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A quiver presentation for Solomon’s descent algebra. (English) Zbl 1203.20004
Summary: The descent algebra \(\Sigma(W)\) is a subalgebra of the group algebra \(\mathbb{Q} W\) of a finite Coxeter group \(W\), which supports a homomorphism with nilpotent kernel and commutative image in the character ring of \(W\). Thus \(\Sigma(W)\) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct \(\Sigma(W)\) 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 \(\Sigma(W)\) and an algorithm for the construction of a quiver presentation for the descent algebra \(\Sigma(W)\) of any given finite Coxeter group \(W\).

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16G20 Representations of quivers and partially ordered sets
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E10 Combinatorial aspects of representation theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)
Full Text: DOI
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