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n the present paper, which is a direct sequel of our papers [10,11,35]joint with Roozbeh Hazrat, we achieve a further dramatic reduction of the gener-ating sets for commutators of relative elementary subgroups in Chevalley groups.Namely, let Φ be a reduced irreducible root system of rank≥2, letRbe a commu-tative ring and letA, Bbe two ideals ofR. We consider subgroups of the ChevalleygroupG(Φ, R) of type Φ overR. The unrelative elementary subgroupE(Φ, A) oflevelAis generated (as a group) by the elementary unipotentsxα(a),α∈Φ,a∈A,of levelA. Its normal closure in the absolute elementary subgroupE(Φ, R) is de-noted byE(Φ, R, A) and is called the relative elementary subgroup of levelA. Themain results of [11,35] consisted in construction of economic generator sets for themutual commutator subgroups [E(Φ, R, A), E(Φ, R, B)], whereAandBare twoideals ofR. It turned out that one can take Stein—Tits—Vaserstein generators ofE(Φ, R, AB), plus elementary commutators of the formyα(a, b) = [xα(a), x−α(b)],wherea∈A,b∈B. Here we improve these results even further, by showing that infact it suffices to engage only elementary commutators corresponding toonelongroot, and that moduloE(Φ, R, AB) the commutatorsyα(a, b) behave as symbols.We discuss also some further variations and applications of these results.
Translated title of the contributionКоммутаторы относительных и настоящих элменетарных подгрупп в группах Шевалле
Original languageEnglish
Number of pages18
JournalProceedings of the Edinburgh Mathematical Society
StateSubmitted - 13 Mar 2020

    Scopus subject areas

  • Mathematics(all)

    Research areas

  • Chevalley groups, elementary subgroups, generation of mixed commutator subgroups, standard commutator formulae

ID: 52306715