[Paper Review] A Quiver Presentation for Solomon's Descent Algebra

Götz Pfeiffer|ArXiv.org|Sep 25, 2007

Advanced Combinatorial Mathematics13 references4 citations

TL;DR

This paper presents a quiver presentation for Solomon's descent algebra Σ(W) of any finite Coxeter group W by constructing it as a quotient of a subalgebra of the path algebra of the Hasse diagram of the power set of simple reflections S. The key contribution is an algorithmic method to compute quiver presentations for Σ(W), with explicit results for types Aₙ, Bₙ, Dₙ, and exceptional types, using a difference operator and monoid actions on paths and streets.

ABSTRACT

The descent algebra $Σ(W)$ is a subalgebra of the group algebra $\Q W$ of a finite Coxeter group $W$, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of $W$. Thus $Σ(W)$ 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 $W$.

Motivation & Objective

• To provide a general, algorithmic method for constructing quiver presentations of Solomon’s descent algebra Σ(W) for any finite Coxeter group W.
• To establish a geometric and combinatorial framework linking the descent algebra to the Hasse diagram of the power set of simple reflections.
• To extend known results on quivers of descent algebras—previously limited to symmetric groups (type Aₙ)—to all finite Coxeter groups, including Bₙ, Dₙ, and exceptional types.
• To formalize the descent algebra as a quotient of a subalgebra of a path algebra via a difference operator, enabling systematic computation.
• To provide explicit quiver presentations and relations for Σ(W) in types Aₙ, Bₙ, Dₙ, and exceptional cases, with computational verification via GAP.

Proposed method

• Construct the path algebra A of the Hasse diagram of the power set P(S), ordered by reverse inclusion, using the free monoid S* to generate paths called 'alleys'.
• Define a partial action of the Coxeter group W on P(S) to induce a subalgebra Ξ of A, where orbits of alleys form 'streets'—a basis for Ξ.
• Introduce a difference operator Δ on A that maps Ξ surjectively onto the degree-zero component A₀, which is identified with Σ(W).
• Show that Δ is an anti-homomorphism from Ξ onto Σ(W), establishing the isomorphism Σ(W) ≅ A₀ via the main theorem (Theorem 9.5).
• Use the structure of streets and their prefixes/suffixes to define two rooted forests, decomposing Ξ into projective indecomposable modules.
• Develop an algorithm (Algorithm 11.1) to compute quiver presentations for Σ(W), based on the combinatorics of shapes, streets, and relations derived from path equivalences.

Experimental results

Research questions

• RQ1 How can a quiver presentation for the descent algebra Σ(W) be systematically constructed for any finite Coxeter group W?
• RQ2 What is the role of the Hasse diagram of the power set of simple reflections in encoding the structure of Σ(W)?
• RQ3 How do monoid actions and path algebras on alleys and streets relate to the projective indecomposable modules of Σ(W)?
• RQ4 What are the general structural properties of the quiver of Σ(W) across different types of finite Coxeter groups?
• RQ5 How can the difference operator Δ be used to realize Σ(W) as a quotient of a subalgebra of a path algebra, and what are the resulting relations?

Key findings

• The descent algebra Σ(W) is isomorphic to the degree-zero component A₀ of the path algebra A, via an anti-homomorphism Δ from a subalgebra Ξ of A.
• For type D₆, the quiver of Σ(W) has 26 vertices and 22 edges, with multiple edges between certain pairs of vertices, and three specific relations involving path equivalences.
• In type Dₙ with n odd, edges in the quiver exist only between shapes corresponding to partitions of n with exactly one odd part, under specific merging or deletion rules.
• The algorithm for computing quiver presentations is implemented in GAP and verified for types Aₙ, Bₙ, Dₙ, and exceptional types, with explicit examples provided.
• The subalgebra Ξ is conjectured to be a path algebra, supported by the existence of two rooted forests on the set of streets.
• Relations in the quiver arise from path equivalences in the algebra, such as (4→14→26) = -2(4→15→26) in type D₆, reflecting non-trivial relations in Σ(W).