The Canonical Complex of the Weak Order

We define and study the canonical complex of a finite semidistributive lattice L. It is the simplicial complex on the join or meet irreducible elements of L which encodes each interval of L by recording the canonical join representation of its bottom element and the canonical meet representation of its top element. This complex behaves properly with respect to lattice quotients of L, in the sense that the canonical complex of a quotient of L is the subcomplex of the canonical complex of L induced by the join or meet irreducibles of L uncontracted in the quotient. We then describe combinatorially the canonical complex of the weak order on permutations in terms of semi-crossing arc bidiagrams, formed by the superimposition of two non-crossing arc diagrams of N. Reading. We provide explicit direct bijections between the semi-crossing arc bidiagrams and the weak order interval posets of G. Châtel, V. Pilaud and V. Pons. Finally, we provide an algorithm to describe the Kreweras maps in any lattice quotient of the weak order in terms of semi-crossing arc bidiagrams.