Ikhlef–Morin-Duchesne seed products

Abstract

Uniform product evaluations of the periodic Temperley–Lieb seed recurrences.

Definition 1.1 (The q-number).

$$\forall q:\mathbb{C},\forall k:\mathbb{Z},\mathrm{qbr}\left(q,k\right)=\frac{q^k-q^{-k}}{q-q^{-1}}$$

Formalization. D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.qbr (✓ std3).

Citation. Yacine Ikhlef and Alexi Morin-Duchesne (2024). Fusion of irreducible modules in the periodic Temperley–Lieb algebra. DOI: 10.21468/SciPostPhys.17.5.132. URL: https://arxiv.org/abs/2312.14837v2.

Commentary.

Equation (2.14), p. 11, writes [k] = (q^k − q^{−k})/(q − q^{−1}). Here k is an integer and q is complex; all divisions are complex divisions.

Definition 1.2 (The symmetric q-power).

$$\forall q:\mathbb{C},\forall k:\mathbb{Z},\mathrm{qcb}\left(q,k\right)=q^k+q^{-k}$$

Formalization. D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.qcb (✓ std3).

Citation. Yacine Ikhlef and Alexi Morin-Duchesne (2024). Fusion of irreducible modules in the periodic Temperley–Lieb algebra. DOI: 10.21468/SciPostPhys.17.5.132. URL: https://arxiv.org/abs/2312.14837v2.

Commentary.

Proposition C.1 (p. 61): “where α_p = q^p + q^{−p} and {m} = q^m + q^{−m}.” The argument k ranges over the integers.

Definition 1.3 (A variable and its inverse).

$$\forall z:\mathbb{C},\mathrm{al}\left(z\right)=z+z^{-1}$$

Formalization. D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.al (✓ std3).

Citation. Yacine Ikhlef and Alexi Morin-Duchesne (2024). Fusion of irreducible modules in the periodic Temperley–Lieb algebra. DOI: 10.21468/SciPostPhys.17.5.132. URL: https://arxiv.org/abs/2312.14837v2.

Commentary.

Proposition C.1 (p. 61): “where X^{(p)}_m(ω, x, y) is a polynomial in the variables α_a = x + x^{−1}, α_b = y + y^{−1} and α_ω = ω + ω^{−1}, that satisfies the recursion relation for m ⩾ p + 1”. The function al sends z to z + z⁻¹.

Definition 1.4 (The positive-depth seed recurrence).

$$\forall q:\mathbb{C},\forall x:\mathbb{C},\forall y:\mathbb{C},\forall w:\mathbb{C},\forall p:\mathbb{N},(\mathrm{seedX}\left(q,x,y,w,p,0\right)=1)\wedge ((\mathrm{seedX}\left(q,x,y,w,p,1\right)=\mathrm{al}\left(x\right)+\frac{\mathrm{al}\left(y\right)\cdot \mathrm{al}\left(w\right)}{\mathrm{qcb}\left(q,\mathrm{Int}.\mathrm{ofNat}(p)+1\right)})\wedge (\forall n:\mathbb{N},\mathrm{let}k:\mathbb{Z}:=\mathrm{Int}.\mathrm{ofNat}(p)+\mathrm{Int}.\mathrm{ofNat}(n)+1;\mathrm{seedX}\left(q,x,y,w,p,n+2\right)=(\mathrm{al}\left(x\right)+\frac{\mathrm{al}\left(q^p\right)\cdot \mathrm{al}\left(y\right)\cdot \mathrm{al}\left(w\right)}{\mathrm{qcb}\left(q,k\right)\cdot \mathrm{qcb}\left(q,k+1\right)})\cdot \mathrm{seedX}\left(q,x,y,w,p,n+1\right)+\frac{(\mathrm{al}{\left(q^p\right)}^2-\mathrm{qcb}{\left(q,k\right)}^2)\cdot (\mathrm{al}{\left(y\right)}^2-\mathrm{qcb}{\left(q,k\right)}^2)\cdot (\mathrm{al}{\left(w\right)}^2-\mathrm{qcb}{\left(q,k\right)}^2)}{(q-q^{-1}{)}^2\cdot \mathrm{qbr}\left(q,2\cdot k-1\right)\cdot \mathrm{qbr}\left(q,2\cdot k+1\right)\cdot \mathrm{qcb}{\left(q,k\right)}^2}\cdot \mathrm{seedX}\left(q,x,y,w,p,n\right)))$$

Formalization. D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.seedX (✓ std3).

Citation. Yacine Ikhlef and Alexi Morin-Duchesne (2024). Fusion of irreducible modules in the periodic Temperley–Lieb algebra. DOI: 10.21468/SciPostPhys.17.5.132. URL: https://arxiv.org/abs/2312.14837v2.

Commentary.

Proposition C.1 (p. 61): “where X^{(p)}m(ω, x, y) is a polynomial in the variables α_a = x + x^{−1}, α_b = y + y^{−1} and α_ω = ω + ω^{−1}, that satisfies the recursion relation for m ⩾ p + 1”. Equations (C.3)–(C.4) give X{m+1} = (α_a + α_pα_bα_ω/({m}{m+1})) X_m + (α_p² − {m}²)(α_b² − {m}²)(α_ω² − {m}²)/((q − q⁻¹)²[2m − 1][2m + 1]{m}²) X_{m−1}, with X_p = 1 and X_{p+1} = α_a + α_bα_ω/{p + 1}. The expression seedX q x y w p n denotes X^{(p)}_{p+n}(w,x,y). The recurrence index is k = p+n+1; Int.ofNat displays each natural-to-integer coercion.

Definition 1.5 (The zero-depth seed recurrence).

$$\forall q:\mathbb{C},\forall x:\mathbb{C},\forall y:\mathbb{C},\forall w:\mathbb{C},(\mathrm{seedSigma}\left(q,x,y,w,0\right)=1)\wedge ((\mathrm{seedSigma}\left(q,x,y,w,1\right)=\mathrm{al}\left(x\right)+\frac{\mathrm{al}\left(y\right)\cdot \mathrm{al}\left(w\right)}{\mathrm{qcb}\left(q,1\right)})\wedge (\forall n:\mathbb{N},\mathrm{let}k:\mathbb{Z}:=\mathrm{Int}.\mathrm{ofNat}(n)+1;\mathrm{seedSigma}\left(q,x,y,w,n+2\right)=(\mathrm{al}\left(x\right)+\frac{2\cdot \mathrm{al}\left(y\right)\cdot \mathrm{al}\left(w\right)}{\mathrm{qcb}\left(q,k\right)\cdot \mathrm{qcb}\left(q,k+1\right)})\cdot \mathrm{seedSigma}\left(q,x,y,w,n+1\right)-\frac{\mathrm{qbr}{\left(q,k\right)}^2\cdot (\mathrm{al}{\left(y\right)}^2-\mathrm{qcb}{\left(q,k\right)}^2)\cdot (\mathrm{al}{\left(w\right)}^2-\mathrm{qcb}{\left(q,k\right)}^2)}{\mathrm{qbr}\left(q,2\cdot k+1\right)\cdot \mathrm{qbr}\left(q,2\cdot k-1\right)\cdot \mathrm{qcb}{\left(q,k\right)}^2}\cdot \mathrm{seedSigma}\left(q,x,y,w,n\right)))$$

Formalization. D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.seedSigma (✓ std3).

Citation. Yacine Ikhlef and Alexi Morin-Duchesne (2024). Fusion of irreducible modules in the periodic Temperley–Lieb algebra. DOI: 10.21468/SciPostPhys.17.5.132. URL: https://arxiv.org/abs/2312.14837v2.

Commentary.

Proposition C.2 (p. 65): “The seed states σ^{(0)}{0,m} with m ⩾ 0 are polynomials in α_a = x + x^{−1}, α_b = y + y^{−1} and α{ab} = w + w^{−1} that satisfy the linear system for m ⩾ 1”. Equations (C.24)–(C.25) give the displayed recurrence and its two initial values, independently of seedX.

Definition 1.6 (The symmetric finite product).

$$\forall q:\mathbb{C},\forall x:\mathbb{C},\forall w:\mathbb{C},\forall epsilon:\mathbb{C},\forall N:\mathbb{N},\mathrm{seedProduct}\left(q,x,w,epsilon,N\right)=(w\cdot x{)}^{-\mathrm{Int}.\mathrm{ofNat}(N)}\cdot \prod _{j\in \mathrm{Finset}.\mathrm{range}(N)}((x\cdot q^{-\mathrm{Int}.\mathrm{ofNat}(N)+1+2\cdot \mathrm{Int}.\mathrm{ofNat}(j)}+epsilon\cdot w)\cdot (epsilon\cdot q^{-\mathrm{Int}.\mathrm{ofNat}(N)+1+2\cdot \mathrm{Int}.\mathrm{ofNat}(j)}+x\cdot w))$$

Formalization. D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.seedProduct (✓ std3).

Citation. Yacine Ikhlef and Alexi Morin-Duchesne (2024). Fusion of irreducible modules in the periodic Temperley–Lieb algebra. DOI: 10.21468/SciPostPhys.17.5.132. URL: https://arxiv.org/abs/2312.14837v2.

Commentary.

Conjecture C.1 (p. 62): “For p ⩾ 1, the function X^{(p)}_m(ω, x, y) specialised to y = εq^m evaluates to X^{(p)}m(ω, x, εq^m) = (ωx)^{p−m} ∏{r=−(m−p−1)/2}^{(m−p−1)/2} (xq^{2r} + εω)(εq^{2r} + xω). (C.5)” The product has N factors, with integer exponent 2r = −N+1+2j for j in Finset.range N. For N=0 the product is empty and equals one. The same expression occurs in (C.26).

Definition 1.7 (Conjectures C.1 and C.2).

$$claim=(\forall q:\mathbb{C},\forall x:\mathbb{C},\forall w:\mathbb{C},\forall epsilon:\mathbb{C},(q\ne 0)⟹((x\ne 0)⟹((w\ne 0)⟹((\forall n:\mathbb{N},(1\le n)⟹(q^n\ne 1))⟹((epsilon=1\vee epsilon=-1)⟹((\forall p:\mathbb{N},\forall m:\mathbb{N},(1\le p)⟹((p\le m)⟹(\mathrm{seedX}\left(q,x,epsilon\cdot q^m,w,p,m-p\right)=\mathrm{seedProduct}\left(q,x,w,epsilon,m-p\right))))\wedge (\forall m:\mathbb{N},\mathrm{seedSigma}\left(q,x,epsilon\cdot q^m,w,m\right)=\mathrm{seedProduct}\left(q,x,w,epsilon,m\right))))))))$$

Formalization. D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.claim (✓ std3).

Citation. Yacine Ikhlef and Alexi Morin-Duchesne (2024). Fusion of irreducible modules in the periodic Temperley–Lieb algebra. DOI: 10.21468/SciPostPhys.17.5.132. URL: https://arxiv.org/abs/2312.14837v2.

Commentary.

Conjecture C.1 (p. 62): “For p ⩾ 1, the function X^{(p)}m(ω, x, y) specialised to y = εq^m evaluates to X^{(p)}m(ω, x, εq^m) = (ωx)^{p−m} ∏{r=−(m−p−1)/2}^{(m−p−1)/2} (xq^{2r} + εω)(εq^{2r} + xω). (C.5)” Conjecture C.2 (p. 65): “The intermediate seed states σ^{(0)}{0,m}(w, x, y) specialised to y = εq^m evaluate to σ^{(0)}{0,m}(w, x, εq^m) = (wx)^{−m} ∏{r=−(m−1)/2}^{(m−1)/2} (xq^{2r} + εw)(εq^{2r} + xw). (C.26)” The domain is nonzero complex q,x,w, with q not a root of unity and ε equal to 1 or −1. The first assertion has 1 ≤ p ≤ m; the second has every natural m. Natural subtraction m−p is truncated subtraction, and agrees with the integer difference under p ≤ m.

Theorem 1.8 (Both product formulas hold).

$$\forall q:\mathbb{C},\forall x:\mathbb{C},\forall w:\mathbb{C},\forall epsilon:\mathbb{C},(q\ne 0)⟹((x\ne 0)⟹((w\ne 0)⟹((\forall n:\mathbb{N},(1\le n)⟹(q^n\ne 1))⟹((epsilon=1\vee epsilon=-1)⟹((\forall p:\mathbb{N},\forall m:\mathbb{N},(1\le p)⟹((p\le m)⟹(\mathrm{seedX}\left(q,x,epsilon\cdot q^m,w,p,m-p\right)=\mathrm{seedProduct}\left(q,x,w,epsilon,m-p\right))))\wedge (\forall m:\mathbb{N},\mathrm{seedSigma}\left(q,x,epsilon\cdot q^m,w,m\right)=\mathrm{seedProduct}\left(q,x,w,epsilon,m\right)))))))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.result (✓ std3). ∎

Resolves. Problems/ikhlef-morin-duchesne-2023-conjectures-c1-c2 (proved) by D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.result.

Source. Repository-derived.

Acknowledgement. Yacine Ikhlef and Alexi Morin-Duchesne (2024). Fusion of irreducible modules in the periodic Temperley–Lieb algebra. DOI: 10.21468/SciPostPhys.17.5.132. URL: https://arxiv.org/abs/2312.14837v2.

Commentary.

The seed recurrence equals a monic Askey–Wilson recurrence after Q=q², h=q^(p+1), a=h/(yw), b=hw/y, c=hyw, d=hy/w, and T=−al(x). A coupled Darboux identity shifts a to aQ. At the terminal degree the coefficient A vanishes because abQ^(N−1)=1. Induction then gives one linear factor at each shift. Symmetric exponents sum to zero, producing seedProduct. The separately defined sigma recurrence equals the p=0 seedX recurrence. The formulas hold uniformly in m and p; intermediate values at a fixed specialization are not asserted to factor.

References

• Truth anchor: D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.al
• Truth anchor: D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.claim
• Truth anchor: D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.qbr
• Truth anchor: D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.qcb
• Truth anchor: D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.result
• Truth anchor: D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.seedProduct
• Truth anchor: D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.seedSigma
• Truth anchor: D5/S3/Quantum/Algebra/PeriodicTemperleyLiebSeedProduct.seedX
• Dependency: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry