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 some computational exercises related to the lectures.
Exploring the available features for reflection groups in Sage¶
Exercise (basic computations + explore the classification)
For all finite Coxeter groups \(W\) (just a few of them for the infinite families):
#. Compute the cardinality of `W`
- Compute the length of the longest element of \(W\)
See CoxeterGroup()
, samples()
Exercise (pictures)
- Construct the root lattice for type \(G_2\) and plot it (see Root Systems, Tutorial: visualizing root systems).
- Draw more pictures, for finite and affine Weyl groups!
Exercise (computing with roots)
Check on examples the property that \(ws_i\) is longer than \(w\) if and only if \(w.\alpha_i\) is a positive root.
Two options with the current implementation in Sage:
- In the crystalographic case, build the root lattice and its Weyl group
- Use the permutation representation
Exercise (enumerative combinatorics for reduced words)
- Count the number of reduced words for the longest element in
\(S_n\) and retrieve the sequence from the Online Encyclopedia
of Integer Sequences, for example by using
oeis
. - Check on computer that this matches with OEIS’s suggestion
about
standard Young tableaux
). - The bijection is known as Edelman-Green’s insertion. Search for
its implementation is Sage (see
search_src()
). - Try with other types.
Around Piotr’s lectures¶
Exercises
- Draw the (truncated) Cayley graph for Gamma = 3,3,3
- Implement the twist operation
- Implement the twist-rigidity test
- Implement listing all applicable twists
- Compute all Coxeter systems that can be obtained from a given
Coxeter system by applying twists (see
RecursivelyEnumeratedSet
) - Implement the (truncated) Davis complex
Around Vic’s lectures¶
Exercise (product formula for inversions)
- Check the product formula for the inversions statistic in the
- symmetric group;
- Retrieve the analogue product formula for some other reflection groups.
Exercise (other product formula)
- Implement a function that, given a polynomial \(\prod(1-q^{d_i})\) in expanded form, recovers the \(d_i\) (see exercise 2 in Vic’s exercise sheet);
- Use it to recover the degrees, exponents, and coexponents for a couple reflection groups from their Molien formula, and check the product formula of the lectures (see Computing Molien-type sums for reflection groups).