t_orbit_space_b31.html

The T-orbit space of the Weyl group B3

Orbit spaces of Weyl groups acting on compact tori: a unified and explicit polynomial description

Evelyne Hubert, Tobias Metzlaff, Cordian Riener 2024
https://epubs.siam.org/doi/10.1137/23M158173X

Download this Maple worksheet at https://github.com/TobiasMetzlaff/GeneralizedChebyshev/blob/main/Maple%20Worksheets/t_orbit_space_b3.mw

Download the Maple package "GeneralizedChebyshev" at https://github.com/TobiasMetzlaff/GeneralizedChebyshev/blob/main/GeneralizedChebyshev.mpl

> restart:
interface(warnlevel=0):
with(LinearAlgebra):
read("GeneralizedChebyshev.mpl"):
with(GeneralizedChebyshev);

(1)

> Type,n := B,3;

(2)

The T-orbit space is the set of points $z=(z_1,z_2,z_3) \in \mathbb{R}^3$ with $H(z) \succeq 0$ (positive semi-definite).

Here, $H \in \mathbb{Q}[z_1,z_2,z_3]^{3 \times 3}$ is a 3-by-3 matrix polynomial.

In the standard monomial basis:

> HermiteMatrix(Type,n)

rtable(1 .. 3, 1 .. 3, [[`+`(`-`(`*`(9, `*`(`^`(z[1], 2)))), `*`(6, `*`(z[2])), 3), `+`(`-`(`*`(27, `*`(`^`(z[1], 3)))), `*`(27, `*`(z[1], `*`(z[2]))), `-`(`*`(24, `*`(`^`(z[3], 2)))), `*`(12, `*`(z[1... (3)

In the basis of generalized Chebyshev polynomials of the first kind associated to B3 (it is the same formula for all cases):

> THermiteMatrix(Type,n)

rtable(1 .. 3, 1 .. 3, [[`+`(`-`(T[2, 0, 0]), T[0, 0, 0]), `+`(T[1, 0, 0], `-`(T[3, 0, 0])), `+`(`-`(T[4, 0, 0]), T[0, 0, 0])], [`+`(T[1, 0, 0], `-`(T[3, 0, 0])), `+`(`-`(T[4, 0, 0]), T[0, 0, 0]), `+`... (4)

To compute the generalized Chebyshev polynomials in the standard monomial basis (for, e.g., the second entry, there is a scaling factor $3/4$ to make the formula as nice as possible.):

> TPoly(Type,[1,0,0]) - TPoly(Type,[3,0,0])

(5)

For questions, please contact "math@tobiasmetzlaff.com".