reflection-groups-exercises.rst

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`

1. Compute the length of the longest element of W

See :func:`~sage.combinat.root_system.coxeter_group.CoxeterGroup`, :meth:`~sage.combinat.root_system.cartan_type.CartanTypeFactory.samples`

Exercise (pictures)

1. Construct the root lattice for type G_2 and plot it (see :ref:`sage.combinat.root_system`, :ref:`sage.combinat.root_system.plot`).
2. Draw more pictures, for finite and affine Weyl groups!

Exercise (computing with roots)

1. 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)

1. 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 :obj:`oeis`.
2. Check on computer that this matches with OEIS's suggestion about :class:`standard Young tableaux <StandardTableaux>`).
3. The bijection is known as Edelman-Green's insertion. Search for its implementation is Sage (see :func:`search_src`).
4. Try with other types.

Around Piotr's lectures

Exercises

1. Draw the (truncated) Cayley graph for Gamma = 3,3,3
2. Implement the twist operation
3. Implement the twist-rigidity test
4. Implement listing all applicable twists
5. Compute all Coxeter systems that can be obtained from a given Coxeter system by applying twists (see :class:`RecursivelyEnumeratedSet`)
6. Implement the (truncated) Davis complex

Around Vic's lectures

Exercise (product formula for inversions)

1. Check the product formula for the inversions statistic in the

   symmetric group;

2. Retrieve the analogue product formula for some other reflection groups.

Exercise (other product formula)

1. 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);
2. 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 :ref:`demo-reflection-groups-molien`).