Alternating subgroups of Coxeter groups. (Q942172)

The problem of extending combinatorial identities on symmetric groups to other groups, including alternating groups, was posed by Foata and others. A solution for the alternating group \(A_n\) of the symmetric group \(S_n\) was given in [\textit{D. Bernstein} and \textit{A. Regev}, Sémin. Lothar. Comb. 53, B53b (2005; Zbl 1065.05005)] and [\textit{A. Regev} and \textit{Y. Roichman}, Adv. Appl. Math. 33, No. 4, 676-709 (2004; Zbl 1057.05004)], which is based on a Coxeter-like presentation of \(A_n\). The goal of the present paper is to explore whether combinatorial properties of Coxeter groups may be extended to their alternating subgroups, using Coxeter-like presentations. For a Coxeter system \((W,S)\), its alternating subgroup \(W^+\) is the kernel of the sign character that sends every \(s\in S\) to \(-1\). An exercise from \textit{N. Bourbaki} [Elements of mathematics. Lie groups and Lie algebras. Chapters 4-6. Transl. from the French by Andrew Pressley. Berlin: Springer (2002; Zbl 0983.17001)] gives a simple presentation for \(W^+\) after one chooses a generator \(s_0\in S\). The present paper explores the combinatorial properties of this presentation, distinguishing different levels of generality regarding the chosen generator \(s_0\).