Descent algebra

In algebra, Solomon's descent algebra of a Coxeter group is a subalgebra of the integral group ring of the Coxeter group, introduced by Solomon (1976).

The descent algebra of the symmetric group

In the special case of the symmetric group S_n, the descent algebra is given by the elements of the group ring such that permutations with the same descent set have the same coefficients. (The descent set of a permutation σ consists of the indices i such that σ(i) > σ(i+1).) The descent algebra of the symmetric group S_n has dimension 2^n-1. It contains the peak algebra as a left ideal.

References

• Solomon, Louis (1976), "A Mackey formula in the group ring of a Coxeter group", J. Algebra, 41 (2): 255–264, doi:10.1016/0021-8693(76)90182-4, MR 0444756