Rhombic tilings of polygons and classes of reduced words in Coxeter groups
type: Item
instance of: scholarly article
title: Rhombic tilings of polygons and classes of reduced words in Coxeter groups
zbMATH Open document ID: 0867.05019
author: Serge Elnitsky
published in: Journal of Combinatorial Theory. Series A
publication date: 1997-07-09T00:00:00Z
review text: This paper presents and explores bijections between equivalence classes of reduced words, rhombic tilings of (possibly non-convex) polygons and lattice paths. In particular, Theorem 2.2 establishes, for every permutation \(\sigma\) of \(n\) elements, a bijection between the set of rhombic tilings of a certain \((2n)\)-gon \(X(\sigma)\) and the equivalence classes of reduced decompositions of \(\sigma\) modulo commuting non-adjacent transpositions. This extends a known correspondence in the special case when \(\sigma =w_0\) is the order-reversing permutation---the only case when \(X(\sigma)\) is convex. A further correspondence between rhombic tilings of certain centrally-symmetric 8-gons (of side lengths \(a,b,1,1\)) and lattice paths provides the tools for the proof of a formula for the number of such tilings, conjectured by Kuperberg and Propp, together with a \(q\)-analogue. Finally, extensions for signed permutations of type \(B_n\) and \(D_n\) are presented.
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 05A05
zbMATH DE Number: 980661
zbMATH Keywords: higher Bruhat orders
zbMATH Keywords: reduced words
zbMATH Keywords: rhombic tilings
zbMATH Keywords: polygons
zbMATH Keywords: lattice paths
zbMATH Keywords: signed permutations
reviewed by: Günter M. Ziegler
MaRDI profile type: Publication
OpenAlex ID: W2053480094
DOI: 10.1006/JCTA.1997.2723
Recommended article: Q3576678
Recommended article: Rhombic tilings and Bott–Samelson varieties
Recommended article: Some combinatorial aspects of reduced words in finite Coxeter groups
Recommended article: Reduced words in affine Coxeter groups
Recommended article: Tiling with polyominoes and combinatorial group theory
Recommended article: Infinite Reduced Words and the Tits Boundary of a Coxeter Group
Recommended article: Symmetric polyomino tilings, tribones, ideals, and Gröbner bases
Recommended article: Free arrangements and rhombic tilings
Recommended article: Q4331195
Recommended article: Words, tilings and combinatorial spectra
description: scientific article; zbMATH DE number 980661