Coxeter Pop-Tsack Torsing

Given a finite irreducible Coxeter group W with a fixed Coxeter element c , we define the Coxeter pop-tsack torsing operator Pop T : W → W by Pop T ( w ) = w · π T ( w ) − 1 , where π T ( w ) is the join in the noncrossing partition lattice NC( w,c ) of the set of reflections lying weakly below w in the absolute order. This definition serves as a “Bessis dual” version of the first author’s notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack sorting map on symmetric groups. We show that if W is coincidental or of type D , then the identity element of W is the unique periodic point of Pop T and the maximum size of a forward orbit of Pop T is the Coxeter number h of W . In each of these types, we obtain a natural lift from W to the dual braid monoid of W . We also prove that W is coincidental if and only if it has a unique forward orbit of size h . For arbitrary W , we show that the forward orbit of c − 1 under Pop T has size h and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit.