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## Exercices

### The degenerate Hecke Algebra

The Iwahori-Hecke algebra Hn(q) is the C-algebra generated by elements Ti for i<n with the relations: 2 T_i^2=(q-1)T_i+q for 1≤i ≤n-1,
T_iT_j=T_jT_i for |i-j|>1,
T_iT_i+1T_i =T_i+1T_iT_i+1T for 1≤i ≤n-2, The 0-Hecke algebra is obtained by setting q=0 in these relations. Then, the first relation becomes Ti2=-Ti. Let πi := 1+Ti.

- Define a parameterized domain `perm(n)` for the permutations of n. Note that this may be automatic in the near future.
- Define a parameterized domain `HeckePi(n)` for the Hecke algebra in the basis (πσ)σ∈ Sn with coefficient the rational (`Dom::rational`).
- Define the elementary generator πi. Check some braid relations.
- Define a parameterized domain `HeckeT(n)` for the Hecke algebra in the basis (Tσ)σ∈ Sn with coefficient the rational (`Dom::rational`).
- Check for small n that sum and alternating sum of the Tσ are two idempotents.
- Define the change of basis between the two versions. This can be done by hands or using the Bruhat order (`permutations::smallerBruhat`). Make the conversion automatic.
- We need to compute with different coefficients, in particular we often need to compute with unknown coefficients taken out `Dom::ExpressionField()`. Modify the two parameterized domains `HeckePi(n)` and `HeckeT(n)` so that they accept a ring for second parameter with `Dom::ExpressionField()` as default value.

As a application we want now to compute the radical of Hn(0). We use a characterization from Dickson. theoremTheorem[section] Let A be a sub algebra of Mn(F) for a field F of zero characteristic. Then x∈A is in the radical of A if and only if the trace of each y in the two sided (resp. left, right) ideal generated by x (AxA, resp. Ax, xA) is zero. For finite dimensional algebra A, we realize A as a matrix algebra by considering any faithful representation, for example the regular one.

- Let g := ∑σ∈ Sn aσ Πσ be the generic element of Hn(0). Compute g and the generating family (gΠμ)μ∈ Sn for the ideal.
- Write a function `trace` that compute the trace of the endomorphism x∈Hn(0) →x e where e∈Hn(0).
- Using Dickson criterion, compute the equation of the radical of Hn(0).
- What is the dimension of the radical, of the quotient ?
- Write a function `radNormal` that normalize an element of Hn(0) modulo its radical.

Suppose that A is an algebra with a basis B. In MuPAD, A is a certain domain `Alg`. The basis B of A is indexed by the element of the domain `Alg::basisIndices`. Like all combinatorial domain this domain may have a list method which can be called by `Alg::basisIndices::list`.

- Write a function `GetBasis` to compute a basis of an algebra or more generally of a free module.
- Write a function `Generic Element` to compute a generic element;
- Write a function `Radical` to compute the radical of an algebra.

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MuPAD Combinat, an open source algebraic combinatorics package