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The two-eigenvalue problem and density of Jones representation of braid groups. (English) Zbl 1045.20027
The paper centres on the ‘two-eigenvalue problem’. This concerns the classification of unitary representations of a compact Lie group \(G\) generated topologically by conjugates of an element \(g\) which is represented by a matrix with just two distinct eigenvalues whose ratio is not \(\pm 1\).
The first section gives an account of the cases where the centre of \(G\) has codimension \(\geq 1\) in \(G\), and a list of the known classification where \(G\) modulo its centre is finite. The remainder of the paper is mainly concerned with unitary representations of the braid groups \(B_n\), where the elementary braids \(\sigma_i\) take the place of the element \(g\) above. Again representations with just two eigenvalues are studied, which then induce representations of the Hecke algebras \(H_n(q)\) for suitable \(q\in\mathbb{C}\).
The authors consider the irreducible representations of \(H_n(q)\) in \(U(N)\) corresponding to different partitions \(\lambda\) of \(n\), and look for values of \(q=\exp(2\pi i/r)\in\mathbb{C}\) such that the image of \(B_n\) is ‘standard’ in the sense that the identity component of its closure in \(U(N)\) is covered by \(\text{SU}(N)\), \(\text{Sp}(N)\) or \(\text{Spin}(N)\). The cases where the image fails to be standard are listed, and correspond to cases with known finite image.
The authors go on to study the evaluations of the Jones polynomial of a knot at \(q=\exp(2\pi i/r)\) and the distribution of these values over knots which are the closure of an \(n\)-braid \(\beta\) with \(k\) crossings, using the fact that the Jones polynomial of a closed braid can be found from the traces of the irreducible representations of \(\beta\in H_n(q)\) corresponding to partitions with just two parts.
The final section looks at density results for representations of the pure mapping class group of a surface of genus \(g\) with \(n\) boundary components arising from Chern-Simons theory with \(r=5\) and \(G=\text{SO}(3)\).

20F36 Braid groups; Artin groups
57M25 Knots and links in the \(3\)-sphere (MSC2010)
20C08 Hecke algebras and their representations
81P68 Quantum computation
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