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arXiv · 2111.11553

The canonical complex of the weak order

Doriann Albertin·Vincent Pilaud
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Abstract

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. Ch\^atel, 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.

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Doriann Albertin, Vincent Pilaud. 2021-11-22. The canonical complex of the weak order. https://doi.org/10.1007/s11083-022-09605-7

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