This is a Maple 2024 (https://maplesoft.com) package for computations with generalized Chebyshev polynomials associated to the simple complex Lie algebras (https://en.wikipedia.org/wiki/Simple_Lie_algebra). 

The main purpose of this package is to produce matrices for semi-definite programs that appear in polynomial optimization [4]. 
Beyond that, it features several functionalities for computations with Weyl groups of simple Lie type and polynomial descriptions for the orbit space [5]. 

If you are specifically interested in Weyl groups acting on minuscule weights: 
This is covered in a Maple 2008 worksheet by Michael Singer (https://singer.math.ncsu.edu/papers/minuscule/). 



How to use this package: 

Open/Create a Maple worksheet. 
Download the file 'GeneralizedChebsyhev.mpl' and place a copy in the same folder as your worksheet. 
In your worksheet, type 

>    read("GeneralizedChebyshev.mpl"): 
>    with(GeneralizedChebyshev);

(The first time executing this command can output an error. In this case, save and restart Maple.) 

The worksheet 'GeneralizedChebsyhevHelp.mw' is a guide through the available commands of the package. 



Mathematical Background: 

A Euclidean reflection group W that leaves a full-dimensional lattice Omega invariant is called Weyl group. 
The reflections can be defined through a crystallographic root system, which is a set of points with "nice" properties in the sense of [1,2,3]. 
The invariant lattice Omega is spanned by the fundamental weights omega_1...omega_n of the root system and also called weight lattice. 
A theorem from multiplicative invariant theory states that those elements of the group ring Q[Omega], which are invariant under the induced action of W, form a polynomial algebra: 
 
(*) $\mathbb{Q}[\Omega]^\mathcal{W} = \mathbb{Q}[\theta_{\omega_1}, ..., \theta_{\omega_n}],$

where for every weight $\mu\in\Omega$ we define the "generalized cosine" 

    $\theta_\mu := \frac{1}{\vert\mathcal{W}\vert} \sum\limits_{s \in \mathcal{W}} e^{s(\mu)}$,
    
which is simply the averaging sum over all orbit points and in particular invariant. 
The property (*) allows to define the generalized Chebyshev polynomial (of the first kind) associated to a weight, namely the unique multivariate $T_\mu \in \mathbb{Q}[z_1, ..., z_n]$, such that 

    $T_\mu(\theta_{\omega_1}, ..., \theta_{\omega_n}) = \theta_\mu$.

(Why is it called "generalized"? Because this extends the univariate Chebyshev polynomials defined by $T_k ((x+x^{-1})/2) = (x^k + x^{-k})/2$.) 
These polynomials form a basis of $\mathbb{Q}[z_1, ..., z_n]$ and are orthogonal on the orbit space of W, that is, on the basic semi-algebraic set 

    $Z := { (\theta_{\mu_1}, ..., \theta_{\mu_n})(u) \vert u in \mathbb{R}^n }$.

Here, $\theta_\mu$ becomes a function in $u$ by setting $e^{\mu}(u) := \exp(-2 \pi i \langle \mu, u \rangle )$. 
Those are the special functions associated to root systems, that is, periodic W-invariant trigonometric polynomials, see [6] for more. 
In [5], we have constructed a Hermite matrix polynomial H with the property

    $Z = \{ z \in \mathbb{R}^n \vert H(z) \mbox{ is positive semi-definite} \}$.
    
The matrix entries of H are given through a closed formula that is available as a command in the package. 
Alternatively, one can use a "Procesi-Schwarz-type approach", which is described in section 4 of [7], which we conjecture to be applicable for multiplicative actions in [5]. 

Any root system can be decomposed into irreducible components which classify the 7 families of simple Lie algebras: 

    A (n>=1)    B    C (n>=2)    D (n>=4)    E (n=6,7,8)    F (n=4)    G (n=2)

Any semi-simple Lie algebra admits a root system that decomposes into orthogonal, irreducible components which are of one of the above "simple Lie types". Hence, we only need to consider the latter. 



Problems/Questions:

Feel free to contact 'tobias.metzlaff@rptu.de'. 



Literature:

-Books:

[1] Bourbaki: Groupes et algèbres de Lie.
https://link.springer.com/book/10.1007/978-3-540-34491-9

[2] J. E. Humphreys: Introduction to Lie algebras and representation theory.
https://link.springer.com/book/10.1007/978-1-4612-6398-2

[3] R. Kane: Reflection groups and invariant theory.
https://link.springer.com/chapter/10.1007/978-1-4757-3542-0_1


-Our work on the subject:

[4] E. Hubert, T. Metzlaff, P. Moustrou and C. Riener: Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs.
https://link.springer.com/article/10.1007/s10107-024-02149-1

[5] E. Hubert, T. Metzlaff, and C. Riener: Orbit Spaces of Weyl Groups Acting on Compact Tori: A Unified and Explicit Polynomial Description.
https://epubs.siam.org/doi/10.1137/23M158173X


-An introduction to the numerical aspects of Fourier analysis with special functions of root systems:

[6] H. Munthe-Kaas, M. Nome and B. N. Ryland: Through the Kaleidoscope: Symmetries, Groups and Chebyshev-Approximations from a Computational Point of View.
https://www.cambridge.org/core/books/abs/foundations-of-computational-mathematics-budapest-2011/through-the-kaleidoscope-symmetries-groups-and-chebyshevapproximations-from-a-computational-point-of-view/5216EE38DB87E5688221552CD99BA9A6


-The "Procesi-Schwarz-type approach":

[7] C. Procesi and G. Schwarz: Inequalities defining orbit spaces.
https://link.springer.com/article/10.1007/BF01388587