Mathematics – Combinatorics
Scientific paper
2004-05-25
Mathematics
Combinatorics
24 pages, 8 figures
Scientific paper
Louis Solomon showed that the group algebra of the symmetric group $\mathfrak{S}_{n}$ has a subalgebra called the descent algebra, generated by sums of permutations with a given descent set. In fact, he showed that every Coxeter group has something that can be called a descent algebra. There is also a commutative, semisimple subalgebra of Solomon's descent algebra generated by sums of permutations with the same number of descents: an "Eulerian" descent algebra. For any Coxeter group that is also a Weyl group, Paola Cellini proved the existence of a different Eulerian subalgebra based on a modified definition of descent. We derive the existence of Cellini's subalgebra for the case of the symmetric group and of the hyperoctahedral group using a variation on Richard Stanley's theory of $P$-partitions.
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