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

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