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Failure of Jones-Wenzl generation at a root of unity

Abstract

Three two-strand Jones-Wenzl projectors at q = exp(pi i/6) give a corner not generated by its projected boundary generators.

Definition 1.1 (The Temperley-Lieb relations).

Formalization. D5/S3/Quantum/Algebra/JonesWenzlGenerationRefutation.rel (✓ std3).

Citation. Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Commentary.

The relation on the free complex algebra has three alternatives: a generator squares to nu times itself, adjacent generators satisfy the triple-product relation, and nonadjacent generators commute. The free generators have indices in Fin (n - 1), whose natural values start at zero. Natural subtraction is truncated, so this also defines the relation for n = 0 and n = 1. These are the presented-algebra relations of Flores and Peltola, arXiv:1811.12364v2, WJalgebra_gen-sec1-intro.tex lines 118–131. The maps Fin.val display the inclusions of finite indices into the natural numbers.

Definition 1.2 (The presented algebra).

Formalization. D5/S3/Quantum/Algebra/JonesWenzlGenerationRefutation.TL (✓ std3).

Citation. Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Commentary.

The quotient of the free complex algebra by these relations is the Temperley-Lieb algebra. The source identifies this presentation with its diagram algebra at lines 118–131; the coefficient nu is an arbitrary complex number.

Definition 1.3 (One-indexed algebra generators).

Formalization. D5/S3/Quantum/Algebra/JonesWenzlGenerationRefutation.gen (✓ std3).

Citation. Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Commentary.

For 1 <= i < n, the generator is the quotient image of the free generator with finite index i - 1. For all other natural indices it is zero. This total convention retains the source’s one-indexed generators at lines 119–144. The two implications give the two branches of the dependent conditional. The dot in the finite-index constructor suppresses only its proof of i - 1 < n - 1; that proof exists under the displayed condition and does not affect the quotient element.

Definition 1.4 (Laurent quantum integers).

Formalization. D5/S3/Quantum/Algebra/JonesWenzlGenerationRefutation.qint (✓ std3).

Citation. Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Commentary.

For natural k, the quantum integer is the Laurent polynomial with exponents k - 1 - 2j for 0 <= j < k. Both natural indices are explicitly cast to integers before subtraction and exponentiation. For nonzero q with q squared different from 1 this equals the quotient in equation Qinteger, lines 160–168. At q = 1 and q = -1 the Laurent polynomial supplies the continuous values k and (-1) to the power k - 1 times k, respectively; the empty sum for k = 0 is zero. This convention makes the recursion meaningful at the two parameters where the source sets pbar to infinity.

Definition 1.5 (The root-order bound).

Formalization. D5/S3/Quantum/Algebra/JonesWenzlGenerationRefutation.pbar (✓ std3).

Citation. Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Commentary.

The value is infinity at q = 1 or q = -1. Otherwise it is the order of q squared when that order is nonzero, and infinity when orderOf returns zero for infinite order. The finite order is explicitly included in ENat. For q = exp(pi i p’/p) with coprime positive p and p’, the order of q squared is p. Thus this is the root-order convention of equation MinPower, lines 170–179; the definition is total even at q = 0.

Definition 1.6 (Embedded Jones-Wenzl recursion).

Formalization. D5/S3/Quantum/Algebra/JonesWenzlGenerationRefutation.jw (✓ std3).

Citation. Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Commentary.

The recursion takes place in TL n (-q - q inverse), with an offset a for a box starting at strand a + 1. It starts at P zero = 1. The zero quantum integer makes P one = 1, after which the step is precisely P at k + 1 = P at k + ([k]/[k + 1]) times P at k U at a + k P at k. This agrees with the embedded recursion at lines 270–285. The constant motive of Nat.rec is displayed as Function.const on the natural numbers. The formula defines a total recursion; its projector interpretation uses the usual size and root-order restrictions.

Definition 1.7 (The composite projector).

Formalization. D5/S3/Quantum/Algebra/JonesWenzlGenerationRefutation.compProj (✓ std3).

Citation. Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Commentary.

The list σ specifies consecutive box sizes. For each zero-based list index i, the preceding sizes sum to the offset and List.getD reads the entry at i with zero as its default. List.prod multiplies these embedded projectors in list order and gives 1 for an empty list. For positive admissible sizes this is the side-by-side composite projector in lines 385–402.

Definition 1.8 (Projected boundary generators).

Formalization. D5/S3/Quantum/Algebra/JonesWenzlGenerationRefutation.jwGen (✓ std3).

Citation. Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Commentary.

Sandwich the ambient generator at the sum of the first i box sizes between two copies of the composite projector. For 1 <= i < the length of σ this is the source’s one-indexed Jones-Wenzl generator in equation EGenerators-00, lines 507–516. The function itself is defined for every natural i.

Definition 1.9 (The generation assertion).

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

Citation. Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Commentary.

Conjecture 1.2, lines 544–545, extends item 1 of Theorem 1.1 from the total-size bound at line 502 to the maximum-box-size bound. The displayed proposition is its generation assertion: for every nonzero complex q and every list σ of positive sizes individually below pbar q, every sandwich of an arbitrary ambient element t belongs to the unital complex subalgebra generated by the composite projector and the projected boundary generators. Each size is cast from the natural numbers to ENat in the root-order comparison. Algebra.adjoin uses the ambient unit; the corner algebra in lines 403–406 uses the composite projector as its unit. Failure of membership even in the displayed ambient unital subalgebra suffices to disprove generation in the corner, and hence also minimal generation. The list (0) is excluded by positivity; the empty list is allowed.

Theorem 1.10 (Three boxes give a counterexample).

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

Resolves. Problems/flores-peltola-2018-jones-wenzl-generation-refutation (refuted) by D5/S3/Quantum/Algebra/JonesWenzlGenerationRefutation.result.

Source. Repository-derived.

Acknowledgement. Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Commentary.

Set r = sqrt 3 in the real numbers and include it in the complex numbers. Take q = exp(pi i/6) and σ = [2,2,2]. The exponential gives q = r/2 + i/2, q inverse = r/2 - i/2, q nonzero, and -q - q inverse = -r. The square of q is a primitive sixth root, so pbar q = 6. The Laurent sums give [0] = 0, [1] = 1, [2] = r. The recursion therefore makes the composite projector P = (1 + r inverse U₁)(1 + r inverse U₃)(1 + r inverse U₅), and its generating set is exactly {P, PU₂P, PU₄P}. Let W = U₂U₁U₄U₃U₂U₅U₄ and let Z = PWP.

Use the quadratic rational algebra with basis 1, omega and relation omega squared = 3. Five explicit nine-by-nine matrices A₀ through A₄ satisfy Aᵢ squared = -omega Aᵢ, AᵢAⱼAᵢ = Aᵢ for adjacent indices, and AᵢAⱼ = AⱼAᵢ for nonadjacent indices. All entries are rational pairs in this basis, so these equalities are exact finite computations. An explicit matrix B equals (1 + (omega/3)A₀)(1 + (omega/3)A₂)(1 + (omega/3)A₄).

The exact computations also give a vector v = (0, 3 - omega, 0, 0, 0, 0, 0, 0, 2) with vB = v, (vA₁)B = (-omega/3)v, and (vA₃)B = (-omega/3)v. Seven successive multiplications by A₁, A₀, A₃, A₂, A₁, A₄, A₃ are specified by eight exact vectors. The sixth coordinate, numbered from zero, of the last vector multiplied by B is -2 omega/3.

The rational algebra homomorphism sending omega to the complex inclusion of r exists because r squared = 3. Apply it entrywise to the matrices. Their three relation families then give a complex algebra homomorphism from TL 6 (-r) to the nine-by-nine complex matrices, sending U at i + 1 to the image of Aᵢ. It sends P to the image of B, since the image of omega/3 is r inverse. The seven vector identities consequently give the sixth coordinate of v times the image of WP as -2 omega/3 before applying the scalar homomorphism.

Let ell be the complex vector obtained from v. Algebra homomorphisms commute with vector-matrix multiplication and scalar multiplication, so ell is fixed by the image of P and is a common left eigenvector of the images of PU₂P and PU₄P, with eigenvalue -r/3. The sixth coordinate of ell times the image of Z is -2r/3, which is nonzero.

The matrices M satisfying ell M = c ell for some complex c form a unital complex subalgebra: scalar matrices have this property, sums add the scalars, and products multiply them. The images of P, PU₂P and PU₄P all lie in it. Its inverse image under the algebra homomorphism therefore contains the subalgebra generated by {P, PU₂P, PU₄P}. But ell has sixth coordinate zero, so every matrix in this subalgebra has zero sixth coordinate after multiplication on the left by ell. The nonzero value for Z proves that Z is outside the generated subalgebra.

Finally, all three entries of σ are positive and 2 < 6 = pbar q. Applying the asserted universal generation property to this q, this list and t = W would place Z in that generated subalgebra, a contradiction. The total size is 6, so this example does not contradict the source’s theorem under its strict total-size bound.

References