Top Element for Finite

Proposition

(i) If $W$ is finite, there exists an element $w_0\in W$ such that $w\le w_0$ for all $w\in W$. (ii) Conversely, suppose that $(W,S)$ has an element $x$ such that $D_L(x)=S$. Then, $W$ is finite and $x=w_0$.

Proposition

The top element $w_0$ of a finite group has the following properties: (i) $w_0^2=e$. (ii) $\ell (w{w}_0)=\ell (w_0)-\ell (w)$, for all $w\in W$. (iii) $T_L(w{w}_0)=T\setminus T_L(w)$, for all $w\in W$. (iv) $\ell (w_0)=|T|$.

Corollary

(i) $\ell (w_{0}w)=\ell (w_0)-\ell (w)$, for all $w\in W$. (ii) $\ell (w_{0}w{w}_0)=\ell (w)$, for all $w\in W$.

Proposition

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

Theorem

Suppose that $(W,S)$ is irreducible and $|S|\ge 3$. If $\phi :W\to W$ is an automorphism of Bruhat order and $\phi (s)=s$ for all $s\in S$, then either $\phi (x)=x$ for all $x\in W$ or $\phi (x)=x^{-1}$ for all $x\in W$.

Corollary

If $(W,S)$ is irreducible and $|S|\ge 3$, then the automorphism group of Bruhat order is generated by the diagram automorphisms and the mapping $x\mapsto x^{-1}$