Yang-Baxter type equations and posets of maximal chains

Ruth Lawrence

Abstract: The usual Yang-Baxter equation may be viewed as a commutativity relation on faces of a permutahedron. These polyhedra are related via extension posets to certain arrangements of hyperplanes and their vertices are in 1-1 correspondence with maximal chains in the Boolean poset B_n. 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 in S_{a_1}x...xS_{a_k} is 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.

Keywords: Yang-Baxter equation, uniform extension posets, symmetric group, maximal chains, zonotope decompositions, hyperplane arrangements

Length: 37 pages

Reference: J. Comb. Th.79 (1997) 68-104. MR1449750 (98g:05161) (review by Volkmar Welker .)

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Last updated on April 15th, 2018.