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.
Bibliographical noteFunding Information:
The author thanks IHES for their generous hospitality while the first version of this paper was being typed and for support from a Raymond and Beverly Sackler Fellowship. Much of the present version was written while the author was a visiting researcher at the Hebrew University in Jerusalem, whom the author would like to thank for making their facilities freely available. She also wishes to thank P. Hanlon for teaching her about posets, J. Stembridge for some useful discussions, and the referee for having suggested a number of improvements in the presentation of this paper. This work is supported in part by NSF grant DMS-9626544.