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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 CoxeterGroup(), samples()

Exercise (pictures)

  1. Construct the root lattice for type \(G_2\) and plot it (see Root Systems, Tutorial: visualizing root systems).
  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 oeis.
  2. Check on computer that this matches with OEIS’s suggestion about standard Young tableaux).
  3. The bijection is known as Edelman-Green’s insertion. Search for its implementation is Sage (see 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 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 Computing Molien-type sums for reflection groups).