In algebra, Solomon's descent algebra of a

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

is a subalgebra of the integral group ring of the Coxeter group, introduced by .

The descent algebra of the symmetric group

In the special case of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

''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 In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group ''S'n'', studied by . It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for permutati ...

as a

left ideal In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even n ...

References

* Reflection groups {{group-theory-stub