A ‘Second Orthogonality Relation’ for Characters of Brauer Algebras

Ram, Arun

doi:10.1006/eujc.1996.0132

Cited by 15 publications

(16 citation statements)

References 7 publications

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“…Ram that was cited in [DS94] for the relevant representation theory of these groups has now appeared as [Ram95] -see also [Ram97]). We outline the relevant arguments in §8.…”

Section: Eigenvalues Of Random Matrices 2617mentioning

confidence: 99%

“…The relevant moment formulae were established in [DS94] using the thesis work of Arun Ram that has now been published in [Ram95,Ram97]. We refer to these papers for background and details.…”

Section: Orthogonal and Symplectic Matricesmentioning

confidence: 99%

“…Observe that if K is odd, then E[p λ (M n )] = 0; whereas if K = 2m is even and n ≥ 2K, then Theorem 2.8 of [Ram97] gives Multiplying p j and p k , integrating over the orthogonal group, and using the orthogonality of characters now yields (8.2). A very similar argument handles the case when n is even.…”

Section: Orthogonal and Symplectic Matricesmentioning

confidence: 99%

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Linear functionals of eigenvalues of random matrices

Diaconis

Evans

2001

Trans. Amer. Math. Soc.

227

194

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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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“…Ram that was cited in [DS94] for the relevant representation theory of these groups has now appeared as [Ram95] -see also [Ram97]). We outline the relevant arguments in §8.…”

Section: Eigenvalues Of Random Matrices 2617mentioning

confidence: 99%

Section: Orthogonal and Symplectic Matricesmentioning

confidence: 99%

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Linear functionals of eigenvalues of random matrices

Diaconis

Evans

2001

Trans. Amer. Math. Soc.

227

194

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“…On the other hand, the treatment of the cases K = O n and K = Sp n makes use of less familiar material, namely, Ram's work [9,10] on the characters of Brauer algebras. Nowadays there exist alternative methods to prove these moment formulae.…”

Section: Introductionmentioning

confidence: 99%

On the Diaconis-Shahshahani Method in Random Matrix Theory

Stolz¹

2005

J Algebr Comb

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If is a random variable with values in a compact matrix group K , then the traces Tr( j ) ( j ∈ N) are real or complex valued random variables. As a crucial step in their approach to random matrix eigenvalues, Diaconis and Shahshahani computed the joint moments of any fixed number of these traces if is distributed according to Haar measure and if K is one of U n , O n or Sp n , where n is large enough. In the orthogonal and symplectic cases, their proof is based on work of Ram on the characters of Brauer algebras.The present paper contains an alternative proof of these moment formulae. It invokes classical invariant theory (specifically, the tensor forms of the First Fundamental Theorems in the sense of Weyl) to reduce the computation of matrix integrals to a counting problem, which can be solved by elementary means.

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“…Indeed, the answer to their computations turns out to be expressible as values of characters of the Brauer algebra. Those were evaluated by Ram [20,21], and are given by a rather complicated-looking function g in [9,Theorem 4]. …”

Section: Introductionmentioning

confidence: 99%

Averages over classical Lie groups, twisted by characters

Dehaye

2007

Journal of Combinatorial Theory, Series A

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We compute E G ( i tr(g λ i )), where g ∈ G = Sp(2n) or SO(m) (m = 2n, 2n+1) with Haar measure. This was first obtained by Diaconis and Shahshahani [Persi Diaconis, Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994) 49-62. Studies in applied probability], but our proof is more self-contained and gives a combinatorial description for the answer. We also consider how averages of general symmetric functions E G Φ n are affected when we introduce a character χ G λ into the integrand. We show that the value of E G χ G λ Φ n /E G Φ n approaches a constant for large n. More surprisingly, the ratio we obtain only changes with Φ n and λ and is independent of the Cartan type of G. Even in the unitary case, Bump and Diaconis [Daniel Bump, Persi Diaconis, Toeplitz minors, J. Combin. Theory Ser. A 97 (2) (2002) 252-271. Erratum for the proof of Theorem 4 available at http://sporadic.stanford.edu/bump/correction.ps and in a third reference in the abstract] have obtained the same ratio. Finally, those ratios can be combined with asymptotics for E G Φ n due to Johansson [Kurt Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (3) (1997) 519-545] and provide asymptotics for E G χ G λ Φ n .

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A ‘Second Orthogonality Relation’ for Characters of Brauer Algebras

Cited by 15 publications

References 7 publications

Linear functionals of eigenvalues of random matrices

Linear functionals of eigenvalues of random matrices

On the Diaconis-Shahshahani Method in Random Matrix Theory

Averages over classical Lie groups, twisted by characters

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