchange column
Q ( − , y s ) = U s + V s − U s + 1 − V s + 1 − 1 s > 1 y s − 1 + ( 1 − 1 i = s + 1 − 1 j = s + 1 ) y s + 1 . (119) Q(-,y_s)=U_s+V_s-U_{s+1}-V_{s+1}-\mathbf{1}_{s>1}y_{s-1}+(1-\mathbf{1}_{i=s+1}-\mathbf{1}_{j=s+1})y_{s+1}. \tag*{(119)} Q ( − , y s ) = U s + V s − U s + 1 − V s + 1 − 1 s > 1 y s − 1 + ( 1 − 1 i = s + 1 − 1 j = s + 1 ) y s + 1 . (119) Each terminal condition i = s + 1 i=s+1 i = s + 1 j = s + 1 j=s+1 j = s + 1 y s + 1 y_{s+1} y s + 1
Set α s = 1 − 1 i = s + 1 − 1 j = s + 1 \alpha_s=1-\mathbf{1}_{i=s+1}-\mathbf{1}_{j=s+1} α s = 1 − 1 i = s + 1 − 1 j = s + 1 s < m − 1 s<m-1 s < m − 1 α s = 1 \alpha_s=1 α s = 1 s = m − 1 s=m-1 s = m − 1 i ≠ j i\ne j i = j − 1 -1 − 1 i = j i=j i = j − U 1 − V 1 + U 2 + V 2 − α 1 y 2 -U_1-V_1+U_2+V_2-\alpha_1y_2 − U 1 − V 1 + U 2 + V 2 − α 1 y 2 y 1 ′ = [ U 2 V 2 y 1 − 1 ] + [ U 1 V 1 y 2 y 1 − 1 ] y'_1=[U_2V_2y_1^{-1}]+[U_1V_1y_2y_1^{-1}] y 1 ′ = [ U 2 V 2 y 1 − 1 ] + [ U 1 V 1 y 2 y 1 − 1 ] U s , V s U_s,V_s U s , V s
Q c u r r e n t ( y s , − ) = U s + 1 + V s + 1 − e y s − 1 ′ − α s y s + 1 . Q_{\mathrm{current}}(y_s,-)=U_{s+1}+V_{s+1}-e_{y'_{s-1}}-\alpha_sy_{s+1}. Q current ( y s , − ) = U s + 1 + V s + 1 − e y s − 1 ′ − α s y s + 1 . Thus the nonterminal recurrence is y s ′ = [ U s + 1 V s + 1 y s − 1 ] + [ y s − 1 ′ y s + 1 y s − 1 ] y'_s=[U_{s+1}V_{s+1}y_s^{-1}]+[y'_{s-1}y_{s+1}y_s^{-1}] y s ′ = [ U s + 1 V s + 1 y s − 1 ] + [ y s − 1 ′ y s + 1 y s − 1 ] m ≥ 3 m\ge3 m ≥ 3
y m − 1 ′ = [ U m V m y m 1 i = j y m − 1 − 1 ] + [ y m − 2 ′ y m − 1 − 1 ] . y'_{m-1}=[U_mV_my_m^{1_{i=j}}y_{m-1}^{-1}]+[y'_{m-2}y_{m-1}^{-1}]. y m − 1 ′ = [ U m V m y m 1 i = j y m − 1 − 1 ] + [ y m − 2 ′ y m − 1 − 1 ] . The second term no longer contains y m y_m y m m = 2 m=2 m = 2 [ U 1 V 1 y 1 − 1 ] [U_1V_1y_1^{-1}] [ U 1 V 1 y 1 − 1 ] (26) . Every bracket contains coordinates of one current seed and uses its Weyl normalization; in particular, products of y s , y s ′ y_s,y'_s y s , y s ′
For a uniform termwise formula, let 1 ≤ k ≤ m 1\le k\le m 1 ≤ k ≤ m k k k
[ U k V k y k − 1 − 1 y k 1 k = i + 1 1 k = j − 1 ] , y 0 = 1. (120) [U_kV_ky_{k-1}^{-1}y_k^{1_{k=i+1}1_{k=j-1}}],\qquad y_0=1. \tag*{(120)} [ U k V k y k − 1 − 1 y k 1 k = i + 1 1 k = j − 1 ] , y 0 = 1. (120) A nonterminal term has two denominator factors, a single terminal one, and a double terminal acquires y m y_m y m V k V_k V k § 4.2 fixes every coefficient to one. The row/column convention preserves the two exchange-neighbor sets and their equal-sign lateral-side identification used in neutral-support gluing.
Quadrilateral cutting
Lemma A.4. The exponent map defined in (A.6) is the canonical quadrilateral cutting map (27) .
Proof. Glue triangles U , V U,V U , V U ( a , b , 0 ) U(a,b,0) U ( a , b , 0 ) V ( b , a , 0 ) V(b,a,0) V ( b , a , 0 ) U U U V V V
e ( a , b , c ) E 0 = e U ( a , b , c ) + e V ( n − a , a , 0 ) , representative in U , e_{(a,b,c)}E_0=e_U(a,b,c)+e_V(n-a,a,0),\qquad\text{representative in }U, e ( a , b , c ) E 0 = e U ( a , b , c ) + e V ( n − a , a , 0 ) , representative in U , e ( a , b , c ) E 0 = e V ( a , b , c ) + e U ( n − a , a , 0 ) , representative in V . (121) e_{(a,b,c)}E_0=e_V(a,b,c)+e_U(n-a,a,0),\qquad\text{representative in }V. \tag*{(121)} e ( a , b , c ) E 0 = e V ( a , b , c ) + e U ( n − a , a , 0 ) , representative in V . (121) The two definitions agree on the seam as the sum of its two copies. We verify H E 0 = D H c u t HE_0=DH_{\mathrm{cut}} H E 0 = D H cut H − 1 = K / n H^{-1}=K/n H − 1 = K / n (7) , this makes E 0 E_0 E 0
In [20 Appendix A(260)–(261)] , an interior H-row has value + 1 +1 + 1
( a + 1 , b , c − 1 ) , ( a − 1 , b + 1 , c ) , ( a , b − 1 , c + 1 ) (a+1,b,c-1),(a-1,b+1,c),(a,b-1,c+1) ( a + 1 , b , c − 1 ) , ( a − 1 , b + 1 , c ) , ( a , b − 1 , c + 1 ) and value − 1 -1 − 1
( a − 1 , b , c + 1 ) , ( a + 1 , b − 1 , c ) , ( a , b + 1 , c − 1 ) . (a-1,b,c+1),(a+1,b-1,c),(a,b+1,c-1). ( a − 1 , b , c + 1 ) , ( a + 1 , b − 1 , c ) , ( a , b + 1 , c − 1 ) . The boundary row a = 0 a=0 a = 0
e ( 0 , b , c ) + e ( 1 , b , c − 1 ) − e ( 0 , b + 1 , c − 1 ) − e ( 1 , b − 1 , c ) . (122) e_{(0,b,c)}+e_{(1,b,c-1)}-e_{(0,b+1,c-1)}-e_{(1,b-1,c)}. \tag*{(122)} e ( 0 , b , c ) + e ( 1 , b , c − 1 ) − e ( 0 , b + 1 , c − 1 ) − e ( 1 , b − 1 , c ) . (122) Cyclic rotation gives the cut-triangle H-row on the seam side c = 0 c=0 c = 0
H ( a , b , 0 ) = e ( a , b , 0 ) + e ( a , b − 1 , 1 ) − e ( a + 1 , b − 1 , 0 ) − e ( a − 1 , b , 1 ) . (123) H_{(a,b,0)}=e_{(a,b,0)}+e_{(a,b-1,1)}-e_{(a+1,b-1,0)}-e_{(a-1,b,1)}. \tag*{(123)} H ( a , b , 0 ) = e ( a , b , 0 ) + e ( a , b − 1 , 1 ) − e ( a + 1 , b − 1 , 0 ) − e ( a − 1 , b , 1 ) . (123) For a nonseam source row, its original triangle retains the same H-row. The other component is the signed sum of e ( n − a , a , 0 ) e(n-a,a,0) e ( n − a , a , 0 ) a − 1 , a , a + 1 a-1,a,a+1 a − 1 , a , a + 1 a = 0 a=0 a = 0 D H c u t DH_{\mathrm{cut}} D H cut
On seam U ( a , b , 0 ) = V ( b , a , 0 ) U(a,b,0)=V(b,a,0) U ( a , b , 0 ) = V ( b , a , 0 ) a , b > 0 a,b>0 a , b > 0 a + b = n a+b=n a + b = n
+ e U ( a , b − 1 , 1 ) − e U ( a − 1 , b , 1 ) + e V ( b , a − 1 , 1 ) − e V ( b − 1 , a , 1 ) . (124) +e_U(a,b-1,1)-e_U(a-1,b,1)+e_V(b,a-1,1)-e_V(b-1,a,1). \tag*{(124)} + e U ( a , b − 1 , 1 ) − e U ( a − 1 , b , 1 ) + e V ( b , a − 1 , 1 ) − e V ( b − 1 , a , 1 ) . (124) By (A.6), its U-component is
e U ( a , b − 1 , 1 ) − e U ( a − 1 , b , 1 ) + e U ( a , b , 0 ) − e U ( a + 1 , b − 1 , 0 ) , e_U(a,b-1,1)-e_U(a-1,b,1)+e_U(a,b,0)-e_U(a+1,b-1,0), e U ( a , b − 1 , 1 ) − e U ( a − 1 , b , 1 ) + e U ( a , b , 0 ) − e U ( a + 1 , b − 1 , 0 ) , exactly (A.8); its V-component similarly gives the other cut H-row. Therefore every row satisfies H E 0 = D H c u t HE_0=DH_{\mathrm{cut}} H E 0 = D H cut a = 1 a=1 a = 1 b = 1 b=1 b = 1 n = 2 n=2 n = 2
With A p q = g U ( n − p , q , p − q ) A_{pq}=g_U(n-p,q,p-q) A pq = g U ( n − p , q , p − q ) F q p = g V ( q , n − p , p − q ) F_{qp}=g_V(q,n-p,p-q) F q p = g V ( q , n − p , p − q ) 3.2 3.2 3.2 (27) . The canonical map is based, so its Weyl coefficient is one. This proves (27) .
Endpoint and topology cases
Table 4.
Zhili College, Tsinghua University, Beijing 100084, China
tangqy24@mails.tsinghua.edu.cn
