An exceptional four-dimensional unitary Hecke Yang–Baxter operator

Abstract

A five-word Pauli formula gives an exact \(16\times16\) unitary Yang–Baxter operator in the class \([e^{i\pi/3},1/2,4]\), the smallest-dimensional unresolved member of the sole remaining existence family in Lechner's classification up to braid-character equivalence. Its spectral projection has unnormalized partial traces \(2I_4\), yielding the \(\eta=1/2\) Markov trace and a faithful ordinary localization of the \(\mathcal C(\mathfrak{sl}_3,6)\) Jones–Wenzl tower.

1. Result

Galindo, Hong, and Rowell studied this representation sequence in 2012. They ruled out ordinary local dimension two and supplied quasi- and \((3,1)\)-generalized localizations. In March 2026, Lechner's two-eigenvalue classification left \([e^{i\pi/3},1/2,2m]\) as its one unresolved existence family and identified base dimension four as the first open case.

The construction shows that this sequence is not the possible counterexample to the localization conjecture contemplated in the 2012 paper. Lechner's general observation also implies that the associated Yang–Baxter endomorphism has Jones index \(4\) and is standard braided.

2. The exact construction

On \(V\otimes V\cong(\mathbb C^2)^{\otimes4}\), use \(I,X,Z\) and \(J=XZ\), with a four-letter word denoting a tensor product. Set

The first four Pauli words combine into one reflection \[ M=\frac{-ZIZZ-ZIJJ-JIZJ+JIJZ}{2}, \] while \(E=XIXX\) is an anticommuting reflection. Thus \(H(\alpha,\beta)=\alpha M+\beta E\) is an involution on the circle \(\alpha^2+\beta^2=1\). The complete Pauli expansion shows that the braid relation selects exactly \(\beta^2=1/3\), \(\alpha^2=2/3\). The coefficients are forced; the only search-discovered ingredient was the sparse five-word support.

These two identities immediately give the Yang–Baxter equation for \(R\).

3. Why this is a full localization

The \((-1)\)-spectral projection is \(P=(I-H)/2\). Exact calculation gives

For normalized matrix traces \(\tau_n=4^{-n}\operatorname{Tr}\), the partial trace is precisely the Markov step

With the conjugate-linear involution \(e_i^*=e_i\), the maps are unital \(*\)-representations and obey \(\rho_{n+1}(\iota_n(x))=\rho_n(x)\otimes I_4\). Therefore \(\tau_n\circ\rho_n\) is the Markov trace defining \(H_n(3,6)\). Faithfulness of ordinary matrix trace makes \(\ker\rho_n\) exactly the trace annihilator; the induced injections are compatible with the tower maps.

Dimension one cannot carry the generator, and dimension two is excluded by the 2012 theorem. In dimension three, the only allowed Markov parameters force a Temperley–Lieb or complementary Temperley–Lieb quotient. The two corresponding relations are nonzero in \(H_3(3,6)\), each with exceptional trace norm \(1/18\), so neither candidate can be faithful. Hence four is minimal.

4. Independent exact verification

• verify_exact.py — standard-library arithmetic in \(\mathbb Q(\sqrt2,\sqrt3,i)\); checks the complete \(16\times16\), \(32\times32\), and \(64\times64\) identities exactly.
• verify_tensor_words.py — no matrices; checks all 18 surviving generic six-letter Pauli words and their polynomial coefficients.
• verify_supplied.py — the original SymPy verifier, preserved byte for byte for provenance.
• verification_output.txt — frozen output from the successful release run.

Minimal dependency-free check

cd exceptional_ybe_d4
python3 verify_exact.py
python3 verify_tensor_words.py

After swapping the two qubits inside every ququart, one qubit is a global spectator and the active \(8\times8\) operator satisfies the standard \((3,2)\)-generalized Yang–Baxter equation. The global conjugacy shows that this generalized representation has the same kernel as the faithful ordinary one, so it also gives a faithful \((3,2)\)-localization. No equivalence with the earlier quaternionic \((3,1)\) operator is asserted.

5. Related work, novelty, and limitations

A focused primary-source, scholarly-index, web, and exact-code search found no earlier ordinary four-dimensional localization or equivalent Pauli formula. The result therefore appears new and appears to answer the open class in arXiv:2603.20158. Absolute priority cannot be proved: differently normalized, unindexed, unpublished, paywalled, or concurrent work may exist.

Tensoring with identities gives examples in every base dimension \(4m\). It does not settle dimensions \(6,10,14,\ldots\), classify all solutions, prove uniqueness within the Pauli ansatz, or identify a tower intertwiner with the older quaternionic construction.

AI-assistance and verification disclosure

OpenAI language models were used extensively in exploration, proof development, software generation, verification design, literature organization, and drafting. Alec Kriebel directed the work, selected the released claims, and is responsible for the decision to publish these materials. Model output was not treated as evidence by itself. The sparse support arose from an AI-assisted numerical search whose original code and seeds were not retained; no mathematical claim relies on that search.

Suggested citation

Alec Kriebel, “An exceptional four-dimensional unitary Hecke Yang–Baxter operator,” version 1.1.0, 27 July 2026. Versioned release · Project page.