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

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 iₖ(ΓW)·|W|\,/\,2ᵏ, where ΓW is the Coxeter graph of W and iₖ(ΓW) is the number of independent sets of size k of ΓW when W is finite. Specializing to Aₙ, we recover the characterizations and enumerations of Boolean intervals in the weak order of Aₙ given in arXiv:2306.14734. We provide the analogous results for types Cₙ and Dₙ, including the related generating functions and additional connections to well-known integer sequences.