This document is one of More SageMath Tutorials. You may edit it on github. $$\def\NN{\mathbb{N}}$$ $$\def\ZZ{\mathbb{Z}}$$ $$\def\QQ{\mathbb{Q}}$$ $$\def\RR{\mathbb{R}}$$ $$\def\CC{\mathbb{C}}$$

# Exercise sheet¶

This sheet contains a few additional exercises related to the lectures.

Exercise: parabola in projective space

Plot a parabola in 3D, and illustrate that it degenerates into an ellipse when looking tangentially.

Hint: see parametric_plot3d() and the options aspect_ratio, frame, and viewer='threejs' of show().

A solution:

sage: var('u')
sage: p = parametric_plot3d((u, -u^2, 0), (u,-40,40), boundary_style=None)
sage: p.show(viewer="threejs", frame=False)


Research problem

Define the operators

$D_{q,k} = (1+qx_1\partial_1)\partial_1^k+\cdots+ (1+qx_n\partial_n)\partial_n^k$

acting on the polynomial ring $$\mathbb{Q}[x_1,\dots,x_n]$$. At $$q=0$$, the operators degenerate to the symmetric powersums, seen as differential operators. Their joint zeroes form the space of harmonic polynomials, which is of dimension $$n!$$, carries the graded regular representation of $$S_n$$, etc.

Conjecture [Wood with successive refinements by Hivert & T., D’Aderrio & Mocci, Bergeron & Borie & T.]:

• The same holds for $$q$$-harmonic polynomials, defined as the joint zeroes of the operators $$D_{q,k}, k\geq 1$$.
• Exceptions: $$q=-a/b$$ for $$a,b \in \mathbb{NN}$$ with $$1\leq a \leq n \leq b$$.
• This extends to Coxeter groups $$G(m,p,n)$$ and diagonal harmonics.

Many things have been tried, but I (Nicolas) believe nobody tried to use the Cherednik algebra to tackle this problem.

References:

• arXiv:1010.4985 On a conjecture of Hivert and Thiéry about Steenrod operators Michele D’Adderio, Luca Moci
• arXiv:1011.3654 Deformed diagonal h`armonic polynomials for complex reflection groups François Bergeron, Nicolas Borie, Nicolas M.Thiéry
• arXiv:0812.3566 Harmonics for Deformed Steenrod Operators Francois Bergeron, Adriano Garsia, Nolan Wallach
• arXiv:0812.3056 Deformation of symmetric functions and the rational Steenrod algebra Florent Hivert, Nicolas M. Thiéry