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On an Identity due to Bump and Diaconis, and Tracy and Widom

  Published:2011-02-10
 Printed: Jun 2011
  • Paul-Olivier Dehaye,
    Merton College, University of Oxford, United Kingdom
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Abstract

A classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener-Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump-Diaconis-Tracy-Widom identity is a differentiated version of the classical Jacobi-Trudi identity.
Keywords: Toeplitz matrices, Jacobi-Trudi identity, Szegő limit theorem, Heine identity, Wiener-Hopf factorization Toeplitz matrices, Jacobi-Trudi identity, Szegő limit theorem, Heine identity, Wiener-Hopf factorization
MSC Classifications: 47B35, 05E05, 20G05 show english descriptions Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
Symmetric functions and generalizations
Representation theory
47B35 - Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
05E05 - Symmetric functions and generalizations
20G05 - Representation theory
 

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