An algebraic identity and the Jacobi–Trudi formula

Abstract

The first Jacobi–Trudi identity expresses Schur polynomials as determinants of matrices, the entries of which are complete homogeneous polynomials. The Schur polynomials were defined by Cauchy in 1815 as the quotients of determinants constructed from certain partitions. The Schur polynomials have become very important because of their close relationship with the irreducible characters of the symmetric groups and the general linear groups, as well as due to their numerous applications in combinatorics. The Jacobi–Trudi identity was first formulated by Jacobi in 1841 and proved by Nicola Trudi in 1864. Since then, this identity and its numerous generalizations have been the focus of much attention due to the important role which they play in various areas of mathematics, including mathematical physics, representation theory, and algebraic geometry. Various proofs of the Jacobi–Trudi identity, which are based on different ideas (in particular, a natural combinatorial proof using Young tableaux), have been found. The paper contains a short simple proof of the first Jacobi–Trudi identity and discusses its relationship with other well-known polynomial identities.