A note on the subword complexes in Coxeter groups

TL;DR

This paper investigates the algebraic and combinatorial properties of subword complexes in Coxeter groups, establishing new results on their Stanley--Reisner ideals, shelling orders, and invariants, including conditions for complete intersection rings.

Contribution

It proves that the Stanley--Reisner ideal of the Alexander dual of subword complexes has linear quotients and introduces a shelling order, linking algebraic invariants to combinatorial structures.

Findings

Stanley--Reisner ideal has linear quotients in lex order

Shelling order on facets of subword complexes established

Stanley--Reisner ring can be a complete intersection for certain complexes

Abstract

We prove that the Stanley--Reisner ideal of the Alexander dual of the subword complexes in Coxeter groups has linear quotients with respect to the lexicographical order of the minimal monomial generators. As a consequence, we obtain a shelling order on the facets of the subword complex. We relate some invariants of the subword complexes or of their dual with invariants of the word. For a particular class of subword complexes, we prove that the Stanley--Reisner ring is a complete intersection ring.