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

Canadian mathematical bulletin

Paul-Olivier Dehaye¹ 

[1] Merton College, University of Oxford, United Kingdom

关键词: Toeplitz matrices;    Jacobi-Trudi identity;    Szegő limit theorem;    Heine identity;    Wiener-Hopf factorization;   

DOI  :  10.4153/CMB-2011-011-5

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

A classical question for a Toeplitz matrix with given symbol is how tocompute asymptotics for the determinants of its reductions to finiterank. One can also consider how those asymptotics are affected whenshifting 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 Widomused 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.

【 授权许可】

Unknown