An exterior product identity for Schur functions (Q1921353)

Let \(A\) be a matrix of \(\text{Mat}_n(k)\), the set of all \(n\times n\)-matrices with entries taken from the commutative ring \(k\). Let \(\bigwedge^n\text{Mat}_n(k)\) be the \(n\)th exterior power of \(\text{Mat}_n(k)\) as an \(n^2\)-dimensional free \(k\)-module. A coordinate-free characterization of the Schur functions of (eigenvalues of) \(A\), with \(\lambda=(\lambda_1,\lambda_2,\dots,\lambda_n)\in\mathbb{Z}^n\) is presented: \[ A^{\lambda_1+n-1}\wedge\cdots\wedge A^{\lambda_n+n-n}=s_\lambda(A)A^{n-1}\wedge\cdots\wedge A\wedge I. \] For \(A=\text{diag}(x_1,x_2,\dots,x_n)\) this becomes the usual definition of the Schur functions. The above identity is used to derive new identities and to simplify the proofs of old identities involving Schur functions and linear recurrent sequences. Its place in algebra and Lie theory is discussed.