Symmetric polynomials in the symplectic alphabet and the change of variables \(z_j = x_j + x_j^{-1}\)

Per Alexandersson¹ ; Luis Angel González-Serrano² ; Egor Maximenko³ ; Mario Alberto Moctezuma-Salazar²

¹Dept. of Mathematics, Stockholm University
²Escuela Superior de Física y Matemáticas del Instituto Politécnico Nacional
³Instituto Politécnico Nacional

The electronic journal of combinatorics, Tome 28 (2021) no. 1

Résumé

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 $\Phi_n(P)$ and show that $\Phi_n$ is an epimorphism of algebras. We compute $\Phi_n(P)$ 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 polynomials of the form $\Phi_n(\operatorname{s}_{\lambda/\mu}^{(2n)})$, where $\operatorname{s}_{\lambda/\mu}^{(2n)}$ is a skew Schur polynomial in $2n$ variables, arise naturally in the study of the minors of symmetric banded Toeplitz matrices, when the generating symbol is a palindromic Laurent polynomial, and its roots can be written as $x_1,\ldots,x_n,x^{-1}_1,\ldots,x^{-1}_n$. Trench (1987) and Elouafi (2014) found efficient formulas for the determinants of symmetric banded Toeplitz matrices. We show that these formulas are equivalent to the result of Ciucu and Krattenthaler (2009) about the factorization of the characters of classical groups.

Détail

DOI : 10.37236/9354

Classification : 05E05, 05E10, 15B05, 15B57
Mots-clés : Toeplitz determinants, Chebyshev polynomials