An exterior product identity for Schur functions Original Research Article

Let A be a matrix in Matn(k), where k is a commutative ring. Let ΛnMatn(k) be the nth exterior power of Matn(k) as an n2-dimensional free k-module. We present a coordinate-free characterisation of the Schur functions of (eigenvalues of) A, sλ(A), with λ = (λ1,…, λn) ε Zn: Aλ1 + n − 1 Λ … Λ Aλn + n − n = sλ(A)An − 1 Λ … Λ A Λ I. This becomes the usual definition of the Schur functions when A = diag(x1,…,xn). A coordinate version of this identity was found earlier by A. Kilikauskas. We show how the “master identity” above may be used to derive new identities, and simplify the proofs of old identities involving Schur functions and linear recurrent sequences. We also discuss its place in algebra and Lie theory.