YANG-BAXTER TYPE EQUATIONS AND POSETS OF MAXIMAL-CHAINS

This paper addresses the problem of constructing higher dimensional ve rsions of the Yang-Baxter equation from a purely combinatorial perspec tive. The usual Yang-Baxter equation may be viewed as the commutativit y constraint on the two-dimensional faces of a permutahedron, a polyhe dron which is related to the extension poset of a certain arrangement of hyperplanes and whose vertices are in 1-1 correspondence with maxim al chains in the Boolean poser R-n. In this paper, similar constructio ns are performed in one dimension higher, the associated algebraic rel ations replacing the Yang-Baxter equation being similar to the permuta hedron equation. The geometric structure of the poser of maximal chain s;in S(al)x...xS(ak) is discussed in some derail, and cell types are F ound to be classified by a poser of ''partitions of partitions'' in mu ch the same way as those for permutahedra are classified by ordinary p artitions. (C) 1997 Academic Press.