Introduction
The positivity problem. Fomin and Zele-vinsky introduced cluster algebras in [9 ] , motivated by total positivity and dual canonical bases. A cluster is a transcendence basis, and mutation replaces one of its variables by a rational expression determined by a binomial exchange relation. Their Laurent phenomenon states that every cluster variable is a Laurent polynomial in the variables of any cluster. They also conjectured that the coefficients of these expansions are nonnegative integer combinations of elements of the coefficient group. The caterpillar argument of [10 Theorem 2.1] gives a criterion for the Laurent property in polynomial exchange patterns. In [11 ] , Fomin and Zelevinsky developed principal coefficients, F F F
Rank-two formulas provided explicit positive expansions and positive bases. Lee, Li, and Zelevinsky [16 ] constructed the greedy basis using compatible pairs on Dyck paths. Lee and Schiffler [17 ] proved Laurent positivity for all skew-symmetric cluster algebras. Gross, Hacking, Keel, and Kontsevich [15 ] established the skew-symmetrizable case through cluster scattering diagrams and theta functions. Their construction identifies cluster monomials with theta functions and derives Laurent positivity from positive wall functions.
Chekhov and Shapiro introduced generalized cluster algebras in a 2011 preprint, published as [7 ] , to describe Teichmüller spaces of surfaces with orbifold points. The orbifold mutations lead to quadratic exchange polynomials; their algebraic definition allows homogeneous exchange polynomials of arbitrary prescribed degrees. The coefficient tuples mutate with the exchange matrix. They proved the generalized Laurent phenomenon in Theorem 2.5 and established positivity for the motivating orbifold transformations. Conjecture 5.1 asks for positivity for reciprocal exchange polynomials with positive coefficients. We use the corrected third arXiv version of their paper.
The rank-two and coefficient theories developed alongside this conjecture. Rupel [25 ] extended greedy bases to generalized rank-two algebras with monic palindromic exchange polynomials with positive coefficients, using compatible gradings on maximal Dyck paths. Nakanishi [19 ] established formulas in terms of c c c g g g F F F [20 ] related these data to left and right ordinary companion cluster algebras and recovered the companion variables by specializations of generalized F F F
Cheung, Kelley, and Musiker [8 ] constructed scattering diagrams and theta functions for reciprocal generalized cluster algebras. Mou [18 ] developed generalized scattering diagrams by factoring the exchange polynomials into binomials and proved positivity in the factor parameters. The original exchange coefficients give a different positivity question. If
P ( z ) = ∏ i ( 1 + α i z ) , P(z) = \prod_i (1+\alpha_i z), P ( z ) = i ∏ ( 1 + α i z ) , then those coefficients are the elementary symmetric functions e s ( α ) e_s(\alpha) e s ( α )
α 1 2 + α 2 2 = e 1 ( α ) 2 − 2 e 2 ( α ) . \alpha_1^2+\alpha_2^2=e_1(\alpha)^2-2e_2(\alpha). α 1 2 + α 2 2 = e 1 ( α ) 2 − 2 e 2 ( α ) . Thus a positive expression in the factor parameters can have a negative coefficient when written in the original exchange generators.
Burcroff, Lee, and Mou [4 , 5 ] resolved this coefficient positivity problem. Their result was announced in 2024; the PNAS article and the detailed proof appeared in 2025. They introduced tight gradings on Dyck paths and used them to count rank-two wall coefficients in the original polynomial coefficients. Their two proofs [5 Sections 5.4–5.5] use shadowed gradings and the one-bending property of broken lines; the second compares the shadowed-grading and broken-line recursions directly. They then proved positivity for several incoming walls, passed to arbitrary rank, and obtained Laurent positivity and positive theta-function structure constants. Their Laurent theorem [5 Theorem 7.9 and Section 7.2] covers arbitrary finite rank, skew-symmetrizable exchange matrices, frozen variables, and nonreciprocal exchange polynomials. Theorem 2.1 below gives another proof of this Laurent-positivity statement.
Quiver moduli provide a related description of the local wall coefficients. Reineke and Weist [23 ] expressed tropical-vertex functions through Euler characteristics of stable bipartite quiver moduli. Burcroff, Lee, Mou, Musiker, and Reineke [6 ] combined these formulas with tight gradings to study coefficient formulas and Weyl symmetry. On the structural side, Ramos and Whiting [22 ] realized generalized cluster algebras of geometric type as subquotients of ordinary cluster algebras. Akagi and Nakanishi [1 ] extended the comparison to arbitrary semifields and composite cluster patterns; their Remark 6.5 discusses the distinction between positivity in factor parameters and in exchange coefficients. In the quantum setting, Ye [27 ] constructed scattering diagrams for reciprocal data with a specified compatibility normalization and proved positivity in the factor parameters.
1.2. The attached algebra. Our local construction uses a finite algebra with a particle-permutation action. Fix coprime integers a , b > 0 a,b>0 a , b > 0 p , q p,q p , q p a − q b = 1 pa-qb=1 p a − q b = 1 k ≥ 1 k \ge1 k ≥ 1 a 1 , … , a k a , b 1 , … , b k b a_1,\ldots,a_{ka},b_1,\ldots,b_{kb} a 1 , … , a k a , b 1 , … , b k b T T T
a i p ∏ j = 1 k b ( a i − b j ) = T b , ( − b j ) q ∏ i = 1 k a ( a i − b j ) = T a . (1) a_i^p\prod_{j=1}^{kb}(a_i-b_j)=T^b,\qquad(-b_j)^q\prod_{i=1}^{ka}(a_i-b_j)=T^a. \tag*{(1)} a i p j = 1 ∏ k b ( a i − b j ) = T b , ( − b j ) q i = 1 ∏ k a ( a i − b j ) = T a . (1) The group G k = S k a × S k b G_k=S_{ka}\times S_{kb} G k = S k a × S k b T = 0 T=0 T = 0 C [ [ T ] ] \mathbb{C}[[T]] C [[ T ]] D k p , q D_k^{p,q} D k p , q § 3 treats the same construction with any positive integer multiplicity on the mixed factors.
For finite alphabets α \alpha α β \beta β f a , b ( U ) f_{a,b}(U) f a , b ( U ) ( a , b ) (a,b) ( a , b ) U U U ∏ i ( 1 + α i z ) \prod_i(1+\alpha_i z) ∏ i ( 1 + α i z ) ∏ j ( 1 + β j z ) \prod_j(1+\beta_j z) ∏ j ( 1 + β j z )
[ U k ] f a , b ( U ) = ω α ω β ch ( D k p , q ) . (2) [U^k]f_{a,b}(U)=\omega_\alpha\omega_\beta\operatorname{ch}\left(D_k^{p,q}\right). \tag*{(2)} [ U k ] f a , b ( U ) = ω α ω β ch ( D k p , q ) . (2) Here ch \operatorname{ch} ch ω \omega ω h j h_j h j e j e_j e j ( p , q ) (p,q) ( p , q ) p a − q b = 1 pa-qb=1 p a − q b = 1 k k k D k p , q D_k^{p,q} D k p , q
[ U k ] f a , b ( U ) = ∑ λ ⊢ k a , μ ⊢ k b n λ μ e λ ( α ) e μ ( β ) , n λ μ ∈ Z ≥ 0 . (3) [U^k]f_{a,b}(U)=\sum_{\lambda\vdash ka,\,\mu\vdash kb}n_{\lambda\mu}e_\lambda(\alpha)e_\mu(\beta),\qquad n_{\lambda\mu}\in\mathbb{Z}_{\ge0}. \tag*{(3)} [ U k ] f a , b ( U ) = λ ⊢ k a , μ ⊢ k b ∑ n λ μ e λ ( α ) e μ ( β ) , n λ μ ∈ Z ≥ 0 . (3) where the indices are partitions and e λ = ∏ i e λ i e_\lambda=\prod_i e_{\lambda_i} e λ = ∏ i e λ i n λ μ n_{\lambda\mu} n λ μ [6 Theorem 4.10] . Equation (2) derives them from the attached factor by an equivariant algebraic construction.
Products of linear forms are classical objects of Gale duality [2 ] . Toric quantum Stanley–Reisner presentations and finite factors selected by valuations occur in González–Woodward [14 ] . The construction here identifies the full symmetric-group character of the attached factor of (1) with a wall coefficient. A deformation separating the core and attached particles produces an induced-character identity, uniformly over the positive Bézout pairs. This identity is compared with a framed Harder–Narasimhan factorization to prove (2) .
The proof includes a cut degeneration, the required finite-algebra comparisons, and the equivariant deformation. Rimmer studied regular equivariant deformations and their permutation representations [24 Corollary 6.2.8 and Section 8.1] . Here polynomial Hermite interpolation constructs the deformation and controls the directions transverse to the coordinate-equality strata. The thin-quiver quotients and alternation have antecedents in Franzen [12 ] , and the polynomial factorization is a form of the Chow–Hall factorization of Franzen and Reineke [13 ] . We adapt the separated-center proof and formal transport arguments of [21 Sections 3–4] . The passage from two inputs to arbitrary rank follows the scattering-diagram reduction of [5 Section 9] ; we give its labeled induction and mutation comparison with integral ambient normals. The generalized Laurent phenomenon [7 ] , convex-polytope shellability [3 ] , and standard completion and regular-sequence results [26 ] supply the background facts.
Finally, § 14 constructs finite rational algebras of dimensions n λ μ n_{\lambda\mu} n λ μ
Generalized cluster patterns
We write N = Z ≥ 0 \mathbb{N}=\mathbb{Z}_{\ge0} N = Z ≥ 0 [ c ] + = max ( c , 0 ) [c]_+=\max(c,0) [ c ] + = max ( c , 0 ) P \mathbb{P} P Z [ P ] \mathbb{Z}[\mathbb{P}] Z [ P ] N [ P ] \mathbb{N}[\mathbb{P}] N [ P ] P \mathbb{P} P
Degree-separated exchange matrices
Fix n n n N − n N-n N − n x 1 , … , x N x_1,\ldots,x_N x 1 , … , x N N × n N \times n N × n B ~ = ( b i j ) \widetilde{B}=(b_{ij}) B = ( b ij ) r 1 , … , r n r_1,\ldots,r_n r 1 , … , r n q i = ( q i , 0 , … , q i , r i ) q_i=(q_{i,0},\ldots,q_{i,r_i}) q i = ( q i , 0 , … , q i , r i ) P \mathbb{P} P n × n n \times n n × n B B B
D B = − B t D , D = diag ( δ 1 , … , δ n ) , δ i ∈ Z > 0 . DB=-B^{\mathsf{t}}D,\qquad D=\operatorname{diag}(\delta_1,\ldots,\delta_n),\qquad\delta_i\in\mathbb{Z}_{>0}. D B = − B t D , D = diag ( δ 1 , … , δ n ) , δ i ∈ Z > 0 . The matrix is degree-separated: the usual physical exchange matrix is B diag ( r i ) B\operatorname{diag}(r_i) B diag ( r i ) i i i r i r_i r i D p h y s D_{\mathrm{phys}} D phys D p h y s diag ( r i ) − 1 D_{\mathrm{phys}}\operatorname{diag}(r_i)^{-1} D phys diag ( r i ) − 1
For a mutable index k k k
M k + = ∏ i = 1 N x i [ b i k ] + , M k − = ∏ i = 1 N x i [ − b i k ] + . M_k^+=\prod_{i=1}^{N}x_i^{[b_{ik}]_+},\qquad M_k^-=\prod_{i=1}^{N}x_i^{[-b_{ik}]_+}. M k + = i = 1 ∏ N x i [ b ik ] + , M k − = i = 1 ∏ N x i [ − b ik ] + . Mutation replaces x k x_k x k
x k x k ′ = ∑ s = 0 r k q k , s ( M k + ) r k − s ( M k − ) s , x i ′ = x i ( i ≠ k ) . (4) x_kx_k'=\sum_{s=0}^{r_k}q_{k,s}(M_k^+)^{r_k-s}(M_k^-)^s,\qquad x_i'=x_i\quad(i\ne k). \tag*{(4)} x k x k ′ = s = 0 ∑ r k q k , s ( M k + ) r k − s ( M k − ) s , x i ′ = x i ( i = k ) . (4) The matrix changes by
b i j ′ = { − b i j , i = k or j = k , b i j + r k ( [ b i k ] + b k j + b i k [ − b k j ] + ) , i , j ≠ k . (5) b_{ij}'=
\begin{cases}
-b_{ij},&i=k\text{ or }j=k,\\
b_{ij}+r_k\bigl([b_{ik}]_+b_{kj}+b_{ik}[-b_{kj}]_+\bigr),&i,j\ne k.
\end{cases} \tag*{(5)} b ij ′ = { − b ij , b ij + r k ( [ b ik ] + b k j + b ik [ − b k j ] + ) , i = k or j = k , i , j = k . (5) Frozen rows obey the same rule. The degrees and symmetrizer remain fixed.
Coefficients and representatives
Put q ˉ i , s = q i , s / q i , 0 \bar{q}_{i,s}=q_{i,s}/q_{i,0} q ˉ i , s = q i , s / q i , 0
q k , s ′ = q k , r k − s , q_{k,s}'=q_{k,r_k-s}, q k , s ′ = q k , r k − s , q ˉ j , s ′ = { q ˉ j , s q k , 0 − s b k j , b k j ≥ 0 , q ˉ j , s q k , r k − s b k j , b k j ≤ 0 , ( j ≠ k ) . (6) \bar{q}_{j,s}'=
\begin{cases}
\bar{q}_{j,s}q_{k,0}^{-sb_{kj}},&b_{kj}\ge0,\\
\bar{q}_{j,s}q_{k,r_k}^{-sb_{kj}},&b_{kj}\le0,
\end{cases}
\qquad(j\ne k). \tag*{(6)} q ˉ j , s ′ = { q ˉ j , s q k , 0 − s b k j , q ˉ j , s q k , r k − s b k j , b k j ≥ 0 , b k j ≤ 0 , ( j = k ) . (6) For j ≠ k j\ne k j = k q j , 0 ′ ∈ P q_{j,0}'\in\mathbb{P} q j , 0 ′ ∈ P (4) –(6) are the Chekhov–Shapiro rules, with the order of each coefficient tuple chosen to agree with (4) . Mutation reverses the exchange polynomial at the mutated index, including for nonreciprocal coefficient tuples.
A pattern assigns seeds to the vertices of the regular labeled n n n canonical representative used below takes q j , 0 ′ = q j , 0 q_{j,0}'=q_{j,0} q j , 0 ′ = q j , 0 j ≠ k j\ne k j = k
The generalized cluster algebra is the Z [ P ] \mathbb{Z}[\mathbb{P}] Z [ P ] Frac ( Z [ P ] ) ( x 1 , … , x N ) \operatorname{Frac}(\mathbb{Z}[\mathbb{P}])(x_1,\ldots,x_N) Frac ( Z [ P ]) ( x 1 , … , x N )
Theorem 2.1 (Laurent positivity). In a Chekhov–Shapiro generalized cluster pattern of arbitrary finite rank, every cluster monomial has a Laurent expansion in the variables of any seed with coefficients in N [ P ] \mathbb{N}[\mathbb{P}] N [ P ]
Take the exchange coefficients of a chosen seed to be independent symbols, with invertible endpoints, and use the canonical representative rooted at that seed. Every such Laurent expansion has coefficients in
R + = N [ q i , 0 ± 1 , q i , r i ± 1 , q i , s ( 1 ≤ i ≤ n , 0 < s < r i ) ] . (7) R_+ = \mathbb{N}\left[q_{i,0}^{\pm1},q_{i,r_i}^{\pm1},q_{i,s}\ (1 \le i \le n,\ 0 < s < r_i)\right]. \tag*{(7)} R + = N [ q i , 0 ± 1 , q i , r i ± 1 , q i , s ( 1 ≤ i ≤ n , 0 < s < r i ) ] . (7) The first assertion follows from the universal second assertion and the rescaling argument below. If the endpoints of the reference exchange polynomials equal one, (7) is the polynomial semiring in their intermediate coefficients. An arbitrary permitted normalization may introduce inverses of intermediate coefficients; the invariant assertion for such a representative is positivity in N [ P ] \mathbb{N}[\mathbb{P}] N [ P ]
We use the following Laurent theorem.
Theorem 2.2 (Chekhov–Shapiro Laurent phenomenon, [7 Theorem 2.5] ). Every variable in a generalized cluster pattern is a Laurent polynomial over Z [ P ] \mathbb{Z}[\mathbb{P}] Z [ P ]
The theorem applies to the coefficient rule (6) , including nonreciprocal tuples. Its free-abelian coefficient-group formulation in [7 ] gives the statement above: a finite mutation word involves finitely many coefficient elements and common scales, whose subgroup in P \mathbb{P} P f f f δ f \delta_f δ f δ i \delta_i δ i
b i f = − δ f b f i / δ i ( i ≤ n ) , b f g = 0 ( f , g > n ) . b_{if}=-\delta_f b_{fi}/\delta_i\quad(i\le n),\qquad b_{fg}=0\quad(f,g>n). b i f = − δ f b f i / δ i ( i ≤ n ) , b f g = 0 ( f , g > n ) . The resulting square matrix is integral and skew-symmetrizable by diag ( δ 1 , … , δ N ) \operatorname{diag}(\delta_1,\ldots,\delta_N) diag ( δ 1 , … , δ N )
Remark 2.3. Zero intermediate coefficients, or intermediate coefficients that are positive sums in another coefficient group, are obtained by specializing the universal polynomial expression after Theorem 2.1 is proved. Endpoints are specialized to units. Since no intermediate coefficient is inverted in (7) , these specializations preserve both the exchange relation and positivity.
Finite Gale algebras
Fix relatively prime positive integers a a a b b b m ≥ 1 m\ge1 m ≥ 1 p p p q q q
p a − q b = 1. (8) pa-qb=1. \tag*{(8)} p a − q b = 1. (8) Such a positive pair exists, since one can add a sufficiently large multiple of ( b , a ) (b,a) ( b , a ) G 0 = { 1 } G_0=\{1\} G 0 = { 1 } k ≥ 1 k\ge1 k ≥ 1
A = k a , B 0 = k b , G k = S A × S B 0 . A=ka,\qquad B_0=kb,\qquad G_k=S_A\times S_{B_0}. A = k a , B 0 = k b , G k = S A × S B 0 . The two symmetric groups act by permuting the coordinates of their respective colors. Consider the square polynomial system
a i p ∏ j = 1 B 0 ( a i − b j ) m = T b ( 1 ≤ i ≤ A ) , (9) a_i^p\prod_{j=1}^{B_0}(a_i-b_j)^m=T^b\qquad(1\le i\le A), \tag*{(9)} a i p j = 1 ∏ B 0 ( a i − b j ) m = T b ( 1 ≤ i ≤ A ) , (9) ( − b j ) q ∏ i = 1 A ( a i − b j ) m = T a ( 1 ≤ j ≤ B 0 ) . (10) (-b_j)^q\prod_{i=1}^{A}(a_i-b_j)^m=T^a\qquad(1\le j\le B_0). \tag*{(10)} ( − b j ) q i = 1 ∏ A ( a i − b j ) m = T a ( 1 ≤ j ≤ B 0 ) . (10) All factors selected from this system will be full scheme-theoretic factors. In particular, a multiple solution contributes its entire Artin local algebra.
A numerical identity and the valuation criterion
Set
N A = p + m k b , N B = q + m k a , M k = max { k a N A , k b N B } . N_A = p + mkb,\qquad N_B = q + mka,\qquad M_k = \max\{kaN_A,kbN_B\}. N A = p + mk b , N B = q + mk a , M k = max { k a N A , k b N B } . For integers 0 ≤ u ≤ k a 0 \le u \le ka 0 ≤ u ≤ k a 0 ≤ v ≤ k b 0 \le v \le kb 0 ≤ v ≤ k b
Δ = b u − a v , D 0 = N A u − N B v . \Delta= bu-av,\qquad D_0 = N_Au-N_Bv. Δ = b u − a v , D 0 = N A u − N B v . Equation (8) gives
b D 0 = N A Δ + v . (11) bD_0 = N_A\Delta+v. \tag*{(11)} b D 0 = N A Δ + v . (11) Consequently D 0 D_0 D 0 Δ \Delta Δ Δ ≠ 0 \Delta\ne0 Δ = 0 Δ < 0 \Delta< 0 Δ < 0 N A Δ + v ≤ − N A + k b < 0 N_A\Delta+v \le-N_A+kb < 0 N A Δ + v ≤ − N A + k b < 0 Δ = 0 \Delta= 0 Δ = 0 ( u , v ) = ℓ ( a , b ) (u,v)=\ell(a,b) ( u , v ) = ℓ ( a , b ) 1 ≤ ℓ ≤ k 1 \le\ell\le k 1 ≤ ℓ ≤ k D 0 = ℓ D_0=\ell D 0 = ℓ D 0 D_0 D 0 ∣ D 0 ∣ ≤ M k |D_0| \le M_k ∣ D 0 ∣ ≤ M k u u u v v v
First replace the right sides of (9) –(10) by independent nonzero parameters. This gives the polynomial moment map
( a , b ) ⟼ ( a i p ∏ j ( a i − b j ) m , ( − b j ) q ∏ i ( a i − b j ) m ) i , j . (a,b)\longmapsto\left(a_i^p\prod_j(a_i-b_j)^m,\,(-b_j)^q\prod_i(a_i-b_j)^m\right)_{i,j}. ( a , b ) ⟼ ( a i p j ∏ ( a i − b j ) m , ( − b j ) q i ∏ ( a i − b j ) m ) i , j . Its inverse image of the target torus is finite over that torus. To see this, apply the valuative criterion. Let R R R ν \nu ν ν \nu ν u u u v v v ν \nu ν
ν { p u − q v + m u ( B 0 − v ) − m v ( A − u ) } = ν D 0 . \nu\{pu-qv+mu(B_0-v)-mv(A-u)\}=\nu D_0. ν { p u − q v + m u ( B 0 − v ) − m v ( A − u )} = ν D 0 . The right side of the quotient is a unit. This contradicts D 0 ≠ 0 D_0 \ne0 D 0 = 0 R R R
For the central system, fix an extension of the T T T C ( ( T ) ) ‾ \overline{\mathbb{C}((T))} C (( T )) val ( T ) = 1 \operatorname{val}(T)=1 val ( T ) = 1
ν D 0 = val ( T ) Δ . (12) \nu D_0=\operatorname{val}(T)\Delta. \tag*{(12)} ν D 0 = val ( T ) Δ. (12) The sign information in (11) excludes ν < 0 \nu< 0 ν < 0 Δ ≠ 0 \Delta\ne0 Δ = 0 Δ , D 0 \Delta,D_0 Δ , D 0
ν = ∣ Δ ∣ ∣ D 0 ∣ ≥ 1 M k . (13) \nu=\frac{|\Delta|}{|D_0|}\ge\frac{1}{M_k}. \tag*{(13)} ν = ∣ D 0 ∣ ∣Δ∣ ≥ M k 1 . (13) Every mixed difference then has valuation at least ν \nu ν
The possible nonzero centers
Consider a cluster with common nonzero residue c c c u u u v v v 0 = val ( T ) ( b u − a v ) 0=\operatorname{val}(T)(bu-av) 0 = val ( T ) ( b u − a v ) b u − a v = 0 bu-av=0 b u − a v = 0 ℓ ( a , b ) \ell(a,b) ℓ ( a , b ) k ( a , b ) k(a,b) k ( a , b )
The framing contribution to the limiting quotient for this cluster is ( − 1 ) q b ℓ c (-1)^{qb}\ell^{c} ( − 1 ) q b ℓ c h ( a , b ) h(a,b) h ( a , b ) ( − 1 ) m a b ℓ h (-1)^{mab}\ell^{h} ( − 1 ) mab ℓ h
ℓ c = ( − 1 ) q b ℓ + m a b ℓ ( k − ℓ ) = σ k ℓ , σ k = ( − 1 ) q b + m a b ( k − 1 ) . \ell^{c}=(-1)^{qb\ell+mab\ell(k-\ell)}=\sigma_{k}^{\ell},\qquad\sigma_{k}=(-1)^{qb+mab(k-1)}. ℓ c = ( − 1 ) q b ℓ + mab ℓ ( k − ℓ ) = σ k ℓ , σ k = ( − 1 ) q b + mab ( k − 1 ) . The last equality uses the evenness of ℓ ( ℓ − 1 ) \ell(\ell-1) ℓ ( ℓ − 1 )
P k ( z ) = ∏ h = 1 k ( z h − σ k h ) . (14) P_{k}(z)=\prod_{h=1}^{k}(z^{h}-\sigma_{k}^{h}). \tag*{(14)} P k ( z ) = h = 1 ∏ k ( z h − σ k h ) . (14) Every nonzero center is a root of P k P_{k} P k P k ( 0 ) ≠ 0 P_{k}(0)\ne0 P k ( 0 ) = 0
Proposition 3.1. The horizontal closure of (3.2)–(3.3), completed at T = 0 T=0 T = 0 C [ [ T ] ] \mathbb{C}[[T]] C [[ T ]] G k G_{k} G k
B k p , q , C k p , q , D k p , q . B_{k}^{p,q},\qquad C_{k}^{p,q},\qquad D_{k}^{p,q}. B k p , q , C k p , q , D k p , q . Their G k G_{k} G k B 0 p , q = C 0 p , q = D 0 p , q = C B_{0}^{p,q}=C_{0}^{p,q}=D_{0}^{p,q}=\mathbb{C} B 0 p , q = C 0 p , q = D 0 p , q = C
Proof. By horizontal closure we mean that the equation ideal is saturated with respect to T T T T T T T T T (12) excludes a point at infinity of the projective horizontal closure above T = 0 T=0 T = 0 (14) then shows that the special support is finite. The projective closure is consequently quasi-finite near this fiber and is finite there. Its affine horizontal closure has the same germ. Flatness of Noetherian completion preserves the injectivity of multiplication by T T T C [ [ T ] ] \mathbb{C}[[T]] C [[ T ]]
The special support is contained in the finite set of tuples whose coordinates belong to { 0 } ∪ { P k = 0 } \{0\}\cup\{P_{k}=0\} { 0 } ∪ { P k = 0 } G k G_{k} G k C [ [ T ] ] \mathbb{C}[[T]] C [[ T ]]
Cut degenerations and chamber algebras
Gale duality for complete intersections uses products of linear forms [2 ] . Positive-valuation factors obtained by completion and parameter inversion also occur in the toric construction of González–Woodward [14 ] .
Let Q = ( V , E ) Q=(V,E) Q = ( V , E )
∂ : Z E ⟶ Z 0 V , e ⟼ e h e a d ( e ) − e t a i l ( e ) , \partial:\mathbb{Z}^{E}\longrightarrow\mathbb{Z}_{0}^{V},\qquad e\longmapsto e_{\mathrm{head}(e)}-e_{\mathrm{tail}(e)}, ∂ : Z E ⟶ Z 0 V , e ⟼ e head ( e ) − e tail ( e ) , where Z 0 V \mathbb{Z}_{0}^{V} Z 0 V
L = { ( d h e a d ( e ) − d t a i l ( e ) ) e ∈ E : d ∈ Z V } , N = Z E / L . (15) L=\left\{(d_{\mathrm{head}(e)}-d_{\mathrm{tail}(e)})_{e\in E}:d\in\mathbb{Z}^{V}\right\},\qquad N=\mathbb{Z}^{E}/L. \tag*{(15)} L = { ( d head ( e ) − d tail ( e ) ) e ∈ E : d ∈ Z V } , N = Z E / L . (15) The coboundary ∂ ∗ : Z V / Z 1 ⟶ Z E \partial^{*}:\mathbb{Z}^{V}/\mathbb{Z}\mathbf{1}\longrightarrow\mathbb{Z}^{E} ∂ ∗ : Z V / Z 1 ⟶ Z E ∂ \partial ∂ L L L ± 1 \pm1 ± 1 L L L N N N n = ∣ E ∣ − ∣ V ∣ + 1 n=|E|-|V|+1 n = ∣ E ∣ − ∣ V ∣ + 1 ν e \nu_e ν e e e e N N N η 1 , … , η n \eta_1,\ldots,\eta_n η 1 , … , η n N ∗ N^* N ∗
ℓ j ( w ) = ∑ e ∈ E ⟨ η j , ν e ⟩ w e . (16) \ell_j(\mathbf{w})=\sum_{e\in E}\langle\eta_j,\nu_e\rangle w_e. \tag*{(16)} ℓ j ( w ) = e ∈ E ∑ ⟨ η j , ν e ⟩ w e . (16) The common zero space of these forms is L C L_{\mathbb{C}} L C w e = h head ( e ) − h tail ( e ) w_e=h_{\operatorname{head}(e)}-h_{\operatorname{tail}(e)} w e = h head ( e ) − h tail ( e ) h h h ℓ j \ell_j ℓ j L C L_{\mathbb{C}} L C
Fix a rational vector θ ∈ Q 0 V \theta\in\mathbb{Q}_0^V θ ∈ Q 0 V θ ( S ) : = ∑ v ∈ S θ v ≠ 0 \theta(S):=\sum_{v\in S}\theta_v\ne0 θ ( S ) := ∑ v ∈ S θ v = 0 S ⊂ V S\subset V S ⊂ V ϖ \varpi ϖ ∂ ϖ = θ \partial\varpi=\theta ∂ ϖ = θ S S S Out ( S ) \operatorname{Out}(S) Out ( S )
E S ( w ) = ∏ e ∈ Out ( S ) w e , J θ = ( E S : θ ( S ) < 0 ) . E_S(\mathbf{w})=\prod_{e\in\operatorname{Out}(S)}w_e,\qquad J_\theta=(E_S:\theta(S)<0). E S ( w ) = e ∈ Out ( S ) ∏ w e , J θ = ( E S : θ ( S ) < 0 ) . The exact initial ideal
Let I L ⊂ C [ w ] I_L\subset\mathbb{C}[\mathbf{w}] I L ⊂ C [ w ] w l + − w l − \mathbf{w}^{l^+}-\mathbf{w}^{l^-} w l + − w l − l ∈ L l\in L l ∈ L C [ w ] → C [ N ] \mathbb{C}[\mathbf{w}]\to\mathbb{C}[N] C [ w ] → C [ N ] w e ↦ X ν e w_e\mapsto X^{\nu_e} w e ↦ X ν e
Lemma 4.1. One has in ϖ ( I L ) = J θ \operatorname{in}_{\varpi}(I_L)=J_\theta in ϖ ( I L ) = J θ
Proof. For the potential d = 1 S d=1_S d = 1 S θ ( S ) < 0 \theta(S)<0 θ ( S ) < 0 J θ ⊆ in ϖ ( I L ) J_\theta\subseteq\operatorname{in}_{\varpi}(I_L) J θ ⊆ in ϖ ( I L )
For the opposite inclusion, shift an integral potential by a constant and decompose it into its nonempty proper threshold sets:
d = ∑ h = min d + 1 max d 1 S h , S h = { v : d v ≥ h } . d=\sum_{h=\min d+1}^{\max d}1_{S_h},\qquad S_h=\{v:d_v\ge h\}. d = h = m i n d + 1 ∑ m a x d 1 S h , S h = { v : d v ≥ h } . The decomposition of l = ∂ ∗ d l=\partial^*d l = ∂ ∗ d θ ⋅ d > 0 \theta\cdot d>0 θ ⋅ d > 0 S h S_h S h w l + \mathbf{w}^{l^+} w l + θ ⋅ d < 0 \theta\cdot d<0 θ ⋅ d < 0 d d d − d -d − d θ ⋅ d = 0 \theta\cdot d=0 θ ⋅ d = 0 l ≠ 0 l\ne0 l = 0 J θ J_\theta J θ
Finally, group a polynomial in I L I_L I L N N N J θ J_\theta J θ I L I_L I L □ \square □
Choose e ≥ 1 e\ge1 e ≥ 1 e ϖ e\varpi e ϖ e θ e\theta e θ T = t e T=t^e T = t e l = ∂ ∗ d l=\partial^*d l = ∂ ∗ d θ ⋅ d ≥ 0 \theta\cdot d\ge0 θ ⋅ d ≥ 0
w l + − t e θ ⋅ d w l − . (17) \mathbf{w}^{l^+}-t^{e\theta\cdot d}\mathbf{w}^{l^-}. \tag*{(17)} w l + − t e θ ⋅ d w l − . (17) Reverse the two terms when necessary, and retain a tied binomial unchanged. The ideal of these homogenizations, denoted by I θ h I_\theta^h I θ h I L I_L I L N N N t t t C [ t ] \mathbb{C}[t] C [ t ] C [ w ] / J θ \mathbb{C}[\mathbf{w}]/J_\theta C [ w ] / J θ C [ t ] \mathbb{C}[t] C [ t ]
A finite basis for the cut quotient
The flow polytope
P θ = { u ∈ R E : u e ≥ 0 , ∂ u = θ } P_\theta=\{u\in\mathbb{R}^E:u_e\ge0,\partial u=\theta\} P θ = { u ∈ R E : u e ≥ 0 , ∂ u = θ } has dimension n n n ϖ \varpi ϖ S S S θ ( S ) = 0 \theta(S)=0 θ ( S ) = 0 T 0 T_0 T 0
Restriction gives an integral isomorphism
ker ∂ ⟶ Z E ∖ T 0 . \ker\partial\longrightarrow\mathbb{Z}^{E\setminus T_0}. ker ∂ ⟶ Z E ∖ T 0 . One prescribes the complementary coordinates and solves for the tree coordinates using its unimodular reduced incidence matrix. Locally at the vertex, all tree coordinates remain strictly positive. The local feasible cone is consequently the positive orthant in the coordinates E ∖ T 0 E\setminus T_0 E ∖ T 0 P θ P_\theta P θ
The nonface ideal of this complex is exactly J θ J_\theta J θ I I I d d d
d head ( e ) − d tail ( e ) ≥ 0 ( e ∉ I ) , θ ⋅ d < 0. d_{\operatorname{head}(e)}-d_{\operatorname{tail}(e)}\ge0 \quad(e\notin I), \qquad\theta\cdot d<0. d head ( e ) − d tail ( e ) ≥ 0 ( e ∈ / I ) , θ ⋅ d < 0. After a shift and multiplication by a positive integer, decompose d d d θ \theta θ S S S θ ( S ) < 0 \theta(S)<0 θ ( S ) < 0 Out ( S ) ⊆ I \operatorname{Out}(S)\subseteq I Out ( S ) ⊆ I S S S θ ( S ) = inflow ( S ) ≥ 0 \theta(S)=\operatorname{inflow}(S)\ge0 θ ( S ) = inflow ( S ) ≥ 0
For n > 0 n>0 n > 0 J θ J_\theta J θ E ∖ T 0 E\setminus T_0 E ∖ T 0 [3 ] . Let F 1 , … , F t F_1,\ldots,F_t F 1 , … , F t R j ⊆ F j R_j\subseteq F_j R j ⊆ F j j j j [ R j , F j ] [R_j,F_j] [ R j , F j ] w R = ∏ e ∈ R w e w_R=\prod_{e\in R}w_e w R = ∏ e ∈ R w e R ⊆ E R\subseteq E R ⊆ E
On each facet the coefficient matrix of (16) is invertible, by the tree-complement isomorphism: under N ∗ = ker ∂ N^*=\ker\partial N ∗ = ker ∂ T 0 T_0 T 0 ± 1 \pm1 ± 1 A 0 = C [ ℓ 1 , … , ℓ n ] A_0=\mathbb{C}[\ell_1,\ldots,\ell_n] A 0 = C [ ℓ 1 , … , ℓ n ] Δ j \Delta_j Δ j j j j C [ Δ j ] \mathbb{C}[\Delta_j] C [ Δ j ]
C [ Δ j ] ⟶ C [ Δ j − 1 ] \mathbb{C}[\Delta_j]\longrightarrow\mathbb{C}[\Delta_{j-1}] C [ Δ j ] ⟶ C [ Δ j − 1 ] is w R j C [ w e : e ∈ F j ] w_{R_j}\mathbb{C}[w_e:e\in F_j] w R j C [ w e : e ∈ F j ] F j F_j F j R j R_j R j A 0 A_0 A 0 w R j w_{R_j} w R j
C [ w ] / J θ = ⨁ j = 1 t A 0 w R j as an A 0 -module . (18) \mathbb{C}[\mathbf{w}]/J_\theta=\bigoplus_{j=1}^{t}A_0w_{R_j}\quad\text{as an }A_0\text{-module}. \tag*{(18)} C [ w ] / J θ = j = 1 ⨁ t A 0 w R j as an A 0 -module . (18) Independence follows by reducing a relation to the preceding partial complex and then using the free rank-one kernel. In particular the ℓ j \ell_j ℓ j
H θ = C [ w ] / ( J θ , ℓ 1 , … , ℓ n ) (19) H_\theta=\mathbb{C}[\mathbf{w}]/(J_\theta,\ell_1,\ldots,\ell_n) \tag*{(19)} H θ = C [ w ] / ( J θ , ℓ 1 , … , ℓ n ) (19) is finite dimensional. If n = 0 n=0 n = 0 w e w_e w e J θ J_\theta J θ H θ = C H_\theta=\mathbb{C} H θ = C
The complete positive factor
Proposition 4.2. Let a finite group G G G Q Q Q θ \theta θ T = t e T=t^e T = t e C [ [ t ] ] \mathbb{C}[[t]] C [[ t ]] H θ H_\theta H θ
ℓ j ( w ) = 0 , w ∂ ∗ d = t e θ ⋅ d ( d ∈ Z V / Z 1 ) (20) \ell_j(\mathbf{w})=0,\qquad\mathbf{w}^{\partial^*d}=t^{e\theta\cdot d}\quad(d\in\mathbb{Z}^{V}/\mathbb{Z}1) \tag*{(20)} ℓ j ( w ) = 0 , w ∂ ∗ d = t e θ ⋅ d ( d ∈ Z V / Z 1 ) (20) on which all arrow coordinates have positive valuation. Its special and generic fibers have the same ungraded G G G
Proof. Average ϖ \varpi ϖ G G G e e e C [ t ] \mathbb{C}[t] C [ t ] ( t , w ) (t,\mathbf{w}) ( t , w ) ℓ j \ell_j ℓ j S θ S_\theta S θ t t t t t t C [ w ] / J θ \mathbb{C}[\mathbf{w}]/J_\theta C [ w ] / J θ (18) and flatness of this completion show that ℓ 1 , … , ℓ n \ell_1,\ldots,\ell_n ℓ 1 , … , ℓ n ( t , ℓ 1 , … , ℓ n ) (t,\ell_1,\ldots,\ell_n) ( t , ℓ 1 , … , ℓ n ) t t t [26 ] .
The arrow ideal of S θ / t S θ = H θ S_\theta/tS_\theta=H_\theta S θ / t S θ = H θ ( t , w ) (t,\mathbf{w}) ( t , w ) t t t S θ S_\theta S θ (18) . Successive reduction modulo t t t S θ S_\theta S θ C [ [ t ] ] \mathbb{C}[[t]] C [[ t ]] t t t t t t S θ S_\theta S θ
Put
A ′ = C [ [ t ] ] [ w ] / ( I θ h , ℓ 1 , … , ℓ n ) . A'=\mathbb{C}[[t]][\mathbf{w}]/(I_\theta^h,\ell_1,\ldots,\ell_n). A ′ = C [[ t ]] [ w ] / ( I θ h , ℓ 1 , … , ℓ n ) . Its ( t , w ) (t,\mathbf{w}) ( t , w ) S θ S_\theta S θ C [ [ t ] ] \mathbb{C}[[t]] C [[ t ]] A ′ A' A ′ J J J J / J 2 = 0 J/J^2=0 J / J 2 = 0
A ′ ≃ S θ × A o t h e r ′ . (21) A'\simeq S_\theta\times A'_{\mathrm{other}}. \tag*{(21)} A ′ ≃ S θ × A other ′ . (21) After inverting t t t E ≠ ∅ E\ne\varnothing E = ∅ t t t E = ∅ E=\varnothing E = ∅ C ( ( t ) ) \mathbb{C}((t)) C (( t )) (20) .
On S θ S_\theta S θ ( t , w ) (t,\mathbf{w}) ( t , w ) C ( ( t ) ) \mathbb{C}((t)) C (( t )) S θ S_\theta S θ S θ [ t − 1 ] S_\theta[t^{-1}] S θ [ t − 1 ]
The two neighboring chambers Apply the construction to the graph with vertices A 1 , … , A k a A_1,\ldots,A_{k_a} A 1 , … , A k a B 1 , … , B k b B_1,\ldots,B_{k_b} B 1 , … , B k b ∞ \infty ∞
B j ⟶ A i ( m copies ) , ∞ ⟶ A i ( p copies ) , B j ⟶ ∞ ( q copies ) . B_j\longrightarrow A_i\ (m\text{ copies}),\qquad\infty\longrightarrow A_i\ (p\text{ copies}),\qquad B_j\longrightarrow\infty\ (q\text{ copies}). B j ⟶ A i ( m copies ) , ∞ ⟶ A i ( p copies ) , B j ⟶ ∞ ( q copies ) . Set the potential at ∞ \infty ∞ a i − b j a_i-b_j a i − b j a i a_i a i − b j -b_j − b j
θ 0 = ( b on A i , − a on B j , 0 at ∞ ) , κ = ( N A on A i , − N B on B j , − k at ∞ ) . \theta_0=(b\text{ on }A_i,\,-a\text{ on }B_j,\,0\text{ at }\infty),\qquad\kappa=(N_A\text{ on }A_i,\,-N_B\text{ on }B_j,\,-k\text{ at }\infty). θ 0 = ( b on A i , − a on B j , 0 at ∞ ) , κ = ( N A on A i , − N B on B j , − k at ∞ ) . Let θ δ = θ 0 − δ κ \theta_\delta=\theta_0-\delta\kappa θ δ = θ 0 − δ κ δ ± = ± ( 2 M k + 1 ) − 1 \delta_{\pm}=\pm(2M_k+1)^{-1} δ ± = ± ( 2 M k + 1 ) − 1 θ 0 \theta_0 θ 0 1 / ( 2 m k ) 1/(2mk) 1/ ( 2 mk ) b / ( 2 p ) b/(2p) b / ( 2 p ) a / ( 2 q ) a/(2q) a / ( 2 q ) M k ≥ m k M_k\ge mk M k ≥ mk M k ≥ p / b M_k\ge p/b M k ≥ p / b M k ≥ q / a M_k\ge q/a M k ≥ q / a δ ± \delta_{\pm} δ ± θ δ ± \theta_{\delta_{\pm}} θ δ ±
These two characters are generic. It suffices to consider a subset not containing ∞ \infty ∞ ( u , v ) (u,v) ( u , v ) Δ − δ D 0 \Delta-\delta D_0 Δ − δ D 0 Δ ≠ 0 \Delta\ne0 Δ = 0 ∣ δ D 0 ∣ < 1 / 2 |\delta D_0|<1/2 ∣ δ D 0 ∣ < 1/2 Δ \Delta Δ Δ = 0 \Delta=0 Δ = 0 D 0 = ℓ > 0 D_0=\ell>0 D 0 = ℓ > 0 H − = H θ δ − H_-=H_{\theta_{\delta_-}} H − = H θ δ − H + = H θ δ + H_+=H_{\theta_{\delta_+}} H + = H θ δ +
Corollary 4.3. There are isomorphisms of ungraded G k G_k G k
B k p , q ≃ H − , C k p , q ≃ H + . (22) \mathcal{B}_k^{p,q}\simeq H_-,\qquad\mathcal{C}_k^{p,q}\simeq H_+. \tag*{(22)} B k p , q ≃ H − , C k p , q ≃ H + . (22) Proof. Choose a common ramification T = t e T=t^e T = t e val ( t ) = 1 / e \operatorname{val}(t)=1/e val ( t ) = 1/ e w ( 0 ) \mathbf{w}^{(0)} w ( 0 ) θ 0 \theta_0 θ 0 w ϵ ( δ ) = T − δ w ϵ ( 0 ) = t − e δ w ϵ ( 0 ) w_\epsilon^{(\delta)}=T^{-\delta}w_\epsilon^{(0)}=t^{-e\delta}w_\epsilon^{(0)} w ϵ ( δ ) = T − δ w ϵ ( 0 ) = t − eδ w ϵ ( 0 ) ϵ ∈ E \epsilon\in E ϵ ∈ E l = ∂ ∗ d l=\partial^*d l = ∂ ∗ d ∑ ϵ ∈ E l ϵ = κ ⋅ d \sum_{\epsilon\in E}l_\epsilon=\kappa\cdot d ∑ ϵ ∈ E l ϵ = κ ⋅ d
( w ( δ ) ) l = T − δ κ ⋅ d ( w ( 0 ) ) l = T ( θ 0 − δ κ ) ⋅ d = t e θ δ ⋅ d . \left(\mathbf{w}^{(\delta)}\right)^l=T^{-\delta\kappa\cdot d}\left(\mathbf{w}^{(0)}\right)^l=T^{(\theta_0-\delta\kappa)\cdot d}=t^{e\theta_\delta\cdot d}. ( w ( δ ) ) l = T − δ κ ⋅ d ( w ( 0 ) ) l = T ( θ 0 − δ κ ) ⋅ d = t e θ δ ⋅ d . The homogeneous linear relations are preserved, so this scaling is an isomorphism of the generic polynomial Gale algebras for θ 0 \theta_0 θ 0 θ δ \theta_\delta θ δ θ 0 \theta_0 θ 0 δ \delta δ δ − \delta_- δ − δ + \delta_+ δ + (13) makes every arrow valuation greater than δ + \delta_+ δ + (22) . □
Separating the core and attached factors
A square deformation separates the core and attached equations.
A completion lemma over a one-dimensional generic base
Let K K K F 1 , … , F N , h ∈ K [ s , T , z 1 , … , z N ] F_1,\ldots,F_N,h\in K[s,T,z_1,\ldots,z_N] F 1 , … , F N , h ∈ K [ s , T , z 1 , … , z N ]
A 0 = K [ s , T , z , h − 1 ] / ( ( F 1 , … , F N ) : T ∞ ) . A_0=K[s,T,z,h^{-1}]/((F_1,\ldots,F_N):T^\infty). A 0 = K [ s , T , z , h − 1 ] / (( F 1 , … , F N ) : T ∞ ) . Suppose A 0 / T A 0 A_0/TA_0 A 0 / T A 0 K [ s ] K[s] K [ s ]
R = K [ s ] [ [ T ] ] , S = R [ T − 1 ] , A ′ = A 0 ⊗ K [ s , T ] R , B = l i m ← j A 0 / T j A 0 . R=K[s][[T]],\qquad S=R[T^{-1}],\qquad A'=A_0\otimes_{K[s,T]}R,\qquad B=\varprojlim_j A_0/T^jA_0. R = K [ s ] [[ T ]] , S = R [ T − 1 ] , A ′ = A 0 ⊗ K [ s , T ] R , B = j lim A 0 / T j A 0 . The ring here is K [ s ] [ [ T ] ] K[s][[T]] K [ s ] [[ T ]] s s s
Lemma 5.1. The algebra B B B R R R A ′ → B A'\to B A ′ → B T T T B [ T − 1 ] B[T^{-1}] B [ T − 1 ] S S S A 0 A_0 A 0 B [ T − 1 ] B[T^{-1}] B [ T − 1 ] s = 0 s=0 s = 0 s = 1 s=1 s = 1
Proof. The ring R R R T T T K [ s , T ] K[s,T] K [ s , T ] A ′ / T j A ′ ≃ A 0 / T j A 0 A'/T^jA'\simeq A_0/T^jA_0 A ′ / T j A ′ ≃ A 0 / T j A 0 j ≥ 1 j\ge1 j ≥ 1 B B B T T T A ′ A' A ′ K [ s ] K[s] K [ s ] A 0 / T A 0 A_0/TA_0 A 0 / T A 0 T T T B B B R R R A ′ A' A ′ A ′ → B A'\to B A ′ → B (21) , its kernel is a finitely generated idempotent ideal.
T T T R R R 1 − T f 1-Tf 1 − T f f ∈ R f\in R f ∈ R ( T , g ( s ) ) (T,g(s)) ( T , g ( s )) g ∈ K [ s ] g\in K[s] g ∈ K [ s ] ( 0 ) ⊂ ( T ) ⊂ ( T , g ( s ) ) (0)\subset(T)\subset(T,g(s)) ( 0 ) ⊂ ( T ) ⊂ ( T , g ( s )) R R R T T T S S S ( s ) (s) ( s ) T T T B [ T − 1 ] B[T^{-1}] B [ T − 1 ]
S [ z , h − 1 ] / ( F 1 , … , F N ) . S[z,h^{-1}]/(F_1,\ldots,F_N). S [ z , h − 1 ] / ( F 1 , … , F N ) . Let q \mathfrak q q B [ T − 1 ] B[T^{-1}] B [ T − 1 ] p = q ∩ S \mathfrak p=\mathfrak q\cap S p = q ∩ S p \mathfrak p p P \mathfrak P P S [ z , h − 1 ] S[z,h^{-1}] S [ z , h − 1 ]
D = ( S [ z , h − 1 ] ) P D=(S[z,h^{-1}])_{\mathfrak P} D = ( S [ z , h − 1 ] ) P is a regular local ring of dimension N + 1 N+1 N + 1 S p S_{\mathfrak p} S p N N N D / ( F 1 , … , F N ) D/(F_1,\ldots,F_N) D / ( F 1 , … , F N ) π \pi π S p S_{\mathfrak p} S p S S S D / ( F 1 , … , F N , π ) D/(F_1,\ldots,F_N,\pi) D / ( F 1 , … , F N , π ) D D D F 1 , … , F N , π F_1,\ldots,F_N,\pi F 1 , … , F N , π D D D π \pi π D / ( F 1 , … , F N ) D/(F_1,\ldots,F_N) D / ( F 1 , … , F N ) S S S S p S_{\mathfrak p} S p
Choose a finite splitting-field extension K ′ / K K'/K K ′ / K K ′ [ s ] [ [ T ] ] [ T − 1 ] K'[s][[T]][T^{-1}] K ′ [ s ] [[ T ]] [ T − 1 ] s = c s=c s = c S S S K ( ( T ) ) K((T)) K (( T )) c = 0 , 1 c=0,1 c = 0 , 1
The coupled equations and their selected factor
Fix r , l ≥ 0 r,l\geq0 r , l ≥ 0 r + l = k r+l=k r + l = k r ( a , b ) r(a,b) r ( a , b ) x i , y j x_i,y_j x i , y j l ( a , b ) l(a,b) l ( a , b ) u α , v β u_\alpha,v_\beta u α , v β G r × G l G_r\times G_l G r × G l s s s
x i p ∏ j ( x i − y j ) m ∏ β ( s x i − v β ) m = T b , (23) x_i^p\prod_j(x_i-y_j)^m\prod_\beta(sx_i-v_\beta)^m=T^b, \tag*{(23)} x i p j ∏ ( x i − y j ) m β ∏ ( s x i − v β ) m = T b , (23) ( − y j ) q ∏ i ( x i − y j ) m ∏ α ( u α − s y j ) m = T a , (24) (-y_j)^q\prod_i(x_i-y_j)^m\prod_\alpha(u_\alpha-sy_j)^m=T^a, \tag*{(24)} ( − y j ) q i ∏ ( x i − y j ) m α ∏ ( u α − s y j ) m = T a , (24) u α p ∏ β ( u α − v β ) m ∏ j ( u α − s y j ) m = T b , (25) u_\alpha^p\prod_\beta(u_\alpha-v_\beta)^m\prod_j(u_\alpha-sy_j)^m=T^b, \tag*{(25)} u α p β ∏ ( u α − v β ) m j ∏ ( u α − s y j ) m = T b , (25) ( − v β ) q ∏ α ( u α − v β ) m ∏ i ( s x i − v β ) m = T a . (26) (-v_\beta)^q\prod_\alpha(u_\alpha-v_\beta)^m\prod_i(sx_i-v_\beta)^m=T^a. \tag*{(26)} ( − v β ) q α ∏ ( u α − v β ) m i ∏ ( s x i − v β ) m = T a . (26) Each mixed factor is changed in both of its endpoint equations. At s = 1 s=1 s = 1 s = 0 s=0 s = 0
( p ′ , q ′ ) = ( p + m r b , q + m r a ) , p ′ a − q ′ b = 1. (27) (p',q')=(p+mrb,q+mra),\qquad p'a-q'b=1. \tag*{(27)} ( p ′ , q ′ ) = ( p + m r b , q + m r a ) , p ′ a − q ′ b = 1. (27) Its nonzero-center sign is unchanged:
( − 1 ) b q ′ + m a b ( l − 1 ) = ( − 1 ) b q + m a b ( k − 1 ) = σ k . (-1)^{bq'+mab(l-1)}=(-1)^{bq+mab(k-1)}=\sigma_k. ( − 1 ) b q ′ + mab ( l − 1 ) = ( − 1 ) b q + mab ( k − 1 ) = σ k . In particular, the attached sector at the decoupled endpoint is the intrinsic attached sector for this shifted framing.
Let I I I T T T
h = ( ∏ α u α ∏ β v β ) ( ∏ i , β ( s x i − v β ) ∏ α , j ( u α − s y j ) ) ∏ i P k ( x i ) ∏ j P k ( y j ) . (28) h=\left(\prod_{\alpha}u_{\alpha}\prod_{\beta}v_{\beta}\right)\left(\prod_{i,\beta}(sx_i-v_{\beta})\prod_{\alpha,j}(u_{\alpha}-sy_j)\right)\prod_iP_k(x_i)\prod_jP_k(y_j). \tag*{(28)} h = α ∏ u α β ∏ v β i , β ∏ ( s x i − v β ) α , j ∏ ( u α − s y j ) i ∏ P k ( x i ) j ∏ P k ( y j ) . (28) An empty product is one. The closed points of the special support of this localized algebra satisfy
x i = y j = 0 , u α , v β ∈ { P k = 0 } . (29) x_i=y_j=0,\qquad u_{\alpha},v_{\beta}\in\{P_k=0\}. \tag*{(29)} x i = y j = 0 , u α , v β ∈ { P k = 0 } . (29) Indeed, T T T T ≠ 0 T\ne0 T = 0 s s s h h h h h h
If a core cluster of charge i ( a , b ) i(a,b) i ( a , b ) c c c j ( a , b ) j(a,b) j ( a , b ) d d d
( s c − d ) m a b i j ( d − s c ) m a b i j = ( − 1 ) m a b i j . \frac{(sc-d)^{m_{abij}}}{(d-sc)^{m_{abij}}}=(-1)^{m_{abij}}. ( d − sc ) m abij ( sc − d ) m abij = ( − 1 ) m abij . Its reciprocal is the same sign. Interactions within one family give the identical sign. Thus the calculation of (14) applies unchanged: every nonzero center lies in { P k = 0 } \{P_k=0\} { P k = 0 } s s s P k ( x i ) , P k ( y j ) P_k(x_i),P_k(y_j) P k ( x i ) , P k ( y j ) (28) exclude such centers from the core, proving (29) .
The special algebra is finitely generated over C \mathbb{C} C (29) holds modulo its nilradical. On this reduced support, the denominator h h h s s s − v β -v_{\beta} − v β u α u_{\alpha} u α P k ( 0 ) P_k(0) P k ( 0 ) h − 1 h^{-1} h − 1 C [ s ] \mathbb{C}[s] C [ s ] C [ s ] \mathbb{C}[s] C [ s ]
Lemma 5.1 now applies. After inverting T T T s = 0 , 1 s=0,1 s = 0 , 1 T T T (28) have unit residues, so their inverse operators are integral as well. The commutative operator algebra they generate over the valuation ring is finite and complete. The homomorphism from the localized saturated algebra is therefore T T T B B B B B B R R R (29) forces exactly the stated core and attached valuations. Exactness of Noetherian completion on finite modules identifies B / ( s − c ) B B/(s-c)B B / ( s − c ) B A 0 / ( s − c ) A 0 A_0/(s-c)A_0 A 0 / ( s − c ) A 0 Lemma 5.1 gives equality of their characters.
Independent unit parameters and the tensor product
At the decoupled endpoint, the core right sides are u A T b u_A T^b u A T b u B T a u_B T^a u B T a
u A = V − m , u B = U − m , U = ∏ α u α , V = ∏ β ( − v β ) . (30) u_A = V^{-m}, \qquad u_B = U^{-m}, \qquad U = \prod_{\alpha} u_\alpha, \qquad V = \prod_{\beta}(-v_\beta). \tag*{(30)} u A = V − m , u B = U − m , U = α ∏ u α , V = β ∏ ( − v β ) . (30) These are two G l G_l G l
For the general graph above and an integer h ≥ 1 h \ge1 h ≥ 1 R 0 = C [ u 1 ± 1 , … , u h ± 1 ] R_0 = \mathbb{C}[u_1^{\pm1}, \ldots, u_h^{\pm1}] R 0 = C [ u 1 ± 1 , … , u h ± 1 ] χ : L → R 0 ∗ \chi: L \to R_0^* χ : L → R 0 ∗ G G G χ \chi χ (17) by
w l + − χ ( l ) t e θ ⋅ d w l − . w^{l+} - \chi(l)t^{e\theta\cdot d}w^{l-}. w l + − χ ( l ) t e θ ⋅ d w l − . Since the lattice N = Z E / L N = \mathbb{Z}^{E}/L N = Z E / L χ \chi χ Z E \mathbb{Z}^{E} Z E J θ R 0 [ w ] J_\theta R_0[\mathbf{w}] J θ R 0 [ w ] w \mathbf{w} w H θ ⊗ R 0 H_\theta\otimes R_0 H θ ⊗ R 0
Let I θ , χ h I_{\theta,\chi}^{h} I θ , χ h R 0 [ t , w ] / I θ , χ h R_0[t,\mathbf{w}]/I_{\theta,\chi}^{h} R 0 [ t , w ] / I θ , χ h ( t , w ) (t,\mathbf{w}) ( t , w ) S θ , χ S_{\theta,\chi} S θ , χ ( t , w ) (t,\mathbf{w}) ( t , w ) t t t t t t R 0 [ w ] / J θ R_0[\mathbf{w}]/J_\theta R 0 [ w ] / J θ (18) and flatness of this completion make the ℓ j \ell_j ℓ j t t t ℓ j \ell_j ℓ j t t t t t t H θ ⊗ C R 0 H_\theta\otimes_{\mathbb{C}} R_0 H θ ⊗ C R 0 t t t C \mathbb{C} C H θ H_\theta H θ t t t R 0 [ [ t ] ] R_0[[t]] R 0 [[ t ]] t t t H θ → S θ , χ H_\theta\to S_{\theta,\chi} H θ → S θ , χ t t t
S θ , χ ≃ H θ ⊗ C R 0 [ [ t ] ] S_{\theta,\chi} \simeq H_\theta\otimes_{\mathbb{C}} R_0[[t]] S θ , χ ≃ H θ ⊗ C R 0 [[ t ]] as equivariant modules. {#eq:5.9}
The completion map from R 0 [ [ t ] ] [ w ] / ( I θ , χ h , ℓ 1 , … , ℓ n ) R_0[[t]][\mathbf{w}]/(I_{\theta,\chi}^{h},\ell_1,\ldots,\ell_n) R 0 [[ t ]] [ w ] / ( I θ , χ h , ℓ 1 , … , ℓ n )
Choose a common ramification T = t e T=t^e T = t e O = C [ [ t ] ] \mathcal{O}=\mathbb{C}[[t]] O = C [[ t ]] K = C ( ( t ) ) K=\mathbb{C}((t)) K = C (( t )) R 0 = C [ u A ± 1 , u B ± 1 ] R_0=\mathbb{C}[u_A^{\pm1},u_B^{\pm1}] R 0 = C [ u A ± 1 , u B ± 1 ] r ( a , b ) r(a,b) r ( a , b ) r ≥ 1 r\ge1 r ≥ 1
χ ( δ ∗ d ) = u A ∑ i ( d A i − d ∞ ) u B − ∑ j ( d B j − d ∞ ) . \chi(\delta^*d)=u_A^{\sum_i(d_{A_i}-d_{\infty})}u_B^{-\sum_j(d_{B_j}-d_{\infty})}. χ ( δ ∗ d ) = u A ∑ i ( d A i − d ∞ ) u B − ∑ j ( d B j − d ∞ ) . The inverse exponent at the second-color vertices comes from the head-minus-tail incidence convention. At a valued specialization where u A , u B u_A,u_B u A , u B (13) remain valid. The scaling in Corollary 4.3 then identifies the generic positive factor with the generic core of this two-unit system. The full-factor argument in Proposition 4.2 applies to these unit specializations as well. Indeed, specialize the parameters in the units of a finite complete valuation-ring extension. In a full local factor with positive arrow valuations, the arrow multiplication operators generate a finite algebra over that valuation ring and are topologically nilpotent. Evaluation therefore extends continuously to the universal completion. Conversely, the arrows in this completion remain topologically nilpotent after base change. Its specialized idempotent consequently selects exactly the full positive local factors. Denote the universal positive completion by U U U r = 0 r=0 r = 0 U = R 0 [ [ t ] ] \mathcal{U}=R_0[[t]] U = R 0 [[ t ]] Corollary 4.3 give
U ≃ C r p , q ⊗ C R 0 [ [ t ] ] \mathcal{U}\simeq C_r^{p,q}\otimes_{\mathbb{C}}R_0[[t]] U ≃ C r p , q ⊗ C R 0 [[ t ]] as equivariant modules.
Let D \mathcal{D} D (5.5)}, and set D K = D [ t − 1 ] D_K=D[t^{-1}] D K = D [ t − 1 ] l = 0 l=0 l = 0 D = O \mathcal{D}=\mathcal{O} D = O U = V = 1 U=V=1 U = V = 1 U , V U,V U , V D \mathcal{D} D O \mathcal{O} O (5.8)} a continuous homomorphism R 0 [ [ t ] ] → D R_0[[t]]\to\mathcal{D} R 0 [[ t ]] → D
E = U ⊗ R 0 [ [ t ] ] D . \mathcal{E}=\mathcal{U}\otimes_{R_0[[t]]}\mathcal{D}. E = U ⊗ R 0 [[ t ]] D . This is finite free over O \mathcal{O} O
E ≃ C r p , q ⊗ C D \mathcal{E}\simeq C_r^{p,q}\otimes_{\mathbb{C}}\mathcal{D} E ≃ C r p , q ⊗ C D as ( G r × G l ) (G_r\times G_l) ( G r × G l ) O \mathcal{O} O
The substituted units are G l G_l G l G l G_l G l (5.10)} is the tensor-product action.
Now invert t t t A K A_K A K D K D_K D K (5.1)}–{(xref}(5.2)} at s = 0 s=0 s = 0 A K A_K A K D K D_K D K E K = E [ t − 1 ] E_K=E[t^{-1}] E K = E [ t − 1 ] A K A_K A K A K A_K A K D K D_K D K n \mathfrak{n} n D K D_K D K D K / n D_K/\mathfrak{n} D K / n K K K u A , u B u_A,u_B u A , u B D \mathcal{D} D O \mathcal{O} O E K E_K E K n A K \mathfrak{n}A_K n A K e , f e,f e , f e ( 1 − f ) e(1-f) e ( 1 − f ) f ( 1 − e ) f(1-e) f ( 1 − e ) e = f e=f e = f D K D_K D K
The finite free equivariant model D \mathcal{D} D G l G_l G l D l p + m r b , q + m r a D_l^{p+m r b,q+m r a} D l p + m r b , q + m r a (5.10)} gives, for ( g r , g l ) ∈ G r × G l (g_r,g_l)\in G_r\times G_l ( g r , g l ) ∈ G r × G l
Tr ( ( g r , g l ) ∣ E K ) = Tr ( g r ∣ C r p , q ) Tr ( g l ∣ D l p + m r b , q + m r a ) . \operatorname{Tr}((g_r,g_l)\mid E_K)=\operatorname{Tr}(g_r\mid C_r^{p,q})\operatorname{Tr}(g_l\mid D_l^{p+m r b,q+m r a}). Tr (( g r , g l ) ∣ E K ) = Tr ( g r ∣ C r p , q ) Tr ( g l ∣ D l p + m r b , q + m r a ) . Lemma 5.1 transports this character to the coupled sector at s = 1 s=1 s = 1 Proposition 3.1 , so its generic and special characters agree.
Theorem 5.2. For every k ≥ 0 k\geq0 k ≥ 0 (8) , there is an isomorphism of ungraded G k G_k G k
B k p , q ≃ ⨁ r + l = k Ind G r × G l G k ( C r p , q ⊗ D l p + m r b , q + m r a ) . (31) \mathcal{B}_k^{p,q}\simeq\bigoplus_{r+l=k}\operatorname{Ind}_{G_r\times G_l}^{G_k}\left(C_r^{p,q}\otimes D_l^{p+m r b,q+m r a}\right). \tag*{(31)} B k p , q ≃ r + l = k ⨁ Ind G r × G l G k ( C r p , q ⊗ D l p + m r b , q + m r a ) . (31) Proof. For k = 0 k=0 k = 0 k ≥ 1 k\geq1 k ≥ 1 r ( a , b ) r(a,b) r ( a , b ) r r r l ( a , b ) l(a,b) l ( a , b ) (5.1)}–{(xref}(5.4)}, and the decoupled calculation gives the tensor product above. The stabilizer of the chosen subsets is G r × G l G_r\times G_l G r × G l G k G_k G k r r r
Finite zero schemes and Young characters
All representations in this section are ungraded. For a partition λ = ( λ 1 , … , λ r ) \lambda=(\lambda_1,\ldots,\lambda_r) λ = ( λ 1 , … , λ r ) n n n S λ = S λ 1 × ⋯ × S λ r ⊂ S n S_\lambda=S_{\lambda_1}\times\cdots\times S_{\lambda_r}\subset S_n S λ = S λ 1 × ⋯ × S λ r ⊂ S n Ind S λ S n 1 \operatorname{Ind}_{S_\lambda}^{S_n}1 Ind S λ S n 1
Lemma 6.1. Let K K K
V = ⨁ c = 1 t K n c , G = ∏ c = 1 t S n c , V=\bigoplus_{c=1}^{t}K^{n_c},\qquad G=\prod_{c=1}^{t}S_{n_c}, V = c = 1 ⨁ t K n c , G = c = 1 ∏ t S n c , and let G G G F : V ⟶ V F:V\longrightarrow V F : V ⟶ V G G G Z Z Z G G G F − 1 ( 0 ) F^{-1}(0) F − 1 ( 0 )
O ( Z ) ≅ ⨁ λ ( c ) ⊢ n c m λ ( 1 ) , … , λ ( t ) Ind ∏ c S λ ( c ) G 1 , m λ ( 1 ) , … , λ ( t ) ∈ N . (32) \mathcal{O}(Z)\cong\bigoplus_{\lambda^{(c)}\vdash n_c}m_{\lambda^{(1)},\ldots,\lambda^{(t)}}\operatorname{Ind}_{\prod_c S_{\lambda^{(c)}}}^{G}1,\qquad m_{\lambda^{(1)},\ldots,\lambda^{(t)}}\in\mathbb{N}. \tag*{(32)} O ( Z ) ≅ λ ( c ) ⊢ n c ⨁ m λ ( 1 ) , … , λ ( t ) Ind ∏ c S λ ( c ) G 1 , m λ ( 1 ) , … , λ ( t ) ∈ N . (32) Proof. Put N = ∑ c n c N=\sum_c n_c N = ∑ c n c N = 0 N=0 N = 0 N > 0 N>0 N > 0 N N N t c , j t_{c,j} t c , j
P c ( z ) = ∑ j = 0 n c − 1 t c , j z j . P_c(z)=\sum_{j=0}^{n_c-1}t_{c,j}z^j. P c ( z ) = j = 0 ∑ n c − 1 t c , j z j . The equations
F c , i ( x ) + P c ( x c , i ) = 0 ( 1 ≤ c ≤ t , 1 ≤ i ≤ n c ) (33) F_{c,i}(x)+P_c(x_{c,i})=0\qquad(1\leq c\leq t,\ 1\leq i\leq n_c) \tag*{(33)} F c , i ( x ) + P c ( x c , i ) = 0 ( 1 ≤ c ≤ t , 1 ≤ i ≤ n c ) (33) are equivariant, with trivial $G$-action on the parameter space $T=\mathbb{A}_K^N$. We prove first that a nonempty Zariski open subset of $T$ gives finite reduced zero schemes.
Stratify V V V c c c z c , 1 , … , z c , d c z_{c,1},\ldots,z_{c,d_c} z c , 1 , … , z c , d c m c , 1 , … , m c , d c m_{c,1},\ldots,m_{c,d_c} m c , 1 , … , m c , d c d = ∑ c d c d=\sum_c d_c d = ∑ c d c H = ∏ c , l S m c , l H=\prod_{c,l}S_{m_{c,l}} H = ∏ c , l S m c , l
Keep one equation from (6.2) for each equality block. The other equations in that block agree by equivariance. Evaluation of a polynomial of degree at most n c − 1 n_c-1 n c − 1 d c d_c d c d d d T T T N N N T T T N − 1 N-1 N − 1 N − 1 N-1 N − 1
Let R R R d d d R δ z + E z δ t = 0 R\,\delta z+E_z\,\delta t=0 R δ z + E z δ t = 0 E z E_z E z R R R R R R T T T
At each point of the stratum, the full Jacobian is an endomorphism of the H H H V V V H H H m c , l > 1 m_{c,l}>1 m c , l > 1 λ c , l \lambda_{c,l} λ c , l
det Jac ( F + P ) = det R ∏ c , l : m c , l > 1 λ c , l m c , l − 1 . (34) \det\operatorname{Jac}(F+P)=\det R\prod_{c,l:m_{c,l}>1}\lambda_{c,l}^{m_{c,l}-1}. \tag*{(34)} det Jac ( F + P ) = det R c , l : m c , l > 1 ∏ λ c , l m c , l − 1 . (34) If r c r_c r c c c c d c + r c ≤ n c d_c+r_c\le n_c d c + r c ≤ n c P c P_c P c
∏ m c , l = 1 ( z − z c , l ) ∏ m c , l > 1 ( z − z c , l ) 2 , \prod_{m_{c,l}=1}(z-z_{c,l})\prod_{m_{c,l}>1}(z-z_{c,l})^2, m c , l = 1 ∏ ( z − z c , l ) m c , l > 1 ∏ ( z − z c , l ) 2 , of degree d c + r c d_c+r_c d c + r c λ c , l \lambda_{c,l} λ c , l λ c , l = 0 \lambda_{c,l}=0 λ c , l = 0 N − 1 N-1 N − 1
There are finitely many equality partitions. Removing all the bad image closures just described leaves a nonempty open subset T ∘ \mathcal{T}^{\circ} T ∘ (34) . Its affine zero scheme is consequently reduced and zero-dimensional, hence finite.
Choose a parameter vector whose line through the origin is not contained in T ∖ T ∘ \mathcal{T}\setminus\mathcal{T}^{\circ} T ∖ T ∘
F c , i s ( x ) = F c , i ( x ) + s p c ( x c , i ) . F^{s}_{c,i}(x)=F_{c,i}(x)+s\,p_c(x_{c,i}). F c , i s ( x ) = F c , i ( x ) + s p c ( x c , i ) . Over K ( ( s ) ) K((s)) K (( s )) Z Z Z K [ x ] / ( F ) K[x]/(F) K [ x ] / ( F ) G G G g g g ( K [ x ] / ( F ) ) [ g − 1 ] = O ( Z ) (K[x]/(F))[g^{-1}]=\mathcal{O}(Z) ( K [ x ] / ( F )) [ g − 1 ] = O ( Z )
A = K [ [ s ] ] [ x , g − 1 ] / ( F s ) , A ^ = lim j A / s j A . A=K[[s]][x,g^{-1}]/(F^{s}),\qquad\widehat{A}=\lim_{j} A/s^{j}A. A = K [[ s ]] [ x , g − 1 ] / ( F s ) , A = j lim A / s j A . Its special fiber is exactly O ( Z ) \mathcal{O}(Z) O ( Z ) A ^ \widehat{A} A K [ [ s ] ] K[[s]] K [[ s ]] s s s s 2 , … s^2,\ldots s 2 , …
At each point x 0 x_0 x 0 Z Z Z N + 1 N+1 N + 1 F 1 s , … , F N s , s F^{s}_1,\ldots,F^{s}_N,s F 1 s , … , F N s , s K [ [ s , x − x 0 ] ] K[[s,x-x_0]] K [[ s , x − x 0 ]] s s s N N N s s s A / s A A/sA A / s A s A sA s A A ^ \widehat{A} A K [ [ s ] ] K[[s]] K [[ s ]]
The completion map A → A ^ A\to\widehat{A} A → A K [ [ s ] ] K[[s]] K [[ s ]] A A A J J J J / J 2 = 0 J/J^2=0 J / J 2 = 0 A ^ \widehat{A} A A A A s s s
The group G G G K [ [ s ] ] K[[s]] K [[ s ]] G G G E E E Hom G ( E , A ^ ) \operatorname{Hom}_G(E,\widehat{A}) Hom G ( E , A ) K [ [ s ] ] K[[s]] K [[ s ]] K ( ( s ) ) K((s)) K (( s )) G G G (32) .
Regular equivariant deformations and their permutation representations were studied in [24 Corollary 6.2.8 and Section 8.1] . The proof above gives polynomial perturbations with separate degree bounds for each color and retains the selected full local factors.
For an alphabet α \alpha α p r ( α ) = ∑ i α i r p_r(\alpha)=\sum_i\alpha_i^r p r ( α ) = ∑ i α i r r ≥ 1 r\ge1 r ≥ 1 h j ( α ) h_j(\alpha) h j ( α ) j j j e j ( α ) e_j(\alpha) e j ( α ) j j j h 0 = e 0 = 1 h_0=e_0=1 h 0 = e 0 = 1 λ \lambda λ
p λ = ∏ i p λ i , h λ = ∏ i h λ i , e λ = ∏ i e λ i . p_{\lambda}=\prod_i p_{\lambda_i},\qquad h_{\lambda}=\prod_i h_{\lambda_i},\qquad e_{\lambda}=\prod_i e_{\lambda_i}. p λ = i ∏ p λ i , h λ = i ∏ h λ i , e λ = i ∏ e λ i . For a finite S n S_n S n E E E
ch E = 1 n ! ∑ σ ∈ S n tr ( σ ∣ E ) p λ ( σ ) . \operatorname{ch} E=\frac{1}{n!}\sum_{\sigma\in S_n}\operatorname{tr}(\sigma\mid E)\,p_{\lambda(\sigma)}. ch E = n ! 1 σ ∈ S n ∑ tr ( σ ∣ E ) p λ ( σ ) . where λ ( σ ) \lambda(\sigma) λ ( σ ) σ \sigma σ S A × S B S_A\times S_B S A × S B h λ h μ h_\lambda h_\mu h λ h μ
∑ n ≥ 0 h n z n = ∏ i ( 1 − α i z ) − 1 = exp ( ∑ r ≥ 1 p r z r r ) . \sum_{n\geq0}h_nz^n=\prod_i(1-\alpha_i z)^{-1}=\exp\left(\sum_{r\geq1}\frac{p_rz^r}{r}\right). n ≥ 0 ∑ h n z n = i ∏ ( 1 − α i z ) − 1 = exp ( r ≥ 1 ∑ r p r z r ) . identifies h n h_n h n S n S_n S n ι = ω α ω β \iota=\omega_\alpha\omega_\beta ι = ω α ω β ω ( h j ) = e j \omega(h_j)=e_j ω ( h j ) = e j ω ( p j ) = ( − 1 ) j − 1 p j \omega(p_j)=(-1)^{j-1}p_j ω ( p j ) = ( − 1 ) j − 1 p j W W W sgn W \operatorname{sgn}_W sgn W W W W E E E
E sgn W = { u ∈ E : w u = sgn W ( w ) u for every w ∈ W } E^{\operatorname{sgn}_W}=\{u\in E:wu=\operatorname{sgn}_W(w)u\text{ for every }w\in W\} E sgn W = { u ∈ E : w u = sgn W ( w ) u for every w ∈ W } for its sign-isotypic subspace. Since sgn W \operatorname{sgn}_W sgn W dim E sgn W = dim Hom W ( sgn W , E ) \dim E^{\operatorname{sgn}_W}=\dim\operatorname{Hom}_W(\operatorname{sgn}_W,E) dim E sgn W = dim Hom W ( sgn W , E ) C ( ( T ) ) \mathbb{C}((T)) C (( T )) G k = S k a × S k b G_k=S_{k_a}\times S_{k_b} G k = S k a × S k b
ch D k p , q = ∑ λ ⊢ k a , μ ⊢ k b n λ μ p , q h λ ( α ) h μ ( β ) , n λ μ p , q ∈ N . (35) \operatorname{ch}D_k^{p,q}=\sum_{\substack{\lambda\vdash k_a,\ \mu\vdash k_b}}n_{\lambda\mu}^{p,q}h_\lambda(\alpha)h_\mu(\beta),\qquad n_{\lambda\mu}^{p,q}\in\mathbb{N}. \tag*{(35)} ch D k p , q = λ ⊢ k a , μ ⊢ k b ∑ n λ μ p , q h λ ( α ) h μ ( β ) , n λ μ p , q ∈ N . (35) A polynomial Harder–Narasimhan decomposition
The polynomial factorization below is the shuffle form of the Chow–Hall Harder–Narasimhan decomposition and destabilizing-product quotient of [13 Theorems 6.2 and 8.1] . Its separated-center proof is adapted from [21 Sections 3.1–3.2] .
Let I I I a i j ∈ N a_{ij}\in\mathbb{N} a ij ∈ N d ∈ N I d\in\mathbb{N}^I d ∈ N I
S d = Q [ x i , α : i ∈ I , 1 ≤ α ≤ d i ] ∏ i S d i , χ ( d , e ) = ∑ i d i e i − ∑ i , j a i j d i e j . S_d=\mathbb{Q}[x_{i,\alpha}:i\in I,\ 1\leq\alpha\leq d_i]^{\prod_i S_{d_i}},\qquad\chi(d,e)=\sum_i d_i e_i-\sum_{i,j}a_{ij}d_i e_j. S d = Q [ x i , α : i ∈ I , 1 ≤ α ≤ d i ] ∏ i S d i , χ ( d , e ) = i ∑ d i e i − i , j ∑ a ij d i e j . Every variable has polynomial degree one. If f ∈ S d f\in S_d f ∈ S d g ∈ S e g\in S_e g ∈ S e
f ∗ g = ∑ shuffles f ( x ) g ( y ) ∏ i , j ∏ α = 1 d i ∏ β = 1 e j ( y j , β − x i , α ) a i j − δ i j . (36) f*g=\sum_{\text{shuffles}}f(x)g(y)\prod_{i,j}\prod_{\alpha=1}^{d_i}\prod_{\beta=1}^{e_j}(y_{j,\beta}-x_{i,\alpha})^{a_{ij}-\delta_{ij}}. \tag*{(36)} f ∗ g = shuffles ∑ f ( x ) g ( y ) i , j ∏ α = 1 ∏ d i β = 1 ∏ e j ( y j , β − x i , α ) a ij − δ ij . (36) There is one summand for each partition of the variables of every color into the two ordered packets of sizes d i , e i d_i,e_i d i , e i (36) is a polynomial. Expanding three ordered packets proves associativity. Its degree is
deg ( f ∗ g ) = deg f + deg g − χ ( d , e ) (37) \deg(f*g)=\deg f+\deg g-\chi(d,e) \tag*{(37)} deg ( f ∗ g ) = deg f + deg g − χ ( d , e ) (37) for homogeneous inputs whose product is nonzero.
For the zero dimension vector, S 0 = Q S_0=\mathbb{Q} S 0 = Q c , η : Z I → Q c,\eta:\mathbb{Z}^I\to\mathbb{Q} c , η : Z I → Q c ( e i ) > 0 c(e_i)>0 c ( e i ) > 0 μ ( d ) = η ( d ) / c ( d ) \mu(d)=\eta(d)/c(d) μ ( d ) = η ( d ) / c ( d ) d ≠ 0 d\ne0 d = 0 d ≠ 0 d\ne0 d = 0
I d μ = ∑ u + v = d , u , v ≠ 0 μ ( u ) > μ ( d ) S u ∗ S v , B d μ = S d / I d μ . (38) I_d^\mu=\sum_{\substack{u+v=d,\ u,v\ne0\\ \mu(u)>\mu(d)}}S_u\ast S_v,\qquad B_d^\mu=S_d/I_d^\mu. \tag*{(38)} I d μ = u + v = d , u , v = 0 μ ( u ) > μ ( d ) ∑ S u ∗ S v , B d μ = S d / I d μ . (38) The sum is an ordinary homogeneous ideal of S d S_d S d S u ⊗ S v S_u\otimes S_v S u ⊗ S v I d μ I_d^\mu I d μ
Theorem 7.1. Choose homogeneous linear sections B d μ → S d B_d^\mu\to S_d B d μ → S d d ≠ 0 d\ne0 d = 0
⨁ d 1 + ⋯ + d s = d μ ( d 1 ) > ⋯ > μ ( d s ) B d 1 μ ⊗ ⋯ ⊗ B d s μ → ∼ S d . (39) \bigoplus_{\substack{d^1+\cdots+d^s=d\\ \mu(d^1)>\cdots>\mu(d^s)}}B_{d^1}^\mu\otimes\cdots\otimes B_{d^s}^\mu\xrightarrow{\sim}S_d. \tag*{(39)} d 1 + ⋯ + d s = d μ ( d 1 ) > ⋯ > μ ( d s ) ⨁ B d 1 μ ⊗ ⋯ ⊗ B d s μ ∼ S d . (39) All parts d i d^i d i d = 0 d=0 d = 0 j 1 , … , j s j_1,\ldots,j_s j 1 , … , j s ∑ i j i − ∑ i < l χ ( d i , d l ) \sum_i j_i-\sum_{i<l}\chi(d^i,d^l) ∑ i j i − ∑ i < l χ ( d i , d l )
Proof. Surjectivity follows by induction on ∣ d ∣ = ∑ i d i |d|=\sum_i d_i ∣ d ∣ = ∑ i d i (38) , any polynomial is a section representative plus a sum of products with μ ( u ) > μ ( d ) > μ ( v ) \mu(u)>\mu(d)>\mu(v) μ ( u ) > μ ( d ) > μ ( v ) ( c ( d i ) , η ( d i ) ) (c(d^i),\eta(d^i)) ( c ( d i ) , η ( d i ))
The two packets being combined never make up the entire list. The original unstable cut lies strictly above the total endpoint chord, and every subsequent path remains above that cut. A whole list consisting of two nondecreasing slopes would lie below the total chord, a contradiction. Thus lower-particle induction is valid at every replacement. There are only finitely many ordered lists of nonzero subvectors summing to d d d
For injectivity, fix a decreasing target list R = ( r 1 , … , r q ) R=(r_1,\ldots,r_q) R = ( r 1 , … , r q ) j j j z j z_j z j B r j μ B_{r_j}^\mu B r j μ
S d ⟶ ( ⨂ j B r j μ ) [ z 1 , … , z q ] . (40) S_d\longrightarrow\left(\bigotimes_j B_{r_j}^\mu\right)[z_1,\ldots,z_q]. \tag*{(40)} S d ⟶ ( j ⨂ B r j μ ) [ z 1 , … , z q ] . (40) To analyze its value on a rational shuffle expression, extend scalars to Q ( z 1 , … , z q ) \mathbb{Q}(z_1,\ldots,z_q) Q ( z 1 , … , z q ) z j − z i z_j-z_i z j − z i
Consider a decreasing source list D = ( d 1 , … , d p ) D=(d_1,\ldots,d_p) D = ( d 1 , … , d p ) d i j d_{ij} d ij i i i j j j d i d_i d i r j r_j r j ⨂ i , j S d i j \bigotimes_{i,j}S_{d_{ij}} ⨂ i , j S d ij j j j
Write u h j = ∑ i ≤ h d i j u_{hj}=\sum_{i\le h}d_{ij} u hj = ∑ i ≤ h d ij u h j ≠ 0 u_{hj}\ne0 u hj = 0 μ ( u h j ) > μ ( r j ) \mu(u_{hj})>\mu(r_j) μ ( u hj ) > μ ( r j ) r j − u h j ≠ 0 r_j-u_{hj}\ne0 r j − u hj = 0 I r j μ I_{r_j}^{\mu} I r j μ
η ( u h j ) ≤ μ ( r j ) c ( u h j ) , 0 ≤ c ( u h j ) ≤ c ( r j ) . (41) \eta(u_{hj})\le\mu(r_j)c(u_{hj}),\qquad0\le c(u_{hj})\le c(r_j). \tag*{(41)} η ( u hj ) ≤ μ ( r j ) c ( u hj ) , 0 ≤ c ( u hj ) ≤ c ( r j ) . (41) At fixed total charge C C C ∑ j μ ( r j ) t j \sum_j\mu(r_j)t_j ∑ j μ ( r j ) t j 0 ≤ t j ≤ c ( r j ) 0\le t_j\le c(r_j) 0 ≤ t j ≤ c ( r j ) ∑ j t j = C \sum_jt_j=C ∑ j t j = C j < l j<l j < l t j < c ( r j ) t_j<c(r_j) t j < c ( r j ) t l > 0 t_l>0 t l > 0 min { c ( r j ) − t j , t l } \min\{c(r_j)-t_j,t_l\} min { c ( r j ) − t j , t l } t l t_l t l t j t_j t j C C C t j = c ( u h j ) t_j=c(u_{hj}) t j = c ( u hj ) (41) therefore puts every source breakpoint below the target polygon. Concavity of the target puts the entire source path below it. Thus a source polygon not pointwise at most the target polygon has zero restriction.
If two polygon functions agree, their strictly decreasing slopes and segment lengths agree in order. At the first breakpoint, equality in the capacity bound forces the first target packet to be full and all later packets to be empty. Since 0 ≤ d i j ≤ r j 0\le d_{ij}\le r_j 0 ≤ d ij ≤ r j c c c d i = r i d_i=r_i d i = r i (40) is the tensor of translated section representatives multiplied by
K R = ∏ 1 ≤ i < j ≤ q ∏ u ∈ I ∏ v ∈ I ∏ α = 1 ( r i ) u ∏ β = 1 ( r j ) v ( z j − z i + x v , β ( j ) − x u , α ( i ) ) a u v − δ u v . (42) K_R=\prod_{1\le i<j\le q}\prod_{u\in I}\prod_{v\in I}\prod_{\alpha=1}^{(r_i)_u}\prod_{\beta=1}^{(r_j)_v}(z_j-z_i+x_{v,\beta}^{(j)}-x_{u,\alpha}^{(i)})^{a_{uv}-\delta_{uv}}. \tag*{(42)} K R = 1 ≤ i < j ≤ q ∏ u ∈ I ∏ v ∈ I ∏ α = 1 ∏ ( r i ) u β = 1 ∏ ( r j ) v ( z j − z i + x v , β ( j ) − x u , α ( i ) ) a uv − δ uv . (42) This kernel is separately symmetric in every complete target packet and has invertible scalar constant term ∏ 1 ≤ i < j ≤ q ( z j − z i ) − χ ( r i , r j ) \prod_{1\le i<j\le q}(z_j-z_i)^{-\chi(r_i,r_j)} ∏ 1 ≤ i < j ≤ q ( z j − z i ) − χ ( r i , r j )
Take a finite relation between the images of the summands of (39) . Choose a pointwise-minimal source polygon among the nonzero terms and use its full list as the target. All incomparable or larger polygons vanish. A strictly smaller one is absent by minimality; equal polygons with a different full list also vanish. Cancel (42) from the remaining term. The tensor of translated section representatives is then zero. Its vanishing in the completion implies its vanishing before completion, since a nonnegatively graded vector space injects into the product of its graded pieces. Common translation is an automorphism of each polynomial quotient, so inverse translation kills the original tensor coefficient. Remove this list and repeat. This proves injectivity. The grading statement follows from (37) .
For a nonnegatively graded vector space V V V Hilb ( V ; t ) = ∑ j ≥ 0 ( dim V j ) t j \operatorname{Hilb}(V;t)=\sum_{j\ge0}(\dim V_j)t^j Hilb ( V ; t ) = ∑ j ≥ 0 ( dim V j ) t j S d S_d S d
Hilb ( S d ; t ) = ∏ i ∈ I ∏ j = 1 d i ( 1 − t j ) − 1 . \operatorname{Hilb}(S_d;t)=\prod_{i\in I}\prod_{j=1}^{d_i}(1-t^j)^{-1}. Hilb ( S d ; t ) = i ∈ I ∏ j = 1 ∏ d i ( 1 − t j ) − 1 . Theorem 7.1 gives, for d ≠ 0 d\ne0 d = 0
Hilb ( B d μ ; t ) = Hilb ( S d ; t ) \operatorname{Hilb}(B_d^\mu;t)=\operatorname{Hilb}(S_d;t) Hilb ( B d μ ; t ) = Hilb ( S d ; t ) − ∑ d 1 + ⋯ + d s = d , s ≥ 2 d i ≠ 0 , μ ( d 1 ) > ⋯ > μ ( d s ) t − ∑ i < l χ ( d i , d l ) ∏ i = 1 s Hilb ( B d i μ ; t ) . -\sum_{\substack{d^1+\cdots+d^s=d,\ s\ge2\\ d^i\ne0,\ \mu(d^1)>\cdots>\mu(d^s)}}t^{-\sum_{i<l}\chi(d^i,d^l)}\prod_{i=1}^{s}\operatorname{Hilb}(B_{d^i}^\mu;t). − d 1 + ⋯ + d s = d , s ≥ 2 d i = 0 , μ ( d 1 ) > ⋯ > μ ( d s ) ∑ t − ∑ i < l χ ( d i , d l ) i = 1 ∏ s Hilb ( B d i μ ; t ) . Every part in this finite sum has smaller total dimension. Induction therefore proves that Hilb ( B d h ; t ) \operatorname{Hilb}(B_d^h;t) Hilb ( B d h ; t )
Put ⟨ d , e ⟩ = χ ( e , d ) − χ ( d , e ) \langle d,e\rangle=\chi(e,d)-\chi(d,e) ⟨ d , e ⟩ = χ ( e , d ) − χ ( d , e )
X d X e = v ⟨ d , e ⟩ X d + e . (43) X^dX^e=v^{\langle d,e\rangle}X^{d+e}. \tag*{(43)} X d X e = v ⟨ d , e ⟩ X d + e . (43) All series are completed by nonnegative dimension vectors; each fixed dimension involves finitely many decompositions. Coefficients belong to Q ( v ) \mathbb{Q}(v) Q ( v )
S = ∑ d v χ ( d , d ) Hilb ( S d ; v 2 ) X d , H ρ = 1 + ∑ d ≠ 0 μ ( d ) = ρ v χ ( d , d ) Hilb ( B d μ ; v 2 ) X d . \mathcal{S}=\sum_d v^{\chi(d,d)}\operatorname{Hilb}(S_d;v^2)X^d,\qquad H_\rho=1+\sum_{\substack{d\ne0\\\mu(d)=\rho}}v^{\chi(d,d)}\operatorname{Hilb}(B_d^\mu;v^2)X^d. S = d ∑ v χ ( d , d ) Hilb ( S d ; v 2 ) X d , H ρ = 1 + d = 0 μ ( d ) = ρ ∑ v χ ( d , d ) Hilb ( B d μ ; v 2 ) X d . Theorem 7.1 gives S = ∏ ρ ↘ H ρ \mathcal{S}=\prod_\rho^{\searrow}H_\rho S = ∏ ρ ↘ H ρ
χ ( d + e , d + e ) + 2 ( i + j − χ ( d , e ) ) = χ ( d , d ) + 2 i + χ ( e , e ) + 2 j + ⟨ d , e ⟩ , \begin{aligned}
\chi(d+e,d+e)+2(i+j-\chi(d,e))&=\chi(d,d)+2i+\chi(e,e)+2j+\langle d,e\rangle,
\end{aligned} χ ( d + e , d + e ) + 2 ( i + j − χ ( d , e )) = χ ( d , d ) + 2 i + χ ( e , e ) + 2 j + ⟨ d , e ⟩ , which is precisely the weight prescribed by (43) . For any fixed total-dimension cutoff, an ordered factorization by prescribed slopes is unique: the unknown term of each smallest unresolved dimension occurs linearly in its one prescribed factor, and all other terms use smaller dimensions.
The attached algebra and the two-input wall function
Fix an integer m ≥ 1 m\ge1 m ≥ 1 α = ( α 1 , … , α s ) \alpha=(\alpha_1,\ldots,\alpha_s) α = ( α 1 , … , α s ) β = ( β 1 , … , β t ) \beta=(\beta_1,\ldots,\beta_t) β = ( β 1 , … , β t ) I A = { A 1 , … , A s } I_A=\{A_1,\ldots,A_s\} I A = { A 1 , … , A s } I B = { B 1 , … , B t } I_B=\{B_1,\ldots,B_t\} I B = { B 1 , … , B t } I = I A ⊔ I B I=I_A\sqcup I_B I = I A ⊔ I B m m m B B B A A A A ( d ) = ∑ i ∈ I A d i A(d)=\sum_{i\in I_A}d_i A ( d ) = ∑ i ∈ I A d i B ( d ) = ∑ j ∈ I B d j B(d)=\sum_{j\in I_B}d_j B ( d ) = ∑ j ∈ I B d j
χ ( d , e ) = ∑ i d i e i − m B ( d ) A ( e ) , ⟨ d , e ⟩ = m ( B ( d ) A ( e ) − A ( d ) B ( e ) ) . (44) \chi(d,e)=\sum_i d_i e_i-mB(d)A(e),\qquad\langle d,e\rangle=m\bigl(B(d)A(e)-A(d)B(e)\bigr). \tag*{(44)} χ ( d , e ) = i ∑ d i e i − m B ( d ) A ( e ) , ⟨ d , e ⟩ = m ( B ( d ) A ( e ) − A ( d ) B ( e ) ) . (44) The skew form depends only on the two aggregate dimensions. Define the commuting tags by
α d A = ∏ i = 1 s α i d A i , β d B = ∏ j = 1 t β j d B j . \alpha^{d_A}=\prod_{i=1}^{s}\alpha_i^{d_{A_i}},\qquad\beta^{d_B}=\prod_{j=1}^{t}\beta_j^{d_{B_j}}. α d A = i = 1 ∏ s α i d A i , β d B = j = 1 ∏ t β j d B j . They retain the color information when the based monomials are projected to the rank-two aggregate lattice.
Central factors and their two framed chambers
Fix coprime a , b > 0 a,b>0 a , b > 0 γ = ( a , b ) \gamma=(a,b) γ = ( a , b ) θ 0 = b A − a B \theta_0=bA-aB θ 0 = b A − a B − θ 0 / ( A + B ) -\theta_0/(A+B) − θ 0 / ( A + B )
H ( U ; v ) = 1 + ∑ k ≥ 1 ∑ ( A ( d ) , B ( d ) ) = k ( a , b ) v χ ( d , d ) Hilb ( B d μ ; v 2 ) α d A β d B U k . (45) H(U;v)=1+\sum_{k\ge1}\ \sum_{\substack{(A(d),B(d))=k(a,b)}}v^{\chi(d,d)}\operatorname{Hilb}(B_d^\mu;v^2)\alpha^{d_A}\beta^{d_B}U^k. \tag*{(45)} H ( U ; v ) = 1 + k ≥ 1 ∑ ( A ( d ) , B ( d )) = k ( a , b ) ∑ v χ ( d , d ) Hilb ( B d μ ; v 2 ) α d A β d B U k . (45) All monomials in this central sector commute. The series does not involve a choice of framing.
Choose p , q > 0 p,q>0 p , q > 0 p a − q b = 1 pa-qb=1 p a − q b = 1 ∞ \infty ∞ p p p ∞ \infty ∞ A A A q q q B B B ∞ \infty ∞
On framed dimension vectors set
τ = A + B + d ∞ , ϕ = p A − q B , μ ± = − θ 0 ± ϵ ϕ τ . \tau=A+B+d_\infty,\qquad\phi=pA-qB,\qquad\mu_\pm=\frac{-\theta_0\pm\epsilon\phi}{\tau}. τ = A + B + d ∞ , ϕ = p A − q B , μ ± = τ − θ 0 ± ϵ ϕ . Within any fixed finite dimension cutoff, choose a rational ϵ > 0 \epsilon>0 ϵ > 0 − θ 0 / τ -\theta_0/\tau − θ 0 / τ k γ k\gamma k γ ϕ / τ = 1 / ( a + b ) \phi/\tau=1/(a+b) ϕ / τ = 1/ ( a + b ) ( l γ , 1 ) (l\gamma,1) ( l γ , 1 ) ϕ / τ = l / ( l ( a + b ) + 1 ) < 1 / ( a + b ) \phi/\tau=l/(l(a+b)+1)<1/(a+b) ϕ / τ = l / ( l ( a + b ) + 1 ) < 1/ ( a + b ) W W W
W = H T a f t e r = T b e f o r e H . (46) W = H T_{\mathrm{after}} = T_{\mathrm{before}} H. \tag*{(46)} W = H T after = T before H . (46) We verify that these are the chambers used in Corollary 4.3 . In total dimension ( k γ , 1 ) (k\gamma,1) ( k γ , 1 ) θ 0 \theta_0 θ 0 l γ l\gamma l γ μ + \mu_+ μ + (38) . The balanced subset has κ \kappa κ N A l a − N B l b = l ( p a − q b ) = l N_A la-N_B lb=l(pa-qb)=l N A l a − N B l b = l ( p a − q b ) = l − δ l -\delta l − δ l θ 0 − δ κ \theta_0-\delta\kappa θ 0 − δ κ δ > 0 \delta>0 δ > 0 μ − \mu_- μ − (46) is therefore a compatible identity of full formal series.
The thin and colored polynomial quotients
The thin-quiver quotient and Vandermonde comparison below are algebraic versions of the presentation and alternation in [12 Sections 3.1–3.2, Lemma 3.8] . The cut products and the linear forms (16) have integer coefficients. Hence the cut algebra has the rational form
H θ , Q = Q [ w ] / ( J θ , ℓ 1 , … , ℓ n ) , H θ = H θ , Q ⊗ Q C . H_{\theta,\mathbb{Q}}=\mathbb{Q}[\mathbf{w}]/(J_\theta,\ell_1,\ldots,\ell_n), \qquad H_\theta=H_{\theta,\mathbb{Q}}\otimes_{\mathbb{Q}}\mathbb{C}. H θ , Q = Q [ w ] / ( J θ , ℓ 1 , … , ℓ n ) , H θ = H θ , Q ⊗ Q C . The particle actions preserve this form. Scalar extension preserves graded dimensions and commutes with the sign projectors ∣ W ∣ − 1 ∑ w ∈ W sgn W ( w ) w |W|^{-1}\sum_{w\in W}\operatorname{sgn}_W(w)w ∣ W ∣ − 1 ∑ w ∈ W sgn W ( w ) w ( d , 1 ) (d,1) ( d , 1 ) ( A ( d ) , B ( d ) ) = k ( a , b ) (A(d),B(d))=k(a,b) ( A ( d ) , B ( d )) = k ( a , b ) k ≥ 1 k\geq1 k ≥ 1 i i i d i d_i d i P P P x ∞ x_\infty x ∞ W d = ∏ i S d i W_d=\prod_i S_{d_i} W d = ∏ i S d i
R = P W d , Δ d = ∏ i ∏ r < l ( x i , l − x i , r ) , ζ ( d ) = ∑ i ( d i 2 ) . R=P^{W_d}, \qquad\Delta_d=\prod_i\prod_{r<l}(x_{i,l}-x_{i,r}), \qquad\zeta(d)=\sum_i\binom{d_i}{2}. R = P W d , Δ d = i ∏ r < l ∏ ( x i , l − x i , r ) , ζ ( d ) = i ∑ ( 2 d i ) . For the chosen chamber, let J J J P P P
E S = ∏ e : S → S c ( x head ( e ) − x tail ( e ) ) . E_S=\prod_{e:S\to S^c}(x_{\operatorname{head}(e)}-x_{\operatorname{tail}(e)}). E S = e : S → S c ∏ ( x head ( e ) − x tail ( e ) ) . In thin dimension each split is a labeled subset, so its sole shuffle term has kernel E S E_S E S E S P E_SP E S P (38) is P / J P/J P / J H θ , Q [ x ∞ ] H_{\theta,\mathbb{Q}}[x_\infty] H θ , Q [ x ∞ ] x ∞ = 0 x_\infty=0 x ∞ = 0 H θ , Q H_{\theta,\mathbb{Q}} H θ , Q
Let I I I R R R ( d , 1 ) (d,1) ( d , 1 ) e , f e,f e , f E S E_S E S E S Δ e Δ f E_S\Delta_e\Delta_f E S Δ e Δ f P W e × W f P^{W_e\times W_f} P W e × W f Δ d = Δ e Δ f D e , f \Delta_d=\Delta_e\Delta_fD_{e,f} Δ d = Δ e Δ f D e , f D e , f D_{e,f} D e , f
Δ d ∑ σ ∈ W d / ( W e × W f ) σ ( E S h D e , f ) , h ∈ P W e × W f . \Delta_d\sum_{\sigma\in W_d/(W_e\times W_f)}\sigma\left(\frac{E_Sh}{D_{e,f}}\right), \qquad h\in P^{W_e\times W_f}. Δ d σ ∈ W d / ( W e × W f ) ∑ σ ( D e , f E S h ) , h ∈ P W e × W f . The sum is exactly the shuffle expression for that split. Normalizing the alternation operators only multiplies it by nonzero rational scalars and does not change the image. Summing over the forbidden splits yields
J ∩ P sgn W d = Δ d I , R / I → ⋅ Δ d ( P / J ) sgn W d . (47) J\cap P^{\operatorname{sgn}W_d}=\Delta_d I,\qquad R/I\xrightarrow{\ \cdot\Delta_d\ }(P/J)^{\operatorname{sgn}W_d}. \tag*{(47)} J ∩ P sgn W d = Δ d I , R / I ⋅ Δ d ( P / J ) sgn W d . (47) The equality also proves surjectivity, since every alternating polynomial is divisible by Δ d \Delta_d Δ d
The frame variable does not occur in Δ d \Delta_d Δ d (47) commutes with setting x ∞ = 0 x_\infty=0 x ∞ = 0 P b e f o r e ( d ; t ) P_{\mathrm{before}}(d;t) P before ( d ; t ) P a f t e r ( d ; t ) P_{\mathrm{after}}(d;t) P after ( d ; t ) H − H_- H − H + H_+ H +
Hilb ( H − sgn W d ; v 2 ) = v 2 ζ ( d ) P b e f o r e ( d ; v 2 ) , \operatorname{Hilb}\left(H_-^{\operatorname{sgn}W_d};v^2\right)=v^{2\zeta(d)}P_{\mathrm{before}}(d;v^2), Hilb ( H − sgn W d ; v 2 ) = v 2 ζ ( d ) P before ( d ; v 2 ) , Hilb ( H + sgn W d ; v 2 ) = v 2 ζ ( d ) P a f t e r ( d ; v 2 ) . (48) \operatorname{Hilb}\left(H_+^{\operatorname{sgn}W_d};v^2\right)=v^{2\zeta(d)}P_{\mathrm{after}}(d;v^2). \tag*{(48)} Hilb ( H + sgn W d ; v 2 ) = v 2 ζ ( d ) P after ( d ; v 2 ) . (48) For the pure frame set P b e f o r e ( 0 ; t ) = P a f t e r ( 0 ; t ) = 1 P_{\mathrm{before}}(0;t)=P_{\mathrm{after}}(0;t)=1 P before ( 0 ; t ) = P after ( 0 ; t ) = 1 A , B A,B A , B χ t h i n = A + B − m A B \chi_{\mathrm{thin}}=A+B-mAB χ thin = A + B − m A B
χ ( d , d ) − χ t h i n = 2 ζ ( d ) . (49) \chi(d,d)-\chi_{\mathrm{thin}}=2\zeta(d). \tag*{(49)} χ ( d , d ) − χ thin = 2 ζ ( d ) . (49) Thus the Vandermonde degree exactly matches the difference of the two Euler normalizations.
The framing scalar and regularity at one
Let δ = e ∞ \delta=e_\infty δ = e ∞ z = X δ z=X^\delta z = X δ f , g f,g f , g f ⋅ d = p A ( d ) f\cdot d=pA(d) f ⋅ d = p A ( d ) g ⋅ d = q B ( d ) g\cdot d=qB(d) g ⋅ d = q B ( d )
χ e x t ( d + δ , d + δ ) = χ ( d , d ) + 1 − ( f + g ) ⋅ d . \chi_{\mathrm{ext}}(d+\delta,d+\delta)=\chi(d,d)+1-(f+g)\cdot d. χ ext ( d + δ , d + δ ) = χ ( d , d ) + 1 − ( f + g ) ⋅ d . The common frame coordinate contributes Hilbert series ( 1 − v 2 ) − 1 (1-v^2)^{-1} ( 1 − v 2 ) − 1 v / ( 1 − v 2 ) v/(1-v^2) v / ( 1 − v 2 ) v χ ( d , d ) − ( f + g ) ⋅ d v^{\chi(d,d)-(f+g)\cdot d} v χ ( d , d ) − ( f + g ) ⋅ d
⟨ δ , d ⟩ e x t = ( f − g ) ⋅ d , X d + δ = v − ( f − g ) ⋅ d z X d . \langle\delta,d\rangle_{\mathrm{ext}}=(f-g)\cdot d,\qquad X^{d+\delta}=v^{-(f-g)\cdot d}zX^d. ⟨ δ , d ⟩ ext = ( f − g ) ⋅ d , X d + δ = v − ( f − g ) ⋅ d z X d . Moving the frame monomial to the left therefore changes f + g f+g f + g 2 f 2f 2 f z A b e f o r e zA_{\mathrm{before}} z A before z A a f t e r zA_{\mathrm{after}} z A after
A b e f o r e ( U ; v ) = ∑ k ≥ 0 ∑ ( A ( d ) , B ( d ) ) = k ( a , b ) v χ ( d , d ) − 2 p A ( d ) ⋅ P b e f o r e ( d ; v 2 ) α d A β d B U k , \begin{aligned}
A_{\mathrm{before}}(U;v)
&=\sum_{k\geq0}\ \sum_{(A(d),B(d))=k(a,b)}
v^{\chi(d,d)-2pA(d)}\\
&\qquad\cdot P_{\mathrm{before}}(d;v^2)\,\alpha^{d_A}\beta^{d_B}U^k,
\end{aligned} A before ( U ; v ) = k ≥ 0 ∑ ( A ( d ) , B ( d )) = k ( a , b ) ∑ v χ ( d , d ) − 2 p A ( d ) ⋅ P before ( d ; v 2 ) α d A β d B U k , A a f t e r ( U ; v ) = ∑ k ≥ 0 ∑ ( A ( d ) , B ( d ) ) = k ( a , b ) v χ ( d , d ) − 2 p A ( d ) ⋅ P a f t e r ( d ; v 2 ) α d A β d B U k . (50) \begin{aligned}
A_{\mathrm{after}}(U;v)
&=\sum_{k\geq0}\ \sum_{(A(d),B(d))=k(a,b)}
v^{\chi(d,d)-2pA(d)}\\
&\qquad\cdot P_{\mathrm{after}}(d;v^2)\,\alpha^{d_A}\beta^{d_B}U^k. \tag*{(50)}
\end{aligned} A after ( U ; v ) = k ≥ 0 ∑ ( A ( d ) , B ( d )) = k ( a , b ) ∑ v χ ( d , d ) − 2 p A ( d ) ⋅ P after ( d ; v 2 ) α d A β d B U k . (50) Each constant coefficient is one, contributed by the pure frame. Every coefficient is a Laurent polynomial in v v v
For an unframed dimension d d d ( A ( d ) , B ( d ) ) = k γ (A(d),B(d))=k\gamma ( A ( d ) , B ( d )) = k γ ⟨ δ , d ⟩ e x t = p A ( d ) − q B ( d ) = k \langle\delta,d\rangle_{\mathrm{ext}}=pA(d)-qB(d)=k ⟨ δ , d ⟩ ext = p A ( d ) − q B ( d ) = k X d z = v − 2 k z X d X^dz=v^{-2k}zX^d X d z = v − 2 k z X d U z = v − 2 U U_z=v^{-2}U U z = v − 2 U v / ( 1 − v 2 ) v/(1-v^2) v / ( 1 − v 2 ) z z z (46) . The remaining central monomials commute, so the result is
A b e f o r e ( U ; v ) = A a f t e r ( U ; v ) F H ( U ; v ) , F H ( U ; v ) = H ( v − 2 U ; v ) H ( U ; v ) . (51) A_{\mathrm{before}}(U;v)=A_{\mathrm{after}}(U;v)F_H(U;v),\qquad
F_H(U;v)=\frac{H(v^{-2}U;v)}{H(U;v)}. \tag*{(51)} A before ( U ; v ) = A after ( U ; v ) F H ( U ; v ) , F H ( U ; v ) = H ( U ; v ) H ( v − 2 U ; v ) . (51) Because A a f t e r A_{\mathrm{after}} A after F H F_H F H
F o r i g ( U ) = F H ( U ; v ) ∣ v = 1 . (52) F_{\mathrm{orig}}(U)=\left.F_H(U;v)\right|_{v=1}. \tag*{(52)} F orig ( U ) = F H ( U ; v ) ∣ v = 1 . (52) Both H H H F o r i g F_{\mathrm{orig}} F orig
Recovering the whole character
For any finite-dimensional representation E E E G = S A × S B G=S_A\times S_B G = S A × S B α d A β d B \alpha^{d_A}\beta^{d_B} α d A β d B ch E \operatorname{ch} E ch E dim E W d \dim E^{W_d} dim E W d [ G : W d ] [G:W_d] [ G : W d ] d d d W d W_d W d
[ α d A β d B ] ch E = [ G : W d ] ∣ G ∣ ∑ w ∈ W d tr ( w ∣ E ) = dim E W d . [\alpha^{d_A}\beta^{d_B}]\operatorname{ch} E=\frac{[G:W_d]}{|G|}\sum_{w\in W_d}\operatorname{tr}(w\mid E)=\dim E^{W_d}. [ α d A β d B ] ch E = ∣ G ∣ [ G : W d ] w ∈ W d ∑ tr ( w ∣ E ) = dim E W d . Tensoring with the two sign representations gives
[ α d A β d B ] ι ch E = dim Hom W d ( sgn W d , E ) . (53) [\alpha^{d_A}\beta^{d_B}]\iota\operatorname{ch} E=\dim\operatorname{Hom}_{W_d}(\operatorname{sgn}_{W_d},E). \tag*{(53)} [ α d A β d B ] ι ch E = dim Hom W d ( sgn W d , E ) . (53) Permuting the colors permutes the tag variables without changing the shuffle construction. Adjoining a color and then setting its tag to zero retains exactly the dimensions having zero component at that color, and recovers the original construction. Thus the coefficient of U k U^k U k F o r i g F_{\mathrm{orig}} F orig ( k a , k b ) (ka,kb) ( k a , k b ) k a , k b ka,kb k a , k b ι \iota ι
Define the two ordinary character series
B p , q ( U ) = ∑ k ≥ 0 ch B k p , q U k , C p , q ( U ) = ∑ k ≥ 0 ch C k p , q U k . \mathcal{B}^{p,q}(U)=\sum_{k\geq0}\operatorname{ch}B_k^{p,q}U^k,\qquad\mathcal{C}^{p,q}(U)=\sum_{k\geq0}\operatorname{ch}C_k^{p,q}U^k. B p , q ( U ) = k ≥ 0 ∑ ch B k p , q U k , C p , q ( U ) = k ≥ 0 ∑ ch C k p , q U k . At v = 1 v=1 v = 1 (50) disappear. Equations (48) –(53) and Corollary 4.3 identify its before and after coefficients with ι B p , q \iota\mathcal{B}^{p,q} ι B p , q ι C p , q \iota\mathcal{C}^{p,q} ι C p , q (51) becomes
ι B p , q = ( ι C p , q ) F o r i g , B p , q = C p , q F Y , F Y = ι F o r i g . (54) \iota\mathcal{B}^{p,q}=(\iota\mathcal{C}^{p,q})F_{\mathrm{orig}},\qquad\mathcal{B}^{p,q}=\mathcal{C}^{p,q}F_Y,\qquad F_Y=\iota F_{\mathrm{orig}}. \tag*{(54)} ι B p , q = ( ι C p , q ) F orig , B p , q = C p , q F Y , F Y = ι F orig . (54) The equality holds in the completed ring of symmetric functions in both alphabets, because (53) identifies every monomial coefficient in each of the sufficiently large alphabets just described.
Proposition 8.1. For every k ≥ 0 k\geq0 k ≥ 0 p a − q b = 1 pa-qb=1 p a − q b = 1
ch D k p , q = [ U k ] F Y , [ U k ] F o r i g = ι ch D k p , q . (55) \operatorname{ch}D_k^{p,q}=[U^k]F_Y,\qquad[U^k]F_{\mathrm{orig}}=\iota\operatorname{ch}D_k^{p,q}. \tag*{(55)} ch D k p , q = [ U k ] F Y , [ U k ] F orig = ι ch D k p , q . (55) In particular the attached character is independent of ( p , q ) (p,q) ( p , q )
[ U k ] F o r i g = ∑ λ ⊢ k a , μ ⊢ k b n λ μ e λ ( α ) e μ ( β ) , n λ μ ∈ N . (56) [U^k]F_{\mathrm{orig}}=\sum_{\lambda\vdash ka,\ \mu\vdash kb}n_{\lambda\mu}e_\lambda(\alpha)e_\mu(\beta),\qquad n_{\lambda\mu}\in\mathbb{N}. \tag*{(56)} [ U k ] F orig = λ ⊢ k a , μ ⊢ k b ∑ n λ μ e λ ( α ) e μ ( β ) , n λ μ ∈ N . (56) Proof. Induct on k k k ( p , q ) (p,q) ( p , q ) k = 0 k=0 k = 0 Theorem 5.2 gives
ch B k p , q = ∑ r + l = k ( ch C r p , q ) ( ch D l p + m r b , q + m r a ) . \operatorname{ch}B_k^{p,q}=\sum_{r+l=k}(\operatorname{ch}C_r^{p,q})(\operatorname{ch}D_l^{p+mrb,q+mra}). ch B k p , q = r + l = k ∑ ( ch C r p , q ) ( ch D l p + m r b , q + m r a ) . Whenever r > 0 r>0 r > 0 l < k l<k l < k ( p + m r b ) a − ( q + m r a ) b = 1 (p+mrb)a-(q+mra)b=1 ( p + m r b ) a − ( q + m r a ) b = 1 [ U l ] F Y [U^l]F_Y [ U l ] F Y U k U^k U k (54) leaves only r = 0 r=0 r = 0 (55) . Finally apply ι \iota ι (35) to obtain (56) .
Identification with formal two-input reordering
On the aggregate lattice, for w = ( R , S ) w=(R,S) w = ( R , S ) K w = m ( a S − b R ) K_w=m(aS-bR) K w = m ( a S − b R ) (44) , ⟨ γ , w ⟩ = − K w \langle\gamma,w\rangle=-K_w ⟨ γ , w ⟩ = − K w
X w ⟼ X w H ( v − 2 K w U ; v ) H ( U ; v ) . (57) X^w\longmapsto X^w\frac{H(v^{-2K_w}U;v)}{H(U;v)}. \tag*{(57)} X w ⟼ X w H ( U ; v ) H ( v − 2 K w U ; v ) . (57) For K > 0 K>0 K > 0
H ( v − 2 K U ; v ) H ( U ; v ) = ∏ j = 0 K − 1 F H ( v − 2 j U ; v ) . \frac{H(v^{-2K}U;v)}{H(U;v)}=\prod_{j=0}^{K-1}F_H(v^{-2j}U;v). H ( U ; v ) H ( v − 2 K U ; v ) = j = 0 ∏ K − 1 F H ( v − 2 j U ; v ) . For negative K K K K = 0 K=0 K = 0 (51) permits specialization at v = 1 v=1 v = 1 U = x a y b U=x^{a}y^{b} U = x a y b
x R y S ⟼ x R y S F o r i g ( x a y b ) m ( a S − b R ) . (58) x^{R}y^{S}\longmapsto x^{R}y^{S}F_{\mathrm{orig}}(x^{a}y^{b})^{m(aS-bR)}. \tag*{(58)} x R y S ⟼ x R y S F orig ( x a y b ) m ( a S − b R ) . (58) An arrowless single-color axis has series
E v ( z ) = ∑ n ≥ 0 v n 2 z n ∏ j = 1 n ( 1 − v 2 j ) . E_v(z)=\sum_{n\geq0}\frac{v^{n^2}z^n}{\prod_{j=1}^{n}(1-v^{2j})}. E v ( z ) = n ≥ 0 ∑ ∏ j = 1 n ( 1 − v 2 j ) v n 2 z n . Direct comparison of coefficients gives E v ( v − 2 z ) = ( 1 + v − 1 z ) E v ( z ) E_v(v^{-2}z)=(1+v^{-1}z)E_v(z) E v ( v − 2 z ) = ( 1 + v − 1 z ) E v ( z )
∏ i = 1 s E v ( α i X ( 1 , 0 ) ) , ∏ j = 1 t E v ( β j X ( 0 , 1 ) ) . \prod_{i=1}^{s}E_v(\alpha_iX^{(1,0)}),\qquad\prod_{j=1}^{t}E_v(\beta_jX^{(0,1)}). i = 1 ∏ s E v ( α i X ( 1 , 0 ) ) , j = 1 ∏ t E v ( β j X ( 0 , 1 ) ) . The same conjugation calculation gives their classical actions
( x , y ) ⟼ ( x , y P ( x ) m ) , ( x , y ) ⟼ ( x Q ( y ) − m , y ) , (59) (x,y)\longmapsto(x,yP(x)^m),\qquad(x,y)\longmapsto(xQ(y)^{-m},y), \tag*{(59)} ( x , y ) ⟼ ( x , y P ( x ) m ) , ( x , y ) ⟼ ( x Q ( y ) − m , y ) , (59) where P ( z ) = ∏ i ( 1 + α i z ) P(z)=\prod_i(1+\alpha_i z) P ( z ) = ∏ i ( 1 + α i z ) Q ( z ) = ∏ j ( 1 + β j z ) Q(z)=\prod_j(1+\beta_j z) Q ( z ) = ∏ j ( 1 + β j z )
The full polynomial series is the A A A B B B A A A B B B A A A B B B B B B A A A (59) . Uniqueness can also be seen directly degree by degree: at the first unresolved degree an interior monomial belongs to exactly one ray, and its correction appears linearly.
Theorem 8.2 (Two-input positivity). Let
P ( z ) = 1 + c 1 z + ⋯ + c r z r , Q ( z ) = 1 + d 1 z + ⋯ + d s z s P(z)=1+c_1z+\cdots+c_rz^r,\qquad Q(z)=1+d_1z+\cdots+d_sz^s P ( z ) = 1 + c 1 z + ⋯ + c r z r , Q ( z ) = 1 + d 1 z + ⋯ + d s z s have independent coefficients. In the formal reordering of ( x , y ) ⟼ ( x , y P ( x ) ) (x,y)\longmapsto(x,yP(x)) ( x , y ) ⟼ ( x , y P ( x )) ( x , y ) ⟼ ( x Q ( y ) − 1 , y ) (x,y)\longmapsto(xQ(y)^{-1},y) ( x , y ) ⟼ ( x Q ( y ) − 1 , y ) ( a , b ) (a,b) ( a , b )
x R y S ⟼ x R y S f a , b ( x a y b ) a S − b R , f a , b ( 0 ) = 1. x^Ry^S\longmapsto x^Ry^Sf_{a,b}(x^ay^b)^{aS-bR},\qquad f_{a,b}(0)=1. x R y S ⟼ x R y S f a , b ( x a y b ) a S − b R , f a , b ( 0 ) = 1. Every coefficient of f a , b f_{a,b} f a , b N [ c 1 , … , c r , d 1 , … , d s ] \mathbb{N}[c_1,\ldots,c_r,d_1,\ldots,d_s] N [ c 1 , … , c r , d 1 , … , d s ]
Proof. Set m = 1 m=1 m = 1 (58) . Because gcd ( a , b ) = 1 \gcd(a,b)=1 g cd( a , b ) = 1 ( R , S ) ⟼ a S − b R (R,S)\longmapsto aS-bR ( R , S ) ⟼ a S − b R f a , b = F o r i g f_{a,b}=F_{\mathrm{orig}} f a , b = F orig c i , d j c_i,d_j c i , d j e i ( α ) , e j ( β ) e_i(\alpha),e_j(\beta) e i ( α ) , e j ( β ) (56) is a polynomial with nonnegative integer coefficients in those generators. They are algebraically independent for alphabets of sizes r , s r,s r , s □ \square □
Positive formal wall operations
We now pass from Theorem 8.2 to arbitrary rank. We use finite quotients of a graded Lie algebra throughout. The several-input mechanism is established in [5 Section 9.2] ; our formal transport is adapted from [21 Section 4] . We give the details for the present coefficient convention and primitive ambient normals.
The completed Hamiltonian group
Let M M M Ω : M × M → Z \Omega: M \times M \to\mathbb{Z} Ω : M × M → Z C = { p 1 , … , p n } C = \{p_1,\ldots,p_n\} C = { p 1 , … , p n } M M M
C + = ∑ i N p i , ∣ ∑ i a i p i ∣ = ∑ i a i . C^+ = \sum_i \mathbb{N}p_i,\qquad\left|\sum_i a_i p_i\right| = \sum_i a_i. C + = i ∑ N p i , i ∑ a i p i = i ∑ a i . Assume that the span of C C C rad Ω \operatorname{rad}\Omega rad Ω R + R_+ R + R R R K = Frac ( R ) K = \operatorname{Frac}(R) K = Frac ( R )
g C = ∏ a ∈ C + ∖ { 0 } K X a , [ X a , X b ] = Ω ( a , b ) X a + b (60) \mathfrak{g}_C = \prod_{a\in C^+\setminus\{0\}} K X^a,\qquad[X^a,X^b] = \Omega(a,b)X^{a+b} \tag*{(60)} g C = a ∈ C + ∖ { 0 } ∏ K X a , [ X a , X b ] = Ω ( a , b ) X a + b (60) is filtered by root degree. Its quotient in degrees at most d d d K K K G C G_C G C
{ X a , X w } = Ω ( a , w ) X a + w . \{X^a,X^w\} = \Omega(a,w)X^{a+w}. { X a , X w } = Ω ( a , w ) X a + w . The action is faithful: a nonzero coefficient at the first root a a a X w X^w X w Ω ( a , − ) ≠ 0 \Omega(a,-) \ne0 Ω ( a , − ) = 0 C + C^+ C +
If a a a Ω ( a , − ) = δ a n a \Omega(a,-) = \delta_a n_a Ω ( a , − ) = δ a n a δ a > 0 \delta_a > 0 δ a > 0 n a ∈ M ∗ n_a \in M^* n a ∈ M ∗ H = ∑ j ≥ 1 h j X j a H = \sum_{j\ge1} h_j X^{ja} H = ∑ j ≥ 1 h j X j a
exp ( { H , − } ) ( X w ) = X w f ( X a ) n a ( w ) , f ( Z ) = exp ( δ a ∑ j ≥ 1 j h j Z j ) . (61) \exp(\{H,-\})(X^w) = X^w f(X^a)^{n_a(w)},\qquad f(Z) = \exp\left(\delta_a \sum_{j\ge1} jh_j Z^j\right). \tag*{(61)} exp ({ H , − }) ( X w ) = X w f ( X a ) n a ( w ) , f ( Z ) = exp ( δ a j ≥ 1 ∑ j h j Z j ) . (61) We call f f f { H , X w } / X w \{H,X^w\}/X^w { H , X w } / X w n a ( w ) δ a ∑ j j h j X j a n_a(w)\delta_a\sum_j jh_jX^{ja} n a ( w ) δ a ∑ j j h j X j a X j a X^{ja} X j a f f f
Lemma 9.1 (Three-factor decomposition). For h ∈ M R ∗ h \in M^*_{\mathbb{R}} h ∈ M R ∗ g ∈ G C g \in G_C g ∈ G C
g = g + ( h ) g 0 ( h ) g − ( h ) , g = g_+(h)g_0(h)g_-(h), g = g + ( h ) g 0 ( h ) g − ( h ) , where the logarithms of the three factors are supported on roots with positive, zero, and negative h h h h h h
Proof. The three indicated spaces are Lie subalgebras, since signs are preserved by addition within each one. Work first through degree d d d s s s a a a s s s X a X^a X a d d d
For the refinement assertion, take h ′ h' h ′ h h h h h h d d d g 0 ( h ) g_0(h) g 0 ( h ) h ′ h' h ′ g + ( h ) g_+(h) g + ( h ) g − ( h ) g_-(h) g − ( h ) h ′ h' h ′ □ \square □
In a finite quotient, the finitely many hyperplanes a ⊥ a^\perp a ⊥ M R ∗ M_{\mathbb{R}}^* M R ∗ g 0 ( h ) g_0(h) g 0 ( h ) Lemma 9.1 shows that the action from the all-positive chamber to a generic h h h g − ( h ) g_-(h) g − ( h )
Conversely, a consistent central system of such wall factors is recovered from its total product g g g η \eta η C C C h + t η h+t\eta h + t η t t t t = 0 t=0 t = 0 h h h g g g h h h g g g
Put V ( a ) = Ω ( − , a ) ∈ M R ∗ V(a)=\Omega(-,a)\in M_{\mathbb{R}}^* V ( a ) = Ω ( − , a ) ∈ M R ∗ V ( a ) V(a) V ( a ) a ⊥ a^\perp a ⊥ a a a
Lemma 9.2 (Incoming prescription). For each i i i Q i ( Z ) Q_i(Z) Q i ( Z )
Ω ( p i , − ) = δ i n i , log Q i ( Z ) = ∑ j ≥ 1 c i j Z j \Omega(p_i,-)=\delta_i n_i,\qquad\log Q_i(Z)=\sum_{j\geq1}c_{ij}Z^j Ω ( p i , − ) = δ i n i , log Q i ( Z ) = j ≥ 1 ∑ c ij Z j where n i n_i n i δ i > 0 \delta_i>0 δ i > 0 g ∈ G C g\in G_C g ∈ G C
[ X a ] log g 0 ( V ( a ) ) = { c i j / ( δ i j ) , a = j p i , j ≥ 1 , 0 , a lies on no simple ray . (62) [X^a]\log g_0(V(a))=
\begin{cases}
c_{ij}/(\delta_i j), & a=jp_i,\ j\geq1,\\
0, & a\text{ lies on no simple ray}.
\end{cases} \tag*{(62)} [ X a ] log g 0 ( V ( a )) = { c ij / ( δ i j ) , 0 , a = j p i , j ≥ 1 , a lies on no simple ray . (62) At a small generic displacement within a ⊥ a^\perp a ⊥ V ( a ) V(a) V ( a )
Proof. The coefficient of X a X^a X a log g \log g log g (62) with coefficient one. All remaining terms involve smaller roots, by the proof of Lemma 9.1 . Thus (62) determines log g \log g log g
Let λ a ( H ) = [ X a ] H \lambda_a(H)=[X^a]H λ a ( H ) = [ X a ] H ker V ( a ) \ker V(a) ker V ( a ) b + c = a b+c=a b + c = a Ω ( b , a ) = 0 \Omega(b,a)=0 Ω ( b , a ) = 0 Ω ( b , c ) = Ω ( b , a − b ) = 0 \Omega(b,c)=\Omega(b,a-b)=0 Ω ( b , c ) = Ω ( b , a − b ) = 0
λ a ( BCH ( H 1 , H 2 ) ) = λ a ( H 1 ) + λ a ( H 2 ) \lambda_a(\operatorname{BCH}(H_1,H_2))=\lambda_a(H_1)+\lambda_a(H_2) λ a ( BCH ( H 1 , H 2 )) = λ a ( H 1 ) + λ a ( H 2 ) for H 1 , H 2 H_1,H_2 H 1 , H 2 g 0 ( V ( a ) ) g_0(V(a)) g 0 ( V ( a )) a ⊥ a^\perp a ⊥ X a X^a X a a a a Q i Q_i Q i (61) , and all nonsimple incoming functions are one. □ \square □
Two inputs on an ambient lattice
We apply the two-input theorem while retaining the integral normals of M M M
Lemma 9.3. Suppose that
ϕ i ( X w ) = X w P i ( X r i ) n i ( w ) ( i = 1 , 2 ) , \phi_i(X^w)=X^wP_i(X^{r_i})^{n_i(w)}\qquad(i=1,2), ϕ i ( X w ) = X w P i ( X r i ) n i ( w ) ( i = 1 , 2 ) , where n i ∈ M ∗ n_i \in M^* n i ∈ M ∗ n i ( r i ) = 0 n_i(r_i)=0 n i ( r i ) = 0 n i n_i n i Ω ( r i , − ) \Omega(r_i,-) Ω ( r i , − ) P i P_i P i R + R_+ R + M M M R + R_+ R +
Proof. The commuting case is immediate. Otherwise, after relabeling, put
u = n 1 ( r 2 ) > 0 , v = − n 2 ( r 1 ) > 0 , e 1 = r 1 / v , e 2 = r 2 / u . u=n_1(r_2)>0,\qquad v=-n_2(r_1)>0,\qquad e_1=r_1/v,\qquad e_2=r_2/u. u = n 1 ( r 2 ) > 0 , v = − n 2 ( r 1 ) > 0 , e 1 = r 1 / v , e 2 = r 2 / u . In the enlarged lattice M ′ = M + Z e 1 + Z e 2 M'=M+\mathbb{Z}e_1+\mathbb{Z}e_2 M ′ = M + Z e 1 + Z e 2 n 1 n_1 n 1 n 2 n_2 n 2 e 1 , e 2 e_1,e_2 e 1 , e 2 ( 0 , 1 ) (0,1) ( 0 , 1 ) ( − 1 , 0 ) (-1,0) ( − 1 , 0 ) L = Z e 1 + Z e 2 L=\mathbb{Z}e_1+\mathbb{Z}e_2 L = Z e 1 + Z e 2
P 1 ( x v ) , P 2 ( y u ) . P_1(x^v),\qquad P_2(y^u). P 1 ( x v ) , P 2 ( y u ) . Theorem 8.2 , with absent coefficients specialized to zero, therefore gives positive output functions. On a primitive L L L r = a e 1 + b e 2 r=ae_1+be_2 r = a e 1 + b e 2 a n 1 + b n 2 an_1+bn_2 a n 1 + b n 2 n i = λ i Ω ( r i , − ) n_i=\lambda_i\Omega(r_i,-) n i = λ i Ω ( r i , − ) λ i > 0 \lambda_i>0 λ i > 0
λ 1 v = λ 2 u , a n 1 + b n 2 = ( λ 1 v ) Ω ( a e 1 + b e 2 , − ) . \lambda_1v=\lambda_2u,\qquad an_1+bn_2=(\lambda_1v)\Omega(ae_1+be_2,-). λ 1 v = λ 2 u , a n 1 + b n 2 = ( λ 1 v ) Ω ( a e 1 + b e 2 , − ) . which gives the required direction and orientation.
The factorization is also valid on all ambient monomials. To see this, write ∂ n ( X w ) = n ( w ) X w \partial_n(X^w)=n(w)X^w ∂ n ( X w ) = n ( w ) X w
[ X r ∂ n , X s ∂ m ] = X r + s ( n ( s ) ∂ m − m ( r ) ∂ n ) . [X^r\partial_n,X^s\partial_m]=X^{r+s}\bigl(n(s)\partial_m-m(r)\partial_n\bigr). [ X r ∂ n , X s ∂ m ] = X r + s ( n ( s ) ∂ m − m ( r ) ∂ n ) . Every covector generated here lies in the span of n 1 , n 2 n_1,n_2 n 1 , n 2 L L L L L L M ′ M' M ′
The logarithms of the inputs have support in N r 1 + N r 2 \mathbb{N}r_1+\mathbb{N}r_2 N r 1 + N r 2 (61) shows that the output functions preserve it as well. Their nonconstant monomials therefore belong to M M M ρ r ∈ M \rho r\in M ρ r ∈ M n 1 ( ρ r ) = ρ b n_1(\rho r)=\rho b n 1 ( ρ r ) = ρ b n 2 ( ρ r ) = − ρ a n_2(\rho r)=-\rho a n 2 ( ρ r ) = − ρ a a , b a,b a , b ρ ∈ Z \rho\in\mathbb{Z} ρ ∈ Z
M ∩ Q r = Z ( h r ) M\cap\mathbb{Q}r=\mathbb{Z}(hr) M ∩ Q r = Z ( h r ) for some h ∈ Z > 0 h\in\mathbb{Z}_{>0} h ∈ Z > 0
The support condition therefore writes the output function as f ( Z ) = g ( Z h ) f(Z)=g(Z^h) f ( Z ) = g ( Z h ) Z = X r Z=X^r Z = X r g g g f f f a n 1 + b n 2 = t n p r i m an_1+bn_2=tn_{\mathrm{prim}} a n 1 + b n 2 = t n prim M M M t ∈ Z > 0 t\in\mathbb{Z}_{>0} t ∈ Z > 0
X w ⟼ X w ( g ( X h r ) t ) n p r i m ( w ) . X^w\longmapsto X^w\bigl(g(X^{hr})^t\bigr)^{n_{\mathrm{prim}}(w)}. X w ⟼ X w ( g ( X h r ) t ) n prim ( w ) . Thus the function in the primitive ambient ray variable, with primitive integral normal, is g t g^t g t R + R_+ R +
Several inputs and higher rank.
Proposition 9.4. Let a finite collection of incoming polynomial line functions have roots in a common pointed rank-two cone. The lines pass through the origin, and the functions use the primitive integral normals of the ambient lattice. If their constant terms are one and their coefficients belong to R + R_+ R + R + R_+ R +
Proof. We retain the ambient normals, so every two-input operation below is covered by Lemma 9.3. Commuting inputs already form a consistent diagram. Otherwise work in the dual of the two-dimensional root span, whose alternating pairing is nondegenerate.
Give each nonconstant original input monomial an independent coefficient c s c_s c s z s z_s z s F = N S F = \mathbb{N}^{S} F = N S w s w_s w s ρ : F → L \rho: F \to L ρ : F → L z α X ρ ( α ) z^\alpha X^{\rho(\alpha)} z α X ρ ( α ) ∑ s w s α s \sum_s w_s\alpha_s ∑ s w s α s ρ ( α ) ≠ 0 \rho(\alpha) \ne0 ρ ( α ) = 0 α ≠ 0 \alpha\ne0 α = 0
For full-line inputs at one point, the additional outgoing correction functions are determined recursively. At the first unresolved weight, the logarithm of the small-circle product is a finite sum of labeled Hamiltonian monomials. Each monomial has one prescribed outgoing ray. Its correction coefficient enters the circle product linearly with a nonzero primitive-normal conversion factor, so a unique choice cancels that defect. These corrections are central modulo the next weight. This proves existence, uniqueness, and rational-polynomial dependence on the input coefficients. To count input occurrences, multiply all c s c_s c s ϵ \epsilon ϵ ϵ 2 \epsilon^2 ϵ 2 P P P F 0 F_0 F 0 F 0 / P F_0/P F 0 / P
Uniformly over all finite labeled data and positive integer weights, we prove the following assertions simultaneously through weight d d d
the additional central correction functions have nonnegative integral polynomial coefficients in the independent c s c_s c s
after any fixed rational translation of the original lines with distinct parallel lines and no triple original intersection, there is a finite diagram, consistent through weight d d d
Combine rays with the same base point and direction, and read overlapping collinear germs by multiplying their functions. The joints are the ray bases and intersections of nonparallel supports. Both assertions in weight one follow from the first-order calculation above.
Assume both assertions through weight d d d q q q q q q q q q q q q d d d
An additional coefficient of weight d + 1 d+1 d + 1 d d d c s c_s c s
Consider a local label multidegree α \alpha α d + 1 d+1 d + 1
∑ s α s = d + 1 − ∑ s ( w s − 1 ) α s ≤ d . \sum_s \alpha_s = d+1-\sum_s (w_s-1)\alpha_s \le d. s ∑ α s = d + 1 − s ∑ ( w s − 1 ) α s ≤ d . Give all local labels weight one, retaining their geometric exponents and normals. The central induction assertion through weight d d d
If every used label has weight one, it comes from an original line, since every generated ray starts in weight at least two. At most two original lines pass through q q q
At each old joint, assign these weight d + 1 d+1 d + 1 d + 1 d+1 d + 1 d + 2 d+2 d + 2 d + 1 d+1 d + 1 d + 1 d+1 d + 1
To obtain the central assertion, dilate this finite affine diagram by a positive scalar tending to zero. A fixed circle enclosing all joints has product one. Every original line tends to its prescribed central line, and each correction ray tends to the outgoing half-ray in direction − V ( ρ ( α ) ) -V(\rho(\alpha)) − V ( ρ ( α ))
This completes the simultaneous induction. Central uniqueness makes the truncations compatible. Finally specialize c s c_s c s z s = 1 z_s=1 z s = 1
Theorem 9.5. For the incoming prescription of Lemma 9.2 , assume that the coefficients of every Q i Q_i Q i lie in R + R_+ R + Every primitive-normal wall function of the resulting consistent system has coefficients in R + R_+ R +
Proof. We argue in finite root degree. Fix a positive ray a a a a ⊥ a^\perp a ⊥ d d d a ⊥ a^\perp a ⊥ V ( a ) V(a) V ( a ) d d d b ⊥ b^\perp b ⊥ Ω ( b , a ) = 0 \Omega(b,a)=0 Ω ( b , a ) = 0 a ⊥ a^\perp a ⊥ Lemma 9.1 , the change is an ordered rank-two factorization.
Induct on d d d + V ( b ) +V(b) + V ( b ) b b b a a a d d d d − 1 d-1 d − 1 d d d Proposition 9.4 , retaining the primitive normals of M M M d d d Q i Q_i Q i
A coefficient recursion
A transported expression means one fixed finite Laurent polynomial in one chamber, with its values in other chambers obtained by the consistent path operations. These values may be infinite series, but their supports remain in the same finite union S = ⋃ ν = 1 m ν ( m ν + C + ) S = \bigcup_{\nu=1}^{m_\nu}(m_\nu+ C^+) S = ⋃ ν = 1 m ν ( m ν + C + ) w ∈ M w \in M w ∈ M
A w ( F ) = [ X w ] F ( V ( w ) + ε ) , (63) A_w(F) = [X^w]F(V(w) + \varepsilon), \tag*{(63)} A w ( F ) = [ X w ] F ( V ( w ) + ε ) , (63) where a sufficiently small generic displacement is taken at the finite order needed to compute this coefficient. This number does not depend on the displacement. Indeed, a wall separating two such choices has root a a a Ω ( a , w ) = 0 \Omega(a,w)=0 Ω ( a , w ) = 0 w − j a w-ja w − j a a a a X w X^w X w
Proposition 9.6. Assume that the coefficients of the Q i Q_i Q i R + R_+ R + R + R_+ R + R + R_+ R +
Proof. For a generic target h h h h + s V ( w ) h+sV(w) h + s V ( w ) s s s V ( w ) + s − 1 h V(w)+s^{-1}h V ( w ) + s − 1 h X w X^w X w A w ( F ) A_w(F) A w ( F ) a a a Ω ( a , w ) \Omega(a,w) Ω ( a , w ) w − j a w-ja w − j a (61) , the operation on that source consequently involves a nonnegative integral power of the positive wall function of Theorem 9.5 . Write J i ( w − j a i , j ) ∈ R + J_i(w-ja_i,j)\in R_+ J i ( w − j a i , j ) ∈ R + j a i ja_i j a i j > 0 j>0 j > 0 i i i f i f_i f i n i n_i n i
J i ( w − j a i , j ) = [ T j ] f i ( T ) ∣ n i ( w ) ∣ . J_i(w-ja_i,j) = [T^j]f_i(T)^{|n_i(w)|}. J i ( w − j a i , j ) = [ T j ] f i ( T ) ∣ n i ( w ) ∣ . The coefficient adding zero is one. Taking successive differences gives
[ X w ] F ( h ) = A w ( F ) + ∑ i ∑ j > 0 J i ( w − j a i , j ) [ X w − j a i ] F ( h i ) , (64) [X^w]F(h) = A_w(F) + \sum_i\sum_{j>0}J_i(w-ja_i,j)[X^{w-ja_i}]F(h_i), \tag*{(64)} [ X w ] F ( h ) = A w ( F ) + i ∑ j > 0 ∑ J i ( w − j a i , j ) [ X w − j a i ] F ( h i ) , (64) where h i h_i h i w w w C + C^+ C + { u ∈ S : w − u ∈ C + } \{u\in S:w-u\in C^+\} { u ∈ S : w − u ∈ C + } C C C (64) . Induct on this finite partially ordered set, simultaneously for all target chambers. For a minimal element the formula is its incoming coefficient; each subsequent element is a sum of products in R + R_+ R + S S S
Mutation and finite Laurent fibers
The lattice attached to a generalized seed
10.1. The lattice attached to a generalized seed. Let B ~ \widetilde{B} B N N N n n n D = diag ( δ 1 , … , δ n ) D=\operatorname{diag}(\delta_1,\ldots,\delta_n) D = diag ( δ 1 , … , δ n ) B B B n n n I n I_n I n B ^ \widehat{B} B Y 1 , … , Y n Y_1,\ldots,Y_n Y 1 , … , Y n M = Z N ⊕ Z n M=\mathbb{Z}^N\oplus\mathbb{Z}^n M = Z N ⊕ Z n π : Z N → Z n \pi:\mathbb{Z}^N\to\mathbb{Z}^n π : Z N → Z n
p i = − b ^ i , n i ( w ) = − w i , Ω = ( 0 − π T D D π − D B ) . (65) p_i=-\widehat{b}_i,\qquad n_i(w)=-w_i,\qquad
\Omega=\begin{pmatrix}0&-\pi^{\mathsf T}D\\D\pi&-DB\end{pmatrix}. \tag*{(65)} p i = − b i , n i ( w ) = − w i , Ω = ( 0 D π − π T D − D B ) . (65) The matrix is alternating, and direct multiplication gives
Ω ( p i , w ) = δ i n i ( w ) , n i ( p j ) = b i j , Ω ( ∑ i a i p i , e j ) = − δ j a j ( 1 ≤ j ≤ n ) . (66) \Omega(p_i,w)=\delta_i n_i(w),\qquad n_i(p_j)=b_{ij},\qquad
\Omega\left(\sum_i a_i p_i,e_j\right)=-\delta_j a_j\quad(1\leq j\leq n). \tag*{(66)} Ω ( p i , w ) = δ i n i ( w ) , n i ( p j ) = b ij , Ω ( i ∑ a i p i , e j ) = − δ j a j ( 1 ≤ j ≤ n ) . (66) The auxiliary coordinates of the p i p_i p i − I n -I_n − I n (66) proves the required faithfulness on the root span, with arbitrary frozen rows. For the exchange polynomial P i ( Z ) = ∑ s = 0 r i q i , s Z s P_i(Z)=\sum_{s=0}^{r_i}q_{i,s}Z^s P i ( Z ) = ∑ s = 0 r i q i , s Z s Q i = P i / q i , 0 Q_i=P_i/q_{i,0} Q i = P i / q i , 0
ϕ i ( X w ) = X w Q i ( X p i ) n i ( w ) . \phi_i(X^w)=X^wQ_i(X^{p_i})^{n_i(w)}. ϕ i ( X w ) = X w Q i ( X p i ) n i ( w ) . The preceding section constructs the corresponding consistent system. We next compare it with the system of a mutated seed. The support estimate below uses the degree of Q i Q_i Q i
Finiteness in one mutation direction
Fix p ∈ C p \in C p ∈ C Ω ( p , − ) = δ n \Omega(p,-)=\delta n Ω ( p , − ) = δ n δ > 0 \delta>0 δ > 0 Q Q Q r r r u = ∑ c ∈ C ∖ { p } u c c ≠ 0 u=\sum_{c\in C\setminus\{p\}}u_c c\ne0 u = ∑ c ∈ C ∖ { p } u c c = 0 u c ∈ N u_c\in\mathbb{N} u c ∈ N
∣ u ∣ ≠ p = ∑ c ≠ p u c , M σ ( u ) = r ∑ c ≠ p [ − σ n ( c ) ] + + u c ( σ ∈ { + 1 , − 1 } ) . \lvert u\rvert_{\ne p}=\sum_{c\ne p}u_c,\qquad M_\sigma(u)=r\sum_{c\ne p}[-\sigma n(c)]_++u_c\qquad(\sigma\in\{+1,-1\}). ∣ u ∣ = p = c = p ∑ u c , M σ ( u ) = r c = p ∑ [ − σ n ( c ) ] + + u c ( σ ∈ { + 1 , − 1 }) . Lemma 10.1 (Finite fibers). On the open half-space sgn h ( p ) = σ \operatorname{sgn}h(p)=\sigma sgn h ( p ) = σ k p + u kp+u k p + u
0 ≤ k ≤ M σ ( u ) . (67) 0\le k\le M_\sigma(u). \tag*{(67)} 0 ≤ k ≤ M σ ( u ) . (67) Every nonpure wall operation and its inverse consequently has finite Laurent support in the p p p p p p
Proof. The pure wall has function Q Q Q p ⊥ p^\perp p ⊥ N p \mathbb{N}p N p
We induct on ∣ u ∣ ≠ p \lvert u\rvert_{\ne p} ∣ u ∣ = p k k k n ( u ) ≠ 0 n(u)\ne0 n ( u ) = 0 σ = sgn n ( u ) \sigma=\operatorname{sgn}n(u) σ = sgn n ( u ) ( k p + u ) ⊥ (kp+u)^\perp ( k p + u ) ⊥ p p p
h ( p ) + s Ω ( p , k p + u ) = h ( p ) + s δ n ( u ) ( s ≥ 0 ) , h(p)+s\Omega(p,kp+u)=h(p)+s\delta n(u)\qquad(s\ge0), h ( p ) + s Ω ( p , k p + u ) = h ( p ) + sδ n ( u ) ( s ≥ 0 ) , so it does not cross p ⊥ p^\perp p ⊥ k = 0 k=0 k = 0
At any joint off p ⊥ p^\perp p ⊥ p p p p p p p p p (67) is closed under addition on a fixed side. Therefore the coefficient recursion for the local factorization preserves it. Following the finitely many joints proves the upstream bound for the specified k k k k k k
At a generic joint on p ⊥ p^\perp p ⊥ p p p n n n
X k p + w ⟼ X k p + w Q ( X p ) ∣ n ( w ) ∣ . (68) X^{kp+w}\longmapsto X^{kp+w}Q(X^p)^{\lvert n(w)\rvert}. \tag*{(68)} X k p + w ⟼ X k p + w Q ( X p ) ∣ n ( w )∣ . (68) It increases the p p p r ∣ n ( w ) ∣ r\lvert n(w)\rvert r ∣ n ( w )∣
M − σ ( w ) − M σ ( w ) = r ∣ n ( w ) ∣ . M_{-\sigma}(w)-M_\sigma(w)=r\lvert n(w)\rvert. M − σ ( w ) − M σ ( w ) = r ∣ n ( w )∣ . The conjugated product therefore obeys the downstream bound. Ordered refactorization cannot produce a first term outside an additive set containing the support of the product: at such a first term the product coefficient is zero, and every contribution already determined from smaller roots also has permitted support. This proves the downstream bound. Subsequent off-cut joints are handled by the same induction as before. If n ( u ) = 0 n(u)=0 n ( u ) = 0
Finally, an exponential or inverse at positive non-p p p p p p p p p Q Q Q □ \square □
Reversal of an exchange polynomial
Write
P ( Z ) = A + ∑ j = 1 r − 1 q j Z j + B Z r , Q = P / A , τ = B / A , P(Z)=A+\sum_{j=1}^{r-1}q_jZ^j+BZ^r,\qquad Q=P/A,\qquad\tau=B/A, P ( Z ) = A + j = 1 ∑ r − 1 q j Z j + B Z r , Q = P / A , τ = B / A , and define
Q ∨ ( Z ) = τ − 1 Z r Q ( Z − 1 ) , S ( w ) = w − r n ( w ) p , S τ ( X w ) = τ − n ( w ) X S ( w ) . (69) Q^\vee(Z)=\tau^{-1}Z^rQ(Z^{-1}),\qquad S(w)=w-rn(w)p,\qquad S_\tau(X^w)=\tau^{-n(w)}X^{S(w)}. \tag*{(69)} Q ∨ ( Z ) = τ − 1 Z r Q ( Z − 1 ) , S ( w ) = w − r n ( w ) p , S τ ( X w ) = τ − n ( w ) X S ( w ) . (69) Since n ( p ) = 0 n(p)=0 n ( p ) = 0 S S S Ω \Omega Ω Ω ( S u , S v ) \Omega(Su,Sv) Ω ( S u , S v ) Ω ( p , w ) = δ n ( w ) \Omega(p,w)=\delta n(w) Ω ( p , w ) = δ n ( w ) S τ S_\tau S τ S τ S_\tau S τ ϕ p , Q ( X w ) = X w Q ( X p ) n ( w ) \phi_{p,Q}(X^w)=X^wQ(X^p)^{n(w)} ϕ p , Q ( X w ) = X w Q ( X p ) n ( w )
S τ ϕ p , Q = ϕ − p , Q ∨ − 1 . (70) S_\tau\phi_{p,Q}=\phi_{-p,Q^\vee}^{-1}. \tag*{(70)} S τ ϕ p , Q = ϕ − p , Q ∨ − 1 . (70) Indeed, the primitive normal of − p -p − p − n -n − n X w X^w X w
X w ( τ − 1 X − r p Q ( X p ) ) n ( w ) . X^w\left(\tau^{-1}X^{-rp}Q(X^p)\right)^{n(w)}. X w ( τ − 1 X − r p Q ( X p ) ) n ( w ) . This is a rational identity for every integer n ( w ) n(w) n ( w )
Replace the simple-root set by
C ′ = { − p } ∪ { c ′ = c + r [ − n ( c ) ] + p : c ∈ C ∖ { p } } . (71) C'=\{-p\}\cup\{c'=c+r[-n(c)]_+p:c\in C\setminus\{p\}\}. \tag*{(71)} C ′ = { − p } ∪ { c ′ = c + r [ − n ( c ) ] + p : c ∈ C ∖ { p }} . (71) It is again part of a lattice basis. Let L + = id L_+=\operatorname{id} L + = id L − = S L_-=S L − = S (67) , direct calculation gives
L σ ( k p + u ) = ∑ c ≠ p u c c ′ + ( M σ ( u ) − k ) ( − p ) ∈ ( C ′ ) + . (72) L_\sigma(kp+u)=\sum_{c\ne p}u_cc'+(M_\sigma(u)-k)(-p)\in(C')^+. \tag*{(72)} L σ ( k p + u ) = c = p ∑ u c c ′ + ( M σ ( u ) − k ) ( − p ) ∈ ( C ′ ) + . (72) Put b 0 = r max c ≠ p ∣ n ( c ) ∣ b_0=r\max_{c\ne p}|n(c)| b 0 = r max c = p ∣ n ( c ) ∣ b 0 = 0 b_0=0 b 0 = 0 d d d ( b 0 + 1 ) d (b_0+1)d ( b 0 + 1 ) d k ≤ b 0 ∣ u ∣ ≠ p k\le b_0|u|_{\ne p} k ≤ b 0 ∣ u ∣ = p ∣ u ∣ ≠ p |u|_{\ne p} ∣ u ∣ = p
Lemma 10.2 (The mutated formal system). Use the identity on the half-space h ( p ) > 0 h(p)>0 h ( p ) > 0 S τ S_\tau S τ h ( p ) < 0 h(p)<0 h ( p ) < 0 ( − p , Q ∨ ) (-p,Q^\vee) ( − p , Q ∨ ) C ′ C' C ′
Q ∨ on − p , { Q c ( X c ′ ) , n ( c ) ≥ 0 , Q c ( τ − n ( c ) X c ′ ) , n ( c ) ≤ 0 on c ′ . (73) Q^\vee\text{ on }-p,\qquad
\begin{cases}
Q_c(X^{c'}), & n(c)\ge0,\\
Q_c(\tau^{-n(c)}X^{c'}), & n(c)\le0
\end{cases}
\text{ on }c'. \tag*{(73)} Q ∨ on − p , { Q c ( X c ′ ) , Q c ( τ − n ( c ) X c ′ ) , n ( c ) ≥ 0 , n ( c ) ≤ 0 on c ′ . (73) All other incoming coefficients vanish.
Proof. Choose a lattice complement to p p p Z = X p Z=X^p Z = X p K ( Z ) K(Z) K ( Z ) p p p Z Z Z Z Z Z S τ S_\tau S τ Z Z Z
Old consistency holds in this common ring. To verify it, expand K ( Z ) K(Z) K ( Z ) K ( ( Z ) ) K((Z)) K (( Z )) p p p (67) ; a further prescribed coefficient of Z Z Z (70) now gives consistency of the pushed system in the common rational ring. Expanding at Z − 1 = 0 Z^{-1}=0 Z − 1 = 0 (72) , places it in the C ′ C' C ′
It remains to check the incoming data, including their vanishing away from simple roots. Symplecticity gives
V ( S w ) = V ( w ) ∘ S − 1 , n ( S w ) = n ( w ) . V(Sw)=V(w)\circ S^{-1},\qquad n(Sw)=n(w). V ( S w ) = V ( w ) ∘ S − 1 , n ( S w ) = n ( w ) . An incoming direction for c c c n ( c ) > 0 n(c)>0 n ( c ) > 0 S τ S_\tau S τ n ( c ) < 0 n(c)<0 n ( c ) < 0 c ′ c' c ′ (73) . The pure incoming function is Q ∨ Q^\vee Q ∨ (70) . An effective root a ′ a' a ′ n ( a ′ ) > 0 n(a')>0 n ( a ′ ) > 0 a ′ a' a ′ n ( c ) > 0 n(c)>0 n ( c ) > 0 c ′ = c c'=c c ′ = c a ′ a' a ′ n ( a ′ ) < 0 n(a')<0 n ( a ′ ) < 0 S − 1 a ′ S^{-1}a' S − 1 a ′ n n n c ′ = S ( c ) c'=S(c) c ′ = S ( c )
Directions with n ( a ′ ) = 0 n(a')=0 n ( a ′ ) = 0 a ′ a' a ′ ker V ( a ′ ) \ker V(a') ker V ( a ′ )
Incoming coefficients across a mutation.
Proposition 10.3. Let F o l d F_{\mathrm{old}} F old F n e w F_{\mathrm{new}} F new C C C C ′ C' C ′
F o l d = ϕ p , Q ( F n e w ) . F_{\mathrm{old}}=\phi_{p,Q}(F_{\mathrm{new}}). F old = ϕ p , Q ( F new ) . Then
μ ( w ) = w + r [ − n ( w ) ] + + p , A μ ( w ) ′ ( F n e w ) = τ − n ( w ) A w ( F o l d ) . (74) \mu(w)=w+r[-n(w)]_++p,\qquad A'_{\mu(w)}(F_{\mathrm{new}})=\tau^{-n(w)}A_w(F_{\mathrm{old}}). \tag*{(74)} μ ( w ) = w + r [ − n ( w ) ] + + p , A μ ( w ) ′ ( F new ) = τ − n ( w ) A w ( F old ) . (74) The label map μ \mu μ
Proof. Compare the equality in the common rational ring of the preceding proof. Choose h ( c ) < − 1 h(c)<-1 h ( c ) < − 1 c ≠ p c\ne p c = p ∣ h ( p ) ∣ < ( b 0 + 1 ) − 1 |h(p)|<(b_0+1)^{-1} ∣ h ( p ) ∣ < ( b 0 + 1 ) − 1 k p + u kp+u k p + u k ≤ b 0 ∣ u ∣ ≠ p k\le b_0|u|_{\ne p} k ≤ b 0 ∣ u ∣ = p h ( k p + u ) < 0 h(kp+u)<0 h ( k p + u ) < 0 F n e w F_{\mathrm{new}} F new
S τ ( F o l d ) = ϕ − p , Q ∨ − 1 ( F n e w ) . S_\tau(F_{\mathrm{old}})=\phi^{-1}_{-p,Q^\vee}(F_{\mathrm{new}}). S τ ( F old ) = ϕ − p , Q ∨ − 1 ( F new ) . Thus these are the values of the new transported expression on the two sides.
Continue to any required point within one open half-space. Only nonpure operations are crossed. By Lemma 10.1, at each fixed non-p p p p p p p p p Z \mathbb{Z} Z
If n ( w ) > 0 n(w)>0 n ( w ) > 0 w w w n ( w ) < 0 n(w)<0 n ( w ) < 0 S ( w ) S(w) S ( w ) τ − n ( w ) \tau^{-n(w)} τ − n ( w ) n ( w ) = 0 n(w)=0 n ( w ) = 0 w − j p w-jp w − j p n ( w − j p ) = 0 n(w-jp)=0 n ( w − j p ) = 0 X w X^w X w (63) . We may therefore compute it at a small displacement on either side. Both piecewise maps fix its label and coefficient. These cases give (74) . Since n ( μ ( w ) ) = n ( w ) n(\mu(w))=n(w) n ( μ ( w )) = n ( w ) v ↦ v − r [ − n ( v ) ] + + p v\mapsto v-r[-n(v)]_++p v ↦ v − r [ − n ( v ) ] + + p
The complete generalized mutation rule.
We verify that the formal mutation just constructed is the mutation of § 2 , including its coefficient rule. Fix k k k A = q k , 0 A=q_{k,0} A = q k , 0 B ∗ = q k , r k B_*=q_{k,r_k} B ∗ = q k , r k τ = B ∗ / A \tau=B_*/A τ = B ∗ / A
E e k = − e k + r k [ b ^ k ] + , E e i = e i ( i ≠ k ) , E A ( X w ) = A w k X E w . (75) E e_k=-e_k+r_k[\widehat{b}_k]_+,\qquad E e_i=e_i\quad(i\ne k),\qquad E_A(X^w)=A^{w_k}X^{Ew}. \tag*{(75)} E e k = − e k + r k [ b k ] + , E e i = e i ( i = k ) , E A ( X w ) = A w k X E w . (75) Here frozen basis vectors are included among the fixed vectors. The diagonal entry b k k = 0 b_{kk}=0 b k k = 0 E 2 = 1 E^2=1 E 2 = 1
μ k ∗ = ϕ p k , Q k E A . (76) \mu_k^*=\phi_{p_k,Q_k}E_A. \tag*{(76)} μ k ∗ = ϕ p k , Q k E A . (76) For the k k k
A x k ( M k + ) r k Q k ( M k − / M k + ) , \frac{A}{x_k}(M_k^+)^{r_k}Q_k(M_k^-/M_k^+), x k A ( M k + ) r k Q k ( M k − / M k + ) , which is precisely (4) ; the other generators are fixed.
Let F F F n × n n\times n n × n
F k k = − 1 , F k j = r k [ − b k j ] + ( j ≠ k ) . F_{kk}=-1,\qquad F_{kj}=r_k[-b_{kj}]_+\quad(j\ne k). F k k = − 1 , F k j = r k [ − b k j ] + ( j = k ) . The matrix mutation (5) reads
B ^ ′ = E B ^ F . (77) \widehat{B}'=E\widehat{B}F. \tag*{(77)} B ′ = E B F . (77) Its new roots, pulled back by E E E
E p k ′ = − p k , E p j ′ = p j + r k [ − b k j ] + p k ( j ≠ k ) . Ep'_k=-p_k,\qquad Ep'_j=p_j+r_k[-b_{kj}]_+p_k\quad(j\ne k). E p k ′ = − p k , E p j ′ = p j + r k [ − b k j ] + p k ( j = k ) . For j ≠ k j\ne k j = k α = b k j \alpha=b_{kj} α = b k j E A ( X p j ′ ) E_A(X^{p'_j}) E A ( X p j ′ ) A α A^\alpha A α k k k p j ′ p'_j p j ′ α \alpha α (6) , we have
q ˉ j , s = { q ˉ j , s A − s α , α ≥ 0 , q ˉ j , s B − s α , α ≤ 0 , q ˉ j , s = q j , s / q j , 0 . \bar{q}_{j,s}=
\begin{cases}
\bar{q}_{j,s}A^{-s\alpha}, & \alpha\ge0,\\
\bar{q}_{j,s}B^{-s\alpha}, & \alpha\le0,
\end{cases}
\qquad\bar{q}_{j,s}=q_{j,s}/q_{j,0}. q ˉ j , s = { q ˉ j , s A − s α , q ˉ j , s B − s α , α ≥ 0 , α ≤ 0 , q ˉ j , s = q j , s / q j , 0 . After substitution of A α X E p j ′ A^\alpha X^{Ep'_j} A α X E p j ′ α ≥ 0 \alpha\ge0 α ≥ 0 τ − α X E p j ′ \tau^{-\alpha}X^{Ep'_j} τ − α X E p j ′ α ≤ 0 \alpha\le0 α ≤ 0 k k k Q k ∨ Q_k^\vee Q k ∨ (73) .
The new form is Ω ′ = E t Ω E \Omega'=E^{\mathsf t}\Omega E Ω ′ = E t Ω E i ≠ k i\ne k i = k (77) gives
Ω ′ ( p i ′ , w ) = Ω ( p i + r k [ − b k i ] + p k , E w ) = δ i n i ( w ) + r k ( δ i [ b i k ] + − δ k [ − b k i ] + ) n k ( w ) . \begin{aligned}
\Omega'(p'_i,w)=\Omega(p_i+r_k[-b_{ki}]_+p_k,Ew) \\
&=\delta_i n_i(w)+r_k\bigl(\delta_i[b_{ik}]_+-\delta_k[-b_{ki}]_+\bigr)n_k(w).
\end{aligned} Ω ′ ( p i ′ , w ) = Ω ( p i + r k [ − b k i ] + p k , E w ) = δ i n i ( w ) + r k ( δ i [ b ik ] + − δ k [ − b k i ] + ) n k ( w ) . The last coefficient vanishes since D B DB D B i = k i=k i = k (66) . This proves compatibility for arbitrary symmetrizers. It also proves that the construction can be iterated with all frozen rows retained.
This monomial change also identifies incoming coefficients. The equality Ω ′ = E t Ω E \Omega'=E^{\mathsf t}\Omega E Ω ′ = E t Ω E w w w E A E_A E A A w k A^{w_k} A w k E w Ew E w
Arbitrary coefficient representatives
The freedom in the common coefficient scales is accounted for by the following elementary rescaling statement.
Lemma 10.4. Two representatives rooted at the same seed differ, at every seed, by rescaling each cluster variable by an element of the coefficient group. The same holds for their cluster monomials.
Proof. Denote the two representatives by superscripts a and c, and suppose at one seed that, for coefficient-group elements t i t_i t i
x i a = t i x i c , q ‾ j , s a q ‾ j , s c = ∏ i t i s b i j . (78) x_i^a=t_i x_i^c,\qquad\frac{\overline{q}_{j,s}^a}{\overline{q}_{j,s}^c}=\prod_i t_i^{s b_{ij}}. \tag*{(78)} x i a = t i x i c , q j , s c q j , s a = i ∏ t i s b ij . (78) The normalized exchange functions then agree after substitution. Writing λ k = q k , 0 a / q k , 0 c \lambda_k=q_{k,0}^a/q_{k,0}^c λ k = q k , 0 a / q k , 0 c
t i ′ = t i ( i ≠ k ) , t k ′ = λ k t k − 1 ∏ i t i r k [ b i k ] + . (79) t_i'=t_i\quad(i\ne k),\qquad t_k'=\lambda_k t_k^{-1}\prod_i t_i^{r^k[b_{ik}]_+}. \tag*{(79)} t i ′ = t i ( i = k ) , t k ′ = λ k t k − 1 i ∏ t i r k [ b ik ] + . (79) We verify that (78) persists. For j ≠ k j\ne k j = k α = b k j \alpha=b_{kj} α = b k j (77) ,
∏ i ( t i ′ ) s b i j ′ = ( ∏ i t i s b i j ) λ k − s α ∏ i t i s r k b i k [ − α ] + . (80) \prod_i(t_i')^{s b_{ij}'}=\left(\prod_i t_i^{s b_{ij}}\right)\lambda_k^{-s\alpha}\prod_i t_i^{s r^k b_{ik}[-\alpha]_+}. \tag*{(80)} i ∏ ( t i ′ ) s b ij ′ = ( i ∏ t i s b ij ) λ k − s α i ∏ t i s r k b ik [ − α ] + . (80) If α ≥ 0 \alpha\ge0 α ≥ 0 (80) . If α ≤ 0 \alpha\le0 α ≤ 0
q k , r k a q k , r k c = λ k ∏ i t i r k b i k ; \frac{q_{k,r_k}^a}{q_{k,r_k}^c}=\lambda_k\prod_i t_i^{r^k b_{ik}}; q k , r k c q k , r k a = λ k i ∏ t i r k b ik ; the leading-endpoint rule gives exactly the same right side. At k k k
q ‾ k , s a q ‾ k , s c = q ‾ k , r k − s a / q ‾ k , r k − s c q ‾ k , r k a / q ‾ k , r k c = ∏ i t i − s b i k = ∏ i ( t i ′ ) s b i k ′ , \frac{\overline{q}_{k,s}^a}{\overline{q}_{k,s}^c}
=\frac{\overline{q}_{k,r_k-s}^a/\overline{q}_{k,r_k-s}^c}{\overline{q}_{k,r_k}^a/\overline{q}_{k,r_k}^c}
=\prod_i t_i^{-s b_{ik}}
=\prod_i(t_i')^{s b_{ik}'}, q k , s c q k , s a = q k , r k a / q k , r k c q k , r k − s a / q k , r k − s c = i ∏ t i − s b ik = i ∏ ( t i ′ ) s b ik ′ , because b k k = 0 b_{kk}=0 b k k = 0 k k k
Starting with the same reference seed, take t i = 1 t_i=1 t i = 1
It follows that positivity in the universal canonical representative implies positivity in N [ P ] \mathbb{N}[\mathbb{P}] N [ P ] P \mathbb{P} P
Laurent positivity
Lemma 11.1. In its own seed, a cluster monomial c X m cX^m c X m m i ≥ 0 m_i\ge0 m i ≥ 0 c c c c c c m m m
Proof. By (66) , Ω ( p i , m ) = − δ i m i ≤ 0 \Omega(p_i,m)=-\delta_i m_i\le0 Ω ( p i , m ) = − δ i m i ≤ 0 m + C + m+C^+ m + C + w = m + r w=m+r w = m + r 0 ≠ r ∈ C + 0\ne r\in C^+ 0 = r ∈ C + h − h_- h − h − + s V ( w ) h_-+sV(w) h − + s V ( w ) s ≥ 0 s\ge0 s ≥ 0 w w w a a a Ω ( a , w ) > 0 \Omega(a,w)>0 Ω ( a , w ) > 0 m m m w w w
r = ∑ j t j a j , t j > 0 , r=\sum_j t_j a_j,\qquad t_j>0, r = j ∑ t j a j , t j > 0 , with roots crossed by this path. It would follow that Ω ( r , w ) > 0 \Omega(r,w)>0 Ω ( r , w ) > 0
Ω ( r , w ) = Ω ( r , m + r ) = Ω ( r , m ) ≤ 0 , \Omega(r,w)=\Omega(r,m+r)=\Omega(r,m)\le0, Ω ( r , w ) = Ω ( r , m + r ) = Ω ( r , m ) ≤ 0 , a contradiction. The calculation may be made in the finite interval between m m m w w w m m m c c c C + C^+ C +
Proof. of Theorem 2.1 . Choose the reference seed and a finite mutation word leading to the seed of the given cluster monomial. Add the auxiliary frozen rows and variables used in (65) . The generalized Laurent phenomenon, Theorem 2.2 , supplies a finite Laurent expression for the monomial in every intermediate seed.
In the terminal seed, write the cluster monomial as X m X^{m} X m m m m Lemma 11.1 gives the single incoming coefficient one at m m m E A E_A E A (76) gives the rational relation required in Proposition 10.3. The incoming labels are carried by its bijection, and the scalar factors are Laurent monomials in the two endpoint coefficients. Thus, going back along the word, the incoming data in the reference seed still consist of a single positive coefficient. Theorem 9.5 and Proposition 9.6 prove positivity of every coefficient in the reference-seed Laurent expression.
The coefficient semiring is the fixed one in (7) . In the canonical representative, q j , 0 ′ = q j , 0 q'_{j,0}=q_{j,0} q j , 0 ′ = q j , 0 j ≠ k j\ne k j = k (74) . Consequently every positive operation above takes place in that fixed semiring. If the initial endpoints are one, no initial interior coefficient is inverted.
Finally set Y 1 = ⋯ = Y n = 1 Y_1=\cdots=Y_n=1 Y 1 = ⋯ = Y n = 1 Lemma 10.4 , followed by specialization to P \mathbb{P} P □ \square □
Finite algebras for the coefficients
We realize the coefficients in Proposition 8.1 as dimensions of explicitly constructed finite algebras. We work over Q \mathbb{Q} Q
A finite algebra for an elementary coefficient
Fix a , b , m , k , p , q a,b,m,k,p,q a , b , m , k , p , q § 3 , and put A = k a A=ka A = k a B 0 = k b B_0=kb B 0 = k b M = 1 + max ( a , b ) M=1+\max(a,b) M = 1 + max ( a , b ) P A , P B P_A,P_B P A , P B A − 1 A-1 A − 1 B 0 − 1 B_0-1 B 0 − 1
a i p ∏ j ( a i − b j ) m − T b + T M P A ( a i ) = 0 , a_i^p\prod_j(a_i-b_j)^m-T^b+T^M P_A(a_i)=0, a i p j ∏ ( a i − b j ) m − T b + T M P A ( a i ) = 0 , ( − b j ) q ∏ i ( a i − b j ) m − T a + T M P B ( b j ) = 0. (81) (-b_j)^q\prod_i(a_i-b_j)^m-T^a+T^M P_B(b_j)=0. \tag*{(81)} ( − b j ) q i ∏ ( a i − b j ) m − T a + T M P B ( b j ) = 0. (81) The coefficients can be chosen so that the full zero scheme over Q ( T ) ‾ \overline{\mathbb{Q}(T)} Q ( T ) Q ( T ) \mathbb{Q}(T) Q ( T ) T M T^M T M
The attached factor of (81) has the same ungraded G k G_k G k D k p , q D_k^{p,q} D k p , q s s s T T T T T T T ≠ 0 T\ne0 T = 0
T b ( 1 − s T M − b P A ( a i ) ) , T a ( 1 − s T M − a P B ( b j ) ) . T^b(1-sT^{M-b}P_A(a_i)),\qquad T^a(1-sT^{M-a}P_B(b_j)). T b ( 1 − s T M − b P A ( a i )) , T a ( 1 − s T M − a P B ( b j )) . The parenthesized factors have residue one. Multiplication of the equations in any equal-center cluster therefore gives exactly the balance and center equations proved in § 3 . Every nonzero center lies in the same fixed finite root set of
P k ( z ) = ∏ h = 1 k ( z h − σ k h ) , σ k = ( − 1 ) b q + m a b ( k − 1 ) . P_k(z)=\prod_{h=1}^{k}\left(z^h-\sigma_k^h\right),\qquad\sigma_k=(-1)^{bq+mab(k-1)}. P k ( z ) = h = 1 ∏ k ( z h − σ k h ) , σ k = ( − 1 ) b q + mab ( k − 1 ) . The inverted coordinates take only nonzero values there. Their inverses are consequently integral on the reduced special support, and then integral on the whole special algebra by nilpotence of its nilradical. The special algebra is finite over Q [ s ] \mathbb{Q}[s] Q [ s ]
Lemma 5.1 makes the completed algebra a finite direct factor over Q [ s ] [ [ T ] ] \mathbb{Q}[s][[T]] Q [ s ] [[ T ]] T T T Q [ [ T ] ] [ T − 1 ] \mathbb{Q}[[T]][T^{-1}] Q [[ T ]] [ T − 1 ]
Its endpoint fibers are the complete attached factors. Indeed, after inverting T T T s s s s = 0 , 1 s=0,1 s = 0 , 1 Q [ [ T ] ] \mathbb{Q[[T]]} Q [[ T ]] s = 0 s=0 s = 0 s = 1 s=1 s = 1
For partitions λ = ( λ 1 , … , λ r ) ⊢ A \lambda=(\lambda_1,\ldots,\lambda_r)\vdash A λ = ( λ 1 , … , λ r ) ⊢ A μ = ( μ 1 , … , μ t ) ⊢ B 0 \mu=(\mu_1,\ldots,\mu_t)\vdash B_0 μ = ( μ 1 , … , μ t ) ⊢ B 0 u 1 , … , u r u_1,\ldots,u_r u 1 , … , u r v 1 , … , v t v_1,\ldots,v_t v 1 , … , v t I λ μ I_{\lambda\mu} I λ μ Q [ T , u , v ] \mathbb{Q}[T,u,v] Q [ T , u , v ]
u i p ∏ j = 1 t ( u i − v j ) m μ j − T b + T M P A ( u i ) = 0 , 1 ≤ i ≤ r , u_i^p\prod_{j=1}^{t}(u_i-v_j)^{m\mu_j}-T^b+T^M P_A(u_i)=0,\qquad1\leq i\leq r, u i p j = 1 ∏ t ( u i − v j ) m μ j − T b + T M P A ( u i ) = 0 , 1 ≤ i ≤ r , ( − v j ) q ∏ i = 1 r ( u i − v j ) m λ i − T a + T M P B ( v j ) = 0 , 1 ≤ j ≤ t . (82) (-v_j)^q\prod_{i=1}^{r}(u_i-v_j)^{m\lambda_i}-T^a+T^M P_B(v_j)=0,\qquad1\leq j\leq t. \tag*{(82)} ( − v j ) q i = 1 ∏ r ( u i − v j ) m λ i − T a + T M P B ( v j ) = 0 , 1 ≤ j ≤ t . (82) Set
Δ = ∏ i < i ′ ( u i − u i ′ ) ∏ j < j ′ ( v j − v j ′ ) , Z = ∏ i u i ∏ j v j , \Delta=\prod_{i<i'}(u_i-u_{i'})\prod_{j<j'}(v_j-v_{j'}),\qquad Z=\prod_i u_i\prod_j v_j, Δ = i < i ′ ∏ ( u i − u i ′ ) j < j ′ ∏ ( v j − v j ′ ) , Z = i ∏ u i j ∏ v j , and define, in the order displayed,
J λ μ = I λ μ : ( T Δ ) ∞ , K λ μ = ( J λ μ mod T ) : Z ∞ , R λ μ = Q [ u , v ] / K λ μ . (83) J_{\lambda\mu}=I_{\lambda\mu}:(T\Delta)^\infty,\qquad K_{\lambda\mu}=(J_{\lambda\mu}\mathbin{\operatorname{mod}} T):Z^\infty,\qquad R_{\lambda\mu}=\mathbb{Q}[u,v]/K_{\lambda\mu}. \tag*{(83)} J λ μ = I λ μ : ( T Δ ) ∞ , K λ μ = ( J λ μ mod T ) : Z ∞ , R λ μ = Q [ u , v ] / K λ μ . (83) The first saturation selects the exact equality pattern on the generic fiber and takes its full horizontal closure. The second selects nonzero centers. Distinct generic blocks can collide at T = 0 T=0 T = 0
Let m d ( λ ) m_d(\lambda) m d ( λ ) d d d λ \lambda λ
W λ μ = ∏ d S m d ( λ ) × ∏ d S m d ( μ ) . W_{\lambda\mu}=\prod_d S_{m_d(\lambda)}\times\prod_d S_{m_d(\mu)}. W λ μ = d ∏ S m d ( λ ) × d ∏ S m d ( μ ) . It permutes blocks of equal size and acts on R λ μ R_{\lambda\mu} R λ μ
Proposition 12.1. The algebra R λ μ R_{\lambda\mu} R λ μ e λ ( α ) e μ ( β ) e_\lambda(\alpha)e_\mu(\beta) e λ ( α ) e μ ( β ) k ( a , b ) k(a,b) k ( a , b )
n λ μ = dim Q R λ μ W λ μ . (84) n_{\lambda\mu}=\dim_{\mathbb{Q}}R_{\lambda\mu}^{W_{\lambda\mu}}. \tag*{(84)} n λ μ = dim Q R λ μ W λ μ . (84) Proof. Over a splitting field of Q ( ( T ) ) \mathbb{Q}((T)) Q (( T )) G k G_k G k G k G_k G k S λ × S μ S_\lambda\times S_\mu S λ × S μ N G k ( S λ × S μ ) / ( S λ × S μ ) = W λ μ N_{G_k}(S_\lambda\times S_\mu)/(S_\lambda\times S_\mu)=W_{\lambda\mu} N G k ( S λ × S μ ) / ( S λ × S μ ) = W λ μ G k G_k G k
After the first saturation in (83) , T T T
Q [ T , u , v , Z − 1 ] / J λ μ \mathbb{Q}[T,u,v,Z^{-1}]/J_{\lambda\mu} Q [ T , u , v , Z − 1 ] / J λ μ in T T T Q [ u , v ] / ( J λ μ mod T ) \mathbb{Q}[u,v]/(J_{\lambda\mu}\mathbin{\operatorname{mod}} T) Q [ u , v ] / ( J λ μ mod T ) Z Z Z Z Z Z Z Z Z Z Z Z
Q [ u , v ] / ( ( J λ μ mod T ) : Z ∞ ) = R λ μ . \mathbb{Q}[u,v]/((J_{\lambda\mu}\mathbin{\operatorname{mod}} T):Z^\infty)=R_{\lambda\mu}. Q [ u , v ] / (( J λ μ mod T ) : Z ∞ ) = R λ μ . The generator-lifting argument of Lemma 5.1 makes the completion finite over Q [ [ T ] ] \mathbb{Q}[[T]] Q [[ T ]] T T T Q [ [ T ] ] \mathbb{Q}[[T]] Q [[ T ]] G k G_k G k n λ μ n_{\lambda\mu} n λ μ
The invariant algebra itself has explicit equations. For each group of equal λ \lambda λ u i u_i u i μ \mu μ v j v_j v j u , v u,v u , v K λ μ K_{\lambda\mu} K λ μ R λ μ W λ μ R_{\lambda\mu}^{W_{\lambda\mu}} R λ μ W λ μ
Effective choices
12.2. Effective choices. The generic perturbations can be chosen by a terminating rational procedure. Enumerate integer coefficient tuples for P A , P B P_A,P_B P A , P B Q ( T ) \mathbb{Q}(T) Q ( T )
( F 1 , … , F A + B 0 , det Jac F ) = ( 1 ) , (F_1,\ldots,F_{A+B_0},\det\operatorname{Jac}F)=(1), ( F 1 , … , F A + B 0 , det Jac F ) = ( 1 ) , where F F F (81) . A successful test says that every affine zero has invertible Jacobian, so the zero scheme is finite and reduced. The nonempty generic open set contains an integer tuple: after clearing denominators, a nonzero polynomial in T T T T T T
A finite formula for a whole Laurent expansion
12.3. A finite formula for a whole Laurent expansion. Fix a mutation word and a cluster monomial in its final seed, using the universal canonical representative of Theorem 2.1 . Append the auxiliary frozen variables Y 1 , … , Y n Y_1,\ldots,Y_n Y 1 , … , Y n F ^ \widehat{F} F p i p_i p i ( p i ) Y j = − δ i j (p_i)_{Y_j}=-\delta_{ij} ( p i ) Y j = − δ ij g g g η \eta η k k k u u u F F F (75) . The inverse update is
g o l d = E u − r k [ − u k ] + + p k , η o l d = η n e w q k , 0 [ u k ] + q k , r k [ − u k ] + . (85) g_{\mathrm{old}}=E u-r_k[-u_k]_++p_k,\qquad
\eta_{\mathrm{old}}=\eta_{\mathrm{new}}q_{k,0}^{[u_k]_+}q_{k,r_k}^{[-u_k]_+}. \tag*{(85)} g old = E u − r k [ − u k ] + + p k , η old = η new q k , 0 [ u k ] + q k , r k [ − u k ] + . (85) It follows directly by inverting the incoming-coefficient identity in § 10 ; in particular all factors in η \eta η
The positive recurrence implies that every transported support lies in g + ∑ i N p i g+\sum_i\mathbb{N}p_i g + ∑ i N p i Theorem 2.2 . Its transports have support in a finite union of translates of the free positive root cone. In each finite interval below a target exponent outside g + ∑ i N p i g+\sum_i \mathbb{N}p_i g + ∑ i N p i
Compute the lowest valuation of the final rational expression in each Y j Y_j Y j b ^ i k \widehat b_{ik} b ik ℓ i \ell_i ℓ i Y j Y_j Y j Y Y Y k k k
ℓ k ′ = − ℓ k + r k min { ∑ i [ b ^ i k ] + + ℓ i , ∑ i [ − b ^ i k ] + + ℓ i } . (86) \ell'_k=-\ell_k+r_k\min\left\{\sum_i[\widehat b_{ik}]_++\ell_i,\sum_i[-\widehat b_{ik}]_++\ell_i\right\}. \tag*{(86)} ℓ k ′ = − ℓ k + r k min { i ∑ [ b ik ] + + ℓ i , i ∑ [ − b ik ] + + ℓ i } . (86) Endpoint coefficients have Y j Y_j Y j L j L_j L j Y j Y_j Y j
An exponent g + ∑ i k i p i g+\sum_i k_i p_i g + ∑ i k i p i
0 ≤ k j ≤ K j : = g Y j − L j . (87) 0\le k_j\le K_j:=g_{Y_j}-L_j. \tag*{(87)} 0 ≤ k j ≤ K j := g Y j − L j . (87) This gives a finite box using only the matrix and coefficient mutations, the incoming update, and the valuation recursion.
For 1 ≤ i ≤ n 1\le i\le n 1 ≤ i ≤ n 1 ≤ s ≤ r i 1\le s\le r_i 1 ≤ s ≤ r i q ˉ i , s \bar q_{i,s} q ˉ i , s § 2 . Write k = ( k 1 , … , k n ) \mathbf{k}=(k_1,\ldots,k_n) k = ( k 1 , … , k n ) K = ( K 1 , … , K n ) \mathbf{K}=(K_1,\ldots,K_n) K = ( K 1 , … , K n ) β = ( β i , s ) 1 ≤ i ≤ n , 1 ≤ s ≤ r i \beta=(\beta_{i,s})_{1\le i\le n,\,1\le s\le r_i} β = ( β i , s ) 1 ≤ i ≤ n , 1 ≤ s ≤ r i Q \mathbb{Q} Q A k , β A_{\mathbf{k},\beta} A k , β
F ^ = η ∑ 0 ≤ k ≤ K X g + ∑ i k i p i ∑ β i , s ≥ 0 ∑ s s β i , s = k i ( 1 ≤ i ≤ n ) dim Q A k , β ∏ i , s q ˉ i , s β i , s . (88) \widehat F=\eta\sum_{0\le\mathbf{k}\le\mathbf{K}}X^{g+\sum_i k_i p_i}\sum_{\substack{\beta_{i,s}\ge0\\ \sum_s s\beta_{i,s}=k_i\ (1\le i\le n)}}\dim_{\mathbb{Q}}A_{\mathbf{k},\beta}\prod_{i,s}\bar q_{i,s}^{\beta_{i,s}}. \tag*{(88)} F = η 0 ≤ k ≤ K ∑ X g + ∑ i k i p i β i , s ≥ 0 ∑ s s β i , s = k i ( 1 ≤ i ≤ n ) ∑ dim Q A k , β i , s ∏ q ˉ i , s β i , s . (88) Keep every initial coefficient separately labeled. Construct the wall operations only through total root degree ∑ j K j \sum_j K_j ∑ j K j Q \mathbb{Q} Q Proposition 9.4 . Once these wall coefficients are fixed, the recurrence of Proposition 9.6 strictly decreases root degree. These three finite procedures construct A k , β A_{\mathbf{k},\beta} A k , β
All geometric choices can be made over Q \mathbb{Q} Q
The displayed weight constraint follows because the initial coefficient q ˉ i , s \bar q_{i,s} q ˉ i , s s p i sp_i s p i Y 1 = ⋯ = Y n = 1 Y_1=\cdots=Y_n=1 Y 1 = ⋯ = Y n = 1 (88) . If several lifted exponents project to the same old exponent, sum their finitely many coefficient algebras by a direct product. Thus column dependencies, nonprimitive columns, and zero projected roots are covered. The rational algorithm constructs the coefficient algebras over the universal independent parameters; positive specialization then gives the coefficient-group statement.
Zhili College, Tsinghua University, Beijing 100084, China
Email address: tangqy24@mails.tsinghua.edu.cn
