On an algebraic identity and formula Jacobi—Trudi

Abstract

The first Jacobi—Trudi identity expresses Schur polynomials as certain determinants of matrices whose entries are complete homogeneous polynomials. The definition of Schur polynomials was given by Cauchy in 1815 as a quotient of certain determinants defined by an integer partition with at most n non-zero parts. Schur functions became very important because of their close relationship with the irreducible characters of both the symmetric groups and the general linear groups, and for their combinatorial applications. The Jacobi—Trudi identity was first stated 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 they play in various areas of mathematics including mathematical physics, representation theory, and algebraic geometry, and various proofs based on different ideas (in particular, a natural combinatorial proof using Young tableaux techniques) have been found. In our paper, we give a short and simple proof of the first Jacobi—Trudi identity and discuss its relationship with some other well-known polynomial identities. Refs 3.