Published paper
Abstract
A slight modification of the Burau representation of the braid group is shown to be unitary relative to an explicitly defined Hermitian form. This provides information about the image of the representation and a tool for studying its faithfulness.
Role in dependence graphs
Proof-critical source
On -real and -complex numbers
This paper is included only for the following marked statement:
- Main theorem and invariant Hermitian form · Proc. Amer. Math. Soc. 90 (1984), VOR pp. 199-202Constructs the invariant Hermitian form for the reduced Burau representation and identifies its definiteness region.
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Audit summary
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Exact reviewed source
AMS version of record · Proceedings of the American Mathematical Society 90 (1984), no. 2, 199–202
Craig C. Squier. The Burau representation is unitary. Proceedings of the American Mathematical Society 90 (1984), no. 2, 199–202.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The modified Burau representation preserves the explicitly displayed Hermitian form over the involutive Laurent-polynomial ring, and its unit-circle specializations are unitary whenever that form is nonsingular. The rank-two specialization and definiteness range used in the q-real dependency are consistent with Squier's matrix after a change of basis and harmless scalar normalization.
Main theoremNo issue found
The invariant Hermitian form is stated over the correct involution
Printed pages 199–201 · version of record
The coefficient involution sends the Laurent parameter to its inverse, so on the unit circle it becomes complex conjugation. The displayed Gram matrix is Hermitian for this involution, and the modified Burau generators preserve it. Where the determinant is nonzero, this is precisely a unitary representation relative to that form.
AMS version of record ↗Rank-two specializationCorrectly specialized
The focal form is the expected specialization
Printed pages 200–202; focal arXiv:2508.08440v2 PDF page 40
For , deleting the invariant line and changing to the two standard Temperley-Lieb/Burau basis vectors turns Squier's form into the focal Gram matrix up to a positive scalar. Its determinant vanishes at the sixth-root endpoints and has the printed sign on the intervening arc, matching the positive-semidefinite range used in Proposition 7.3.
02Proofs2 reported findingsNo proof issues found
The proof is an explicit generator calculation. Each modified Burau generator preserves the form, the determinant identifies the degenerate parameters, and the signature argument on unit-circle arcs gives the stated unitarity region. No hidden classification theorem is needed.
Generator calculationNo issue found in the reviewed proof
Invariance is verified on every braid generator
Printed pages 200–201 · version of record
Substitution of each generator matrix into reproduces entry by entry. Because the braid group is generated by these matrices, the calculation proves invariance for every braid word; no continuity or density argument is being used in place of algebra.
AMS version of record ↗Definiteness and specializationNo issue found in the reviewed proof
The determinant and signature argument handles endpoints correctly
Printed pages 201–202 · version of record
The determinant is a real trigonometric polynomial after is placed on the unit circle. Its zeros partition the circle into constant-signature arcs. Inside the positive arc the preserved form yields the norm estimate; at endpoints the form is semidefinite and the focal argument obtains its bound by continuity.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.