Tests of the formulas for the palindromic project (in Sagemath).

This page is a complement of the paper
P. Alexandersson, L. A. González-Serrano, E. A. Maximenko, M. A. Moctezuma-Salazar:
Symmetric polynomials in the symplectic alphabet and their expression via Dickson–Zhukovsky variables.
Preprint: arXiv:1912.12725 [math.CO].

Here are some programs in Sagemath that verify our formulas for small values of parameters. The maximal values of the parameters are selected in such a manner that each program runs less than 5 minutes on modern computers.

We denote by «pol» the tests in the polynomial field, i.e. with polynomial variables, by «numer» the random tests with rational numbers, and by «sym» the tests in the ring of the symmetric functions (in the Schur basis).

Auxiliary facts

• Formulas for Chebyshov polynomials: pol, numer.
• Sums of x[j] ^ m / proddif(j, x): pol, numer.
• Recursive formulas for elem, homz, and powersum: pol, numer.
• Dual Jacobi–Trudi formula for the symplectic and orthogonal Schur functions: sym.

Bialternant formulas with Chebyshov polynomials

• Bialternant formulas for the symplectic and orthogonal Schur polynomials over the even or odd symplectic alphabet, in terms of Chebyshov polynomials: pol, numer.

Formulas for elemz, homz, powersumz, and their odd analogs

• Various formulas for elemz: pol, numer.
• Various formulas for homz: pol, numer.
• Various formulas for powersumz: pol, numer.
• Various formulas for elemzodd, homzodd, powersumzodd: pol, numer.

Analogs of Cauchy identities

• Analogs of Cauchy identities for schurz and schurzodd: pol, numer.

Minors of banded symmetric Toeplitz matrices

• Minor as schurz: pol.
• Decomposition found by Trench (1987), Ciucu and Krattenthaler (2009), and Elouafi (2014): pol, numer.