Götz Pfeiffer,

A Quiver Presentation for Solomon's Descent Algebra.

Abstract.

The descent algebra

Σ(W)

is a subalgebra of the group algebra of a finite Coxeter group

W

, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of

W

. Thus

Σ(W)

is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct

Σ(W)

as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of

S

, the set of simple reflections in

W

. From this construction we obtain some general information about the quiver of

Σ(W)

and an algorithm for the construction of a quiver presentation for the descent algebra

Σ(W)

of any given finite Coxeter group

W

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