A characterization of the simply-laced FC-finite Coxeter groups. (Q1880354)

Let \((W,S)\) be a Coxeter system. An element of \(W\) is called fully commutative element (or \(w\) is fully commutative) if any reduced expression of \(w\) can be converted into any other reduced expression of \(w\) by exchanging adjacent commuting generators several times. A Coxeter group is called `FC-finite' if the number of its fully commutative elements is finite. For \(s,t\in S\), one denotes the order of \(st\) by \(m(st)\). If \(\{m(st)\mid s,t\in S\}\subset\{1,2,3\}\), then one calls \((W,S)\) (resp. \(W\)) a simply-laced Coxeter system (resp. a simply-laced Coxeter group). For two elements \(w\) and \(x\) of \(W\), we say that \(w\) covers \(x\) if \(x<w\) and \(l(x)=l(w)-1\). We call a Coxeter group bi-full if its fully commutative elements coincide with its fully covering elements. In this paper the authors show that the bi-full Coxeter groups are the ones of type \(A_n\), \(D_n\), \(E_n\) with no restriction on \(n\). In other words, Coxeter groups of type \(E_9,E_{10},\dots\) are also bi-full. According to a result of \textit{C. K. Fan} [Ph.D. Thesis, MIT (1995)], a Coxeter group is a simple-laced FC-finite Coxeter group if and only if it is a bi-full Coxeter group.