Bargain hunting in a Coxeter group (Q6621956)

Let \(W=\langle T \rangle\) be a group with generating set \(T\). Any cost function \(\$: T \rightarrow \mathbb{R}_{>0}\), can be extended to all elements \(w \in W\) by minimizing over all decompositions of \(w\) into products of elements of \(T\), that is \(\$(w)= \min\{\$(t_{1})+ \dots +\$(t_{k}) \mid t_{1}, \dots, t_{k} \in T, \, t_{1}\cdot \ldots \cdot t_{k}=w \}\). \textit{T. K. Petersen} and the second author [J. Comb. 6, No. 1--2, 145--178 (2015; Zbl 1317.20040)], considered the situation in which the symmetric group \(S_{n}\) is generated by the set \(T=\{(i j)\}\) of all transpositions and the cost function is \(\$ ( (i,j) )= | j- i |\). They showed that for this function, the cost of a permutation is half of its total displacement: \(\$(w)=\frac{1}{2} \sum_{i=1}^{n} | w(i)-i |\).\N\NIn the paper under review, the authors generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection \(t\) be the distance between the integers transposed by \(t\) in the combinatorial representation of the group. They show that the cost of arbitrary elements of Weil groups of finite types \(\mathsf{A}\), \(\mathsf{B}\), \(\mathsf{D}\) and affine types \(\widetilde{\mathsf{A}}\), \(\widetilde{\mathsf{B}}\), \(\widetilde{\mathsf{C}}\), \(\widetilde{\mathsf{D}}\) can be computed directly from the elements themselves using a simple, intrinsic formula (see Theorem 3.1).