Cofactors and eigenvectors of banded Toeplitz matrices: Trench formulas via skew Schur polynomials

带状Toeplitz矩阵行列式与积和式的构造性递推

Constructive recurrences for determinants and permanents of banded Toeplitz matrices

针对带状Toeplitz矩阵，提出两种构造性Laplace展开方法，得到行列式与积和式的极小阶递推，并精确分类状态图，推广至循环情形。

Symmetric polynomials in the symplectic alphabet and their expression via Dickson--Zhukovsky variables

Given a symmetric polynomial $P$ in $2n$ variables, there exists a unique symmetric polynomial $Q$ in $n$ variables such that \[ P(x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1}) =Q(x_1+x_1^{-1},\ldots,x_n+x_n^{-1}). \] We denote this polynomial $Q$ by $Φ_n(P)$ and show that $Φ_n$ is an epimorphism of algebras. We compute $Φ_n(P)$ for several families of symmetric polynomials $P$: symplectic and orthogonal Schur polynomials, elementary symmetric polynomials, complete homogeneous polynomials, and power sums. Some of these formulas were already found by Elouafi (2014) and Lachaud (2016). The polynomial…