Characters and Schur indices of the unitary reflection group [321] 3
1975; Mathematical Sciences Publishers; Volume: 58; Issue: 2 Linguagem: Inglês
10.2140/pjm.1975.58.309
ISSN1945-5844
Autores Tópico(s)Advanced Operator Algebra Research
ResumoThe group G considered in this paper is the finite irreducible unitary reflection group in 6 dimensions denoted by [3 21] 3 in the notation of H. S. M. Coxeter.A character table for G is constructed.There are 169 irreducible characters and it is shown that each character has Schur index 1 over Q.Furthermore, each character has values in the field Q(V^3) and this field is a splitting field for G.The concept of a reflection in Euclidean space was generalized by G. C. Shephard [9], who first wrote [21; 3] 3 for [321] 3 .A reflection in unitary space is a linear transformation of finite period with the property that all but one of its characteristic values are equal to 1.While reflections in Euclidean space must have period 2, a reflection in unitary space may have period m for any integer m > 1. Shephard and J. A. Todd [10] studied the finite groups generated by unitary reflections and classified the irreducible groups.The ordinary finite reflection groups, studied extensively by H. S. M. Coxeter (see [2] for references), are special cases of the finite unitary reflection groups.The group G considered in this paper is the largest finite primitive irreducible unitary reflection group which is not a Euclidean group.It is generated by reflections of period 2 and has order 2 9 3 7 5 7.The characters of all the other finite irreducible unitary reflection groups will be studied in a forthcoming paper by this author.
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