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Alternating subgroups of Coxeter groups. (English) Zbl 1211.20035
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 [D. Bernstein and A. Regev, Sémin. Lothar. Comb. 53, B53b (2005; Zbl 1065.05005)] and [A. Regev and 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 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 \(s_0\).

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
05A15 Exact enumeration problems, generating functions
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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