Yang-Baxter type equations and posets of maximal chains
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 \(P_1\). The Hasse diagram of \(P_2\) is given by the vertices and edges of a convex polyhedron, whose two-dimensional faces yield the ``commutativity constraints'' generalizing the Yang-Baxter equations. This web of relationships is clarified by a ``schematic diagram'' at the start of the paper. There are unfortunate complications in carrying out this program. In particular, the usual notion of uniform extension poset is modified in several different ways in order to accomplish the different objectives. The author suggests that there may be fundamental combinatorial obstructions to completing this program that are reflections of the difficulties encountered in other attempts to generalize the Yang-Baxter equations.
scientific article; zbMATH DE number 1076895