Yang–Baxter Type Equations and Posets of Maximal Chains

This paper addresses the problem of constructing higher dimensional versions of the Yang–Baxter equation from a purely combinatorial perspective. The usual Yang–Baxter equation may be viewed as the commutativity constraint on the two-dimensional faces of a permutahedron, a polyhedron which is related to the extension poset of a certain arrangement of hyperplanes and whose vertices are in 1–1 correspondence with maximal chains in the Boolean poset Bn. In this paper, similar constructions are performed in one dimension higher, the associated algebraic relations replacing the Yang–Baxter equation being similar to the permutahedron equation. The geometric structure of the poset of maximal chains inSa1×…×Sakis discussed in some detail, and cell types are found to be classified by a poset of “partitions of partitions” in much the same way as those for permutahedra are classified by ordinary partitions.