  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.
