18 research outputs found
Noncommutative crossing partitions
We define and study noncommutative crossing partitions which are a
generalization of non-crossing partitions. By introducing a new cover relation
on binary trees, we show that the partially ordered set of noncommutative
crossing partitions is a graded lattice. This new lattice contains the Kreweras
lattice, the lattice of non-crossing partitions, as a sublattice. We calculate
the M\"obius function, the number of maximal chains and the number of
-chains in this new lattice by constructing an explicit -labeling on the
lattice. By use of the -labeling, we recover the classical results on the
Kreweras lattice. We characterize two endomorphism on the Kreweras lattice, the
Kreweras complement map and the involution defined by Simion and Ullman, in
terms of the maps on the noncommutative crossing partitions. We also establish
relations among three combinatorial objects: labeled -ary trees,
-chains in the lattice, and -Dyck tilings.Comment: 45 page
A_k Generalization of the O(1) Loop Model on a Cylinder: Affine Hecke Algebra, q-KZ Equation and the Sum Rule
We study the A_k generalized model of the O(1) loop model on a cylinder. The
affine Hecke algebra associated with the model is characterized by a vanishing
condition, the cylindric relation. We present two representations of the
algebra: the first one is the spin representation, and the other is in the
vector space of states of the A_k generalized model. A state of the model is a
natural generalization of a link pattern. We propose a new graphical way of
dealing with the Yang-Baxter equation and -symmetrizers by the use of the
rhombus tiling. The relation between two representations and the meaning of the
cylindric relations are clarified. The sum rule for this model is obtained by
solving the q-KZ equation at the Razumov-Stroganov point.Comment: 43 pages, 22 figures, LaTeX, (ver 2) Introduction rewritten and
Section 4.3 adde