Bruhat Intervals in the Infinite Symmetric Group are Cohen-Macaulay

TL;DR

This paper proves that certain algebraic structures associated with Bruhat intervals in the infinite symmetric group are Cohen-Macaulay, extending finite group results to an infinite-dimensional setting.

Contribution

It introduces an infinite-dimensional analogue of Cohen-Macaulay properties for Bruhat intervals in the infinite symmetric group, generalizing finite group results.

Findings

Stanley-Reisner ring of the order complex is Cohen-Macaulay

Extends finite symmetric group results to infinite case

Provides algebraic properties of Bruhat intervals in $S__$

Abstract

We show that the (non-Noetherian) Stanley-Reisner ring of the order complex of certain intervals in the Bruhat order on the infinite symmetric group $S_\infty$ of all auto-bijections of $\mathbb{N}$ is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. This gives an infinite-dimensional version of results due to Edelman, Bj\"{o}rner, and Kind and Kleinschmidt for finite symmetric groups $S_n$.