TY - JOUR

T1 - Symmetric polynomials in the symplectic alphabet and the change of variables zj = xj + xj -1

AU - Alexandersson, Per

AU - González-Serrano, Luis Angel

AU - Maximenko, Egor A.

AU - Moctezuma-Salazar, Mario Alberto

N1 - Funding Information:
The contribution of the first author has been funded by the Swedish Research Council (Vetenskapsrådet), grant 2015-05308. The contribution of the second, third, and fourth authors has been supported by IPN-SIP projects (Instituto Politécnico Nacional, Mexico), CONACYT scholarships (Mexico), and CONACYT (Mexico) project “Ciencia de Fron-tera” FORDECYT-PRONACES/61517/2020. We are grateful to the anonymous referee for various improvements, especially in the title and the introduction. Eduardo Camps Moreno explained us some necessary facts about algebraically independent generating subsets and suggested [17, Chapter 5].
Funding Information:
The contribution of the first author has been funded by the Swedish Research Council (Vetenskapsr?det), grant 2015-05308. The contribution of the second, third, and fourth authors has been supported by IPN-SIP projects (Instituto Polit?cnico Nacional, Mexico), CONACYT scholarships (Mexico), and CONACYT (Mexico) project \Ciencia de Frontera? FORDECYT-PRONACES/61517/2020. We are grateful to the anonymous referee for various improvements, especially in the title and the introduction. Eduardo Camps Moreno explained us some necessary facts about algebraically independent generating subsets and suggested [17, Chapter 5].
Publisher Copyright:
© The authors.

PY - 2021

Y1 - 2021

N2 - Given a symmetric polynomial P in 2n variables, there exists a unique symmetric polynomial Q in n variables such that P(x1,…,xn, x-1 1,….,x-1 n) = Q(x1 + x-1 1,…, xn + x-1 n). We denote this polynomial Q by Φn(P) and show that Φn is an epimorphism of algebras. We compute Φn(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 Φn(s(2n) λ/µ), where s(2n) λ/µ 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 x1,…, xn, x-1 1,. …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.

AB - Given a symmetric polynomial P in 2n variables, there exists a unique symmetric polynomial Q in n variables such that P(x1,…,xn, x-1 1,….,x-1 n) = Q(x1 + x-1 1,…, xn + x-1 n). We denote this polynomial Q by Φn(P) and show that Φn is an epimorphism of algebras. We compute Φn(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 Φn(s(2n) λ/µ), where s(2n) λ/µ 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 x1,…, xn, x-1 1,. …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.

UR - http://www.scopus.com/inward/record.url?scp=85103284134&partnerID=8YFLogxK

U2 - 10.37236/9354

DO - 10.37236/9354

M3 - Artículo

AN - SCOPUS:85103284134

VL - 28

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

M1 - P1.56

ER -