# 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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