Weak Order

Weak Order

(i) $u{\le }_{R}w$ means that $w=u{s}_1{s}_2\dots s_k$, for some $s_i\in S$ such that $\ell (u{s}_1{s}_2\dots s_i)=\ell (u)+i$, $0\le i\le k$. (ii) $u{\le }_{L}w$ means that $w=s_k{s}_{k-1}\dots s_{1}u$, for some $s_i\in S$ such that $\ell (s_i{s}_{i-1}\dots s_{1}u)=\ell (u)+i$, $0\le i\le k$.

Proposition

(i) There is a one-to-one correspondence between reduced decompositions of $w$ and maximal chains in the interval $[e,w{]}_R$. (ii) $u{\le }_{R}w⟺\ell (u)+\ell (u^{-1}w)=\ell (w)$. (iii) If $W$ is finite, then $w{\le }_R{w}_0$ for all $w\in W$. (iv) Weak order satisfies the “prefix property” $\begin{array}{rl}u{\le }_{R}w⟺& \text{there exist reduced expressions}\\ & u=s_1{s}_2\dots s_k\text{ and }w=s_1{s}_2\dots s_k{s}_1^{\prime }{s}_2^{\prime }\dots s_q^{\prime }.\end{array}$ (v) Weak order satisfies a “chain property” analogous to the chain property in Bruhat order. (vi) Suppose $s\in D_L(u)\cap D_L(w)$. Then, $u{\le }_{R}w⟺su{\le }_{R}sw$. $\square$.

Proposition

$u{\le }_{R}w⟺T_L(u)\subseteq T_L(w)$.

Corollary

The mapping $w\mapsto T_L(w)$ provides an order-preserving and rank-preserving embedding of $W$, as a poset under weak order, into the lattice of all finite subsets of $T$. $\square$

Weak order on a finite Coxeter group shares with Bruhat order the symmetries induced by the top element.

Proposition

For weak order on a finite $W$_, the following hold (i) $w\mapsto w_{0}w$ and $w\mapsto w{w}_0$ are antiautomorphisms. (ii) $w\mapsto w_{0}w{w}_0$ is an automorphism

Proposition

If $u{\le }_{R}w$, then $[u,w{]}_R\cong [e,u^{-1}w{]}_R$

An element $z$ in a poset is said to be the meet (or, greatest lower bound) of a subset $A$ if (i) $z\le y$ for all $y\in A$, and (ii) $u\le y$ for all $y\in A$ implies that $u\le z$. The meet, if it exists, is clearly unique. It is then denoted $\bigwedge A$, or if $A=\{x,y\}$, simply $x\wedge y$. A poset $L$ for which every nonempty subset has a meet is called a complete meet-semilattice. Such a poset has a minimum element $\hat{0}=\bigwedge L$.

Theorem

Weak order on $W$ is a complete meet-semilattice.