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
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