Bases for the Bruhat-Chevalley order on all finite Coxeter groups
According to the definition by \textit{A. Lascoux} and \textit{M. P. Schützenberger}, a base for the Bruhat-Chevalley order \(\leq\) on a Coxeter group \(W\) is a minimal subset \(B\) of \(W\) such that if \(({\mathcal P}(B),\subseteq)\) is the partially ordered set of all subsets of \(B\) then the map \(W\to{\mathcal P}(B)\), \(w\mapsto\{b\in B\mid b\leq w\}\), is an isomorphism of partially ordered sets onto its image [see Electron. J. Combin. 3, No.~2, R27 (1996) = J. Comb. 3, No.~2, 633-667 (1996; Zbl 0885.05111)]. The existence of a base allows us to transform the Bruhat-Chevalley order relation to a purely Boolean condition, the latter provides an efficient and practical criterion for determining the relation \(v\leq w\) in \(W\). An element \(w\in W\) is called a bi-Grassmannian if each of the sets \({\mathcal L}(w)=\{s\in S\mid\ell(sw)<\ell(w)\}\), \({\mathcal R}(w)=\{s\in S\mid\ell(ws)<\ell(w)\}\) consists of precisely one element. The basic result of [loc. cit.] states that the base \(B\) of \((W,\leq)\) is contained in the set of all bi-Grassmannians of \(W\). In the present paper, the authors determine the base for any finite Coxeter group \(W\). In doing this, the authors adopt another (equivalent) definition of the base \(B\) of \(W\) in [loc. cit.]: \(B\) is the set of all elements \(w\in W\) which cannot be obtained as the supremum of a subset of \(W\) not containing \(w\). The above result of [loc. cit.] is crucial in the present work. In the classical cases (i.e., types \(A\), \(B\) and \(D\)), the authors use a coding of the elements of \(W\), which is given in terms of products of minimal right coset representatives along a naturally chosen chain of parabolic subgroups of \(W\). In the exceptional cases, they implement an algorithm in the computer algebra systems GAP and CHEVIE.
scientific article; zbMATH DE number 1097709