Abstract
We generalize the Tamari lattice by extending the notions of $231$-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients of the symmetric group $\mathfrak{S}_{n}$. We show bijectively that these three objects are equinumerous. We show how to extend these constructions to parabolic quotients of any finite Coxeter group. The main ingredient is a certain aligned condition of inversion sets; a concept which can in fact be generalized to any reduced expression of any element in any (not necessarily finite) Coxeter group.
Highlights
1.1 Parabolic Tamari LatticesThe Tamari lattice Tn was introduced by D
In the second part of this article, we define analogues of the sets SJn(231), NCJn and NNJn for all finite Coxeter groups, and we study in which cases we retain the property that these sets are equinumerous
Comparing the presentations (1) and (3), we see that the symmetric group with the generating set of all adjacent transpositions forms a Coxeter system, and the reflections are all conjugates of the adjacent transpositions
Summary
The Tamari lattice Tn was introduced by D. The weak order Weak(Sn) on the group of permutations Sn is the oriented Cayley graph of Sn, using the generating set S of adjacent transpositions. Wachs realized Tn as a sublattice of Weak(Sn) by considering the subset Sn(231) ⊆ Sn of 231-avoiding permutations, whose inversions sets they characterize as “compressed” [7, Section 9]. We generalize the Tamari lattice from the symmetric group to its parabolic quotients. We specify a subset SJn(231) ⊆ SJn by introducing a generalized notion of 231-avoidance, dependent on J, which can again be seen as a “compressed” condition on inversion sets. For J ⊆ S, the restriction Weak(SJn) to SJn(231) is a lattice, which we denote TnJ. TnJ is not generally a sublattice of Weak(SJn), it is a lattice quotient of Weak(SJn).
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