Canonical Join Representations and Join-irreducible elements of Garside shadows in Coxeter groups

In this article, we establish some new combinatorial properties of elements in Coxeter groups. Firstly, we generalise Reading’s theorem on the canonical join representations of elements in finite Coxeter groups to all finitely generated Coxeter groups. Secondly, we show that for any element $x$ in a Coxeter group $W$ and root $β$ in its inversion set $Φ(x)$, the set of elements $y \in W$ satisfying $Φ(x) \cap Φ(y) = { β} $ is convex in the weak order and admits a unique minimal representative. This is strongly connected to determining the cone type of elements of $W$ and leads to efficient computational methods to determine whether arbitrary elements of $W$ have the same cone type.

💡 Research Summary

This paper presents a comprehensive study of two fundamental combinatorial structures in Coxeter groups: canonical join representations and cone types. It generalizes known results from finite to all finitely generated Coxeter groups and provides deep structural insights with significant algorithmic implications.

The first main result (Theorem 1) generalizes Reading’s theorem on canonical join representations. In finite Coxeter groups, Reading showed that the canonical join representation of an element w is the join of unique minimal length elements j_β, each corresponding to a right-descent root β in Φ_R(w). This proof relied on geometric properties of hyperplane arrangements specific to finite groups. This paper provides a uniform proof that extends this result to all finitely generated Coxeter groups, including infinite ones. The key to the generalization is leveraging the combinatorial structure of the short inversion poset, bypassing the need for finite-geometry arguments. This establishes that the lattice-theoretic property of having a canonical join representation is inherent to the weak order of any Coxeter group.

The second major contribution revolves around the structure of cone types. A cone type T(w) consists of all elements v such that the product wv is length-additive. Understanding cone types is crucial for the automatic structure of Coxeter groups. The paper investigates the sets of “witnesses” for boundary roots of a cone type. For an element x and a root β in its inversion set Φ(x), a witness y is an element such that Φ(x) ∩ Φ(y) = {β}. Theorem 2 proves that for any such x and β, if a witness exists, the set of all witnesses is convex in the weak order and contains a unique element y_β of minimal length. This minimal witness satisfies Φ_R(y_β) = {β}.

This leads to Corollary 1, which lifts the result to the level of cone types. For a given cone type T and a boundary root β ∈ ∂T, there exists a unique minimal length element y_β such that for every x in the cone type part Q(T) (elements whose cone type is T), Φ(x) ∩ Φ(y_β) = {β}. This y_β is a gate for the witness set. The paper then connects these minimal witnesses, called “tight gates” (Γ0 = {g ∈ Γ | |Φ_R(g)| = 1}), to the lattice structure of the set of all gates Γ (the unique minimal elements of each cone type part Q(T)). Theorem 3 shows that Γ0 is precisely the set of join-irreducible elements of the poset (Γ, ⪯). A crucial consequence (Corollary 2) is that Γ0 is closed under taking suffixes.

The structural theory has direct computational applications. Corollary 5.2 states that the set of tight gates Γ0 completely determines the set of all cone types T. More importantly, the suffix-closed property of Γ0 leads to an efficient algorithm (Algorithm 6.1) for computing Γ0. Starting from the identity, the algorithm iteratively expands the set by right-multiplying existing tight gates by simple reflections corresponding to short inversions. This algorithm often allows the computation of Γ0 without first computing the much larger set of all gates Γ or all cone types T, providing a practical method for determining whether two arbitrary group elements have the same cone type. Computational experiments using SageMath demonstrate the efficiency of this approach, especially as the rank of the Coxeter group increases.

In summary, this work deepens the combinatorial understanding of Coxeter groups by unifying and extending the theory of canonical join representations and by revealing the fine-grained structure of cone types through convex, gated subsets and tight gates. It successfully bridges theoretical lattice theory and geometric group theory with practical algorithmic design.