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arXiv · 2403.07989

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

Ben Adenbaum·Jennifer Elder·Pamela E. Harris·J. Carlos Martínez Mori
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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 ${i_k(\Gamma_W)\cdot|W|}\,/\,2^{k}$, where $\Gamma_W$ is the Coxeter graph of $W$ and $i_k(\Gamma_W)$ is the number of independent sets of size $k$ of $\Gamma_W$ 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.

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Ben Adenbaum, Jennifer Elder, Pamela E. Harris, J. Carlos Martínez Mori. 2024-03-12. Boolean intervals in the weak Bruhat order of a finite Coxeter group. https://arxiv.org/abs/2403.07989

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