On an identity due to Bump and Diaconis, and Tracy and Widom (Q2882471)

Let \(\sigma:{\mathbb T}\to{\mathbb C}\) be a sufficiently smooth function with the Fourier coefficients \(\{d_k\}_{k\in{\mathbb Z}}\). Consider the finite Toeplitz matrices \(M_n(\sigma)=(d_{i-j})_{n\times n}\) and their minors \(M_n^{\lambda\mu}(\sigma)=(d_{\lambda_i-\mu_j-i+j})_{n\times n}\), where \(\lambda\) and \(\mu\) are partitions of length less than or equal to \(n\), and put \(R^{\lambda\mu}(\sigma)=\lim\limits_{n\to\infty}\frac{M_n^{\lambda\mu}(\sigma)}{M_n(\sigma)}\).NEWLINENEWLINE\textit{C. A. Tracy} and \textit{H. Widom} [``On the limit of some Toeplitz-like determinants'', SIAM J. Matrix Anal. Appl. 23, No. 4, 1194--1196 (2002; Zbl 1010.47019)] obtained the asymptotics \(R^{\lambda\mu}(\sigma)\) as determinants involving the Fourier coefficients in the so-called Wiener-Hopf factorization of \(\sigma\). We will denote that expression by \(TW^{\lambda\mu}(\sigma)\). On the other hand, \textit{D. Bump} and \textit{P. Diaconis} [``Toeplitz minors'', J. Comb. Theory, Ser. A 97, No. 2, 252--271 (2002; Zbl 1005.47030)] obtained another expression for \(R^{\lambda\mu}(\sigma)\) involving Laguerre polynomials and sums over symmetric groups. Their expression will be denoted by \(BD^{\lambda\mu}(\sigma)\). Thus, one has the Bump-Diaconis-Tracy-Widom identity: \(TW^{\lambda\mu}(\sigma)=R^{\lambda\mu}(\sigma)=BD^{\lambda\mu}(\sigma)\) for sufficiently smooth functions \(\sigma\).NEWLINENEWLINEThe author proves that \(TW^{\lambda\mu}(\sigma)=BD^{\lambda\mu}(\sigma)\) directly, without relying on Toeplitz determinants, that is, on \(R^{\lambda\mu}(\sigma)\). He shows that the above identity is a differential version of the classical Jacobi-Trudi identity.