Abstract
A canonical join representation is a certain minimal "factorization" of an element in a finite lattice $L$ analogous to the prime factorization of an integer from number theory. The expression $\bigvee A =w$ is the canonical join representation of $w$ if $A$ is the unique lowest subset of $L$ satisfying $\bigvee A=w$ (where "lowest" is made precise by comparing order ideals under containment). Canonical join representations appear in many familiar guises, with connections to comparability graphs and noncrossing partitions. When each element in $L$ has a canonical join representation, we define the canonical join complex to be the abstract simplicial complex of subsets $A$ such that $\bigvee A$ is a canonical join representation. We characterize the class of finite lattices whose canonical join complex is flag, and show how the canonical join complex is related to the topology of $L$.
Highlights
Before we give the technical background for our main results, we describe several familiar examples in which the combinatorics of canonical join representations appear
We close the paper be raising some questions about the canonical join graph of a finite semidistributive lattice
Which graphs can be realized as the canonical join graph of some finite semidistributive lattice?
Summary
We consider a finite join-semidistributive lattice L whose canonical join complex is not flag. If L is crosscut-simplicial the order complex of each open interval (x, y) in L is either contractible or homotopy equivalent to a sphere with dimension two less than the number of atoms in [x, y] (see [17, Theorem 3.7]). Suppose that L is a finite join-semidistributive lattice and its canonical join complex is flag. Since the weak order on any finite Coxeter group W and the lattice of torsion classes for Λ of finite representation type are both examples of finite semidistributive lattices (see [8, Lemma 9] and [13, Theorem 4.5]), we obtain the following two applications of Theorem 1: Corollary 6. The canonical join complex of tors(Λ) is flag
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