Monomial bases for the primitive complex Shephard groups of rank three

For the group algebra of each primitive non-Coxeter Shephard group of rank three, we construct a monomial basis and its explicit multiplication table. First, we find a Gröbner-Shirshov basis for the Shephard group of type L 2 . Then, since each of the groups of types L 3 and M 3 has a parabolic subg...

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Bibliographic Details
Published inCommunications in algebra Vol. 50; no. 2; pp. 889 - 902
Main Authors Kim, SungSoon, Lee, Dong-il
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 01.02.2022
Taylor & Francis Ltd
Subjects
Agent Orange
Complex reflection group
Gröbner-Shirshov basis
Mathematical tables
Mathematics
Multiplication
Pilot projects
Primary: 20F55
Secondary: 05E15
Shephard group
Subgroups
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Summary:For the group algebra of each primitive non-Coxeter Shephard group of rank three, we construct a monomial basis and its explicit multiplication table. First, we find a Gröbner-Shirshov basis for the Shephard group of type L 2 . Then, since each of the groups of types L 3 and M 3 has a parabolic subgroup isomorphic to the group of type L 2 , by using the sets of minimal right coset representatives of L 2 in L 3 and M 3 , respectively, we apply the Gröbner-Shirshov basis technique to find the monomial bases for the Shephard groups of rank three L 3 and M 3 . From this, we obtain the operation tables between the elements in each of the groups. Also we explicitly show that the group of type M 3 has a subgroup of index 2 isomorphic to the group of type L 3 .
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2021.1976200