Introduction
Motivation and the Frobenius Program
Recent work on the arithmetic properties of integrable connections has uncovered relations between Frobenius action and classical spectral geometry. Etingof and Varchenko [3 , 4 ] initiated this study for periodic pencils of flat connections and their p p p K K K [8 ] showed that full-cycle difference transport at roots of unity has the spectrum of quantum multiplication at powered parameters. The p p p [13 ] provide another instance of the arithmetic structure of these enumerative functions.
In the present paper we shall examine Frobenius action from the point of view of the trigonometric Ruijsenaars–Schneider (tRS) system [12 , 11 ] associated with the cotangent bundle to the complete flag variety T ∗ F l n T^*\mathrm{Fl}_n T ∗ Fl n [14 ] enter the quantum/classical duality between Bethe equations and tRS level sets. Throughout the paper, the difference operators act on Kähler parameters, while the equivariant parameters a 1 , … , a n a_1,\ldots,a_n a 1 , … , a n
Our construction is also motivated by the qKZ/tRS correspondence of the author and Zeitlin [9 ] . In that work a weighted sum of components of a geometric qKZ solution produces a tRS eigenfunction, which is identified with a vertex function in Theorem 5.11. This gives a geometric form of the Matsuo–Cherednik correspondence [10 , 1 ] . It also has a natural interpretation through 3 d 3d 3 d T ∗ F l n T^*\mathrm{Fl}_n T ∗ Fl n [2 ] (Theorem 1 and Section 2.7). On the integrable-system side, the tRS model enjoys the corresponding self 3 d 3d 3 d [6 ] (§ 2.4 ). The projection identities of [9 ] (Proposition 5.5 and Theorems 5.6, 5.10) act in equivariant variables. Here we work with the Kähler difference equations and use the vertex eigenfunction theorem of [5 ] (Theorem 2.10) as our geometric input. The normalization is related to that of [2 ] (§ 2.6 , Propositions 1–3) below. For a review of quantum/classical duality and the relation between quantum geometry and integrable systems, see [6 ] .
In the qKZ/tRS construction of [9 ] , one starts with a fundamental matrix of the quantum difference connection. Its columns are independent vector-valued solutions; its rows are coordinates in a chosen basis, indexed for a complete flag by σ ∈ S n \sigma\in S_n σ ∈ S n τ \tau τ I σ , τ I_{\sigma,\tau} I σ , τ V τ = ∑ σ I σ , τ V_\tau= \sum_\sigma I_{\sigma,\tau} V τ = ∑ σ I σ , τ τ \tau τ V = ∑ σ I σ V = \sum_\sigma I_\sigma V = ∑ σ I σ
At a primitive root of unity, products of tRS Hamiltonians contain many terms with partial shifts. The Newton–Adams combinations cancel these terms and leave only full turns around the coordinate orbits. We prove this operator identity in arbitrary rank and then apply it to joint eigenfunctions. For the finite-sum construction, we start with the integral representation of the vertex and replace integration in each auxiliary variable by a weighted sum over x , q x , … , q p − 1 x x,qx,\ldots,q^{p-1}x x , q x , … , q p − 1 x T ∗ P 1 T^*\mathbb{P}^1 T ∗ P 1
Kähler Shifts and tRS Hamiltonians
Let us first fix the geometric conventions and the normalization of the difference operators. We use the polarization convention of [8 ] , T X = P + ℏ − 1 P ∗ TX = P + \hbar^{-1}P^* T X = P + ℏ − 1 P ∗ ℏ 1 / 2 \hbar^{1/2} ℏ 1/2
q = ω p primitive root of unity of order p ≥ 1 ℏ fiber dilation weight a 1 , … , a n torus equivariant parameters z 1 , … , z n − 1 K a ¨ hler parameters in the geometric connection x r , υ tautological roots and internal summation variables. \begin{aligned}
q &= \omega_p && \text{primitive root of unity of order } p \ge1 \\
\hbar&&& \text{fiber dilation weight} \\
a_1,\ldots,a_n &&& \text{torus equivariant parameters} \\
z_1,\ldots,z_{n-1} &&& \text{Kähler parameters in the geometric connection} \\
x_{r,\upsilon} &&& \text{tautological roots and internal summation variables.}
\end{aligned} q ℏ a 1 , … , a n z 1 , … , z n − 1 x r , υ = ω p primitive root of unity of order p ≥ 1 fiber dilation weight torus equivariant parameters K a ¨ hler parameters in the geometric connection tautological roots and internal summation variables. The order p p p
Introduce homogeneous Kähler coordinates by
z r = q ζ r ζ r + 1 , 1 ≤ r < n . (1) z_r = q\frac{\zeta_r}{\zeta_{r+1}}, \qquad1 \le r < n. \tag*{(1)} z r = q ζ r + 1 ζ r , 1 ≤ r < n . (1) Their simultaneous scale supplies the central character of the GL ( n ) \operatorname{GL}(n) GL ( n ) ζ i ↦ q ζ i \zeta_i \mapsto q\zeta_i ζ i ↦ q ζ i p i p_i p i
p i ζ j = q δ i j ζ j p i , p i z r = q δ i r − δ i , r + 1 z r p i , p i a j = a j p i . (2) p_i\zeta_j=q^{\delta_{ij}}\zeta_jp_i,\qquad p_i z_r=q^{\delta_{ir}-\delta_{i,r+1}}z_rp_i,\qquad p_i a_j=a_jp_i. \tag*{(2)} p i ζ j = q δ ij ζ j p i , p i z r = q δ i r − δ i , r + 1 z r p i , p i a j = a j p i . (2) For a subset I I I p I = ∏ i ∈ I p i p_I=\prod_{i\in I}p_i p I = ∏ i ∈ I p i
H r = ∑ ∣ I ∣ = r b I p I , b I ( ζ ; ℏ ) = ℏ − r ( n − r ) / 2 ∏ i ∈ I j ∉ I ℏ ζ i − ζ j ζ i − ζ j . (3) H_r=\sum_{\lvert I\rvert=r}b_Ip_I,\qquad b_I(\zeta;\hbar)=\hbar^{-r(n-r)/2}\prod_{\substack{i\in I\\j\notin I}}\frac{\hbar\zeta_i-\zeta_j}{\zeta_i-\zeta_j}. \tag*{(3)} H r = ∣ I ∣ = r ∑ b I p I , b I ( ζ ; ℏ ) = ℏ − r ( n − r ) /2 i ∈ I j ∈ / I ∏ ζ i − ζ j ℏ ζ i − ζ j . (3) In particular, H 0 = 1 H_0=1 H 0 = 1 H n = p 1 ⋯ p n H_n=p_1\cdots p_n H n = p 1 ⋯ p n
For generic q q q V τ V_\tau V τ
H r V τ = e r ( a 1 , … , a n ) V τ , H n V τ = ( a 1 ⋯ a n ) V τ . (4) H_rV_\tau=e_r(a_1,\ldots,a_n)V_\tau,\qquad H_nV_\tau=(a_1\cdots a_n)V_\tau. \tag*{(4)} H r V τ = e r ( a 1 , … , a n ) V τ , H n V τ = ( a 1 ⋯ a n ) V τ . (4) These equations were proved in [5 ] (Theorem 2.10). The comparison with the modified vertex of [2 ] is given in § 3.1 . Notice that the homogeneous character is needed to retain the eigenvalue a 1 ⋯ a n a_1\cdots a_n a 1 ⋯ a n
Orbit Graphs and Finite Sums
Let us describe the finite graphs which appear in the calculation. At q p = 1 q^p=1 q p = 1 q q q p p p q m ζ ∘ q^m\zeta^\circ q m ζ ∘ p i p_i p i n n n b I p I b_Ip_I b I p I I I I b I b_I b I p i p p_i^p p i p
The finite vertex sum uses the same construction for its internal variables. Fix the Kähler parameters and choose nonzero initial roots x α ∘ x_\alpha^\circ x α ∘ R = n ( n − 1 ) / 2 R=n(n-1)/2 R = n ( n − 1 ) /2 p R p^R p R T α T_\alpha T α Φ α \Phi_\alpha Φ α
Φ α ( x ) Φ β ( T α x ) = Φ β ( x ) Φ α ( T β x ) . \Phi_\alpha(x)\Phi_\beta(T_\alpha x)=\Phi_\beta(x)\Phi_\alpha(T_\beta x). Φ α ( x ) Φ β ( T α x ) = Φ β ( x ) Φ α ( T β x ) . Thus the weight of a path is unchanged when two adjacent steps are interchanged. To obtain periodic weights on the finite grid, the product along every coordinate cycle must also equal one. These conditions are the Bethe equations at the parameters obtained by Frobenius action, B p , α = 1 \mathcal{B}_{p,\alpha}=1 B p , α = 1 B p , α \mathcal{B}_{p,\alpha} B p , α p p p α \alpha α (119) ; for one internal variable we write B p \mathcal{B}_p B p § 2 .
With these conditions imposed, a weighted total difference cancels after summation: consecutive terms cancel along each coordinate line, and the last term agrees with the first. This elementary observation gives the finite functional and its full-cycle multipliers. The classical tRS/Bethe correspondence then determines the eigenvalues after Frobenius action. In rank one we shall also use the same calculation to verify the ordinary tRS equations and the first-order component system.
From Component Solutions to Vertex Functions
Let us describe the integral meaning of the component presentation. For a difference connection p i Ψ = M i Ψ p_i\Psi=M_i\Psi p i Ψ = M i Ψ Ψ \Psi Ψ [14 ] (§ 3.2 –§ 3.6 ): the weight-function identities ensure flatness, and nonzero closed weighted sums give monodromy eigensections. For the multiplicative qKZ equations, see [14 ] (Sections 5.4–5.5, in particular Corollary 5.7 and equation (5.19)).
In our notation, S Γ [ f ] S_{\Gamma}[f] S Γ [ f ] K K K T ∗ F l n T^*F\!l_n T ∗ F l n f f f x r , u x_{r,u} x r , u (114) , and a contour or residue prescription Γ \Gamma Γ [5 ] (equations (24) –(27) ) for the vertex integral. The cycle, phase, homogeneous character, and normalization remain fixed while the insertion varies. We choose one such cycle and suppress its label, writing S [ f ] S[f] S [ f ]
The explicit components of the finite presentation are descendant periods for this common choice. Set L r = ∏ u = 1 r x r , u L_r=\prod_{u=1}^{r}x_{r,u} L r = ∏ u = 1 r x r , u L 0 = 1 L_0=1 L 0 = 1 β r = L r / L r − 1 \beta_r=L_r/L_{r-1} β r = L r / L r − 1 a i a_i a i
w σ ( x ; a ) = ∏ r = 1 n − 1 ∏ s = r + 1 n β r − a σ ( s ) a σ ( r ) − a σ ( s ) , σ ∈ S n . (5) w_{\sigma}(x;a)=\prod_{r=1}^{n-1}\prod_{s=r+1}^{n}\frac{\beta_r-a_{\sigma(s)}}{a_{\sigma(r)}-a_{\sigma(s)}},\qquad\sigma\in S_n. \tag*{(5)} w σ ( x ; a ) = r = 1 ∏ n − 1 s = r + 1 ∏ n a σ ( r ) − a σ ( s ) β r − a σ ( s ) , σ ∈ S n . (5) Successive Lagrange interpolation gives ∑ σ w σ = 1 \sum_{\sigma}w_{\sigma}=1 ∑ σ w σ = 1 S S S
V = S [ 1 ] = ∑ σ ∈ S n I σ , I σ = S [ w σ ] . (6) V=S[1]=\sum_{\sigma\in S_n}I_{\sigma},\qquad I_{\sigma}=S[w_{\sigma}]. \tag*{(6)} V = S [ 1 ] = σ ∈ S n ∑ I σ , I σ = S [ w σ ] . (6) Thus I Γ = ( S Γ [ w σ ] ) σ ∈ S n I_{\Gamma}=(S_{\Gamma}[w_{\sigma}])_{\sigma\in S_n} I Γ = ( S Γ [ w σ ] ) σ ∈ S n σ \sigma σ Γ \Gamma Γ S [ 1 ] S[1] S [ 1 ] (128) in place of the contour integral, following the discrete integration construction of [14 ] (§ 3.2 ). The phase and its normalization are described in § 3.8 .
This presentation allows us to apply the operator identities directly to the component sum. Once V V V § 2 . For arbitrary flags, the descendant decomposition and the full-cycle equations for each I σ I_{\sigma} I σ § 3.11 .
Bloch Solutions and Full-Cycle Transport
A difference equation with periodic coefficients may have solutions which acquire a multiplier after one period. Such solutions are called Bloch solutions. To see this in our setting, write ζ ( m ) = q m ζ \zeta(m)=q^m\zeta ζ ( m ) = q m ζ
I ( m + e i ) = M i ( ζ ( m ) ) I ( m ) , m ∈ Z n . (7) I(m+e_i)=M_i(\zeta(m))I(m),\qquad m\in\mathbb{Z}^n. \tag*{(7)} I ( m + e i ) = M i ( ζ ( m )) I ( m ) , m ∈ Z n . (7) The product of p p p λ i \lambda_i λ i
I ( m + p e i ) = λ i I ( m ) , V ( m + p e i ) = λ i V ( m ) . (8) I(m+pe_i)=\lambda_i I(m),\qquad V(m+pe_i)=\lambda_i V(m). \tag*{(8)} I ( m + p e i ) = λ i I ( m ) , V ( m + p e i ) = λ i V ( m ) . (8) For example, the scalar equation f ( m + 1 ) = b ( q m ζ 0 ) f ( m ) f(m+1)=b(q^m\zeta_0)f(m) f ( m + 1 ) = b ( q m ζ 0 ) f ( m )
f ( m + p ) = ( ∏ d = 0 p − 1 b ( q d ζ 0 ) ) f ( m ) . f(m+p)=\left(\prod_{d=0}^{p-1}b(q^d\zeta_0)\right)f(m). f ( m + p ) = ( d = 0 ∏ p − 1 b ( q d ζ 0 ) ) f ( m ) . Taking q = − 1 q=-1 q = − 1 b ( z ) = γ ( 1 − c z ) b(z)=\gamma(1-cz) b ( z ) = γ ( 1 − cz ) λ = γ 2 ( 1 − c 2 z 0 2 ) \lambda=\gamma^2(1-c^2z_0^2) λ = γ 2 ( 1 − c 2 z 0 2 ) p i p p_i^p p i p
Main Results
The symmetric functions of powered spectral parameters can be expressed as integral polynomials in the original elementary symmetric functions. Define these Newton–Adams polynomials by
A p , r ( e 1 ( y ) , … , e n ( y ) ) = e r ( y 1 p , … , y n p ) . (9) A_{p,r}(e_1(y),\ldots,e_n(y))=e_r(y_1^p,\ldots,y_n^p). \tag*{(9)} A p , r ( e 1 ( y ) , … , e n ( y )) = e r ( y 1 p , … , y n p ) . (9) For n = 2 n=2 n = 2 r = 1 r=1 r = 1 P p ( E , D ) P_p(E,D) P p ( E , D ) P p ( u + v , u v ) = u p + v p P_p(u+v,uv)=u^p+v^p P p ( u + v , uv ) = u p + v p H r [ p ] H_r^{[p]} H r [ p ]
Main Theorem. Let n , p ≥ 1 n,p\geq1 n , p ≥ 1 q q q p p p (3) satisfy A p , r ( H 1 , … , H n ) = H r [ p ] A_{p,r}(H_1,\ldots,H_n)=H_r^{[p]} A p , r ( H 1 , … , H n ) = H r [ p ]
H r [ p ] = ℏ − p r ( n − r ) / 2 ∑ ∣ I ∣ = r ∏ i ∈ I j ∉ I ℏ p ζ i p − ζ j p ζ i p − ζ j p p I p , 1 ≤ r ≤ n . (10) H_r^{[p]}=\hbar^{-pr(n-r)/2}\sum_{\lvert I\rvert=r}\prod_{\substack{i\in I\\j\notin I}}\frac{\hbar^p\zeta_i^p-\zeta_j^p}{\zeta_i^p-\zeta_j^p}p_I^p,\qquad1\leq r\leq n. \tag*{(10)} H r [ p ] = ℏ − p r ( n − r ) /2 ∣ I ∣ = r ∑ i ∈ I j ∈ / I ∏ ζ i p − ζ j p ℏ p ζ i p − ζ j p p I p , 1 ≤ r ≤ n . (10) If H r V = e r ( a 1 , … , a n ) V H_rV=e_r(a_1,\ldots,a_n)V H r V = e r ( a 1 , … , a n ) V r r r
H r [ p ] V = e r ( a 1 p , … , a n p ) V , 1 ≤ r ≤ n . (11) H_r^{[p]}V=e_r(a_1^p,\ldots,a_n^p)V,\qquad1\leq r\leq n. \tag*{(11)} H r [ p ] V = e r ( a 1 p , … , a n p ) V , 1 ≤ r ≤ n . (11) Thus the joint tRS eigenvalue equations are compatible with Frobenius action for every complete flag and every root order.
The main achievement is the cancellation identity in the formal difference algebra. The classical tRS/Bethe correspondence already determines the spectral level sets after applying the Frobenius action to the parameters. At the operator level, one must also account for all partial shifts in products of Hamiltonians. We prove that their coefficients vanish in the Newton–Adams combinations and calculate the surviving full-cycle coefficients exactly. Theorem 3.4 establishes this in arbitrary rank using affine Hecke centrality and bounds on the shift exponents. For T ∗ P 1 T^*\mathbb{P}^1 T ∗ P 1 p p p Proposition 3.8 gives an additional direct proof at q = − 1 q=-1 q = − 1
Applying the operator identity to a joint eigenfunction gives Corollary 3.5 . The ordinary vertex equations of [5 ] (Theorem 2.10) supply the geometric input, and the Newton–Adams polynomial sends their eigenvalues to e r ( a p ) e_r(a^p) e r ( a p )
There is also a direct quantum K K K [7 ] (Theorem 4.6), the ratios L d / L d − 1 L_d/L_{d-1} L d / L d − 1 (116) , these are the tRS momenta. A geometric Kähler shift inserts L d L_d L d L d p L_d^p L d p [8 ] . The finite functional is constructed by the cyclic summation mechanism of Tarasov–Varchenko [14 ] .
The cancellation used in the original dynamical/tRS correspondence occurs in the auxiliary variables of the vertex representation. Kähler shifts insert tautological classes into the phase, and the identities underlying the scalar eigenvalue equation remove internal twisted differences. Integration over a compatible cycle annihilates these differences. In a finite orbit sum, consecutive weighted terms cancel by reindexing, leaving a boundary term proportional to B p , α − 1 B_{p,\alpha}-1 B p , α − 1 B p , α B_{p,\alpha} B p , α a i a_i a i z r z_r z r x r , u x_{r,u} x r , u q q q ℏ \hbar ℏ (119) . The Bethe equations B p , α = 1 B_{p,\alpha}=1 B p , α = 1 T ∗ P 1 T^*\mathbb{P}^1 T ∗ P 1 Theorem 3.12 proves that a closed weighted orbit annihilates every internal twisted difference.
The cancellation specific to the Frobenius theorem takes place in the external difference operators. Expanding A p , r ( H ) A_{p,r}(H) A p , r ( H ) p I m i p_I^{m_i} p I m i 0 < m i < p 0<m_i<p 0 < m i < p q p = 1 q^p=1 q p = 1 p I p p_I^p p I p b I b_I b I (10) . This identity holds before any operator is applied to a vertex and uses no Bethe equations. Its purpose is to identify the polynomial in quantum Hamiltonians with the actual tRS family after Frobenius action.
For the finite flag sums, internal closure gives explicit Bloch multipliers, and the known tRS/Bethe correspondence places them on the classical level set at the powered parameters. The external operator identity then gives their Frobenius eigenvalue equations in Theorem 3.13 , for every rank and root order. This direct construction and the eigenfunction consequence of the main theorem give compatible realizations of the same Frobenius spectral data.
1.7. Structure of the Paper. In § 2 we study the vertex for T ∗ P 1 T^*\mathbb{P}^1 T ∗ P 1 q = − 1 q=-1 q = − 1 p p p s j = q j s 0 s_j=q^j s_0 s j = q j s 0
§ 3 is devoted to complete flags. We express products of Hamiltonians as weighted sequences of subsets and prove the cancellation of partial shifts using affine Hecke centrality. At q = − 1 q=-1 q = − 1 V = ∑ σ I σ V=\sum_\sigma I_\sigma V = ∑ σ I σ
1.8. Acknowledgements. I would like to thank Andrey Smirnov for useful discussions. I acknowledge support from the Beijing Natural Science Foundation (BJNSF) under grant IS26008. I used ChatGPT-6.1 to study the details of the cancellations after applying the Frobenius action.
The tRS Equations for T*P1
The simplest complete flag variety is P 1 \mathbb{P}^1 P 1 T ∗ P 1 T^*\mathbb{P}^1 T ∗ P 1 a 1 a_1 a 1 a 2 a_2 a 2 ℏ \hbar ℏ T X = P + ℏ − 1 P ∗ TX = P + \hbar^{-1}P^* T X = P + ℏ − 1 P ∗
The difference operators below act on the Kähler variables. We shall use the notation
E = a 1 + a 2 , D = a 1 a 2 , ( b ; q ) d = ∏ j = 0 d − 1 ( 1 − b q j ) , E = a_1 + a_2,\qquad D = a_1a_2,\qquad(b;q)_d = \prod_{j=0}^{d-1}(1-bq^j), E = a 1 + a 2 , D = a 1 a 2 , ( b ; q ) d = j = 0 ∏ d − 1 ( 1 − b q j ) , and write p z p_z p z z ↦ q z z \mapsto qz z ↦ q z p z z = q z p z p_z z = qz p_z p z z = q z p z
The Vertex and Hamiltonians
Let us begin with the relation between the vertex and the two-particle tRS model. The generic-q q q [5 ] (Theorem 2.10). In rank one we can derive the equation from the series itself and describe the character which brings it to the centered tRS normalization.
We choose the Kähler coordinate z z z [8 ] (§2.1) is z / ℏ z/\hbar z /ℏ a i a_i a i { i , j } = { 1 , 2 } \{i,j\} = \{1,2\} { i , j } = { 1 , 2 }
V i b a r e ( z ) = ∑ d ≥ 0 ( ℏ ; q ) d ( ℏ a i / a j ; q ) d ( q ; q ) d ( q a i / a j ; q ) d ( z ℏ ) d . (12) V_i^{\mathrm{bare}}(z)=\sum_{d\geq0}\frac{(\hbar;q)_d(\hbar a_i/a_j;q)_d}{(q;q)_d(qa_i/a_j;q)_d}\left(\frac{z}{\hbar}\right)^d. \tag*{(12)} V i bare ( z ) = d ≥ 0 ∑ ( q ; q ) d ( q a i / a j ; q ) d ( ℏ ; q ) d ( ℏ a i / a j ; q ) d ( ℏ z ) d . (12) The framing parameters enter as coefficients. Introduce a difference character e a i e_{a_i} e a i p z e a i = a i e a i p_z e_{a_i}=a_i e_{a_i} p z e a i = a i e a i V i = e a i V i b a r e V_i=e_{a_i}V_i^{\mathrm{bare}} V i = e a i V i bare
It will be convenient to introduce
R ( y ) = ( y − a 1 ) ( y − a 2 ) , J ( y ) = ( ℏ y − a 1 ) ( ℏ y − a 2 ) . (13) R(y)=(y-a_1)(y-a_2),\qquad J(y)=(\hbar y-a_1)(\hbar y-a_2). \tag*{(13)} R ( y ) = ( y − a 1 ) ( y − a 2 ) , J ( y ) = ( ℏ y − a 1 ) ( ℏ y − a 2 ) . (13) The scalar difference operator associated with the vertex is
L = R ( p z ) − z ℏ J ( p z ) = ( 1 − ℏ z ) p z 2 − E ( 1 − z ) p z + D ( 1 − z / ℏ ) . (14) \mathcal{L}=R(p_z)-\frac{z}{\hbar}J(p_z)=(1-\hbar z)p_z^2-E(1-z)p_z+D(1-z/\hbar). \tag*{(14)} L = R ( p z ) − ℏ z J ( p z ) = ( 1 − ℏ z ) p z 2 − E ( 1 − z ) p z + D ( 1 − z /ℏ ) . (14) Here and below coefficients precede the shifts. The recurrence of (12) is precisely the equation of this operator, as follows. Its relation to the tRS Hamiltonians will be given by (20) .
Proposition 2.1. For generic parameters, the formal series (12) obeys L V i = 0 \mathcal{L}V_i=0 L V i = 0
Proof. Write V i b a r e = ∑ d ≥ 0 v d z d V_i^{\mathrm{bare}}=\sum_{d\geq0}v_dz^d V i bare = ∑ d ≥ 0 v d z d e a i z d e_{a_i}z^d e a i z d d ≥ 1 d\geq1 d ≥ 1
R ( a i q d ) v d = ℏ − 1 J ( a i q d − 1 ) v d − 1 . R(a_iq^d)v_d=\hbar^{-1}J(a_iq^{d-1})v_{d-1}. R ( a i q d ) v d = ℏ − 1 J ( a i q d − 1 ) v d − 1 . Solving for the ratio of consecutive coefficients gives
v d v d − 1 = 1 ℏ ( 1 − ℏ q d − 1 ) ( 1 − ℏ ( a i / a j ) q d − 1 ) ( 1 − q d ) ( 1 − ( a i / a j ) q d ) , \frac{v_d}{v_{d-1}}=\frac{1}{\hbar}\frac{(1-\hbar q^{d-1})(1-\hbar(a_i/a_j)q^{d-1})}{(1-q^d)(1-(a_i/a_j)q^d)}, v d − 1 v d = ℏ 1 ( 1 − q d ) ( 1 − ( a i / a j ) q d ) ( 1 − ℏ q d − 1 ) ( 1 − ℏ ( a i / a j ) q d − 1 ) , which agrees with (12) . The constant term of the equation vanishes since R ( a i ) = 0 R(a_i)=0 R ( a i ) = 0
The phase of the vertex provides another description of this equation. Its ratio under an internal shift is
ϕ ( x ; z ) = z ℏ ∏ i = 1 2 1 − ℏ x / a i 1 − q x / a i = z ℏ J ( x ) R ( q x ) . (15) \phi(x;z)=\frac{z}{\hbar}\prod_{i=1}^{2}\frac{1-\hbar x/a_i}{1-qx/a_i}=\frac{z}{\hbar}\frac{J(x)}{R(qx)}. \tag*{(15)} ϕ ( x ; z ) = ℏ z i = 1 ∏ 2 1 − q x / a i 1 − ℏ x / a i = ℏ z R ( q x ) J ( x ) . (15) For a shift-invariant period functional we therefore have ⟨ R ( x ) ⟩ = ( z / ℏ ) ⟨ J ( x ) ⟩ \langle R(x)\rangle=(z/\hbar)\langle J(x)\rangle ⟨ R ( x )⟩ = ( z /ℏ ) ⟨ J ( x )⟩ x x x (14) . At a root of unity the same argument will be realized by a finite sum, whose boundary term we shall compute explicitly.
For a primitive p p p z p z^p z p ( 1 − q p ) − 1 (1-q^p)^{-1} ( 1 − q p ) − 1 [8 ] (Theorem 1.2). Here we construct finite solutions directly from the rational phase ratios.
To pass to homogeneous Kähler coordinates, let
s = ζ 1 ζ 2 = z q , p i g ( ζ 1 , ζ 2 ) = g ( ζ 1 , … , q ζ i , … , ζ 2 ) . (16) s=\frac{\zeta_1}{\zeta_2}=\frac{z}{q},\qquad p_i g(\zeta_1,\zeta_2)=g(\zeta_1,\ldots,q\zeta_i,\ldots,\zeta_2). \tag*{(16)} s = ζ 2 ζ 1 = q z , p i g ( ζ 1 , ζ 2 ) = g ( ζ 1 , … , q ζ i , … , ζ 2 ) . (16) The shifts p i p_i p i a 1 , a 2 a_1,a_2 a 1 , a 2 ℏ \hbar ℏ
A ℏ ( s ) = 1 − ℏ s ℏ 1 / 2 ( 1 − s ) , B ℏ ( s ) = ℏ − s ℏ 1 / 2 ( 1 − s ) . (17) A_{\hbar}(s)=\frac{1-\hbar s}{\hbar^{1/2}(1-s)},\qquad B_{\hbar}(s)=\frac{\hbar-s}{\hbar^{1/2}(1-s)}. \tag*{(17)} A ℏ ( s ) = ℏ 1/2 ( 1 − s ) 1 − ℏ s , B ℏ ( s ) = ℏ 1/2 ( 1 − s ) ℏ − s . (17) Then the centered Hamiltonians take the form
H 1 = A ℏ ( s ) p 1 + B ℏ ( s ) p 2 , H 2 = p 1 p 2 . (18) H_1=A_{\hbar}(s)p_1+B_{\hbar}(s)p_2,\qquad H_2=p_1p_2. \tag*{(18)} H 1 = A ℏ ( s ) p 1 + B ℏ ( s ) p 2 , H 2 = p 1 p 2 . (18) Let χ \chi χ
p 1 χ = ℏ 1 / 2 χ , p 2 χ = D ℏ − 1 / 2 χ . (19) p_1\chi=\hbar^{1/2}\chi,\qquad p_2\chi=D\hbar^{-1/2}\chi. \tag*{(19)} p 1 χ = ℏ 1/2 χ , p 2 χ = D ℏ − 1/2 χ . (19) By a character we mean a one-dimensional representation of the lifted Kähler shift lattice Z 2 \mathbb{Z}^2 Z 2 ζ 0 \zeta^0 ζ 0 ζ i = q m i ζ i 0 \zeta_i=q^{m_i}\zeta_i^0 ζ i = q m i ζ i 0
χ ( m 1 , m 2 ) = χ ( 0 , 0 ) ℏ m 1 / 2 ( D / ℏ 1 / 2 ) m 2 . \chi(m_1,m_2)=\chi(0,0)\hbar^{m_1/2}(D/\hbar^{1/2})^{m_2}. χ ( m 1 , m 2 ) = χ ( 0 , 0 ) ℏ m 1 /2 ( D / ℏ 1/2 ) m 2 . The constants in (19) are therefore its eigenmultipliers under the two Kähler q q q q q q χ \chi χ ζ 1 , ζ 2 \zeta_1,\zeta_2 ζ 1 , ζ 2 p p p V ( ζ ) = χ ( ζ ) V ( q s ) V(\zeta)=\chi(\zeta)\mathcal{V}(q s) V ( ζ ) = χ ( ζ ) V ( q s ) V \mathcal{V} V p 1 = ℏ 1 / 2 p z p_1=\hbar^{1/2}p_z p 1 = ℏ 1/2 p z p 2 = D ℏ − 1 / 2 p z − 1 p_2=D\hbar^{-1/2}p_z^{-1} p 2 = D ℏ − 1/2 p z − 1
χ − 1 ( H 1 − E ) χ = 1 − ℏ z / q 1 − z / q p z + D 1 − z / ( q ℏ ) 1 − z / q p z − 1 − E = p z − 1 ( 1 − z ) − 1 L . (20) \chi^{-1}(H_1-E)\chi=\frac{1-\hbar z/q}{1-z/q}p_z+D\frac{1-z/(q\hbar)}{1-z/q}p_z^{-1}-E
=\mathit{p}_z^{-1}(1-z)^{-1}\mathcal{L}. \tag*{(20)} χ − 1 ( H 1 − E ) χ = 1 − z / q 1 − ℏ z / q p z + D 1 − z / q 1 − z / ( q ℏ ) p z − 1 − E = p z − 1 ( 1 − z ) − 1 L . (20) The order of multiplication in the last expression is essential. We conclude that
L V = 0 ⟹ H 1 V = E V , H 2 V = D V . (21) \mathcal{L}V=0\quad\Longrightarrow\quad H_1V=EV,\qquad H_2V=DV. \tag*{(21)} L V = 0 ⟹ H 1 V = E V , H 2 V = D V . (21) For the vertex at a fixed point, the character χ e a i \chi_{e_{a_i}} χ e a i ℏ 1 / 2 a i \hbar^{1/2}a_i ℏ 1/2 a i a j / ℏ 1 / 2 a_j/\hbar^{1/2} a j / ℏ 1/2
Bethe Equations from the Eigenvalue Problem
The first nontrivial root of unity is q = − 1 q=-1 q = − 1 H 1 2 − 2 H 2 H_1^2-2H_2 H 1 2 − 2 H 2
Let q = − 1 q=-1 q = − 1 s = − z s=-z s = − z x ≠ 0 x\ne0 x = 0 z ↦ − z z\mapsto-z z ↦ − z R ( x ) R ( − x ) ≠ 0 R(x)R(-x)\ne0 R ( x ) R ( − x ) = 0 μ 0 , μ 1 \mu_0,\mu_1 μ 0 , μ 1 B 2 B_2 B 2
μ 0 = 1 , μ 1 = z ℏ J ( x ) R ( − x ) , \mu_0=1,\qquad\mu_1=\frac{z}{\hbar}\frac{J(x)}{R(-x)}, μ 0 = 1 , μ 1 = ℏ z R ( − x ) J ( x ) , B 2 = ϕ ( x ; z ) ϕ ( − x ; z ) = ( z ℏ ) 2 J ( x ) J ( − x ) R ( x ) R ( − x ) . (22) B_2=\phi(x;z)\phi(-x;z)=\left(\frac{z}{\hbar}\right)^2\frac{J(x)J(-x)}{R(x)R(-x)}. \tag*{(22)} B 2 = ϕ ( x ; z ) ϕ ( − x ; z ) = ( ℏ z ) 2 R ( x ) R ( − x ) J ( x ) J ( − x ) . (22) With e x e_x e x p z e x = x e x p_ze_x=xe_x p z e x = x e x
V x = e x ( 1 + μ 1 ) , V x = χ e x ( 1 + μ 1 ) . (23) \mathcal{V}_x=e_x(1+\mu_1),\qquad V_x=\chi e_x(1+\mu_1). \tag*{(23)} V x = e x ( 1 + μ 1 ) , V x = χ e x ( 1 + μ 1 ) . (23) Since μ 1 ( − z ) = − μ 1 ( z ) \mu_1(-z)=-\mu_1(z) μ 1 ( − z ) = − μ 1 ( z )
p 1 V x = χ e x ℏ 1 / 2 x ( 1 − μ 1 ) , p 2 V x = χ e x D ℏ 1 / 2 x ( 1 − μ 1 ) . (24) p_1V_x=\chi e_x\hbar^{1/2}x(1-\mu_1),\qquad p_2V_x=\chi e_x\frac{D}{\hbar^{1/2}x}(1-\mu_1). \tag*{(24)} p 1 V x = χ e x ℏ 1/2 x ( 1 − μ 1 ) , p 2 V x = χ e x ℏ 1/2 x D ( 1 − μ 1 ) . (24) Acting with the Hamiltonian, we obtain for H 1 V x − E V x H_1V_x-EV_x H 1 V x − E V x
( H 1 − E ) V x χ e x = [ ( 1 − ℏ s ) x + D ( 1 − s / ℏ ) / x ] ( 1 − μ 1 ) 1 − s − E ( 1 + μ 1 ) = R ( x ) + ( z / ℏ ) J ( x ) − μ 1 R ( − x ) − ( z / ℏ ) μ 1 J ( − x ) x ( 1 − s ) = ( 1 − B 2 ) R ( x ) x ( 1 − s ) . (25) \begin{aligned}
\frac{(H_1-E)V_x}{\chi e_x}
=\frac{\left[(1-\hbar s)x+D(1-s/\hbar)/x\right](1-\mu_1)}{1-s}-E(1+\mu_1) \\
&=\frac{R(x)+(z/\hbar)J(x)-\mu_1R(-x)-(z/\hbar)\mu_1J(-x)}{x(1-s)} \\
&=\frac{(1-\mathcal{B}_2)R(x)}{x(1-s)}. \tag*{(25)}
\end{aligned} χ e x ( H 1 − E ) V x = 1 − s [ ( 1 − ℏ s ) x + D ( 1 − s /ℏ ) / x ] ( 1 − μ 1 ) − E ( 1 + μ 1 ) = x ( 1 − s ) R ( x ) + ( z /ℏ ) J ( x ) − μ 1 R ( − x ) − ( z /ℏ ) μ 1 J ( − x ) = x ( 1 − s ) ( 1 − B 2 ) R ( x ) . (25) Notice that the middle terms cancel by μ 1 R ( − x ) = ( z / ℏ ) J ( x ) \mu_1R(-x)=(z/\hbar)J(x) μ 1 R ( − x ) = ( z /ℏ ) J ( x ) − B 2 R ( x ) -\mathcal{B}_2R(x) − B 2 R ( x ) H 2 V x = D V x H_2V_x=DV_x H 2 V x = D V x B 2 = 1 \mathcal{B}_2=1 B 2 = 1
L V x = e x ( 1 − B 2 ) R ( x ) . (26) \mathcal{L}V_x=e_x(1-\mathcal{B}_2)R(x). \tag*{(26)} L V x = e x ( 1 − B 2 ) R ( x ) . (26) In terms of U = x 2 U=x^2 U = x 2 A i = a i 2 A_i=a_i^2 A i = a i 2 Z = z 2 Z=z^2 Z = z 2 B 2 = 1 \mathcal{B}_2=1 B 2 = 1
( U − A 1 ) ( U − A 2 ) = Z ℏ 2 ( U − ℏ − 2 A 1 ) ( U − ℏ − 2 A 2 ) . (27) (U-A_1)(U-A_2)=Z\hbar^2(U-\hbar^{-2}A_1)(U-\hbar^{-2}A_2). \tag*{(27)} ( U − A 1 ) ( U − A 2 ) = Z ℏ 2 ( U − ℏ − 2 A 1 ) ( U − ℏ − 2 A 2 ) . (27) On each branch, x x x z ↦ − z z\mapsto-z z ↦ − z (25) gives H 1 V x = E V x H_1V_x=EV_x H 1 V x = E V x 1 + μ 1 1+\mu_1 1 + μ 1 1 − μ 1 1-\mu_1 1 − μ 1
Let us next consider the square of the Hamiltonian. The relations p 1 s = − s p 1 p_1s=-sp_1 p 1 s = − s p 1 p 2 s = − s p 2 p_2s=-sp_2 p 2 s = − s p 2
H 1 2 = A ℏ ( s ) A ℏ ( − s ) p 1 2 + B ℏ ( s ) B ℏ ( − s ) p 2 2 + [ A ℏ ( s ) B ℏ ( − s ) + B ℏ ( s ) A ℏ ( − s ) ] p 1 p 2 . (28) H_1^2=A_{\hbar}(s)A_{\hbar}(-s)p_1^2+B_{\hbar}(s)B_{\hbar}(-s)p_2^2
+\left[A_{\hbar}(s)B_{\hbar}(-s)+B_{\hbar}(s)A_{\hbar}(-s)\right]p_1p_2. \tag*{(28)} H 1 2 = A ℏ ( s ) A ℏ ( − s ) p 1 2 + B ℏ ( s ) B ℏ ( − s ) p 2 2 + [ A ℏ ( s ) B ℏ ( − s ) + B ℏ ( s ) A ℏ ( − s ) ] p 1 p 2 . (28) The coefficient multiplying the mixed shift reduces to
( 1 − ℏ s ) ( ℏ + s ) + ( ℏ − s ) ( 1 + ℏ s ) ℏ ( 1 − s 2 ) = 2. (29) \frac{(1-\hbar s)(\hbar+s)+(\hbar-s)(1+\hbar s)}{\hbar(1-s^2)}=2. \tag*{(29)} ℏ ( 1 − s 2 ) ( 1 − ℏ s ) ( ℏ + s ) + ( ℏ − s ) ( 1 + ℏ s ) = 2. (29) For the two pure shifts we have
A ℏ ( s ) A ℏ ( − s ) = A ℏ 2 ( s 2 ) , B ℏ ( s ) B ℏ ( − s ) = B ℏ 2 ( s 2 ) . A_{\hbar}(s)A_{\hbar}(-s)=A_{\hbar^2}(s^2),\qquad B_{\hbar}(s)B_{\hbar}(-s)=B_{\hbar^2}(s^2). A ℏ ( s ) A ℏ ( − s ) = A ℏ 2 ( s 2 ) , B ℏ ( s ) B ℏ ( − s ) = B ℏ 2 ( s 2 ) . The mixed term is thus removed by subtracting 2 H 2 2H_2 2 H 2
H 1 2 − 2 H 2 = A ℏ 2 ( s 2 ) p 1 2 + B ℏ 2 ( s 2 ) p 2 2 . (30) H_1^2-2H_2=A_{\hbar^2}(s^2)p_1^2+B_{\hbar^2}(s^2)p_2^2. \tag*{(30)} H 1 2 − 2 H 2 = A ℏ 2 ( s 2 ) p 1 2 + B ℏ 2 ( s 2 ) p 2 2 . (30) On our finite vertex, the eigenmultipliers of p 1 2 p_1^2 p 1 2 p 2 2 p_2^2 p 2 2 ℏ x 2 \hbar x^2 ℏ x 2 D 2 / ( ℏ x 2 ) D^2/(\hbar x^2) D 2 / ( ℏ x 2 )
( H 1 2 − 2 H 2 ) V x = ( 1 − ℏ 2 Z ) U + D 2 ( 1 − Z / ℏ 2 ) / U 1 − Z V x . (31) (H_1^2-2H_2)V_x=\frac{(1-\hbar^2Z)U+D^2(1-Z/\hbar^2)/U}{1-Z}V_x. \tag*{(31)} ( H 1 2 − 2 H 2 ) V x = 1 − Z ( 1 − ℏ 2 Z ) U + D 2 ( 1 − Z / ℏ 2 ) / U V x . (31) By the Bethe equation (27) , the coefficient on the right is a 1 2 + a 2 2 a_1^2+a_2^2 a 1 2 + a 2 2
[ A ℏ 2 ( s 2 ) p 1 2 + B ℏ 2 ( s 2 ) p 2 2 ] V x = ( a 1 2 + a 2 2 ) V x . (32) \left[A_{\hbar^2}(s^2)p_1^2+B_{\hbar^2}(s^2)p_2^2\right]V_x=(a_1^2+a_2^2)V_x. \tag*{(32)} [ A ℏ 2 ( s 2 ) p 1 2 + B ℏ 2 ( s 2 ) p 2 2 ] V x = ( a 1 2 + a 2 2 ) V x . (32) The closed finite vertex therefore solves both the original tRS equations and their precomposition with the Frobenius action. For the latter equations, the full-cycle multipliers and (27) already suffice; the ordinary equation H 1 V x = E V x H_1V_x=EV_x H 1 V x = E V x
Components of the Kähler Connection
The scalar vertex and its first Kähler shift determine a first-order vector equation. We shall choose an interpolation basis in which the sum of the two components recovers the vertex. This construction works for every q q q a 1 ≠ a 2 a_1\ne a_2 a 1 = a 2
w 1 ( y ) = y − a 2 a 1 − a 2 , w 2 ( y ) = a 1 − y a 1 − a 2 , w 1 + w 2 = 1. w_1(y)=\frac{y-a_2}{a_1-a_2},\qquad w_2(y)=\frac{a_1-y}{a_1-a_2},\qquad w_1+w_2=1. w 1 ( y ) = a 1 − a 2 y − a 2 , w 2 ( y ) = a 1 − a 2 a 1 − y , w 1 + w 2 = 1. For a solution V \mathcal{V} V
Φ 1 = p z V − a 2 V a 1 − a 2 , Φ 2 = a 1 V − p z V a 1 − a 2 . (33) \Phi_1=\frac{p_z\mathcal{V}-a_2\mathcal{V}}{a_1-a_2},\qquad\Phi_2=\frac{a_1\mathcal{V}-p_z\mathcal{V}}{a_1-a_2}. \tag*{(33)} Φ 1 = a 1 − a 2 p z V − a 2 V , Φ 2 = a 1 − a 2 a 1 V − p z V . (33) Writing Φ = ( Φ 1 , Φ 2 ) t \Phi=(\Phi_1,\Phi_2)^t Φ = ( Φ 1 , Φ 2 ) t
Φ = B ( V p z V ) , B = 1 a 1 − a 2 ( − a 2 1 a 1 − 1 ) , B − 1 = ( 1 1 a 1 a 2 ) . (34) \Phi=B\begin{pmatrix}\mathcal{V}\\p_z\mathcal{V}\end{pmatrix},\qquad B=\frac{1}{a_1-a_2}\begin{pmatrix}-a_2&1\\a_1&-1\end{pmatrix},\qquad B^{-1}=\begin{pmatrix}1&1\\a_1&a_2\end{pmatrix}. \tag*{(34)} Φ = B ( V p z V ) , B = a 1 − a 2 1 ( − a 2 a 1 1 − 1 ) , B − 1 = ( 1 a 1 1 a 2 ) . (34) The scalar equation gives a Kähler connection on the column ( V , p z V ) t (\mathcal{V},p_z\mathcal{V})^t ( V , p z V ) t
M ( z ) = ( 0 1 − D ( 1 − z / ℏ ) / ( 1 − ℏ z ) E ( 1 − z ) / ( 1 − ℏ z ) ) . (35) M(z)=\begin{pmatrix}0&1\\-D(1-z/\hbar)/(1-\hbar z)&E(1-z)/(1-\hbar z)\end{pmatrix}. \tag*{(35)} M ( z ) = ( 0 − D ( 1 − z /ℏ ) / ( 1 − ℏ z ) 1 E ( 1 − z ) / ( 1 − ℏ z ) ) . (35) In the interpolation basis this becomes
p z Φ = K ( z ) Φ , K ( z ) = B M ( z ) B − 1 . (36) p_z\Phi=K(z)\Phi,\qquad K(z)=BM(z)B^{-1}. \tag*{(36)} p z Φ = K ( z ) Φ , K ( z ) = B M ( z ) B − 1 . (36) An explicit expression follows by setting
r 1 = a 1 ( ℏ a 1 − a 2 ) , r 2 = a 2 ( ℏ a 2 − a 1 ) . r_1=a_1(\hbar a_1-a_2),\qquad r_2=a_2(\hbar a_2-a_1). r 1 = a 1 ( ℏ a 1 − a 2 ) , r 2 = a 2 ( ℏ a 2 − a 1 ) . We find
K ( z ) = ( a 1 0 0 a 2 ) + z ( ℏ − 1 ) ℏ ( 1 − ℏ z ) ( a 1 − a 2 ) ( r 1 r 2 − r 1 − r 2 ) . (37) K(z)=\begin{pmatrix}a_1&0\\0&a_2\end{pmatrix}+\frac{z(\hbar-1)}{\hbar(1-\hbar z)(a_1-a_2)}\begin{pmatrix}r_1&r_2\\-r_1&-r_2\end{pmatrix}. \tag*{(37)} K ( z ) = ( a 1 0 0 a 2 ) + ℏ ( 1 − ℏ z ) ( a 1 − a 2 ) z ( ℏ − 1 ) ( r 1 − r 1 r 2 − r 2 ) . (37) In this basis K ( 0 ) = diag ( a 1 , a 2 ) K(0)=\operatorname{diag}(a_1,a_2) K ( 0 ) = diag ( a 1 , a 2 ) det K ( z ) = D ( 1 − z / ℏ ) / ( 1 − ℏ z ) \det K(z)=D(1-z/\hbar)/(1-\hbar z) det K ( z ) = D ( 1 − z /ℏ ) / ( 1 − ℏ z ) (49) of order p p p
Φ β = e x ∑ j = 0 p − 1 μ j ( x ; z ) w β ( q j x ) , β = 1 , 2 , \Phi_\beta=e_x\sum_{j=0}^{p-1}\mu_j(x;z)w_\beta(q^jx),\qquad\beta=1,2, Φ β = e x j = 0 ∑ p − 1 μ j ( x ; z ) w β ( q j x ) , β = 1 , 2 , V x = χ e x ∑ j = 0 p − 1 μ j ( x ; z ) = χ ( Φ 1 + Φ 2 ) = I 1 + I 2 , I β = χ Φ β . (38) V_x=\chi e_x\sum_{j=0}^{p-1}\mu_j(x;z)=\chi(\Phi_1+\Phi_2)=I_1+I_2,\qquad I_\beta=\chi\Phi_\beta. \tag*{(38)} V x = χ e x j = 0 ∑ p − 1 μ j ( x ; z ) = χ ( Φ 1 + Φ 2 ) = I 1 + I 2 , I β = χ Φ β . (38) Indeed, the formulas for Φ β \Phi_\beta Φ β (33) and the shift relation p z ( e x μ j ) = e x q j x μ j p_z(e_{x\mu_j})=e_xq^jx_{\mu_j} p z ( e x μ j ) = e x q j x μ j
p z Φ − K ( z ) Φ = e x ( 1 − B p ) R ( x ) ( a 1 − a 2 ) ( 1 − ℏ z ) ( 1 − 1 ) . (39) p_z\Phi-K(z)\Phi=\frac{e_x(1-B_p)R(x)}{(a_1-a_2)(1-\hbar z)}\begin{pmatrix}1\\-1\end{pmatrix}. \tag*{(39)} p z Φ − K ( z ) Φ = ( a 1 − a 2 ) ( 1 − ℏ z ) e x ( 1 − B p ) R ( x ) ( 1 − 1 ) . (39) To see this, the residual of the companion equation is ( 0 , L V x / ( 1 − ℏ z ) ) t (0,\mathcal{L}\mathcal{V}_x/(1-\hbar z))^t ( 0 , L V x / ( 1 − ℏ z ) ) t B B B B p = 1 B_p=1 B p = 1 (36) .
After applying the homogeneous gauge, the column I = ( I 1 , I 2 ) t I=(I_1,I_2)^t I = ( I 1 , I 2 ) t
p 1 I = K 1 ( z ) I , p 2 I = K 2 ( z ) I , K 1 ( z ) = ℏ 1 / 2 K ( z ) , K 2 ( z ) = D ℏ 1 / 2 K ( z / q ) − 1 . (40) p_1I=K_1(z)I,\qquad p_2I=K_2(z)I,\qquad K_1(z)=\hbar^{1/2}K(z),\qquad K_2(z)=\frac{D}{\hbar^{1/2}}K(z/q)^{-1}. \tag*{(40)} p 1 I = K 1 ( z ) I , p 2 I = K 2 ( z ) I , K 1 ( z ) = ℏ 1/2 K ( z ) , K 2 ( z ) = ℏ 1/2 D K ( z / q ) − 1 . (40) We work where these matrices are defined and invertible. The argument z / q z/q z / q K 2 K_2 K 2 Φ ( z ) = K ( z / q ) Φ ( z / q ) \Phi(z)=K(z/q)\Phi(z/q) Φ ( z ) = K ( z / q ) Φ ( z / q )
K 2 ( q z ) K 1 ( z ) = K 1 ( z / q ) K 2 ( z ) = D i d . K_2(qz)K_1(z)=K_1(z/q)K_2(z)=D\,\mathrm{id}. K 2 ( q z ) K 1 ( z ) = K 1 ( z / q ) K 2 ( z ) = D id . Hence p 1 p 2 I = D I p_1p_2I=DI p 1 p 2 I = D I V = I 1 + I 2 V=I_1+I_2 V = I 1 + I 2 (20) .
The mechanism can also be expressed directly in terms of the rational weights. Let w = ( w 1 , w 2 ) t w=(w_1,w_2)^t w = ( w 1 , w 2 ) t ϵ = ( 1 , − 1 ) t \epsilon=(1,-1)^t ϵ = ( 1 , − 1 ) t
x w − K ( z ) w = ϕ ( x ; z ) U ( q x ; z ) − U ( x ; z ) , U ( x ; z ) = − R ( x ) ( a 1 − a 2 ) ( 1 − ℏ z ) ϵ . (41) xw-K(z)w=\phi(x;z)U(qx;z)-U(x;z),\qquad U(x;z)=-\frac{R(x)}{(a_1-a_2)(1-\hbar z)}\epsilon. \tag*{(41)} x w − K ( z ) w = ϕ ( x ; z ) U ( q x ; z ) − U ( x ; z ) , U ( x ; z ) = − ( a 1 − a 2 ) ( 1 − ℏ z ) R ( x ) ϵ . (41) The left-hand side equals [ R ( x ) − ( z / ℏ ) J ( x ) ] ϵ / [ ( a 1 − a 2 ) ( 1 − ℏ z ) ] [R(x)-(z/\hbar)J(x)]\epsilon/[(a_1-a_2)(1-\hbar z)] [ R ( x ) − ( z /ℏ ) J ( x )] ϵ / [( a 1 − a 2 ) ( 1 − ℏ z )] ϕ ( x ; z ) R ( q x ) = ( z / ℏ ) J ( x ) \phi(x;z)R(qx)=(z/\hbar)J(x) ϕ ( x ; z ) R ( q x ) = ( z /ℏ ) J ( x )
U 1 = ℏ 1 / 2 U , U 2 = R ( x ) ( a 1 − a 2 ) ℏ 1 / 2 x ( 1 − z / ( q ℏ ) ) ( a 2 − a 1 ) . (42) U_1=\hbar^{1/2}U,\qquad U_2=\frac{R(x)}{(a_1-a_2)\hbar^{1/2}x(1-z/(q\hbar))}\begin{pmatrix}a_2\\-a_1\end{pmatrix}. \tag*{(42)} U 1 = ℏ 1/2 U , U 2 = ( a 1 − a 2 ) ℏ 1/2 x ( 1 − z / ( q ℏ )) R ( x ) ( a 2 − a 1 ) . (42) With ρ 1 = ℏ 1 / 2 x \rho_1=\hbar^{1/2}x ρ 1 = ℏ 1/2 x ρ 2 = D / ( ℏ 1 / 2 x ) \rho_2=D/(\hbar^{1/2}x) ρ 2 = D / ( ℏ 1/2 x ) ρ i w − K i w = ϕ U i ( q x ) − U i ( x ) \rho_iw-K_iw=\phi U_i(qx)-U_i(x) ρ i w − K i w = ϕ U i ( q x ) − U i ( x )
( p i − K i ) I = χ e x ( B p − 1 ) U i ( x ; z ) . (p_i-K_i)I=\chi e_x(B_p-1)U_i(x;z). ( p i − K i ) I = χ e x ( B p − 1 ) U i ( x ; z ) . Thus both Kähler directions have the same Bethe boundary obstruction.
It remains to examine the row ℓ = ( 1 , 1 ) \ell=(1,1) ℓ = ( 1 , 1 ) s = z / q s=z/q s = z / q
ℓ K 1 = ℏ 1 / 2 ( a 1 , a 2 ) , ℓ K 2 = E ( 1 − s ) ℓ − ( 1 − ℏ s ) ( a 1 , a 2 ) ℏ 1 / 2 ( 1 − s / ℏ ) . \ell K_1=\hbar^{1/2}(a_1,a_2),\qquad\ell K_2=\frac{E(1-s)\ell-(1-\hbar s)(a_1,a_2)}{\hbar^{1/2}(1-s/\hbar)}. ℓ K 1 = ℏ 1/2 ( a 1 , a 2 ) , ℓ K 2 = ℏ 1/2 ( 1 − s /ℏ ) E ( 1 − s ) ℓ − ( 1 − ℏ s ) ( a 1 , a 2 ) . The terms involving ( a 1 , a 2 ) (a_1,a_2) ( a 1 , a 2 )
A ℏ ( s ) ℓ K 1 + B ℏ ( s ) ℓ K 2 = E ℓ . (43) A_\hbar(s)\ell K_1+B_\hbar(s)\ell K_2=E\ell. \tag*{(43)} A ℏ ( s ) ℓ K 1 + B ℏ ( s ) ℓ K 2 = E ℓ . (43) Together with the product relation for K 1 , K 2 K_1,K_2 K 1 , K 2
( H 1 − E ) V x = χ e x ( 1 − B p ) R ( x ) x ( 1 − s ) , H 2 V x = D V x . (44) (H_1-E)V_x=\frac{\chi e_x(1-B_p)R(x)}{x(1-s)},\qquad H_2V_x=DV_x. \tag*{(44)} ( H 1 − E ) V x = x ( 1 − s ) χ e x ( 1 − B p ) R ( x ) , H 2 V x = D V x . (44) The Bethe equation therefore closes the component system and the tRS eigenvalue problem simultaneously.
Cyclic Elimination at Arbitrary Order
The Bethe equation can also be recovered by eliminating the intermediate shifts of the vertex. On a finite Kähler orbit, this amounts to computing a determinant, whose vanishing gives the Bethe equation at ( z p , a 1 p , a 2 p , ℏ p ) (z^p,a_1^p,a_2^p,\hbar^p) ( z p , a 1 p , a 2 p , ℏ p )
Let q q q p p p p ≥ 2 p \ge2 p ≥ 2
Z = z p , A i = a i p . Z=z^p,\qquad A_i=a_i^p. Z = z p , A i = a i p . We use the p p p p z z = q z p z p_z z=qz p_z p z z = q z p z p z p = U p_z^p=U p z p = U z z z z , q z , … , q p − 1 z z,qz,\ldots,q^{p-1}z z , q z , … , q p − 1 z U U U L \mathcal{L} L
Proposition 2.2. In the cyclic representation above, the scalar vertex operator has determinant
det L = ( U − A 1 ) ( U − A 2 ) − Z ℏ p ( U − ℏ − p A 1 ) ( U − ℏ − p A 2 ) . (45) \det\mathcal{L}=(U-A_1)(U-A_2)-Z\hbar^p(U-\hbar^{-p}A_1)(U-\hbar^{-p}A_2). \tag*{(45)} det L = ( U − A 1 ) ( U − A 2 ) − Z ℏ p ( U − ℏ − p A 1 ) ( U − ℏ − p A 2 ) . (45) Proof. Conjugating by p z p_z p z z z z q z qz q z z z z p p p z p z^p z p z = 0 z=0 z = 0
det R ( p z ) = ( U − A 1 ) ( U − A 2 ) . \det R(p_z)=(U-A_1)(U-A_2). det R ( p z ) = ( U − A 1 ) ( U − A 2 ) . The coefficient of z p z^p z p − ℏ − 1 diag ( 1 , q , … , q p − 1 ) J ( p z ) -\hbar^{-1}\operatorname{diag}(1,q,\ldots,q^{p-1})J(p_z) − ℏ − 1 diag ( 1 , q , … , q p − 1 ) J ( p z )
( − 1 ) p q p ( p − 1 ) / 2 = − 1. (-1)^p q^{p(p-1)/2}=-1. ( − 1 ) p q p ( p − 1 ) /2 = − 1. We also have det J ( p z ) = ( ℏ p U − A 1 ) ( ℏ p U − A 2 ) \det J(p_z)=(\hbar^pU-A_1)(\hbar^pU-A_2) det J ( p z ) = ( ℏ p U − A 1 ) ( ℏ p U − A 2 ) − Z ℏ − p ( ℏ p U − A 1 ) ( ℏ p U − A 2 ) -Z\hbar^{-p}(\hbar^pU-A_1)(\hbar^pU-A_2) − Z ℏ − p ( ℏ p U − A 1 ) ( ℏ p U − A 2 )
The condition det L = 0 \det\mathcal{L}=0 det L = 0
The Finite-Sum Cancellation for Arbitrary p p p
The finite period realizes the determinant condition through cancellation of adjacent weighted terms. Its boundary is measured by 1 − B p 1-B_p 1 − B p
Keep x ≠ 0 x\ne0 x = 0 x p ≠ a i p x^p\ne a_i^p x p = a i p j ≥ 0 j\ge0 j ≥ 0
μ j ( x ; z ) = ∏ r = 0 j − 1 ϕ ( q r x ; z ) = ( z ℏ ) j ∏ i = 1 2 ( ℏ x / a i ; q ) j ( q x / a i ; q ) j , μ 0 = 1. (46) \mu_j(x;z)=\prod_{r=0}^{j-1}\phi(q^r x;z)=\left(\frac{z}{\hbar}\right)^j\prod_{i=1}^{2}\frac{(\hbar x/a_i;q)_j}{(qx/a_i;q)_j},\qquad\mu_0=1. \tag*{(46)} μ j ( x ; z ) = r = 0 ∏ j − 1 ϕ ( q r x ; z ) = ( ℏ z ) j i = 1 ∏ 2 ( q x / a i ; q ) j ( ℏ x / a i ; q ) j , μ 0 = 1. (46) Here μ j \mu_j μ j j j j x , q x , … , q p − 1 x x,qx,\ldots,q^{p-1}x x , q x , … , q p − 1 x
B p ( x ; z ) : = μ p ( x ; z ) = ∏ r = 0 p − 1 ϕ ( q r x ; z ) . (47) B_p(x;z):=\mu_p(x;z)=\prod_{r=0}^{p-1}\phi(q^r x;z). \tag*{(47)} B p ( x ; z ) := μ p ( x ; z ) = r = 0 ∏ p − 1 ϕ ( q r x ; z ) . (47) A descendant f f f
⟨ f ⟩ x , z : = e x ∑ j = 0 p − 1 μ j ( x ; z ) f ( q j x ) , p z e x = x e x . (48) \langle f\rangle_{x,z}:=e_x\sum_{j=0}^{p-1}\mu_j(x;z)f(q^j x),\qquad p_z e_x=xe_x. \tag*{(48)} ⟨ f ⟩ x , z := e x j = 0 ∑ p − 1 μ j ( x ; z ) f ( q j x ) , p z e x = x e x . (48) where f f f 1 1 1
S x ( z ) = ∑ j = 0 p − 1 μ j ( x ; z ) , V x = e x S x = ⟨ 1 ⟩ x , z . (49) S_x(z)=\sum_{j=0}^{p-1}\mu_j(x;z),\qquad\mathcal{V}_x=e_xS_x=\langle1\rangle_{x,z}. \tag*{(49)} S x ( z ) = j = 0 ∑ p − 1 μ j ( x ; z ) , V x = e x S x = ⟨ 1 ⟩ x , z . (49) The unit period therefore has the same phase weights as every descendant, with no further factor in the internal variable. Its cycle closes when B p = 1 B_p=1 B p = 1 Φ β = ⟨ w β ⟩ x , z \Phi_\beta=\langle w_\beta\rangle_{x,z} Φ β = ⟨ w β ⟩ x , z V x = χ V x V_x=\chi\mathcal{V}_x V x = χ V x μ j ( x ; q z ) = q j μ j ( x ; z ) \mu_j(x;qz)=q^j\mu_j(x;z) μ j ( x ; q z ) = q j μ j ( x ; z )
p z m V x = e x ∑ j = 0 p − 1 μ j ( x ; z ) ( q j x ) m , m ≥ 0. (50) p_z^m\mathcal{V}_x=e_x\sum_{j=0}^{p-1}\mu_j(x;z)(q^jx)^m,\qquad m\geq0. \tag*{(50)} p z m V x = e x j = 0 ∑ p − 1 μ j ( x ; z ) ( q j x ) m , m ≥ 0. (50) These moments allow us to apply the scalar vertex operator to (49) . The result is the arbitrary-order version of (26) , relating the boundary of the finite period to the determinant condition (45) .
Proposition 2.3. The finite cyclic period satisfies
L V x = e x ( 1 − B p ) ( x − a 1 ) ( x − a 2 ) . (51) \mathcal{L}V_x=e_x(1-B_p)(x-a_1)(x-a_2). \tag*{(51)} L V x = e x ( 1 − B p ) ( x − a 1 ) ( x − a 2 ) . (51) On the stated regular locus, it is therefore a solution of the vertex equation if and only if B p = 1 B_p=1 B p = 1
Proof. The consecutive weights μ j \mu_j μ j
μ j + 1 R ( q j + 1 x ) = z ℏ μ j J ( q j x ) . \mu_{j+1}R(q^{j+1}x)=\frac{z}{\hbar}\mu_jJ(q^jx). μ j + 1 R ( q j + 1 x ) = ℏ z μ j J ( q j x ) . Substituting (50) into the operator, we see that consecutive terms cancel and only the endpoints remain:
e x − 1 L V x = ∑ j = 0 p − 1 μ j R ( q j x ) − ∑ j = 0 p − 1 μ j + 1 R ( q j + 1 x ) = R ( x ) − B p R ( q p x ) = ( 1 − B p ) R ( x ) . \begin{aligned}
e_x^{-1}\mathcal{L}V_x=\sum_{j=0}^{p-1}\mu_jR(q^jx)-\sum_{j=0}^{p-1}\mu_{j+1}R(q^{j+1}x) \\
&=R(x)-B_pR(q^px)=(1-B_p)R(x).
\end{aligned} e x − 1 L V x = j = 0 ∑ p − 1 μ j R ( q j x ) − j = 0 ∑ p − 1 μ j + 1 R ( q j + 1 x ) = R ( x ) − B p R ( q p x ) = ( 1 − B p ) R ( x ) . Finally, x p ≠ a i p x^p\ne a_i^p x p = a i p R ( x ) ≠ 0 R(x)\ne0 R ( x ) = 0
We can perform the calculation directly with the tRS Hamiltonian. On each summand the two shifts act by p 1 ( χ e x μ j ) = ℏ 1 / 2 q j x χ e x μ j p_1(\chi e_x\mu_j)=\hbar^{1/2}q^jx\chi e_x\mu_j p 1 ( χ e x μ j ) = ℏ 1/2 q j x χ e x μ j p 2 ( χ e x μ j ) = D ( ℏ 1 / 2 q j x ) − 1 χ e x μ j p_2(\chi e_x\mu_j)=D(\hbar^{1/2}q^jx)^{-1}\chi e_x\mu_j p 2 ( χ e x μ j ) = D ( ℏ 1/2 q j x ) − 1 χ e x μ j
( H 1 − E ) V x χ e x = 1 1 − s ∑ j = 0 p − 1 μ j [ ( 1 − ℏ s ) q j x + D ( 1 − s / ℏ ) q j x − E ( 1 − s ) ] = 1 x ( 1 − s ) ∑ j = 0 p − 1 [ q − j μ j R ( q j x ) − q − ( j + 1 ) μ j + 1 R ( q j + 1 x ) ] = ( 1 − B p ) R ( x ) x ( 1 − s ) . (52) \begin{aligned}
\frac{(H_1-E)V_x}{\chi e_x}
=\frac{1}{1-s}\sum_{j=0}^{p-1}\mu_j\left[(1-\hbar s)q^jx+\frac{D(1-s/\hbar)}{q^jx}-E(1-s)\right] \\
&=\frac{1}{x(1-s)}\sum_{j=0}^{p-1}\left[q^{-j}\mu_jR(q^jx)-q^{-(j+1)}\mu_{j+1}R(q^{j+1}x)\right] \\
&=\frac{(1-B_p)R(x)}{x(1-s)}. \tag*{(52)}
\end{aligned} χ e x ( H 1 − E ) V x = 1 − s 1 j = 0 ∑ p − 1 μ j [ ( 1 − ℏ s ) q j x + q j x D ( 1 − s /ℏ ) − E ( 1 − s ) ] = x ( 1 − s ) 1 j = 0 ∑ p − 1 [ q − j μ j R ( q j x ) − q − ( j + 1 ) μ j + 1 R ( q j + 1 x ) ] = x ( 1 − s ) ( 1 − B p ) R ( x ) . (52) The second equality follows by expanding R , J R,J R , J μ j + 1 R ( q j + 1 x ) = ( z / ℏ ) μ j J ( q j x ) \mu_{j+1}R(q^{j+1}x)=(z/\hbar)\mu_jJ(q^jx) μ j + 1 R ( q j + 1 x ) = ( z /ℏ ) μ j J ( q j x ) s = z / q s=z/q s = z / q q − p = 1 q^{-p}=1 q − p = 1 B p R ( x ) B_pR(x) B p R ( x )
The other Hamiltonian satisfies H 2 V x = D V x H_2V_x = DV_x H 2 V x = D V x B p = 1 B_p = 1 B p = 1
The cyclotomic product of the factors in (46) gives
B p = Z ℏ p ∏ i = 1 2 1 − ℏ p x p / A i 1 − x p / A i . (53) B_p = \frac{Z}{\hbar^p}\prod_{i=1}^{2}\frac{1-\hbar^p x^p/A_i}{1-x^p/A_i}. \tag*{(53)} B p = ℏ p Z i = 1 ∏ 2 1 − x p / A i 1 − ℏ p x p / A i . (53) Here A i = a i p A_i = a_i^p A i = a i p i = 1 , 2 i=1,2 i = 1 , 2 Z = z p Z = z^p Z = z p A ℏ ( s ) A_\hbar(s) A ℏ ( s ) A j A_j A j U = x p U = x^p U = x p B p = 1 B_p = 1 B p = 1 (45) . Thus elimination of the intermediate Kähler shifts and closure of the internal cycle give the same equation.
It remains to check that the cyclic section is nonzero. On a free orbit with z ≠ 0 z \ne0 z = 0
S x ( q m z ) = ∑ j = 0 p − 1 q m j μ j ( x ; z ) , 0 ≤ m < p . S_x(q^m z) = \sum_{j=0}^{p-1}q^{mj}\mu_j(x;z), \qquad0 \le m < p. S x ( q m z ) = j = 0 ∑ p − 1 q mj μ j ( x ; z ) , 0 ≤ m < p . The inverse discrete Fourier transform recovers μ 0 = 1 \mu_0 = 1 μ 0 = 1 p p p (39) –(40) wherever the matrices are defined and invertible.
The Operator Cancellation for Arbitrary p p p
The cancellation in the square of the two-particle Hamiltonian extends to a universal Newton identity at every root of unity. We shall derive it from the orbit graph, independently of the vertex or the Bethe equations. For our finite vertex the resulting Frobenius equation follows both from the ordinary eigenvalue problem and from the full-cycle multipliers together with the Bethe equation at powered parameters.
Recall that on a regular branch of B p = 1 B_p = 1 B p = 1
V x = χ V x = χ e x ∑ j = 0 p − 1 μ j ( x ; z ) = χ ( Φ 1 + Φ 2 ) = I 1 + I 2 . (54) V_x = \chi\mathcal{V}_x = \chi e_x \sum_{j=0}^{p-1}\mu_j(x;z) = \chi(\Phi_1+\Phi_2)=I_1+I_2. \tag*{(54)} V x = χ V x = χ e x j = 0 ∑ p − 1 μ j ( x ; z ) = χ ( Φ 1 + Φ 2 ) = I 1 + I 2 . (54) Its components obey (40) . By (20) , their sum satisfies H 1 V x = ( a 1 + a 2 ) V x H_1V_x = (a_1+a_2)V_x H 1 V x = ( a 1 + a 2 ) V x H 2 V x = a 1 a 2 V x H_2V_x = a_1a_2V_x H 2 V x = a 1 a 2 V x
The full-cycle Kähler shifts act as
p 1 p V x = ℏ p / 2 U V x , p 2 p V x = D p ℏ − p / 2 U − 1 V x . (55) p_1^pV_x=\hbar^{p/2}UV_x,\qquad p_2^pV_x=D^p\hbar^{-p/2}U^{-1}V_x. \tag*{(55)} p 1 p V x = ℏ p /2 U V x , p 2 p V x = D p ℏ − p /2 U − 1 V x . (55) These equations hold for lifted sections, or equivalently on Bloch fibres. Generic multipliers do not give single-valued functions on the finite coordinate orbit.
Introduce the Newton polynomials in two variables by
P 0 ( X , Y ) = 2 , P 1 ( X , Y ) = X , P m ( X , Y ) = X P m − 1 ( X , Y ) − Y P m − 2 ( X , Y ) . (56) P_0(X,Y)=2,\qquad P_1(X,Y)=X,\qquad P_m(X,Y)=XP_{m-1}(X,Y)-YP_{m-2}(X,Y). \tag*{(56)} P 0 ( X , Y ) = 2 , P 1 ( X , Y ) = X , P m ( X , Y ) = X P m − 1 ( X , Y ) − Y P m − 2 ( X , Y ) . (56) Their defining property is P m ( b 1 + b 2 , b 1 b 2 ) = b 1 m + b 2 m P_m(b_1+b_2,b_1b_2)=b_1^m+b_2^m P m ( b 1 + b 2 , b 1 b 2 ) = b 1 m + b 2 m
P p ( H 1 , H 2 ) = A ℏ p ( s p ) p 1 p + B ℏ p ( s p ) p 2 p . (57) P_p(H_1,H_2)=A_{\hbar p}(s^p)p_1^p+B_{\hbar p}(s^p)p_2^p. \tag*{(57)} P p ( H 1 , H 2 ) = A ℏ p ( s p ) p 1 p + B ℏ p ( s p ) p 2 p . (57) We prove the universal operator identity (57) by a calculation on the q q q p = 1 p=1 p = 1 p ≥ 3 p\ge3 p ≥ 3 j ∈ Z / p Z j\in\mathbb{Z}/p\mathbb{Z} j ∈ Z / p Z q j s q^j s q j s A j = A ℏ ( q j s ) A_j=A_{\hbar}(q^j s) A j = A ℏ ( q j s ) B j = B ℏ ( q j s ) B_j=B_{\hbar}(q^j s) B j = B ℏ ( q j s )
j → α A j j + 1 , j → β B j j − 1 , j \xrightarrow{\alpha A_j} j+1,\qquad j \xrightarrow{\beta B_j} j-1, j α A j j + 1 , j β B j j − 1 , where α , β \alpha,\beta α , β f ( s ) f(s) f ( s ) diag ( f ( q j s ) ) \operatorname{diag}(f(q^j s)) diag ( f ( q j s ))
p 1 = α T , p 2 = β T − 1 , ( T v ) j = v j + 1 , H ( s ) = α diag ( A j ) T + β diag ( B j ) T − 1 . (58) \begin{aligned}
p_1&=\alpha T,\qquad p_2=\beta T^{-1},\qquad(Tv)_j=v_{j+1},\\
H(s)&=\alpha\operatorname{diag}(A_j)T+\beta\operatorname{diag}(B_j)T^{-1}. \tag*{(58)}
\end{aligned} p 1 H ( s ) = α T , p 2 = β T − 1 , ( T v ) j = v j + 1 , = α diag ( A j ) T + β diag ( B j ) T − 1 . (58) One checks the shift relations p 1 s = q s p 1 p_1s= qsp_1 p 1 s = q s p 1 p 2 s = q − 1 s p 2 p_2s=q^{-1}sp_2 p 2 s = q − 1 s p 2 H 2 = α β H^2=\alpha\beta H 2 = α β
Matrix powers of H H H det ( X − H ) \det(X-H) det ( X − H )
Disjoint two-cycles form a matching. We index the undirected edge { j , j + 1 } \{j,j+1\} { j , j + 1 } j ∈ Z / p Z j\in\mathbb{Z}/p\mathbb{Z} j ∈ Z / p Z { p − 1 , 0 } \{p-1,0\} { p − 1 , 0 } j → j + 1 j\to j+1 j → j + 1 j + 1 → j j+1\to j j + 1 → j α β \alpha\beta α β
w j = A j B j + 1 = ( 1 − ℏ q j s ) ( ℏ − q j + 1 s ) ℏ ( 1 − q j s ) ( 1 − q j + 1 s ) . (59) w_j=A_jB_{j+1}=\frac{(1-\hbar q^j s)(\hbar-q^{j+1}s)}{\hbar(1-q^j s)(1-q^{j+1}s)}. \tag*{(59)} w j = A j B j + 1 = ℏ ( 1 − q j s ) ( 1 − q j + 1 s ) ( 1 − ℏ q j s ) ( ℏ − q j + 1 s ) . (59) For 0 ≤ k ≤ ⌊ p / 2 ⌋ 0\le k\le\lfloor p/2\rfloor 0 ≤ k ≤ ⌊ p /2 ⌋
M p , k = { M ⊂ Z / p Z : ∣ M ∣ = k , M ∩ ( M + 1 ) = ∅ } . (60) \mathcal{M}_{p,k}=\{M\subset\mathbb{Z}/p\mathbb{Z}:|M|=k,\ M\cap(M+1)=\varnothing\}. \tag*{(60)} M p , k = { M ⊂ Z / p Z : ∣ M ∣ = k , M ∩ ( M + 1 ) = ∅ } . (60) Here M + 1 = { j + 1 : j ∈ M } M+1=\{j+1:j\in M\} M + 1 = { j + 1 : j ∈ M } p p p M M M k k k { j , j + 1 } \{j,j+1\} { j , j + 1 } j ∈ M j\in M j ∈ M M p , k \mathcal{M}_{p,k} M p , k k k k
The permutation with these k k k p − 2 k p-2k p − 2 k ( − α β ) k X p − 2 k ∏ j ∈ M w j (-\alpha\beta)^kX^{p-2k}\prod_{j\in M}w_j ( − α β ) k X p − 2 k ∏ j ∈ M w j ∏ j ∈ M w j \prod_{j\in M}w_j ∏ j ∈ M w j M ∈ M p , k M\in\mathcal{M}_{p,k} M ∈ M p , k (64) .
Lemma 2.4. Let p ≥ 3 p\ge3 p ≥ 3 q q q primitive p p p (59) and matchings (60) , we have
S p , k ( s ) : = ∑ M ∈ M p , k ∏ j ∈ M w j = p p − k ( p − k k ) , 0 ≤ k ≤ ⌊ p / 2 ⌋ . (61) S_{p,k}(s):=\sum_{M\in\mathcal{M}_{p,k}}\prod_{j\in M}w_j
=\frac{p}{p-k}\binom{p-k}{k},\qquad0\le k\le\lfloor p/2\rfloor. \tag*{(61)} S p , k ( s ) := M ∈ M p , k ∑ j ∈ M ∏ w j = p − k p ( k p − k ) , 0 ≤ k ≤ ⌊ p /2 ⌋ . (61) For k = 0 k=0 k = 0 M = ∅ M=\varnothing M = ∅ 1 1 1
Proof. Since a matching uses a vertex at most once, the denominator of its weight contains each factor 1 − q j s 1-q^j s 1 − q j s
N p , k ( s ) : = ( 1 − s p ) S p , k ( s ) = ∑ M ∈ M p , k ∏ j ∈ M ( 1 − ℏ q j s ) ( ℏ − q j + 1 s ) ℏ ∏ v ∉ M ∪ ( M + 1 ) ( 1 − q v s ) . (62) \begin{aligned}
N_{p,k}(s)&:=(1-s^p)S_{p,k}(s)\\
&=\sum_{M\in\mathcal{M}_{p,k}}\prod_{j\in M}\frac{(1-\hbar q^j s)(\hbar-q^{j+1}s)}{\hbar}\prod_{v\notin M\cup(M+1)}(1-q^v s). \tag*{(62)}
\end{aligned} N p , k ( s ) := ( 1 − s p ) S p , k ( s ) = M ∈ M p , k ∑ j ∈ M ∏ ℏ ( 1 − ℏ q j s ) ( ℏ − q j + 1 s ) v ∈ / M ∪ ( M + 1 ) ∏ ( 1 − q v s ) . (62) is a polynomial of degree at most p p p N p , k ( q s ) = N p , k ( s ) N_{p,k}(qs)=N_{p,k}(s) N p , k ( q s ) = N p , k ( s ) q q q p p p
The constant coefficient of each summand in (62) is one, and its degree-p p p
( − 1 ) p − 2 k q 0 + 1 + ⋯ + ( p − 1 ) = − 1 , (-1)^{p-2k}q^{0+1+\cdots+(p-1)}=-1, ( − 1 ) p − 2 k q 0 + 1 + ⋯ + ( p − 1 ) = − 1 , by q p ( p − 1 ) / 2 = ( − 1 ) p − 1 q^{p(p-1)/2}=(-1)^{p-1} q p ( p − 1 ) /2 = ( − 1 ) p − 1 N p , k ( s ) N_{p,k}(s) N p , k ( s ) 1 − s p 1-s^p 1 − s p
( p − k k ) + ( p − k − 1 k − 1 ) = p p − k ( p − k k ) . \binom{p-k}{k}+\binom{p-k-1}{k-1}=\frac{p}{p-k}\binom{p-k}{k}. ( k p − k ) + ( k − 1 p − k − 1 ) = p − k p ( k p − k ) . The first term counts matchings without the edge { p − 1 , 0 } \{p-1,0\} { p − 1 , 0 } k = 0 k=0 k = 0 (61) . When p p p k = p / 2 k=p/2 k = p /2
We can now evaluate the determinant of (58) using Lemma 2.4. A fixed vertex contributes X X X − α β w j -\alpha\beta w_j − α β w j ( − 1 ) p − 1 (-1)^{p-1} ( − 1 ) p − 1 p p p ( − 1 ) p (-1)^p ( − 1 ) p
det ( X − H ( s ) ) = ∑ k = 0 ⌊ p / 2 ⌋ ( − 1 ) k S p , k ( s ) ( α β ) k X p − 2 k − α p ∏ j = 0 p − 1 A j − β p ∏ j = 0 p − 1 B j . (63) \det(X-H(s))=\sum_{k=0}^{\lfloor p/2\rfloor}(-1)^kS_{p,k}(s)(\alpha\beta)^kX^{p-2k}
-\alpha^p\prod_{j=0}^{p-1}A_j-\beta^p\prod_{j=0}^{p-1}B_j. \tag*{(63)} det ( X − H ( s )) = k = 0 ∑ ⌊ p /2 ⌋ ( − 1 ) k S p , k ( s ) ( α β ) k X p − 2 k − α p j = 0 ∏ p − 1 A j − β p j = 0 ∏ p − 1 B j . (63) For even p p p (61) are the coefficients of the Newton polynomial
P p ( X , Y ) = ∑ k = 0 ⌊ p / 2 ⌋ ( − 1 ) k p p − k ( p − k k ) Y k X p − 2 k . (64) P_p(X,Y)=\sum_{k=0}^{\lfloor p/2\rfloor}(-1)^k\frac{p}{p-k}\binom{p-k}{k}Y^kX^{p-2k}. \tag*{(64)} P p ( X , Y ) = k = 0 ∑ ⌊ p /2 ⌋ ( − 1 ) k p − k p ( k p − k ) Y k X p − 2 k . (64) The same expression follows from the recurrence (56) . We therefore identify the matching part of the determinant with the Newton corrections. Cayley–Hamilton will translate this determinant calculation into cancellation of mixed walks in the operator polynomial. No decomposition of an individual walk into disjoint dimers is involved.
The weights of the two winding cycles are the cyclotomic products
∏ j = 0 p − 1 A j = 1 − ℏ p s p ℏ p / 2 ( 1 − s p ) = A ℏ p ( s p ) , \prod_{j=0}^{p-1}A_j=\frac{1-\hbar^p s^p}{\hbar^{p/2}(1-s^p)}=A_{\hbar p}(s^p), j = 0 ∏ p − 1 A j = ℏ p /2 ( 1 − s p ) 1 − ℏ p s p = A ℏ p ( s p ) , ∏ j = 0 p − 1 B j = ℏ p − s p ℏ p / 2 ( 1 − s p ) = B ℏ p ( s p ) . (65) \prod_{j=0}^{p-1}B_j=\frac{\hbar^p-s^p}{\hbar^{p/2}(1-s^p)}=B_{\hbar p}(s^p). \tag*{(65)} j = 0 ∏ p − 1 B j = ℏ p /2 ( 1 − s p ) ℏ p − s p = B ℏ p ( s p ) . (65) Combining (61) –(65) gives the characteristic polynomial
det ( X − H ( s ) ) = P p ( X , α β ) − A ℏ p ( s p ) α p − B ℏ p ( s p ) β p . (66) \det(X-H(s))=P_p(X,\alpha\beta)-A_{\hbar p}(s^p)\alpha^p-B_{\hbar p}(s^p)\beta^p. \tag*{(66)} det ( X − H ( s )) = P p ( X , α β ) − A ℏ p ( s p ) α p − B ℏ p ( s p ) β p . (66) For p = 2 p=2 p = 2
H ( s ) = ( 0 α A 0 + β B 0 α A 1 + β B 1 0 ) . H(s)=\begin{pmatrix}0 & \alpha A_0+\beta B_0 \\ \alpha A_1+\beta B_1 & 0\end{pmatrix}. H ( s ) = ( 0 α A 1 + β B 1 α A 0 + β B 0 0 ) . The characteristic polynomial in this case reads
X 2 − α 2 A 0 A 1 − β 2 B 0 B 1 − α β ( A 0 B 1 + B 0 A 1 ) . X^2-\alpha^2 A_0A_1-\beta^2 B_0B_1-\alpha\beta(A_0B_1+B_0A_1). X 2 − α 2 A 0 A 1 − β 2 B 0 B 1 − α β ( A 0 B 1 + B 0 A 1 ) . Here (29) makes the mixed coefficient 2 2 2 (65) evaluates the pure products. This proves (2.55) for p = 2 p=2 p = 2 P 2 ( X , Y ) = X 2 − 2 Y P_2(X,Y)=X^2-2Y P 2 ( X , Y ) = X 2 − 2 Y
It remains to recover the operator relation from this polynomial. For fixed primitive root q q q H ( s ) H(s) H ( s ) p × p p\times p p × p C ( s , α , β , ℏ 1 / 2 ) \mathbb{C}(s,\alpha,\beta,\hbar^{1/2}) C ( s , α , β , ℏ 1/2 )
χ H ( X ) = det ( X id − H ( s ) ) = X p + c 1 X p − 1 + ⋯ + c p . \chi_H(X)=\det(X\operatorname{id}-H(s))=X^p+c_1X^{p-1}+\cdots+c_p. χ H ( X ) = det ( X id − H ( s )) = X p + c 1 X p − 1 + ⋯ + c p . The coefficients c j c_j c j
χ H ( H ( s ) ) = H ( s ) p + c 1 H ( s ) p − 1 + ⋯ + c p id = 0. \chi_H(H(s))=H(s)^p+c_1H(s)^{p-1}+\cdots+c_p\operatorname{id}=0. χ H ( H ( s )) = H ( s ) p + c 1 H ( s ) p − 1 + ⋯ + c p id = 0. In this substitution, each power of X X X id \operatorname{id} id X X X
( X id − H ( s ) ) adj ( X id − H ( s ) ) = χ H ( X ) id . (X\operatorname{id}-H(s))\operatorname{adj}(X\operatorname{id}-H(s))=\chi_H(X)\operatorname{id}. ( X id − H ( s )) adj ( X id − H ( s )) = χ H ( X ) id . Beginning with the highest power, the coefficient equations express the adjugate coefficients successively as polynomials in H ( s ) H(s) H ( s ) (66) , it becomes
P p ( H ( s ) , α β id ) = [ A ℏ p ( s p ) α p + B ℏ p ( s p ) β p ] id . (67) P_p(H(s),\alpha\beta\operatorname{id})=\left[A_{\hbar^p}(s^p)\alpha^p+B_{\hbar^p}(s^p)\beta^p\right]\operatorname{id}. \tag*{(67)} P p ( H ( s ) , α β id ) = [ A ℏ p ( s p ) α p + B ℏ p ( s p ) β p ] id . (67) The matrix representing H 2 H_2 H 2 α β id \alpha\beta\operatorname{id} α β id (57) in the representation (58) . The independence of α , β \alpha,\beta α , β p 1 p = α p p_1^p=\alpha^p p 1 p = α p p 2 p = β p p_2^p=\beta^p p 2 p = β p f ( s ) p 1 a p 2 b f(s)p_1^ap_2^b f ( s ) p 1 a p 2 b
α a β b diag ( f ( q j s ) ) T a − b . \alpha^a\beta^b\operatorname{diag}(f(q^js))T^{a-b}. α a β b diag ( f ( q j s )) T a − b . Different pairs ( a , b ) (a,b) ( a , b ) α , β \alpha,\beta α , β s s s
We can now apply (57) to V x V_x V x
[ A ℏ p ( s p ) p 1 p + B ℏ p ( s p ) p 2 p ] V x = ( a 1 p + a 2 p ) V x . (68) \left[A_{\hbar^p}(s^p)p_1^p+B_{\hbar^p}(s^p)p_2^p\right]V_x=(a_1^p+a_2^p)V_x. \tag*{(68)} [ A ℏ p ( s p ) p 1 p + B ℏ p ( s p ) p 2 p ] V x = ( a 1 p + a 2 p ) V x . (68) The full-cycle multipliers evaluate the two shifts explicitly:
[ A ℏ p ( Z ) p 1 p + B ℏ p ( Z ) p 2 p ] V x = ( 1 − ℏ p Z ) U + D p ( 1 − Z / ℏ p ) / U 1 − Z V x . (69) \left[A_{\hbar^p}(Z)p_1^p+B_{\hbar^p}(Z)p_2^p\right]V_x=
\frac{(1-\hbar^p Z)U+D^p(1-Z/\hbar^p)/U}{1-Z}V_x. \tag*{(69)} [ A ℏ p ( Z ) p 1 p + B ℏ p ( Z ) p 2 p ] V x = 1 − Z ( 1 − ℏ p Z ) U + D p ( 1 − Z / ℏ p ) / U V x . (69) Here s p = z p = Z s^p=z^p=Z s p = z p = Z Z = 1 Z=1 Z = 1 a 1 p + a 2 p a_1^p+a_2^p a 1 p + a 2 p ( a 1 p + a 2 p ) V x (a_1^p+a_2^p)V_x ( a 1 p + a 2 p ) V x (69) . The difference is
( U − A 1 ) ( U − A 2 ) − Z ℏ p ( U − ℏ − p A 1 ) ( U − ℏ − p A 2 ) U ( 1 − Z ) V x (70) \frac{(U-A_1)(U-A_2)-Z\hbar^p(U-\hbar^{-p}A_1)(U-\hbar^{-p}A_2)}{U(1-Z)}V_x \tag*{(70)} U ( 1 − Z ) ( U − A 1 ) ( U − A 2 ) − Z ℏ p ( U − ℏ − p A 1 ) ( U − ℏ − p A 2 ) V x (70) and vanishes by (45) on B p = 1 B_p=1 B p = 1 (55) gives p 1 p p 2 p V x = D p V x p_1^p p_2^p V_x=D^pV_x p 1 p p 2 p V x = D p V x H r V x = e r ( a 1 , a 2 ) V x H_rV_x=e_r(a_1,a_2)V_x H r V x = e r ( a 1 , a 2 ) V x K i K_i K i
The ordinary eigenvalue equations give the same Frobenius equations, as in the general statement of Corollary 3.5 . The calculation from cycle multipliers and powered Bethe roots extends to complete flags in Theorem 3.13 . In the two-particle case we have, in addition, established nonvanishing of the finite vertex and constructed its first-order component system (40) .
Complete Flags: Cycles and Newton–Adams Cancellation
Let us now consider the n n n § 2 . The shifts p i p_i p i ζ i ↦ q ζ i \zeta_i\mapsto q\zeta_i ζ i ↦ q ζ i q q q p p p p p p
The proof uses centrality in the Hecke algebra and the bounds on shift exponents. For each surviving full-cycle shift there is only one contributing subset sequence, which determines its coefficient. This gives the operator identity in every rank and at every root order, and hence its eigenfunction consequence in Corollary 3.5 . At order two we shall also give a direct proof by pairing the collision residues of a signed bipartition sum.
For the finite vertex sums we also need the internal grid of auxiliary roots x r , s x_{r,s} x r , s 1 ≤ s ≤ r < n 1\le s\le r<n 1 ≤ s ≤ r < n R = n ( n − 1 ) / 2 R=n(n-1)/2 R = n ( n − 1 ) /2 x r , s 0 x^0_{r,s} x r , s 0 q m r , s x r , s 0 q^{m_{r,s}}x^0_{r,s} q m r , s x r , s 0 m r , s ∈ Z / p Z m_{r,s}\in\mathbb{Z}/p\mathbb{Z} m r , s ∈ Z / p Z p R p^R p R ℏ \hbar ℏ
The external operators act on the Kähler parameters of these sums, leaving the internal roots fixed. The tRS/Bethe correspondence identifies their full-cycle multipliers with classical tRS momenta at the powered parameters. Combining this fact with the operator identity gives Theorem 3.13 . Thus Bethe closure provides finite sums to which the operator identity applies, whereas the identity itself and its consequence for joint eigenfunctions hold independently of closure.
The section is organized as follows. Subsections 3.1–3.3 introduce the shift algebra, Adams polynomials, and subset sequences. Hecke centrality in Subsection 3.4 then gives the operator identity and its eigenfunction consequence in Subsection 3.5. Subsection 3.6 supplies the direct order-two proof, and Subsection 3.7 identifies the classical spectral fiber through Bloch multipliers. Starting with Subsection 3.8, we turn to the internal flag phase and its Kähler transport, impose the Bethe closure conditions, and construct the finite sums satisfying the Frobenius equations of Theorem 3.13. Finally, Subsection 3.11 expresses the unit-insertion sum through explicit descendant components.
Parameters and the Formal Shift Algebra
We begin with the normalization of the Hamiltonians. Let q = ω p q=\omega_p q = ω p p ≥ 1 p\ge1 p ≥ 1 ζ 1 , … , ζ n \zeta_1,\ldots,\zeta_n ζ 1 , … , ζ n a 1 , … , a n a_1,\ldots,a_n a 1 , … , a n ℏ ≠ 0 \hbar\ne0 ℏ = 0 x r , s x_{r,s} x r , s 1 ≤ s ≤ r < n 1\le s\le r<n 1 ≤ s ≤ r < n
The geometric Kähler parameters and their shifts are
z r = q ζ r ζ r + 1 , p i ζ j = q δ i j ζ j p i , p i z r = q δ i r − δ i , r + 1 z r p i . (71) z_r=q\frac{\zeta_r}{\zeta_{r+1}},\qquad p_i\zeta_j=q^{\delta_{ij}}\zeta_jp_i,\qquad p_i z_r=q^{\delta_{ir}-\delta_{i,r+1}}z_rp_i. \tag*{(71)} z r = q ζ r + 1 ζ r , p i ζ j = q δ ij ζ j p i , p i z r = q δ i r − δ i , r + 1 z r p i . (71) Notice that p 1 ⋯ p n p_1\cdots p_n p 1 ⋯ p n z r z_r z r
Treating ℏ 1 / 2 \hbar^{1/2} ℏ 1/2 ζ i \zeta_i ζ i
K = C ( ℏ 1 / 2 , ζ 1 , … , ζ n ) , D q = K ⋊ Z n , [ p i , p j ] = 0. (72) \mathbb{K}=\mathbb{C}(\hbar^{1/2},\zeta_1,\ldots,\zeta_n),\qquad\mathcal{D}_q=\mathbb{K}\mathbin{\rtimes}\mathbb{Z}^n,\qquad[p_i,p_j]=0. \tag*{(72)} K = C ( ℏ 1/2 , ζ 1 , … , ζ n ) , D q = K ⋊ Z n , [ p i , p j ] = 0. (72) The cross relations are given by (71) . The fixed a i a_i a i ∑ m c m ( ζ ) p m \sum_m c_m(\zeta)p^m ∑ m c m ( ζ ) p m p i p p_i^p p i p 1 1 1
Z i = ζ i p , Π i = p i p . (73) Z_i=\zeta_i^p,\qquad\Pi_i=p_i^p. \tag*{(73)} Z i = ζ i p , Π i = p i p . (73) Then the center is
Z ( D q ) = C ( ℏ 1 / 2 , Z 1 , … , Z n ) [ Π 1 ± 1 , … , Π n ± 1 ] . Z(\mathcal{D}_q)=\mathbb{C}(\hbar^{1/2},Z_1,\ldots,Z_n)[\Pi_1^{\pm1},\ldots,\Pi_n^{\pm1}]. Z ( D q ) = C ( ℏ 1/2 , Z 1 , … , Z n ) [ Π 1 ± 1 , … , Π n ± 1 ] . Indeed, commutation with the ζ i \zeta_i ζ i p p p p i p_i p i p i p = 1 p_i^p=1 p i p = 1
For I ⊆ { 1 , … , n } I\subseteq\{1,\ldots,n\} I ⊆ { 1 , … , n } p I = ∏ i ∈ I p i p_I=\prod_{i\in I}p_i p I = ∏ i ∈ I p i
H r = ∑ ∣ I ∣ = r b I ( ζ ; ℏ ) p I , b I ( ζ ; ℏ ) = ℏ − r ( n − r ) / 2 ∏ i ∈ I j ∉ I ℏ ζ i − ζ j ζ i − ζ j , 0 ≤ r ≤ n . (74) H_r=\sum_{\lvert I\rvert=r}b_I(\zeta;\hbar)p_I,\qquad b_I(\zeta;\hbar)=\hbar^{-r(n-r)/2}\prod_{\substack{i\in I\\j\notin I}}\frac{\hbar\zeta_i-\zeta_j}{\zeta_i-\zeta_j},\qquad0\le r\le n. \tag*{(74)} H r = ∣ I ∣ = r ∑ b I ( ζ ; ℏ ) p I , b I ( ζ ; ℏ ) = ℏ − r ( n − r ) /2 i ∈ I j ∈ / I ∏ ζ i − ζ j ℏ ζ i − ζ j , 0 ≤ r ≤ n . (74) We have H 0 = 1 H_0=1 H 0 = 1 H n = p 1 ⋯ p n H_n=p_1\cdots p_n H n = p 1 ⋯ p n H r H_r H r ζ i = q d ζ j \zeta_i=q^d\zeta_j ζ i = q d ζ j D q \mathcal{D}_q D q Z i ≠ Z j Z_i\ne Z_j Z i = Z j i ≠ j i\ne j i = j
We use commuting Cherednik operators Y 1 , … , Y n Y_1,\ldots,Y_n Y 1 , … , Y n
Rad ( e r ( Y 1 , … , Y n ) ) = H r , 0 ≤ r ≤ n . (75) \operatorname{Rad}(e_r(Y_1,\ldots,Y_n))=H_r,\qquad0\le r\le n. \tag*{(75)} Rad ( e r ( Y 1 , … , Y n )) = H r , 0 ≤ r ≤ n . (75) Here Rad denotes the radial part, defined below. This convention fixes the common scalar in the Y i Y_i Y i
For generic q q q (4) by [5 ] (Theorem 2.10, equations (21) –(23) ). To compare with the modified vertex V ~ \widetilde V V [ D ] [D] [ D ] u i = a i − 1 u_i=a_i^{-1} u i = a i − 1 z r v e r t = z r / ℏ z_r^{\mathrm{vert}}=z_r/\hbar z r vert = z r /ℏ ζ r / ζ r + 1 = ( ℏ / q ) z r v e r t \zeta_r/\zeta_{r+1}=(\hbar/q)z_r^{\mathrm{vert}} ζ r / ζ r + 1 = ( ℏ/ q ) z r vert ℏ − r ( n − 1 ) / 2 H r V ~ = e r ( a ) V ~ \hbar^{-r(n-1)/2}H_r\widetilde V=e_r(a)\widetilde V ℏ − r ( n − 1 ) /2 H r V = e r ( a ) V
p i χ = ℏ − ( n − 1 ) / 2 χ , V = χ V ~ . (76) p_i\chi=\hbar^{-(n-1)/2}\chi,\qquad V=\chi\widetilde V. \tag*{(76)} p i χ = ℏ − ( n − 1 ) /2 χ , V = χ V . (76) Then H r V = e r ( a ) V H_rV=e_r(a)V H r V = e r ( a ) V p 1 ⋯ p n V = ( a 1 ⋯ a n ) V p_1\cdots p_nV=(a_1\cdots a_n)V p 1 ⋯ p n V = ( a 1 ⋯ a n ) V [9 ] , reviewed in the introduction. These generic-q q q q = ω p q=\omega_p q = ω p
The Adams Operation and its Shift Support
We shall express symmetric functions of the powered spectral variables through the original commuting Hamiltonians. For this purpose define the integral Newton–Adams polynomials by
A p , r ( e 1 ( y ) , … , e n ( y ) ) = e r ( y 1 p , … , y n p ) , 0 ≤ r ≤ n . (77) A_{p,r}\bigl(e_1(y),\ldots,e_n(y)\bigr)=e_r\left(y_1^p,\ldots,y_n^p\right),\qquad0\le r\le n. \tag*{(77)} A p , r ( e 1 ( y ) , … , e n ( y ) ) = e r ( y 1 p , … , y n p ) , 0 ≤ r ≤ n . (77) The fundamental theorem of symmetric polynomials determines A p , r ∈ Z [ h 1 , … , h n ] A_{p,r}\in\mathbb{Z}[h_1,\ldots,h_n] A p , r ∈ Z [ h 1 , … , h n ] A p , 0 = 1 A_{p,0}=1 A p , 0 = 1 A p , n = h n p A_{p,n}=h_n^p A p , n = h n p A p , 1 A_{p,1} A p , 1 P p P_p P p § 2 . For example,
P 2 ( u , v ) = u 2 − 2 v , P 3 ( u , v ) = u 3 − 3 u v , P 4 ( u , v ) = u 4 − 4 u 2 v + 2 v 2 . P_2(u,v)=u^2-2v,\qquad P_3(u,v)=u^3-3uv,\qquad P_4(u,v)=u^4-4u^2v+2v^2. P 2 ( u , v ) = u 2 − 2 v , P 3 ( u , v ) = u 3 − 3 uv , P 4 ( u , v ) = u 4 − 4 u 2 v + 2 v 2 . These polynomials involve no division by p p p
H ( u ) = ∑ j = 0 n ( − u ) j H j . (78) \mathcal{H}(u)=\sum_{j=0}^{n}(-u)^jH_j. \tag*{(78)} H ( u ) = j = 0 ∑ n ( − u ) j H j . (78) The elementary identity
∏ d = 0 p − 1 ( ∑ j = 0 n ( − q d u ) j e j ( y ) ) = ∏ i = 1 n ( 1 − u p y i p ) = ∑ r = 0 n ( − 1 ) r u p r e r ( y 1 p , … , y n p ) (79) \prod_{d=0}^{p-1}\left(\sum_{j=0}^{n}(-q^d u)^j e_j(y)\right)=\prod_{i=1}^{n}(1-u^p y_i^p)=\sum_{r=0}^{n}(-1)^r u^{pr}e_r\left(y_1^p,\ldots,y_n^p\right) \tag*{(79)} d = 0 ∏ p − 1 ( j = 0 ∑ n ( − q d u ) j e j ( y ) ) = i = 1 ∏ n ( 1 − u p y i p ) = r = 0 ∑ n ( − 1 ) r u p r e r ( y 1 p , … , y n p ) (79) therefore gives
∏ d = 0 p − 1 H ( q d u ) = ∑ r = 0 n ( − 1 ) r u p r A p , r ( H 1 , … , H n ) . (80) \prod_{d=0}^{p-1}\mathcal{H}(q^d u)=\sum_{r=0}^{n}(-1)^r u^{pr}A_{p,r}(H_1,\ldots,H_n). \tag*{(80)} d = 0 ∏ p − 1 H ( q d u ) = r = 0 ∑ n ( − 1 ) r u p r A p , r ( H 1 , … , H n ) . (80) Here we have used the commutativity of the H j H_j H j
Lemma 3.1. The polynomial A p , r A_{p,r} A p , r p r pr p r deg h j = j \deg h_j=j deg h j = j p p p 1 1 1 h r p h_r^p h r p r ≥ 1 r\ge1 r ≥ 1 A p , r ( H ) A_{p,r}(H) A p , r ( H )
0 ≤ m i ≤ p , ∑ i = 1 n m i = p r . (81) 0\le m_i\le p,\qquad\sum_{i=1}^{n}m_i=pr. \tag*{(81)} 0 ≤ m i ≤ p , i = 1 ∑ n m i = p r . (81) Proof. We extract the coefficient of u p r u^{pr} u p r (79) . Each of its p p p h j h_j h j h 0 = 1 h_0=1 h 0 = 1 h r p h_r^p h r p A p , r A_{p,r} A p , r
( − 1 ) p r − r q r p ( p − 1 ) / 2 = 1 , (-1)^{pr-r}q^{rp(p-1)/2}=1, ( − 1 ) p r − r q r p ( p − 1 ) /2 = 1 , since q p ( p − 1 ) / 2 = ( − 1 ) p − 1 q^{p(p-1)/2}=(-1)^{p-1} q p ( p − 1 ) /2 = ( − 1 ) p − 1 H j H_j H j j j j p p p p p p □ \square □
Subset Sequences on the External Orbit
The cyclotomic product has a useful combinatorial interpretation. Each term in its expansion selects a subset at each step, and we can group the resulting sequences according to their endpoints. Let I = ( I 0 , … , I p − 1 ) \mathcal{I}=(I_0,\ldots,I_{p-1}) I = ( I 0 , … , I p − 1 ) { 1 , … , n } \{1,\ldots,n\} { 1 , … , n }
k d = ∑ e < d 1 I e , m ( I ) = k p , ζ ⟼ q k d ζ . (82) k_d=\sum_{e<d}\mathbf{1}_{I_e},\qquad m(\mathcal{I})=k_p,\qquad\zeta\longmapsto q^{k_d}\zeta. \tag*{(82)} k d = e < d ∑ 1 I e , m ( I ) = k p , ζ ⟼ q k d ζ . (82) Here m i m_i m i i i i
w ( I ; ζ ) = ( − 1 ) ∣ m ( I ) ∣ q ∑ d = 0 p − 1 d ∣ I d ∣ ∏ d = 0 p − 1 b I d ( q k d ζ ; ℏ ) , (83) w(\mathcal{I};\zeta)=(-1)^{|m(\mathcal{I})|}q^{\sum_{d=0}^{p-1}d|I_d|}\prod_{d=0}^{p-1}b_{I_d}(q^{k_d}\zeta;\hbar), \tag*{(83)} w ( I ; ζ ) = ( − 1 ) ∣ m ( I ) ∣ q ∑ d = 0 p − 1 d ∣ I d ∣ d = 0 ∏ p − 1 b I d ( q k d ζ ; ℏ ) , (83) C m ( ζ ) = ∑ I : m ( I ) = m w ( I ; ζ ) , ∣ m ∣ = ∑ i m i . (84) C_m(\zeta)=\sum_{\mathcal{I}:m(\mathcal{I})=m}w(\mathcal{I};\zeta),\qquad|m|=\sum_i m_i. \tag*{(84)} C m ( ζ ) = I : m ( I ) = m ∑ w ( I ; ζ ) , ∣ m ∣ = i ∑ m i . (84) Here normal ordering means moving all rational functions of the positions to the left and all shifts to the right. The rule is
p k f ( ζ ) = f ( q k ζ ) p k , q k ζ = ( q k 1 ζ 1 , … , q k n ζ n ) . p^k f(\zeta)=f(q^k\zeta)p^k,\qquad q^k\zeta=(q^{k_1}\zeta_1,\ldots,q^{k_n}\zeta_n). p k f ( ζ ) = f ( q k ζ ) p k , q k ζ = ( q k 1 ζ 1 , … , q k n ζ n ) . After the first d d d p k d p^{k_d} p k d p k d b I d ( ζ ; ℏ ) = b I d ( q k d ζ ; ℏ ) p k d p^{k_d}b_{I_d}(\zeta;\hbar)=b_{I_d}(q^{k_d}\zeta;\hbar)p^{k_d} p k d b I d ( ζ ; ℏ ) = b I d ( q k d ζ ; ℏ ) p k d
∏ d = 0 p − 1 H ( q d u ) = ∑ m ∈ { 0 , … , p } n u ∣ m ∣ C m ( ζ ) p m . (85) \prod_{d=0}^{p-1}\mathcal{H}(q^d u)=\sum_{m\in\{0,\ldots,p\}^n}u^{|m|}C_m(\zeta)p^m. \tag*{(85)} d = 0 ∏ p − 1 H ( q d u ) = m ∈ { 0 , … , p } n ∑ u ∣ m ∣ C m ( ζ ) p m . (85) Equation (80) implies C m = 0 C_m=0 C m = 0 p ∤ ∣ m ∣ p\nmid|m| p ∤ ∣ m ∣ C m = 0 C_m=0 C m = 0 0 < m i < p 0<m_i<p 0 < m i < p p = 2 p=2 p = 2
Root-of-Unity Centrality
Let us recall the part of the type-GL(n n n [15 ] (Sections 3.1–3.4). There are two separately commuting families of invertible elements. The coordinates X 1 , … , X n X_1,\ldots,X_n X 1 , … , X n ζ 1 , … , ζ n \zeta_1,\ldots,\zeta_n ζ 1 , … , ζ n Y 1 , … , Y n Y_1,\ldots,Y_n Y 1 , … , Y n Y i Y_i Y i (75) .
The Hecke generators T i T_i T i s i s_i s i q q q ( T i − v ) ( T i + v − 1 ) = 0 (T_i-v)(T_i+v^{-1})=0 ( T i − v ) ( T i + v − 1 ) = 0 v 2 = ℏ v^2=\hbar v 2 = ℏ
T i f ( X ) − ( s i f ) ( X ) T i = ( v − v − 1 ) f ( X ) − ( s i f ) ( X ) 1 − X α i , (86) T_i f(X)-(s_i f)(X)T_i=(v-v^{-1})\frac{f(X)-(s_i f)(X)}{1-X^{\alpha_i}}, \tag*{(86)} T i f ( X ) − ( s i f ) ( X ) T i = ( v − v − 1 ) 1 − X α i f ( X ) − ( s i f ) ( X ) , (86) where α i \alpha_i α i X α i X^{\alpha_i} X α i X X X Y Y Y
In the localized Dunkl realization these identities hold in the formal algebra D q ⋊ S n \mathcal{D}_q\rtimes S_n D q ⋊ S n
L = ∑ w ∈ S n L w w ⟼ Rad ( L ) = ∑ w ∈ S n L w , L w ∈ D q . (87) L=\sum_{w\in S_n}L_w w \longmapsto\operatorname{Rad}(L)=\sum_{w\in S_n}L_w,\qquad L_w\in\mathcal{D}_q. \tag*{(87)} L = w ∈ S n ∑ L w w ⟼ Rad ( L ) = w ∈ S n ∑ L w , L w ∈ D q . (87) The permutations are removed as in the action on symmetric functions, but every lattice shift is retained as a formal symbol. On symmetric functions of the commuting Y i Y_i Y i Rad ( e r ( Y ) ) = H r \operatorname{Rad}(e_r(Y))=H_r Rad ( e r ( Y )) = H r g ( ζ ) g(\zeta) g ( ζ )
Rad ( [ L , g ] ) = [ Rad ( L ) , g ] . \operatorname{Rad}([L,g])=[\operatorname{Rad}(L),g]. Rad ([ L , g ]) = [ Rad ( L ) , g ] . Thus a central element has a radial part which commutes with symmetric coordinate functions.
Here q q q [15 ] is X δ = q − 2 v 2 X_\delta=q^{-2}v^2 X δ = q − 2 v 2 p i p p_i^p p i p X 1 p , … , X n p X_1^p,\ldots,X_n^p X 1 p , … , X n p
Lemma 3.2. The operator A p , r ( H ) A_{p,r}(H) A p , r ( H ) ζ 1 , … , ζ n \zeta_1,\ldots,\zeta_n ζ 1 , … , ζ n
Proof. First assume that the Hecke parameter is generic and consider a symmetric Laurent polynomial f f f X 1 p , … , X n p X_1^p,\ldots,X_n^p X 1 p , … , X n p X i X_i X i q q q p p p (86) vanishes, so f f f f f f
Applying Fourier duality shows that every symmetric Laurent polynomial in Y 1 p , … , Y n p Y_1^p,\ldots,Y_n^p Y 1 p , … , Y n p Y Y Y
Rad ( e r ( Y 1 p , … , Y n p ) ) = Rad ( A p , r ( e 1 ( Y ) , … , e n ( Y ) ) ) \operatorname{Rad}\bigl(e_r(Y_1^p,\ldots,Y_n^p)\bigr)
=\operatorname{Rad}\bigl(A_{p,r}(e_1(Y),\ldots,e_n(Y))\bigr) Rad ( e r ( Y 1 p , … , Y n p ) ) = Rad ( A p , r ( e 1 ( Y ) , … , e n ( Y )) ) = A p , r ( H ) . =A_{p,r}(H). = A p , r ( H ) . The required commutation follows by applying (87) to the commutator with a symmetric coordinate function.
The argument has taken place in the formal difference–reflection algebra, so distinct lattice shifts remain distinct upon specialization. To specialize the Hecke parameter, write the commutator using (74) . Its coefficients are Laurent polynomial in ℏ 1 / 2 \hbar^{1/2} ℏ 1/2 q q q ℏ 1 / 2 \hbar^{1/2} ℏ 1/2
Remark 3.3. The restrictions on root orders in some geometric results of [15 ] are not needed here. Our proof uses the Bernstein relations, affine invariance, and Fourier duality, and applies to every p p p H r H_r H r
Cancellation of All Intermediate Shifts
We can now combine the commutation relation with the bounds of Lemma 3.1 . Introduce the operators
H r [ p ] = ℏ − p r ( n − r ) / 2 ∑ ∣ I ∣ = r ∏ i ∈ I j ∉ I ℏ p Z i − Z j Z i − Z j Π I , Π I = ∏ i ∈ I Π i . (88) H_r^{[p]}=\hbar^{-pr(n-r)/2}\sum_{\lvert I\rvert=r}\prod_{\substack{i\in I\\j\notin I}}\frac{\hbar^p Z_i-Z_j}{Z_i-Z_j}\Pi_I,\qquad\Pi_I=\prod_{i\in I}\Pi_i. \tag*{(88)} H r [ p ] = ℏ − p r ( n − r ) /2 ∣ I ∣ = r ∑ i ∈ I j ∈ / I ∏ Z i − Z j ℏ p Z i − Z j Π I , Π I = i ∈ I ∏ Π i . (88) These belong to Z ( D q ) Z(\mathcal{D}_q) Z ( D q )
Theorem 3.4. For every n n n p ≥ 1 p\ge1 p ≥ 1 p p p q q q
A p , r ( H 1 , … , H n ) = H r [ p ] , 0 ≤ r ≤ n . (89) A_{p,r}(H_1,\ldots,H_n)=H_r^{[p]},\qquad0\le r\le n. \tag*{(89)} A p , r ( H 1 , … , H n ) = H r [ p ] , 0 ≤ r ≤ n . (89) Equivalently,
∏ d = 0 p − 1 H ( q d u ) = ∑ r = 0 n ( − 1 ) r u p r H r [ p ] . (90) \prod_{d=0}^{p-1}\mathcal{H}(q^d u)=\sum_{r=0}^{n}(-1)^r u^{pr}H_r^{[p]}. \tag*{(90)} d = 0 ∏ p − 1 H ( q d u ) = r = 0 ∑ n ( − 1 ) r u p r H r [ p ] . (90) The surviving coefficients are obtained by the Frobenius action on coordinates, coupling, and shifts:
( ζ i , ℏ 1 / 2 , p i ) ⟼ ( ζ i p , ℏ p / 2 , p i p ) , (\zeta_i,\hbar^{1/2},p_i)\longmapsto(\zeta_i^p,\hbar^{p/2},p_i^p), ( ζ i , ℏ 1/2 , p i ) ⟼ ( ζ i p , ℏ p /2 , p i p ) , with no extra parity sign.
Proof. For r = 0 r=0 r = 0 r ≥ 1 r\ge1 r ≥ 1 F r = A p , r ( H ) = ∑ m c m ( ζ ) p m F_r=A_{p,r}(H)=\sum_m c_m(\zeta)p^m F r = A p , r ( H ) = ∑ m c m ( ζ ) p m Lemma 3.2 , we have
0 = [ F r , e 1 ( ζ ) ] = ∑ m c m ( ζ ) ( ∑ i ( q m i − 1 ) ζ i ) p m . 0=[F_r,e_1(\zeta)]=\sum_m c_m(\zeta)\left(\sum_i(q^{m_i}-1)\zeta_i\right)p^m. 0 = [ F r , e 1 ( ζ )] = m ∑ c m ( ζ ) ( i ∑ ( q m i − 1 ) ζ i ) p m . The normal-ordered expansion is unique and the ζ i \zeta_i ζ i c m ≠ 0 c_m\ne0 c m = 0 p ∣ m i p\mid m_i p ∣ m i i i i Lemma 3.1 , this leaves only the shifts
p m = p I p , ∣ I ∣ = r . (91) p^m=p_I^p,\qquad\lvert I\rvert=r. \tag*{(91)} p m = p I p , ∣ I ∣ = r . (91) It remains to compute the coefficient for a fixed I I I I I I p p p H j H_j H j p p p I I I I I I H r H_r H r H r p H_r^p H r p
Its coefficient at p 1 I p\mathbf{1}_I p 1 I ∏ d = 0 p − 1 ( 1 − q d w ) = 1 − w p \prod_{d=0}^{p-1}(1-q^d w)=1-w^p ∏ d = 0 p − 1 ( 1 − q d w ) = 1 − w p
∏ d = 0 p − 1 b I ( q d 1 I ζ ; ℏ ) = ℏ − p r ( n − r ) / 2 ∏ i ∈ I j ∉ I ℏ p ζ i p − ζ j p ζ i p − ζ j p . (92) \prod_{d=0}^{p-1} b_I(q^{d\mathbf{1}_I}\zeta;\hbar)=\hbar^{-pr(n-r)/2}\prod_{\substack{i\in I\\j\notin I}}\frac{\hbar^p\zeta_i^p-\zeta_j^p}{\zeta_i^p-\zeta_j^p}. \tag*{(92)} d = 0 ∏ p − 1 b I ( q d 1 I ζ ; ℏ ) = ℏ − p r ( n − r ) /2 i ∈ I j ∈ / I ∏ ζ i p − ζ j p ℏ p ζ i p − ζ j p . (92) The leading cyclotomic factors cancel between numerator and denominator, also at even orders. This proves (89) , and the generating form follows from (80) . As in the centrality lemma, clearing the position denominators extends the result from generic Hecke parameter to every nonzero ℏ 1 / 2 \hbar^{1/2} ℏ 1/2
Corollary 3.5. Let n , p ≥ 1 n,p\ge1 n , p ≥ 1 q q q p p p V V V ζ i p ≠ ζ j p \zeta_i^p\ne\zeta_j^p ζ i p = ζ j p i ≠ j i\ne j i = j V V V a i a_i a i
H r V = e r ( a 1 , … , a n ) V , 1 ≤ r ≤ n , H_rV=e_r(a_1,\ldots,a_n)V,\qquad1\le r\le n, H r V = e r ( a 1 , … , a n ) V , 1 ≤ r ≤ n , then
H r [ p ] V = A p , r ( H 1 , … , H n ) V = e r ( a 1 p , … , a n p ) V , 1 ≤ r ≤ n . (93) H_r^{[p]}V=\mathcal{A}_{p,r}(H_1,\ldots,H_n)V=e_r(a_1^p,\ldots,a_n^p)V,\qquad1\le r\le n. \tag*{(93)} H r [ p ] V = A p , r ( H 1 , … , H n ) V = e r ( a 1 p , … , a n p ) V , 1 ≤ r ≤ n . (93) Proof. Apply the Adams polynomial to the common eigenfunction. Since the Hamiltonians commute, we get the same polynomial in their eigenvalues:
A p , r ( H ) V = A p , r ( e 1 ( a ) , … , e n ( a ) ) V = e r ( a 1 p , … , a n p ) V . \mathcal{A}_{p,r}(H)V=\mathcal{A}_{p,r}(e_1(a),\ldots,e_n(a))V=e_r(a_1^p,\ldots,a_n^p)V. A p , r ( H ) V = A p , r ( e 1 ( a ) , … , e n ( a )) V = e r ( a 1 p , … , a n p ) V . The left side is H r [ p ] V H_r^{[p]}V H r [ p ] V Theorem 3.4 , which proves the assertion.
The corollary applies to any joint solution defined at the chosen root of unity, without imposing Bethe equations or fixing Bloch multipliers. Specializing an infinite vertex series from generic q q q
Corollary 3.6. The sums (3.13) satisfy
C m ( ζ ) = { ( − 1 ) ∣ I ∣ ℏ − p ∣ I ∣ ( n − ∣ I ∣ ) / 2 ∏ i ∈ I , j ∉ I ℏ p ζ i p − ζ j p ζ i p − ζ j p , m = p 1 I , 0 , otherwise . (94) C_m(\zeta)=
\begin{cases}
(-1)^{|I|}\hbar^{-p|I|(n-|I|)/2}\prod_{\substack{i\in I,j\notin I}}\frac{\hbar^p\zeta_i^p-\zeta_j^p}{\zeta_i^p-\zeta_j^p}, & m=p\mathbf{1}_I,\\
0, & \text{otherwise}.
\end{cases} \tag*{(94)} C m ( ζ ) = { ( − 1 ) ∣ I ∣ ℏ − p ∣ I ∣ ( n − ∣ I ∣ ) /2 ∏ i ∈ I , j ∈ / I ζ i p − ζ j p ℏ p ζ i p − ζ j p , 0 , m = p 1 I , otherwise . (94) For m = p 1 I m=p\mathbf{1}_I m = p 1 I I 0 = ⋯ = I p − 1 = I I_0=\cdots=I_{p-1}=I I 0 = ⋯ = I p − 1 = I
Proof. Comparing (85) with (90) gives the stated coefficients. To reach p 1 I p\mathbf{1}_I p 1 I I I I p p p (92) multiplied by ( − 1 ) p ∣ I ∣ q ∣ I ∣ p ( p − 1 ) / 2 = ( − 1 ) ∣ I ∣ (-1)^{p|I|}q^{|I|p(p-1)/2}=(-1)^{|I|} ( − 1 ) p ∣ I ∣ q ∣ I ∣ p ( p − 1 ) /2 = ( − 1 ) ∣ I ∣
It is useful to compare this calculation with cyclic rotation of the subset sequences. Let m = m ( I ) m=m(I) m = m ( I ) R I = ( I 1 , … , I p − 1 , I 0 ) RI=(I_1,\ldots,I_{p-1},I_0) R I = ( I 1 , … , I p − 1 , I 0 ) s = 1 I 0 s=\mathbf{1}_{I_0} s = 1 I 0
w ( R I ; q s ζ ) b I 0 ( ζ ) = q − ∣ m ∣ w ( I ; ζ ) b I 0 ( q m ζ ) . (95) w(RI;q^s\zeta)b_{I_0}(\zeta)=q^{-|m|}w(I;\zeta)b_{I_0}(q^m\zeta). \tag*{(95)} w ( R I ; q s ζ ) b I 0 ( ζ ) = q − ∣ m ∣ w ( I ; ζ ) b I 0 ( q m ζ ) . (95) Thus rotation changes both the starting point and the argument of the last coefficient. At ∣ m ∣ = p r |m|=pr ∣ m ∣ = p r
A Direct Cancellation at Order Two
Consider q = − 1 q=-1 q = − 1
f ℏ ( v , w ) = ℏ − 1 / 2 ℏ v − w v − w , b I ( ζ ; ℏ ) = ∏ i ∈ I , j ∉ I f ℏ ( ζ i , ζ j ) . (96) f_{\hbar}(v,w)=\hbar^{-1/2}\frac{\hbar v-w}{v-w},\qquad b_I(\zeta;\hbar)=\prod_{\substack{i\in I,j\notin I}}f_{\hbar}(\zeta_i,\zeta_j). \tag*{(96)} f ℏ ( v , w ) = ℏ − 1/2 v − w ℏ v − w , b I ( ζ ; ℏ ) = i ∈ I , j ∈ / I ∏ f ℏ ( ζ i , ζ j ) . (96) Lemma 3.7. For N ≥ 1 N\geq1 N ≥ 1 c c c
S N ( v ; c ) = ∑ L ⊆ { 1 , … , N } ( − 1 ) ∣ L ∣ ∏ i ∈ L , j ∉ L ( 1 + c v i v j v i 2 − v j 2 ) (97) S_N(v;c)=\sum_{L\subseteq\{1,\ldots,N\}}(-1)^{|L|}\prod_{\substack{i\in L,j\notin L}}\left(1+c\frac{v_i v_j}{v_i^2-v_j^2}\right) \tag*{(97)} S N ( v ; c ) = L ⊆ { 1 , … , N } ∑ ( − 1 ) ∣ L ∣ i ∈ L , j ∈ / L ∏ ( 1 + c v i 2 − v j 2 v i v j ) (97) vanishes identically. Set S 0 = 1 S_0=1 S 0 = 1
Proof. We argue by induction on N N N S 1 = 0 S_1=0 S 1 = 0 S N S_N S N v 1 v_1 v 1 v 1 = ± v j v_1=\pm v_j v 1 = ± v j v 1 = v 2 v_1=v_2 v 1 = v 2
For the divisor v 1 = − v 2 v_1=-v_2 v 1 = − v 2
a k = − c v 2 v k v 2 2 − v k 2 , P ± = ∏ k = 3 N ( 1 ± a k ) . a_k=-c\frac{v_2v_k}{v_2^2-v_k^2},\qquad P_{\pm}=\prod_{k=3}^{N}(1\pm a_k). a k = − c v 2 2 − v k 2 v 2 v k , P ± = k = 3 ∏ N ( 1 ± a k ) . When 1 ∈ L 1\in L 1 ∈ L 2 ∉ L 2\notin L 2 ∈ / L P + P_+ P + P − P_- P −
( P + − P − ) S N − 2 ( v 3 , … , v N ; c ) . (P_+-P_-)S_{N-2}(v_3,\ldots,v_N;c). ( P + − P − ) S N − 2 ( v 3 , … , v N ; c ) . For N = 2 N=2 N = 2 N > 2 N>2 N > 2 j j j v 1 → ∞ v_1\to\infty v 1 → ∞ S N S_N S N
Proposition 3.8. For every n n n (90) at q = − 1 q=-1 q = − 1 Lemma 3.7 , independently of Hecke centrality.
Proof. For an endpoint m ∈ { 0 , 1 , 2 } n m\in\{0,1,2\}^{n} m ∈ { 0 , 1 , 2 } n D = { i : m i = 2 } D=\{i:m_i=2\} D = { i : m i = 2 } E = { i : m i = 1 } E=\{i:m_i=1\} E = { i : m i = 1 } F = { i : m i = 0 } F=\{i:m_i=0\} F = { i : m i = 0 } I 0 = D ∪ L I_0=D\cup L I 0 = D ∪ L I 1 = D ∪ ( E ∖ L ) I_1=D\cup(E\setminus L) I 1 = D ∪ ( E ∖ L ) L ⊆ E L\subseteq E L ⊆ E D D D F F F L L L
K D , E , F = ∏ d ∈ D , f ∈ F f ℏ ( ζ d , ζ f ) f ℏ ( − ζ d , ζ f ) ∏ d ∈ D , e ∈ E f ℏ ( ζ d , ζ e ) ∏ e ∈ E , f ∈ F f ℏ ( ζ e , ζ f ) . (98) K_{D,E,F}=\prod_{\substack{d\in D,f\in F}}f_{\hbar}(\zeta_d,\zeta_f)f_{\hbar}(-\zeta_d,\zeta_f)
\prod_{\substack{d\in D,e\in E}}f_{\hbar}(\zeta_d,\zeta_e)
\prod_{\substack{e\in E,f\in F}}f_{\hbar}(\zeta_e,\zeta_f). \tag*{(98)} K D , E , F = d ∈ D , f ∈ F ∏ f ℏ ( ζ d , ζ f ) f ℏ ( − ζ d , ζ f ) d ∈ D , e ∈ E ∏ f ℏ ( ζ d , ζ e ) e ∈ E , f ∈ F ∏ f ℏ ( ζ e , ζ f ) . (98) For i ∈ L i\in L i ∈ L j ∈ E ∖ L j\in E\setminus L j ∈ E ∖ L
f ℏ ( ζ i , ζ j ) f ℏ ( ζ j , − ζ i ) = 1 + ( ℏ − ℏ − 1 ) ζ i ζ j ζ i 2 − ζ j 2 . (99) f_{\hbar}(\zeta_i,\zeta_j)f_{\hbar}(\zeta_j,-\zeta_i)
=1+(\hbar-\hbar^{-1})\frac{\zeta_i\zeta_j}{\zeta_i^2-\zeta_j^2}. \tag*{(99)} f ℏ ( ζ i , ζ j ) f ℏ ( ζ j , − ζ i ) = 1 + ( ℏ − ℏ − 1 ) ζ i 2 − ζ j 2 ζ i ζ j . (99) The sign in H ( u ) H ( − u ) \mathcal{H}(u)\mathcal{H}(-u) H ( u ) H ( − u ) ( − 1 ) ∣ I 0 ∣ = ( − 1 ) ∣ D ∣ + ∣ L ∣ (-1)^{|I_0|}=(-1)^{|D|+|L|} ( − 1 ) ∣ I 0 ∣ = ( − 1 ) ∣ D ∣ + ∣ L ∣ u ∣ m ∣ p m u^{|m|}p^m u ∣ m ∣ p m
C m = ( − 1 ) ∣ D ∣ K D , E , F S ∣ E ∣ ( ( ζ e ) e ∈ E ; ℏ − ℏ − 1 ) . (100) C_m=(-1)^{|D|}K_{D,E,F}S_{|E|}\left((\zeta_e)_{e\in E};\hbar-\hbar^{-1}\right). \tag*{(100)} C m = ( − 1 ) ∣ D ∣ K D , E , F S ∣ E ∣ ( ( ζ e ) e ∈ E ; ℏ − ℏ − 1 ) . (100) The lemma makes this coefficient zero when E ≠ ∅ E\ne\varnothing E = ∅ E = ∅ E=\varnothing E = ∅ K D , ∅ , F = b D ( ζ 2 ; ℏ 2 ) K_{D,\varnothing,F}=b_D(\zeta^2;\hbar^2) K D , ∅ , F = b D ( ζ 2 ; ℏ 2 ) (90) at p = 2 p=2 p = 2
Extracting coefficients from H ( u ) H ( − u ) \mathcal{H}(u)\mathcal{H}(-u) H ( u ) H ( − u )
H r [ 2 ] = H r 2 + 2 ∑ j = 1 min ( r , n − r ) ( − 1 ) j H r − j H r + j . (101) H_r^{[2]}=H_r^2+2\sum_{j=1}^{\min(r,n-r)}(-1)^jH_{r-j}H_{r+j}. \tag*{(101)} H r [ 2 ] = H r 2 + 2 j = 1 ∑ m i n ( r , n − r ) ( − 1 ) j H r − j H r + j . (101) For example, for T ∗ F l 3 T^*\mathrm{Fl}_3 T ∗ Fl 3
H 1 [ 2 ] = H 1 2 − 2 H 2 , H 2 [ 2 ] = H 2 2 − 2 H 1 H 3 , H 3 [ 2 ] = H 3 2 . (102) H_1^{[2]}=H_1^2-2H_2,\qquad H_2^{[2]}=H_2^2-2H_1H_3,\qquad H_3^{[2]}=H_3^2. \tag*{(102)} H 1 [ 2 ] = H 1 2 − 2 H 2 , H 2 [ 2 ] = H 2 2 − 2 H 1 H 3 , H 3 [ 2 ] = H 3 2 . (102) Only the shifts p 1 2 p_1^2 p 1 2 p 2 2 p_2^2 p 2 2 p 3 2 p_3^2 p 3 2 p 1 2 p 2 2 p_1^2p_2^2 p 1 2 p 2 2 p 1 2 p 3 2 p_1^2p_3^2 p 1 2 p 3 2 p 2 2 p 3 2 p_2^2p_3^2 p 2 2 p 3 2 (88) . The calculation takes place in characteristic zero, so the coefficient 2 2 2
Bloch Multipliers and the Classical Spectral Fiber
We shall now interpret the powered eigenvalue equations in terms of the classical tRS spectral fiber. Let V V V
p i p V = λ i V , 1 ≤ i ≤ n . (103) p_i^pV=\lambda_iV,\qquad1\le i\le n. \tag*{(103)} p i p V = λ i V , 1 ≤ i ≤ n . (103) Here the λ i \lambda_i λ i
For a symplectic parameter c c c C ( c 1 / 2 ) \mathbb{C}(c^{1/2}) C ( c 1/2 )
h r ( Z , π ; c ) = c − r ( n − r ) / 2 ∑ ∣ I ∣ = r ∏ i ∈ I j ∉ I c Z i − Z j Z i − Z j ∏ i ∈ I π i , h 0 = 1. (104) h_r(Z,\pi;c)=c^{-r(n-r)/2}\sum_{|I|=r}\prod_{\substack{i\in I\\j\notin I}}\frac{cZ_i-Z_j}{Z_i-Z_j}\prod_{i\in I}\pi_i,\qquad h_0=1. \tag*{(104)} h r ( Z , π ; c ) = c − r ( n − r ) /2 ∣ I ∣ = r ∑ i ∈ I j ∈ / I ∏ Z i − Z j c Z i − Z j i ∈ I ∏ π i , h 0 = 1. (104) These are the centered classical tRS Hamiltonians with usual coupling c − 1 c^{-1} c − 1 c = ℏ p c=\hbar^p c = ℏ p
Proposition 3.9. Suppose (103) holds and Z i ≠ Z j Z_i\ne Z_j Z i = Z j for i ≠ j i\ne j i = j Then
H r [ p ] V = A p , r ( H ) V = e r ( a 1 p , … , a n p ) V , 1 ≤ r ≤ n , (105) H_r^{[p]}V=A_{p,r}(H)V=e_r(a_1^p,\ldots,a_n^p)V,\qquad1\le r\le n, \tag*{(105)} H r [ p ] V = A p , r ( H ) V = e r ( a 1 p , … , a n p ) V , 1 ≤ r ≤ n , (105) if and only if
h r ( Z , λ ; ℏ p ) = e r ( a 1 p , … , a n p ) , 1 ≤ r ≤ n . (106) h_r(Z,\lambda;\hbar^p)=e_r(a_1^p,\ldots,a_n^p),\qquad1\le r\le n. \tag*{(106)} h r ( Z , λ ; ℏ p ) = e r ( a 1 p , … , a n p ) , 1 ≤ r ≤ n . (106) Proof. Substitute the full-cycle eigenvalues (103) into (88) . This gives
H r [ p ] V = h r ( Z , λ ; ℏ p ) V . H_r^{[p]}V=h_r(Z,\lambda;\hbar^p)V. H r [ p ] V = h r ( Z , λ ; ℏ p ) V . By Theorem 3.4 , the left side equals A p , r ( H ) V A_{p,r}(H)V A p , r ( H ) V V V V V V V
In particular, once (106) holds, every section in the same Bloch sector satisfies the powered equations. The ordinary joint eigenvalue problem imposes the further conditions
H r V = e r ( a 1 , … , a n ) V , 1 ≤ r ≤ n . (107) H_rV=e_r(a_1,\ldots,a_n)V,\qquad1\le r\le n. \tag*{(107)} H r V = e r ( a 1 , … , a n ) V , 1 ≤ r ≤ n . (107) These give the following special case.
Corollary 3.10. Under (107) and (103) ,
h r ( Z , λ ; ℏ p ) = e r ( a 1 p , … , a n p ) , 1 ≤ r ≤ n . (108) h_r(Z,\lambda;\hbar^p)=e_r(a_1^p,\ldots,a_n^p),\qquad1\le r\le n. \tag*{(108)} h r ( Z , λ ; ℏ p ) = e r ( a 1 p , … , a n p ) , 1 ≤ r ≤ n . (108) Proof. Acting on V V V (89) , the left side gives A p , r ( e 1 ( a ) , … , e n ( a ) ) = e r ( a p ) \mathcal{A}_{p,r}(e_1(a),\ldots,e_n(a))=e_r(a^p) A p , r ( e 1 ( a ) , … , e n ( a )) = e r ( a p ) h r ( Z , λ ; ℏ p ) h_r(Z,\lambda;\hbar^p) h r ( Z , λ ; ℏ p ) V V V
The same statement can be expressed through the spectral polynomial
P ( v ; Z , π , c ) = ∑ r = 0 n ( − 1 ) r h r ( Z , π ; c ) v n − r (109) \mathcal{P}(v;Z,\pi,c)=\sum_{r=0}^{n}(-1)^rh_r(Z,\pi;c)v^{n-r} \tag*{(109)} P ( v ; Z , π , c ) = r = 0 ∑ n ( − 1 ) r h r ( Z , π ; c ) v n − r (109) which satisfies
P ( v ; Z , λ , ℏ p ) = ∏ i = 1 n ( v − a i p ) . (110) \mathcal{P}(v;Z,\lambda,\hbar^p)=\prod_{i=1}^{n}(v-a_i^p). \tag*{(110)} P ( v ; Z , λ , ℏ p ) = i = 1 ∏ n ( v − a i p ) . (110) This is the classical spectral fiber appearing in both the criterion and its ordinary-eigenfunction special case. For finite sums, Bethe closure gives the full-cycle multipliers explicitly. The classical tRS/Bethe correspondence of [7 ] (Theorems 4.6 and 4.9), applied at powered parameters, places them on (106) . It follows that every admissible rational insertion satisfies the powered equations, as we shall prove in Theorem 3.13 ; a nonzero sum is a joint eigenfunction of these operators. This gives a second application of the operator identity, alongside its direct consequence for an ordinary joint eigenfunction in Corollary 3.5 .
Recall that the momenta λ i \lambda_i λ i p p p
The homogeneous normalization has a simple effect on the spectrum. Multiplying a section by b m 1 + ⋯ + m n b^{m_1+\cdots+m_n} b m 1 + ⋯ + m n b a i ba_i b a i r r r b r b^r b r b p b^p b p
Proposition 3.11. For generic distinct nonzero Z i Z_i Z i c ≠ 0 c\ne0 c = 0 s 1 , … , s n s_1,\ldots,s_n s 1 , … , s n s n ≠ 0 s_n\ne0 s n = 0
h r ( Z , π ; c ) = s r , 1 ≤ r ≤ n , (111) h_r(Z,\pi;c)=s_r,\qquad1\le r\le n, \tag*{(111)} h r ( Z , π ; c ) = s r , 1 ≤ r ≤ n , (111) have exactly n ! n! n ! ( C ∗ ) n (\mathbb{C}^*)^n ( C ∗ ) n s r = e r ( a 1 p , … , a n p ) s_r=e_r(a_1^p,\ldots,a_n^p) s r = e r ( a 1 p , … , a n p )
Proof. Since h r h_r h r r r r π \pi π h n = π 1 ⋯ π n h_n=\pi_1\cdots\pi_n h n = π 1 ⋯ π n P n \mathbb{P}^n P n
h r ( Z , π ; c ) − s r w r = 0. (112) h_r(Z,\pi;c)-s_rw^r=0. \tag*{(112)} h r ( Z , π ; c ) − s r w r = 0. (112) At c 1 / 2 = 1 c^{1/2}=1 c 1/2 = 1 e r ( π ) = 0 e_r(\pi)=0 e r ( π ) = 0 r r r π i = 0 \pi_i=0 π i = 0 s r = e r ( d 1 , … , d n ) s_r=e_r(d_1,\ldots,d_n) s r = e r ( d 1 , … , d n ) d i d_i d i n ! n! n !
Exclude the position diagonals and c = 0 c=0 c = 0 n n n 1 ⋅ 2 ⋯ n = n ! 1\cdot2\cdots n=n! 1 ⋅ 2 ⋯ n = n ! h n = s n ≠ 0 h_n=s_n\ne0 h n = s n = 0
The map taking ( a 1 , … , a n ) ∈ ( C ∗ ) n (a_1,\ldots,a_n)\in(\mathbb{C}^{*})^n ( a 1 , … , a n ) ∈ ( C ∗ ) n e r ( a 1 p , … , a n p ) e_r(a_1^p,\ldots,a_n^p) e r ( a 1 p , … , a n p )
We have counted the classical momentum points of the spectral fiber. For the finite functionals associated with closed Bethe orbits, these momenta are the full-cycle multipliers and give the powered equations of Theorem 3.13 . We now turn to the explicit construction of the internal sums and their multipliers.
The Flag Phase and Kähler Transport
Let us describe the finite sums associated with the flag vertex phase. The internal shift ratios determine their weights, while Kähler shifts give the Bloch multipliers and the corresponding tRS momenta.
We denote by V r \mathcal{V}_r V r r r r F l n \mathrm{Fl}_n Fl n T ∗ F l n T^*\mathrm{Fl}_n T ∗ Fl n
W = C n ⊗ O ⟶ V n − 1 ⟶ ⋯ ⟶ V 1 . \mathcal{W}=\mathbb{C}^n\otimes\mathcal{O}\longrightarrow\mathcal{V}_{n-1}\longrightarrow\cdots\longrightarrow\mathcal{V}_1. W = C n ⊗ O ⟶ V n − 1 ⟶ ⋯ ⟶ V 1 . Using the splitting notation for their equivariant K K K
[ V r ] = ∑ u = 1 r x r , u , x n , i = a i , L r = ∏ u = 1 r x r , u , L 0 = 1 , L n = a 1 ⋯ a n . [\mathcal{V}_r]=\sum_{u=1}^{r}x_{r,u},\qquad x_{n,i}=a_i,\qquad L_r=\prod_{u=1}^{r}x_{r,u},\qquad L_0=1,\qquad L_n=a_1\cdots a_n. [ V r ] = u = 1 ∑ r x r , u , x n , i = a i , L r = u = 1 ∏ r x r , u , L 0 = 1 , L n = a 1 ⋯ a n . Here L r L_r L r det V r \det\mathcal{V}_r det V r V r \mathcal{V}_r V r V V V
P = ∑ r = 1 n − 1 ∑ u = 1 r ∑ v = 1 r + 1 x r , u x r + 1 , v − ∑ r = 1 n − 1 ∑ u , v = 1 r x r , u x r , v , T X = P + ℏ − 1 P ∗ . (113) P=\sum_{r=1}^{n-1}\sum_{u=1}^{r}\sum_{v=1}^{r+1}\frac{x_{r,u}}{x_{r+1,v}}-\sum_{r=1}^{n-1}\sum_{u,v=1}^{r}\frac{x_{r,u}}{x_{r,v}},\qquad TX=P+\hbar^{-1}P^*. \tag*{(113)} P = r = 1 ∑ n − 1 u = 1 ∑ r v = 1 ∑ r + 1 x r + 1 , v x r , u − r = 1 ∑ n − 1 u , v = 1 ∑ r x r , v x r , u , T X = P + ℏ − 1 P ∗ . (113) The contribution of L r L_r L r det P \det P det P ( r + 1 ) − ( r − 1 ) = 2 (r+1)-(r-1)=2 ( r + 1 ) − ( r − 1 ) = 2 z # , r = z r / ℏ z_{\#,r}=z_r/\hbar z # , r = z r /ℏ ∣ q ∣ < 1 \lvert q\rvert<1 ∣ q ∣ < 1
φ ( w , q ) = ∏ d ≥ 0 ( 1 − q d w ) , B q ( w ) = φ ( q w , q ) φ ( ℏ w , q ) . \varphi(w,q)=\prod_{d\geq0}(1-q^d w),\qquad B_q(w)=\frac{\varphi(qw,q)}{\varphi(\hbar w,q)}. φ ( w , q ) = d ≥ 0 ∏ ( 1 − q d w ) , B q ( w ) = φ ( ℏ w , q ) φ ( q w , q ) . If P = ∑ w m w w P=\sum_w m_w w P = ∑ w m w w
F ( z , a , x ; q ) = ∏ r = 1 n − 1 exp ( log ( z r / ℏ ) log L r log q ) ∏ w B q ( w ) m w . (114) \mathcal{F}(z,a,x;q)=\prod_{r=1}^{n-1}\exp\left(\frac{\log(z_r/\hbar)\log L_r}{\log q}\right)\prod_w B_q(w)^{m_w}. \tag*{(114)} F ( z , a , x ; q ) = r = 1 ∏ n − 1 exp ( log q log ( z r /ℏ ) log L r ) w ∏ B q ( w ) m w . (114) The rational shift ratios of this phase remain the data of our construction at roots of unity. They define the finite sums below without evaluating the infinite products in (114) at such values of q q q
For generic q q q f f f
S Γ [ f ] = N Γ χ ( ζ ) ∫ Γ F ( z , a , x ; q ) f ( x ) ∏ α = 1 R d x α 2 π i x α , R = n ( n − 1 ) 2 . \mathcal{S}_{\Gamma}[f]=\mathcal{N}_{\Gamma}\chi(\zeta)\int_{\Gamma}\mathcal{F}(z,a,x;q)f(x)\prod_{\alpha=1}^{R}\frac{\mathrm{d}x_{\alpha}}{2\pi\mathrm{i}x_{\alpha}},\qquad R=\frac{n(n-1)}{2}. S Γ [ f ] = N Γ χ ( ζ ) ∫ Γ F ( z , a , x ; q ) f ( x ) α = 1 ∏ R 2 π i x α d x α , R = 2 n ( n − 1 ) . Here Γ \Gamma Γ χ \chi χ N Γ \mathcal{N}_{\Gamma} N Γ S Γ [ 1 ] \mathcal{S}_{\Gamma}[1] S Γ [ 1 ]
Let T r , u T_{r,u} T r , u x r , u x_{r,u} x r , u q q q x = x r , u x=x_{r,u} x = x r , u
Φ r , u = z r ℏ ∏ v = 1 r + 1 1 − ℏ x / x r + 1 , v 1 − q x / x r + 1 , v ∏ v = 1 r − 1 1 − x r − 1 , v / x 1 − ℏ x r − 1 , v / ( q x ) × ∏ 1 ≤ v ≤ r v ≠ u ( 1 − q x / x r , v ) ( 1 − ℏ x r , v / ( q x ) ) ( 1 − ℏ x / x r , v ) ( 1 − x r , v / x ) . (115) \Phi_{r,u}=\frac{z_r}{\hbar}\prod_{v=1}^{r+1}\frac{1-\hbar x/x_{r+1,v}}{1-qx/x_{r+1,v}}\prod_{v=1}^{r-1}\frac{1-x_{r-1,v}/x}{1-\hbar x_{r-1,v}/(qx)}
\times\prod_{\substack{1\leq v\leq r\\v\ne u}}\frac{(1-qx/x_{r,v})(1-\hbar x_{r,v}/(qx))}{(1-\hbar x/x_{r,v})(1-x_{r,v}/x)}. \tag*{(115)} Φ r , u = ℏ z r v = 1 ∏ r + 1 1 − q x / x r + 1 , v 1 − ℏ x / x r + 1 , v v = 1 ∏ r − 1 1 − ℏ x r − 1 , v / ( q x ) 1 − x r − 1 , v / x × 1 ≤ v ≤ r v = u ∏ ( 1 − ℏ x / x r , v ) ( 1 − x r , v / x ) ( 1 − q x / x r , v ) ( 1 − ℏ x r , v / ( q x )) . (115) Indeed, this expression follows by applying B q ( q w ) / B q ( w ) = ( 1 − ℏ w ) / ( 1 − q w ) B_q(qw)/B_q(w)=(1-\hbar w)/(1-qw) B q ( q w ) / B q ( w ) = ( 1 − ℏ w ) / ( 1 − q w ) P P P
We next consider a shift of the geometric Kähler coordinate z r z_r z r L r L_r L r exp ( log ζ n log L n / log q ) \exp(\log\zeta_n\log L_n/\log q) exp ( log ζ n log L n / log q ) L n L_n L n (1) , multiplication by a character with p i p_i p i ℏ ( n + 1 ) / 2 − i \hbar^{(n+1)/2-i} ℏ ( n + 1 ) /2 − i
ρ i ( x ) = ℏ ( n + 1 ) / 2 − i L i L i − 1 , 1 ≤ i ≤ n . (116) \rho_i(x)=\hbar^{(n+1)/2-i}\frac{L_i}{L_{i-1}},\qquad1\leq i\leq n. \tag*{(116)} ρ i ( x ) = ℏ ( n + 1 ) /2 − i L i − 1 L i , 1 ≤ i ≤ n . (116) The shift p i p_i p i z i z_i z i z i − 1 z_{i-1} z i − 1 ∏ i ρ i = a 1 ⋯ a n \prod_i\rho_i=a_1\cdots a_n ∏ i ρ i = a 1 ⋯ a n K K K τ ^ \widehat{\tau} τ τ \tau τ ⋆ \star ⋆ ⋆ γ \star\gamma ⋆ γ γ \gamma γ [7 ] (Theorem 4.6, equation (59)) identifies the tRS momentum on the Bethe level set with the operator
P d q K = ℏ ( n + 1 ) / 2 − d ⋆ ( det V d ⋆ det ( V d − 1 ∨ ) ) , 1 ≤ d ≤ n . (117) P_d^{\mathrm{qK}}=\hbar^{(n+1)/2-d}\star\left(\det\mathcal{V}_d\star\det(\mathcal{V}_{d-1}^{\vee})\right),\qquad1\leq d\leq n. \tag*{(117)} P d qK = ℏ ( n + 1 ) /2 − d ⋆ ( det V d ⋆ det ( V d − 1 ∨ ) ) , 1 ≤ d ≤ n . (117) where V 0 = 0 \mathcal{V}_0=0 V 0 = 0 V n = W \mathcal{V}_n=\mathcal{W} V n = W [7 ] (Theorem 3.4), the two quantum classes have Bethe eigenvalues L d L_d L d L d − 1 − 1 L_{d-1}^{-1} L d − 1 − 1 P d q K P_d^{\mathrm{qK}} P d qK ρ d \rho_d ρ d [7 ] is ℏ − 1 \hbar^{-1} ℏ − 1 ℏ − n / 2 \hbar^{-n/2} ℏ − n /2 a 1 , … , a n a_1,\ldots,a_n a 1 , … , a n B p , α = 1 B_{p,\alpha}=1 B p , α = 1 x p x^p x p ( z p , a p , ℏ p ) (z^p,a^p,\hbar^p) ( z p , a p , ℏ p ) (121) below are therefore quantum multiplication eigenvalues at these powered parameters. This does not require x x x
The internal and external ratios obey the flatness identities
( T β Φ α ) Φ β = ( T α Φ β ) Φ α , (T_\beta\Phi_\alpha)\Phi_\beta=(T_\alpha\Phi_\beta)\Phi_\alpha, ( T β Φ α ) Φ β = ( T α Φ β ) Φ α , ( p j ρ i ) ρ j = ( p i ρ j ) ρ i , ( p i Φ α ) ρ i = ( T α ρ i ) Φ α . (118) (p_j\rho_i)\rho_j=(p_i\rho_j)\rho_i,\qquad(p_i\Phi_\alpha)\rho_i=(T_\alpha\rho_i)\Phi_\alpha. \tag*{(118)} ( p j ρ i ) ρ j = ( p i ρ j ) ρ i , ( p i Φ α ) ρ i = ( T α ρ i ) Φ α . (118) The first relation follows from commuting the internal shifts of the phase. The second follows from the independence of ρ i \rho_i ρ i z \boldsymbol{z} z α = ( r , u ) \alpha=(r,u) α = ( r , u ) q δ i r − δ i , r + 1 q^{\delta_{ir}-\delta_{i,r+1}} q δ i r − δ i , r + 1
Now set q = ω p q=\omega_p q = ω p R = n ( n − 1 ) / 2 R=n(n-1)/2 R = n ( n − 1 ) /2 x r , u x_{r,u} x r , u p R p^R p R
B p , α = ∏ d = 0 p − 1 T α d Φ α , λ i = ρ i p . (119) \mathcal{B}_{p,\alpha}=\prod_{d=0}^{p-1}T_\alpha^d\Phi_\alpha,\qquad\lambda_i=\rho_i^p. \tag*{(119)} B p , α = d = 0 ∏ p − 1 T α d Φ α , λ i = ρ i p . (119) For one internal variable, B p , α \mathcal{B}_{p,\alpha} B p , α B p \mathcal{B}_p B p § 2 . These norms are invariant under both internal and Kähler shifts. For an internal shift this follows from the telescoping product
T β B p , α B p , α = ∏ d = 0 p − 1 T α d + 1 Φ β T α d Φ β = T α p Φ β Φ β = 1. \frac{T_\beta\mathcal{B}_{p,\alpha}}{\mathcal{B}_{p,\alpha}}=\prod_{d=0}^{p-1}\frac{T_\alpha^{d+1}\Phi_\beta}{T_\alpha^d\Phi_\beta}=\frac{T_\alpha^p\Phi_\beta}{\Phi_\beta}=1. B p , α T β B p , α = d = 0 ∏ p − 1 T α d Φ β T α d + 1 Φ β = Φ β T α p Φ β = 1. Here we used flatness. The mixed relation in (118) gives the same calculation for a Kähler shift. Using ∏ d = 0 p − 1 ( 1 − q d w ) = 1 − w p \prod_{d=0}^{p-1}(1-q^dw)=1-w^p ∏ d = 0 p − 1 ( 1 − q d w ) = 1 − w p
B p , ( r , u ) = Φ r , u ∣ q = 1 ( z p , a p , x p ; ℏ p ) . (120) \mathcal{B}_{p,(r,u)}=\Phi_{r,u}\big|_{q=1}\left(\boldsymbol{z}^p,\boldsymbol{a}^p,\boldsymbol{x}^p;\hbar^p\right). \tag*{(120)} B p , ( r , u ) = Φ r , u q = 1 ( z p , a p , x p ; ℏ p ) . (120) Hence B p , ( r , u ) = 1 \mathcal{B}_{p,(r,u)}=1 B p , ( r , u ) = 1
λ i = ℏ p ( ( n + 1 ) / 2 − i ) ( L i L i − 1 ) p , ∏ i = 1 r λ i = ℏ p r ( n − r ) / 2 L r p . (121) \lambda_i=\hbar^{p((n+1)/2-i)}\left(\frac{L_i}{L_{i-1}}\right)^p,\qquad\prod_{i=1}^r\lambda_i=\hbar^{pr(n-r)/2}L_r^p. \tag*{(121)} λ i = ℏ p (( n + 1 ) /2 − i ) ( L i − 1 L i ) p , i = 1 ∏ r λ i = ℏ p r ( n − r ) /2 L r p . (121) In particular, the monodromy of the geometric shift z r ↦ q z r z_r\mapsto qz_r z r ↦ q z r L r p L_r^p L r p ℏ \hbar ℏ (121) come from the centered homogeneous gauge.
Bethe Equations as Closure Conditions
We shall now interpret the Bethe equations as closure conditions for the finite internal grid. They make the phase weights periodic and hence allow twisted total differences to cancel under summation.
The summation variables are the multiplicative tautological roots x r , u x_{r,u} x r , u 1 ≤ u ≤ r < n 1\leq u\leq r<n 1 ≤ u ≤ r < n (114) . There are R = n ( n − 1 ) / 2 R=n(n-1)/2 R = n ( n − 1 ) /2 x α x_\alpha x α α = ( r , u ) \alpha=(r,u) α = ( r , u ) x 0 = ( x r , u 0 ) \boldsymbol{x}^0=(x^0_{r,u}) x 0 = ( x r , u 0 ) 0 0 0 K K K T ∗ F l n T^*\mathrm{Fl}_n T ∗ Fl n
For the internal shifts we fix the Kähler parameters z \boldsymbol{z} z ζ \boldsymbol{\zeta} ζ a \boldsymbol{a} a ℏ \hbar ℏ q = ω p q=\omega_p q = ω p x n , i = a i x_{n,i}=a_i x n , i = a i T r , u T_{r,u} T r , u x r , u x_{r,u} x r , u q x r , u qx_{r,u} q x r , u ζ \boldsymbol{\zeta} ζ z \boldsymbol{z} z a \boldsymbol{a} a ℏ \hbar ℏ
O = { x j = ( q j α x α 0 ) α = 1 R : j ∈ ( Z / p Z ) R } . (122) \mathcal{O}=\left\{\boldsymbol{x}_j=\left(q^{j^\alpha}x^0_\alpha\right)_{\alpha=1}^R:j\in(\mathbb{Z}/p\mathbb{Z})^R\right\}. \tag*{(122)} O = { x j = ( q j α x α 0 ) α = 1 R : j ∈ ( Z / p Z ) R } . (122) The p R p^R p R x 0 \boldsymbol{x}^0 x 0 p p p Φ α : O → C × \Phi_\alpha:\mathcal{O}\to\mathbb{C}^{\times} Φ α : O → C × x → T α x \boldsymbol{x}\to T_\alpha\boldsymbol{x} x → T α x
Φ α ( x ) Φ β ( T α x ) = Φ β ( x ) Φ α ( T β x ) . (123) \Phi_\alpha(\boldsymbol{x})\Phi_\beta(T_\alpha\boldsymbol{x})=\Phi_\beta(\boldsymbol{x})\Phi_\alpha(T_\beta\boldsymbol{x}). \tag*{(123)} Φ α ( x ) Φ β ( T α x ) = Φ β ( x ) Φ α ( T β x ) . (123) Consequently transport on the lifted grid is independent of the order of the steps. After p p p α \alpha α B p , α B_{p,\alpha} B p , α
For a function f f f O \mathcal{O} O
( Δ α f ) ( x ) = Φ α ( x ) f ( T α x ) − f ( x ) . (124) (\Delta_\alpha f)(x) = \Phi_\alpha(x)f(T_\alpha x) - f(x). \tag*{(124)} ( Δ α f ) ( x ) = Φ α ( x ) f ( T α x ) − f ( x ) . (124) Suppose the weights satisfy μ ( T α x ) = Φ α ( x ) μ ( x ) \mu(T_\alpha x) = \Phi_\alpha(x)\mu(x) μ ( T α x ) = Φ α ( x ) μ ( x ) μ ( x ) Δ α f ( x ) \mu(x)\Delta_\alpha f(x) μ ( x ) Δ α f ( x ) f f f
Theorem 3.12. Let O \mathcal{O} O (122) , with nonzero edge factors Φ α : O → C × \Phi_\alpha:\mathcal{O}\to\mathbb{C}^{\times} Φ α : O → C × (123) . The following conditions are equivalent:
B p , α ( x 0 ) = 1 B_{p,\alpha}(x^0)=1 B p , α ( x 0 ) = 1
there is a nonzero weighted evaluation functional
S [ f ] = ∑ x ∈ O μ ( x ) f ( x ) S[f]=\sum_{x\in\mathcal{O}}\mu(x)f(x) S [ f ] = x ∈ O ∑ μ ( x ) f ( x ) such that S [ Δ α f ] = 0 S[\Delta_\alpha f]=0 S [ Δ α f ] = 0 α \alpha α f f f O \mathcal{O} O
Such a functional is unique up to a scalar and has no zero weights. The same criterion is obtained by testing only rational functions regular on the orbit.
Proof. Comparing the coefficient of f ( T α x ) f(T_\alpha x) f ( T α x ) S [ Δ α f ] = 0 S[\Delta_\alpha f]=0 S [ Δ α f ] = 0
μ ( T α x ) = Φ α ( x ) μ ( x ) . (125) \mu(T_\alpha x)=\Phi_\alpha(x)\mu(x). \tag*{(125)} μ ( T α x ) = Φ α ( x ) μ ( x ) . (125) The orbit is connected and all edge factors are nonzero, so one nonzero weight determines nonzero weights at every point. Iteration of (125) around a coordinate loop forces B p , α = 1 B_{p,\alpha}=1 B p , α = 1 B p , α = 1 B_{p,\alpha}=1 B p , α = 1 μ ( x 0 ) = 1 \mu(x^0)=1 μ ( x 0 ) = 1 O \mathcal{O} O S [ Δ α f ] = 0 S[\Delta_\alpha f]=0 S [ Δ α f ] = 0 □ \square □
Geometrically, the Bethe equations make the holonomy of the rank-one local system of internal transport trivial. The weighted orbit is then a closed cycle with coefficients in this local system, and the theorem is a finite version of Stokes’ formula. It gives a general formulation of the cancellation mechanism in [14 ] (Sections 3 and 5), using the Bethe equations and discrete exactness already present there.
To write the weights explicitly, order the internal indices as α = 1 , … , R \alpha=1,\ldots,R α = 1 , … , R
μ j ( z , a ) = ∏ α = 1 R ∏ d = 0 j α − 1 Φ α ( T 1 j 1 ⋯ T α − 1 j α − 1 T α d x 0 ; z , a ) , 0 ≤ j α < p . (126) \mu_j(z,a)=\prod_{\alpha=1}^{R}\prod_{d=0}^{j_\alpha-1}\Phi_\alpha\left(T_1^{j_1}\cdots T_{\alpha-1}^{j_{\alpha-1}}T_\alpha^d x^0;z,a\right),\qquad0\le j_\alpha<p. \tag*{(126)} μ j ( z , a ) = α = 1 ∏ R d = 0 ∏ j α − 1 Φ α ( T 1 j 1 ⋯ T α − 1 j α − 1 T α d x 0 ; z , a ) , 0 ≤ j α < p . (126) Here μ 0 = 1 \mu_0=1 μ 0 = 1 j α j_\alpha j α B p = ∏ d = 0 p − 1 T d Φ B_p=\prod_{d=0}^{p-1}T^d\Phi B p = ∏ d = 0 p − 1 T d Φ
∑ j = 0 p − 1 μ j ( Φ ( T j x 0 ) f ( T j + 1 x 0 ) − f ( T j x 0 ) ) = ( B p − 1 ) f ( x 0 ) . (127) \sum_{j=0}^{p-1}\mu_j\left(\Phi(T^j x^0)f(T^{j+1}x^0)-f(T^j x^0)\right)=(B_p-1)f(x^0). \tag*{(127)} j = 0 ∑ p − 1 μ j ( Φ ( T j x 0 ) f ( T j + 1 x 0 ) − f ( T j x 0 ) ) = ( B p − 1 ) f ( x 0 ) . (127) For several directions we apply this calculation to each coordinate line, keeping the transverse weights fixed. The opposite faces of the weighted cube cancel precisely when their coordinate holonomies are one.
Finite Sums and the Frobenius tRS Equations
The closed weights define finite vertex sums with insertions. We shall use their Kähler multipliers and the classical tRS/Bethe correspondence to obtain the tRS equations precomposed with the Frobenius action.
Let ζ ( m ) = q m ζ 0 \zeta(\mathbf{m})=q^{m}\zeta^{0} ζ ( m ) = q m ζ 0 z ( m ) z(\mathbf{m}) z ( m ) (1) . We keep a \mathbf{a} a x 0 \mathbf{x}^{0} x 0
G ( m ) = ∏ i ρ i ( x 0 ) m i . G(\mathbf{m})=\prod_i \rho_i(\mathbf{x}^{0})^{m_i}. G ( m ) = i ∏ ρ i ( x 0 ) m i . For a solution of B p , α = 1 \mathcal{B}_{p,\alpha}=1 B p , α = 1
S m [ f ] = G ( m ) ∑ j ∈ { 0 , … , p − 1 } R μ j ( z ( m ) , a ) f ( T j x 0 ; ζ ( m ) , a ) , R = n ( n − 1 ) 2 . (128) \mathcal{S}_{m}[f]=G(\mathbf{m})\sum_{j\in\{0,\ldots,p-1\}^{R}}\mu_j(z(\mathbf{m}),\mathbf{a})f(T^j\mathbf{x}^{0};\zeta(\mathbf{m}),\mathbf{a}),\qquad R=\frac{n(n-1)}{2}. \tag*{(128)} S m [ f ] = G ( m ) j ∈ { 0 , … , p − 1 } R ∑ μ j ( z ( m ) , a ) f ( T j x 0 ; ζ ( m ) , a ) , R = 2 n ( n − 1 ) . (128) Here f f f f = 1 f=1 f = 1 S m [ 1 ] \mathcal{S}_{m}[1] S m [ 1 ] μ j \mu_j μ j G G G f f f q q q p i p f = f p_i^pf=f p i p f = f
S m [ Δ α f ] = 0 , \mathcal{S}_{m}[\Delta_\alpha f]=0, S m [ Δ α f ] = 0 , S m + e i [ f ] = S m [ ρ i p i f ] , S m + p e i [ f ] = λ i S m [ f ] . (129) \mathcal{S}_{m+e_i}[f]=\mathcal{S}_{m}[\rho_i p_i f],\qquad\mathcal{S}_{m+pe_i}[f]=\lambda_i\mathcal{S}_{m}[f]. \tag*{(129)} S m + e i [ f ] = S m [ ρ i p i f ] , S m + p e i [ f ] = λ i S m [ f ] . (129) These relations connect the internal graph of vertices x j \mathbf{x}_j x j Φ α \Phi_\alpha Φ α m ∈ Z n \mathbf{m}\in\mathbb{Z}^n m ∈ Z n i i i ρ i \rho_i ρ i (118) makes this action compatible with internal transport and exact differences. Notice that even though ζ ( m + p e i ) = ζ ( m ) \zeta(\mathbf{m}+p e_i)=\zeta(\mathbf{m}) ζ ( m + p e i ) = ζ ( m ) λ i \lambda_i λ i
Theorem 3.13. Let q = ω p q=\omega_p q = ω p p p p x 0 \mathbf{x}^{0} x 0 B p , α = 1 \mathcal{B}_{p,\alpha}=1 B p , α = 1 α \alpha α λ i = ρ i ( x 0 ) p \lambda_i=\rho_i(\mathbf{x}^{0})^p λ i = ρ i ( x 0 ) p 1 ≤ r ≤ n 1\le r\le n 1 ≤ r ≤ n
h r ( ζ p , λ ; ℏ p ) = e r ( a p ) . (130) h_r(\zeta^p,\lambda;\hbar^p)=e_r(\mathbf{a}^p). \tag*{(130)} h r ( ζ p , λ ; ℏ p ) = e r ( a p ) . (130) If f f f p i p f = f p_i^pf=f p i p f = f V f ( m ) = S m [ f ] V_f(\mathbf{m})=\mathcal{S}_{m}[f] V f ( m ) = S m [ f ]
p i p V f = λ i V f , p^p_iV_f=\lambda_iV_f, p i p V f = λ i V f , H r [ p ] V f = A p , r ( H 1 , … , H n ) V f = e r ( a p ) V f . (131) H_r^{[p]}V_f=A_{p,r}(H_1,\ldots,H_n)V_f=e_r(\mathbf{a}^p)V_f. \tag*{(131)} H r [ p ] V f = A p , r ( H 1 , … , H n ) V f = e r ( a p ) V f . (131) These identities require no ordinary tRS eigenvalue equation and remain valid when V f = 0 V_f=0 V f = 0 V f V_f V f
Proof. Equation (120) identifies B p , α = 1 \mathcal{B}_{p,\alpha}=1 B p , α = 1 x p \mathbf{x}^p x p ( z p , a p , ℏ p ) (z^p,\mathbf{a}^p,\hbar^p) ( z p , a p , ℏ p ) [7 ] (Theorems 4.6 and 4.9). With the normalization following (117) , its momenta are given by (121) , which proves (130) without using a difference equation for V f V_f V f z r = ζ r p / ζ r + 1 p z_r=\zeta_r^p/\zeta_{r+1}^p z r = ζ r p / ζ r + 1 p q p = 1 q^p=1 q p = 1
The full-cycle relation in (129) is the first equation of (131) . Since both the coefficients of H r [ p ] H_r^{[p]} H r [ p ] λ i \lambda_i λ i
H r [ p ] V f = ℏ − p r ( n − r ) / 2 ∑ ∣ I ∣ = r ∏ i ∈ I j ∉ I ℏ p ζ i p − ζ j p ζ i p − ζ j p ∏ i ∈ I λ i V f = h r ( ζ p , λ ; ℏ p ) V f . \begin{aligned}
H_r^{[p]}V_f=\hbar^{-pr(n-r)/2}\sum_{\lvert I\rvert=r}\prod_{\substack{i\in I\\j\notin I}}\frac{\hbar^p\zeta_i^p-\zeta_j^p}{\zeta_i^p-\zeta_j^p}\prod_{i\in I}\lambda_iV_f \\
&= h_r(\zeta^p,\lambda;\hbar^p)V_f.
\end{aligned} H r [ p ] V f = ℏ − p r ( n − r ) /2 ∣ I ∣ = r ∑ i ∈ I j ∈ / I ∏ ζ i p − ζ j p ℏ p ζ i p − ζ j p i ∈ I ∏ λ i V f = h r ( ζ p , λ ; ℏ p ) V f . Now (130) and Theorem 3.4 give the second equation. The calculation does not involve division by V f V_f V f
We have thus obtained the Frobenius equations from Bethe closure for every regular insertion, including the unit class and the descendants below. This gives a finite-sum construction alongside the eigenfunction implication of Corollary 3.5 .
Let us illustrate the cancellation for T ∗ F l 3 T^*\mathrm{Fl}_3 T ∗ Fl 3 x 1 , 1 , x 2 , 1 , x 2 , 2 x_{1,1},x_{2,1},x_{2,2} x 1 , 1 , x 2 , 1 , x 2 , 2 p = 2 p=2 p = 2 μ 0 , μ 1 \mu_0,\mu_1 μ 0 , μ 1 Φ 0 , Φ 1 \Phi_0,\Phi_1 Φ 0 , Φ 1 f 0 , f 1 f_0,f_1 f 0 , f 1
μ 0 ( Φ 0 f 1 − f 0 ) + μ 1 ( Φ 1 f 0 − f 1 ) = ( μ 0 Φ 0 − μ 1 ) f 1 + ( μ 1 Φ 1 − μ 0 ) f 0 = 0. \mu_0(\Phi_0f_1-f_0)+\mu_1(\Phi_1f_0-f_1)=(\mu_0\Phi_0-\mu_1)f_1+(\mu_1\Phi_1-\mu_0)f_0=0. μ 0 ( Φ 0 f 1 − f 0 ) + μ 1 ( Φ 1 f 0 − f 1 ) = ( μ 0 Φ 0 − μ 1 ) f 1 + ( μ 1 Φ 1 − μ 0 ) f 0 = 0. The transport relation makes the first coefficient zero, and the Bethe equation Φ 0 Φ 1 = 1 \Phi_0\Phi_1=1 Φ 0 Φ 1 = 1
λ 1 = ℏ p x 1 , 1 p , λ 2 = ( x 2 , 1 x 2 , 2 x 1 , 1 ) p , λ 3 = ℏ − p ( a 1 a 2 a 3 x 2 , 1 x 2 , 2 ) p . \lambda_1=\hbar^p x_{1,1}^p,\qquad\lambda_2=\left(\frac{x_{2,1}x_{2,2}}{x_{1,1}}\right)^p,\qquad\lambda_3=\hbar^{-p}\left(\frac{a_1a_2a_3}{x_{2,1}x_{2,2}}\right)^p. λ 1 = ℏ p x 1 , 1 p , λ 2 = ( x 1 , 1 x 2 , 1 x 2 , 2 ) p , λ 3 = ℏ − p ( x 2 , 1 x 2 , 2 a 1 a 2 a 3 ) p . Explicit Descendant Components
We now express the unit-insertion vertex as a sum of explicit descendant components. The descendants form a partition of unity, and their finite sums satisfy the Frobenius equations of Theorem 3.13 .
In [9 ] (Theorem 5.11, equations (5.25)–(5.26)), stable-envelope insertions give the components of a geometric qKZ solution. We use the same interpretation of components as vertices with insertions, taking interpolation descendants to decompose the finite closed-cycle sum.
With L 0 = 1 L_0=1 L 0 = 1
β r = L r L r − 1 , 1 ≤ r < n . \beta_r=\frac{L_r}{L_{r-1}},\qquad1\le r<n. β r = L r − 1 L r , 1 ≤ r < n . For pairwise distinct equivariant parameters a 1 , … , a n a_1,\ldots,a_n a 1 , … , a n σ ∈ S n \sigma\in S_n σ ∈ S n
w σ ( x ; a ) = ∏ r = 1 n − 1 ∏ s = r + 1 n β r − a σ ( s ) a σ ( r ) − a σ ( s ) . (132) w_\sigma(x;a)=\prod_{r=1}^{n-1}\prod_{s=r+1}^{n}\frac{\beta_r-a_{\sigma(s)}}{a_{\sigma(r)}-a_{\sigma(s)}}. \tag*{(132)} w σ ( x ; a ) = r = 1 ∏ n − 1 s = r + 1 ∏ n a σ ( r ) − a σ ( s ) β r − a σ ( s ) . (132) By a descendant insertion we mean a symmetric Laurent polynomial in the tautological roots, with coefficients in the localized equivariant coefficient field, inserted into the vertex integral or finite sum. Here β r = L r / L r − 1 \beta_r=L_r/L_{r-1} β r = L r / L r − 1 det V r ⊗ ( det V r − 1 ) − 1 \det\mathcal{V}_r\otimes(\det\mathcal{V}_{r-1})^{-1} det V r ⊗ ( det V r − 1 ) − 1 w σ w_\sigma w σ a i a_i a i a σ ( r ) − a σ ( s ) a_{\sigma(r)}-a_{\sigma(s)} a σ ( r ) − a σ ( s ) rational descendant . The expression is symmetric in the roots of each V r V_r V r a i a_i a i S m S_m S m
V ( m ) = S m [ 1 ] , I σ ( m ) = S m [ w σ ] . (133) V(m)=S_m[1],\qquad I_\sigma(m)=S_m[w_\sigma]. \tag*{(133)} V ( m ) = S m [ 1 ] , I σ ( m ) = S m [ w σ ] . (133) Since w σ w_\sigma w σ a i a_i a i Theorem 3.13 to each I σ I_\sigma I σ V V V
H r [ p ] I σ = e r ( a p ) I σ , p ^ r p I σ = λ i I σ . H_r^{[p]}I_\sigma=e_r(\mathbf{a}^p)I_\sigma,\qquad\hat{p}_r^p I_\sigma=\lambda_i I_\sigma. H r [ p ] I σ = e r ( a p ) I σ , p ^ r p I σ = λ i I σ . Proposition 3.14. The functions (132) obey the rational identity
∑ σ ∈ S n w σ = 1. (134) \sum_{\sigma\in S_n}w_\sigma=1. \tag*{(134)} σ ∈ S n ∑ w σ = 1. (134) Their classes give the fixed-point interpolation basis of localized K T ( T ∗ F l n ) K_T(T^*\mathrm{Fl}_n) K T ( T ∗ Fl n ) β 1 , … , β n − 1 \beta_1,\ldots,\beta_{n-1} β 1 , … , β n − 1 deg β r ≤ n − r \deg_{\beta_r}\le n-r deg β r ≤ n − r (129) , we have
V = ∑ σ ∈ S n I σ , I σ = P σ ( p ) V , (135) V=\sum_{\sigma\in S_n}I_\sigma,\qquad I_\sigma=\mathcal{P}_\sigma(p)V, \tag*{(135)} V = σ ∈ S n ∑ I σ , I σ = P σ ( p ) V , (135) where the difference polynomial is given explicitly by
P σ ( p ) = ∏ r = 1 n − 1 ∏ s = r + 1 n ℏ r − ( n + 1 ) / 2 p r − a σ ( s ) a σ ( r ) − a σ ( s ) . (136) \mathcal{P}_\sigma(p)=\prod_{r=1}^{n-1}\prod_{s=r+1}^{n}\frac{\hbar^{r-(n+1)/2}p_r-a_{\sigma(s)}}{a_{\sigma(r)}-a_{\sigma(s)}}. \tag*{(136)} P σ ( p ) = r = 1 ∏ n − 1 s = r + 1 ∏ n a σ ( r ) − a σ ( s ) ℏ r − ( n + 1 ) /2 p r − a σ ( s ) . (136) Thus ∑ σ P σ ( p ) = 1 \sum_\sigma\mathcal{P}_\sigma(p)=1 ∑ σ P σ ( p ) = 1 is an identity of difference operators.
Proof. For a finite set A A A
∑ a ∈ A ∏ c ∈ A ∖ { a } b − c a − c = 1. \sum_{a\in A}\prod_{c\in A\setminus\{a\}}\frac{b-c}{a-c}=1. a ∈ A ∑ c ∈ A ∖ { a } ∏ a − c b − c = 1. Let us fix σ ( 1 ) \sigma(1) σ ( 1 ) (134) . Applying this identity recursively to the remaining parameters and the variables β 2 , … , β n − 1 \beta_2,\ldots,\beta_{n-1} β 2 , … , β n − 1 β 1 \beta_1 β 1 (134) . No relations among the β r \beta_r β r
At the fixed flag τ \tau τ β r = a τ ( r ) \beta_r=a_{\tau(r)} β r = a τ ( r ) σ ≠ τ \sigma\ne\tau σ = τ r r r a τ ( r ) a_{\tau(r)} a τ ( r ) a σ ( r + 1 ) , … , a σ ( n ) a_{\sigma(r+1)},\ldots,a_{\sigma(n)} a σ ( r + 1 ) , … , a σ ( n ) w σ ∣ τ w_\sigma|_\tau w σ ∣ τ w τ ∣ τ = 1 w_\tau|_\tau=1 w τ ∣ τ = 1 n ! n! n ! w σ w_\sigma w σ ∏ r = 1 n − 1 ( n − r + 1 ) = n ! \prod_{r=1}^{n-1}(n-r+1)=n! ∏ r = 1 n − 1 ( n − r + 1 ) = n !
Applying the linear functional to the partition of unity gives the first equality in (135) . Its external transport rule, together with ρ r = ℏ ( n + 1 ) / 2 − r β r \rho_r=\hbar^{(n+1)/2-r}\beta_r ρ r = ℏ ( n + 1 ) /2 − r β r β r \beta_r β r ℏ r − ( n + 1 ) / 2 p r \hbar^{r-(n+1)/2}p_r ℏ r − ( n + 1 ) /2 p r ρ r \rho_r ρ r (136) , and substituting into (134) proves the last assertion. □
For n = 2 n=2 n = 2 ( x − a 2 ) / ( a 1 − a 2 ) (x-a_2)/(a_1-a_2) ( x − a 2 ) / ( a 1 − a 2 ) ( x − a 1 ) / ( a 2 − a 1 ) (x-a_1)/(a_2-a_1) ( x − a 1 ) / ( a 2 − a 1 ) I σ I_\sigma I σ σ \sigma σ Theorem 3.13 , this construction gives
V = ∑ σ ∈ S n I σ , H r [ p ] V = e r ( a p ) V , p i p V = λ i V . V = \sum_{\sigma\in S_n} I_\sigma,\qquad H_r^{[p]}V = e_r(\mathbf{a}^p)V,\qquad p_i^pV = \lambda_iV. V = σ ∈ S n ∑ I σ , H r [ p ] V = e r ( a p ) V , p i p V = λ i V . Each component I σ I_\sigma I σ H r V = e r ( a ) V H_rV=e_r(a)V H r V = e r ( a ) V Theorem 3.4 . This is Corollary 3.5 , valid in every rank and at every root order.
References
Beijing Institute for Mathematical Sciences and Applications, Beijing, China
Email address: peter.koroteev@gmail.com
