On an identity due to ((Bump and Diaconis) and (Tracy and Widom))

Abstract

A classical question for a Toeplitz matrix with given symbol is 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 (Toeplitz minors, J. Combin. Theory Ser. A, 97 (2002), pp. 252--271) 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 (On the limit of some Toeplitz-like determinants, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 1194--1196) 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.