On excess in finite Coxeter groups

For a finite Coxeter group W and w an element of W the excess of w   is defined to be e(w)=min⁡{ℓ(x)+ℓ(y)−ℓ(w)|w=xy,x2=y2=1} where ℓ is the length function on W  . Here we investigate the behaviour of e(w)e(w), and a related concept reflection excess, when restricted to standard parabolic subgroups of W. Also the set of involutions inverting w is studied.