On the decomposition of Brauer's centralizer algebras

Abstract. Let Mn be a random n × n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n → ∞. By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of Mn. For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of Mn are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices.

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“…The question rests on the semi-simplicity of B K (n), a matter which has not been fully resolved (cf. [HW89,DWH99]). …”

Section: Orthogonal and Symplectic Matricesmentioning

confidence: 99%

Linear functionals of eigenvalues of random matrices

Diaconis

Evans

2001

Trans. Amer. Math. Soc.

227

194

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“…It is based on basic results of Hanlon and Wales [3] on the representation theory of the symmetric group (cf. Sagan [10]).…”

Section: The Value Of F Onmentioning

confidence: 99%

“…Hanlon and Wales [3] showed that the eigenvalues and eigenvectors of A(x) can be described as follows. Consider any partition λ = (t 1 , .…”

Section: The Value Of F Onmentioning

confidence: 99%

On the existence of real R-matrices for virtual link invariants

Regts

Schrijver

Sevenster

2017

Abh. Math. Semin. Univ. Hambg.

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Abstract. We characterize the virtual link invariants that can be described as partition function of a real-valued R-matrix, by being weakly reflection positive. Weak reflection positivity is defined in terms of joining virtual link diagrams, which is a specialization of joining virtual link diagram tangles. Basic techniques are the first fundamental theorem of invariant theory, the Hanlon-Wales theorem on the decomposition of Brauer algebras, and the Procesi-Schwarz theorem on inequalities for closed orbits.

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“…The theorem is proved using the decomposition of Brauer algebras as given by Hanlon and Wales [10], the first fundamental theorem of invariant theory, and the characterization of orbit spaces by inequalities of Procesi and Schwarz [16].…”

Section: Corollarymentioning

confidence: 99%

On partition functions for 3-graphs

Regts

Schrijver

Sevenster

2016

Journal of Combinatorial Theory, Series B

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Abstract.A cyclic graph is a graph with at each vertex a cyclic order of the edges incident with it specified. We characterize which real-valued functions on the collection of cubic cyclic graphs are partition functions of a real vertex model (P. de la Harpe, V.F.R. Jones, Graph invariants related to statistical mechanical models: examples and problems, Journal of Combinatorial Theory, Series B 57 (1993) 207-227). They are characterized by 'weak reflection positivity', which amounts to the positive semidefiniteness of matrices based on the 'k-join' of cubic cyclic graphs (for all k ∈ Z + ).Basic tools are the representation theory of the symmetric group and geometric invariant theory, in particular the Hanlon-Wales theorem on the decomposition of Brauer algebras and the ProcesiSchwarz theorem on inequalities defining orbit spaces.

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On the decomposition of Brauer's centralizer algebras

Cited by 64 publications

References 9 publications

Linear functionals of eigenvalues of random matrices

Linear functionals of eigenvalues of random matrices

On the existence of real R-matrices for virtual link invariants

On partition functions for 3-graphs

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