The descent algebra is a subalgebra of the group algebra of a finite Coxeter group , which supports a homomorphism with nilpotent kernel and commutative image in the character ring of . Thus is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of , the set of simple reflections in . From this construction we obtain some general information about the quiver of and an algorithm for the construction of a quiver presentation for the descent algebra of any given finite Coxeter group .
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