Fully commutative elements in the Weyl and affine Weyl groups.
Let \(W\) be a Weyl or an affine Weyl group and let \(W_c\) be the set of fully commutative elements in \(W\). Lusztig's \(a\)-function is an important invariant for an element of \(W\). It is usually difficult to calculate \(a(z)\) for an arbitrary \(z\in W\). However, in this paper the author associates each \(w\in W\) to a digraph \(G(w)\). By using \(G(w)\), the author gives a graph-theoretic description for Lusztig's \(a\)-function on \(W_c\). For any \(w\in W_c\) denote by \(m(w)\) the maximum possible value of \(l(w_J)\) in an expression \(w=x\cdot w_J\cdot y\). The author proves that \(m(w)=a(w)\) for \(w\in W_c\) such that the value \(a(w)\) for \(w\in W_c\) can be computed easily. The author also describes explicitly all the distinguished involutions of \(W_c\). These results verify two conjectures on \(W_c\): one is the conjecture concerning distinguished involutions in \(W\), this was proposed by the author [in Adv. Sci. China, Math. 3, 79-98 (1990), Conjecture 8.10] and the other is the conjecture concerning the left-connectivity of left cells, this was proposed by \textit{G. Lusztig} [in T. Asai et al., Open problems in algebraic groups, in: R. Hotta (ed.), Problems from the conference on Algebraic Groups and Representations. Katata, August 29-September 3, 1983].
scientific article; zbMATH DE number 2136961
scientific article