[Previous] [Next] [Contents]

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.

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. 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.

[Previous] [Next] [Contents]


MuPAD Combinat, an open source algebraic combinatorics package