Boolean intervals in the weak Bruhat order of a finite Coxeter group

Abstract

Given a Coxeter group $W$ with Coxeter system $(W,S)$, where $S$ is finite. We provide a complete characterization of Boolean intervals in the weak order of $W$ uniformly for all Coxeter groups in terms of independent sets of the Coxeter graph. Moreover, we establish that the number of Boolean intervals of rank $k$ in the weak order of $W$ is ${i_k(Γ_W)\cdot|W|}\,/\,2^{k}$, where $Γ_W$ is the Coxeter graph of $W$ and $i_k(Γ_W)$ is the number of independent sets of size $k$ of $Γ_W$ when $W$ is finite. Specializing to $A_n$, we recover the characterizations and enumerations of Boolean intervals in the weak order of $A_n$ given in arXiv:2306.14734. We provide the analogous results for types $C_n$ and $D_n$, including the related generating functions and additional connections to well-known integer sequences.

Figures (11)

• Figure 1: The graph $\Gamma_{A_n}$ associated to $A_n$.
• Figure 2: The induced subgraph $\Gamma_{A_8}[\{2, 3, 5, 8\}]$.
• Figure 3: The graph $\Gamma_{C_n}$ associated to $C_n$.
• Figure 4: Weak order of the Coxeter group of type $C_3$ represented as mirrored permutations in one-line notation.
• Figure 5: Examples of Boolean intervals in the weak order of $C_3$.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Definition 2.1
• proof : Proof of Theorem \ref{['thm: supp']}
• proof : Proof of Theorem \ref{['thm:local_bools_above']}
• proof : Proof of Corollary \ref{['cor:global_bools_below']}
• Example 3.1
• Theorem 3.2
• Lemma 3.3