The Weak Bruhat Order of $\text{S}_\Sigma$, Consistent Sets, and Catalan Numbers

Source abstract

Chains in the weak Bruhat order$\beta$ of $\text{S}_\Sigma$ (the symmetric group on $\Sigma$) belong to the class of subsets of $\text{S}_\Sigma$ over which unrestricted choice necessarily produces transitive relations under pairwise simple majority vote (consistent sets). If for $\text{A} \subset \text{S}_\Sigma$ we let $\text{T}( \text{A} ) \equiv \cup _{\text{p} \in \text{A}} \text{T} ( \text{p} )$ where $\text{T}( \text{p} ) = \{ ( \text{p}_{\text{i}} , \text{p}_{\text{j}} ,\text{p}_{\text{k}} )| \text{i} < \text{j} < \text{k} \}$ and $\Psi ( \text{A} ) \equiv \{ \text{w} \in \text{S}_\Sigma \mid \text{T} ( \text{w} ) \subset \text{T} ( \text{A} ) \}$ the following theorem (among others) is obtained. Theorem. For all${\text{q}} \in {\text{S}}_\Sigma$, if${\text{A}}$is a saturated chain under$\beta$ then $\Psi ( {\text{qA}} )$is an upper semimodular sublattice of cardinality$|\Psi ( {{\text{qA}}} )|\leqq \frac{1}{{|\Sigma | + 1}} (\begin{smallmatrix} 2|\Sigma | \\ |\Sigma | \end{smallmatrix}) \equiv$The$|\Sigma |$th Catalan number. From the Arrow’s Impossibility Theorem point of view, the results obtained here indicate that majority rule produces transitive results if the collection of voters as a whole can be partitioned into no more than $( |\Sigma |^2 + |\Sigma | )/2$ groups which can be ordered according to the level of disagreement they have with respect to a fixed permutation ${\text{p}}$. On the other hand, by viewing ${\text{S}}_\Sigma$ as a Coxeter group a “novel” combinatorial interpretation of the collection of maximal chains that can be obtained from one another by using only one type of Coxeter transformation is obtained.