Yang-Baxter type equations and posets of maximal chains
type: Item
instance of: scholarly article
title: Yang-Baxter type equations and posets of maximal chains
zbMATH Open document ID: 0922.52003
published in: Journal of Combinatorial Theory. Series A
publication date: 1999-10-05T00:00:00Z
review text: The classical Yang-Baxter equations have the form \[ R_{ij}R_{ik}R_{jk}=R_{jk}R_{ik}R_{ij},\qquad R_{ij}R_{kl}=R_{kl}R_{ij}, \] where \(R_{ij}\) denotes the action of \(R\in A\otimes A\) on the \(i\)th and \(j\)th factors of \(A^{\otimes n}.\) Solutions to these equations yield representations of the braid group, and have many other applications in mathematics and physics. In this paper the author proposes a generalization of the Yang-Baxter equations based solely on geometric and combinatorial considerations, and carries out extensive calculations to reveal possible forms of the ``two-dimensional'' analogue. The motivation for the construction is as follows. The Yang-Baxter equations can be seen geometrically as arising from the 2-dimensional faces of a convex polyhedron, the permutohedron, whose vertices are labelled by permutations and whose edges correspond to transpositions. This polyhedron \(P_1\) has hexagonal and square faces, giving rise to the two types of equations displayed above. The face lattice \(C_1\) of the permutohedron is isomorphic to the partition lattice. The vertices and edges of the permutohedron form the Hasse diagram of the uniform extension poset of a (generic) hyperplane arrangement \(X_1\), the poset of extensions of \(X_1\) by a single pseudo-hyperplane in general position to \(X_1.\) The intersection lattice of \(X_1\) is a truncation \(C_0\) of the Boolean lattice, which in turn is the face lattice of a cube. Finally, the vertices and edges of the cube form the Hasse diagram of the Boolean lattice \(P_0,\) whose poset of maximal chains is isomorphic to the uniform extension poset \(P_1\) realized by the permutohedron. To extend these relationships, the author studies the poset \(P_2\) of maximal chains in \(P_1\), attempting to realize \(P_2\) as the uniform extension poset of a certain hyperplane arrangement \(X_2,\) which should have intersection lattice isomorphic to the face lattice \(C_1\) of the permutohedron realizing \(P_1\). …
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1076895
zbMATH Keywords: Yang-Baxter equations
zbMATH Keywords: arrangement
zbMATH Keywords: uniform extension poset
zbMATH Keywords: convex polyhedron
zbMATH Keywords: permutohedron
zbMATH Keywords: intersection lattice
zbMATH Keywords: poset of maximal chains
author: Ruth Lawrence
reviewed by: Michael Falk
MaRDI profile type: Publication
full work available at URL: jcta.1997.2775
OpenAlex ID: W1997734925
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DOI: 10.1006/JCTA.1997.2775
description: scientific article; zbMATH DE number 1076895