Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T15:01:55.433Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 November 2018

Paul-Olivier Dehaye
Paul-Olivier Dehaye*
Affiliation:
Merton College, University of Oxford, United Kingdome-mail: paul-olivier.dehaye@math.ethz.ch
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

47B3505E0520G05Toeplitz matricesJacobi–Trudi identitySzegö limit theoremHeine identityWiener–Hopf factorization
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 2 , 01 June 2011 , pp. 255 - 269
Copyright
Copyright © Canadian Mathematical Society 2011

References

[BS99] Böttcher, A. and Silbermann, B., Introduction to large truncated Toeplitz matrices. Universitext, Springer-Verlag, New York, 1999.Google Scholar
[Bum04] Bump, D., Lie groups. Graduate Texts in Mathematics, 225, Springer-Verlag, New York, 2004.Google Scholar
[BD02] Bump, D. and Diaconis, P., Toeplitz minors. J. Combin. Theory Ser. A 97(2002), no. 2, 252–271. doi:10.1006/jcta.2001.3214Google Scholar
[Deh06] Dehaye, P.-O., Averages over compact Lie groups, twisted by Weyl characters and application to moments of derivatives of characteristic polynomials. Ph. D. thesis, Stanford University, 2006. http://www.math.ethz.ch/»pdehaye/permanent/thesis.pdf Google Scholar
[Deh07] Dehaye, P.-O., Averages over classical Lie groups, twisted by characters. J. Combin. Theory Ser. A 114(2004), no. 7, 1278–1292. doi:10.1016/j.jcta.2007.01.008Google Scholar
[FJ04] Fauser, B. and Jarvis, P. D., A Hopf laboratory for symmetric functions. J. Phys. A 37(2004), no. 5, 1633–1663. doi:10.1088/0305-4470/37/5/012Google Scholar
[FJK] Fauser, B., Jarvis, P. D., and King, R. C., A Hopf algebraic approach to the theory of group branchings. arXiv:math-ph/0508034v1.Google Scholar
[FJKW06] Fauser, B., Jarvis, P. D., King, R. C., and Wybourne, B. G., New branching rules induced by plethysm. J. Phys. A 39(2006), no. 11, 2611–2655. doi:10.1088/0305-4470/39/11/006Google Scholar
[Lyo03] Lyons, R., Szegő limit theorems. Geom. Funct. Anal. 13(2003), no. 3, 574–590. doi:10.1007/s00039-003-0423-xGoogle Scholar
[Mac95] Macdonald, I. G., Symmetric functions and Hall polynomials. Second ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.Google Scholar
[Sag01] Sagan, B. E., The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Second ed., Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001.Google Scholar
[TW02] Tracy, C. A. and Widom, H., On the limit of some Toeplitz-like determinants. SIAM J. Matrix Anal. Appl. 23(2002), no. 4, 1194–1196 (electronic). doi:10.1137/S0895479801395367Google Scholar