In algebra, Solomon's descent algebra of a
Coxeter group is a subalgebra of the
integral group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given ...
of the Coxeter group, introduced by .
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 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.
References
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Reflection groups
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