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Clifford classes for isoclinic groups
Authors:Alexandre Turull
Affiliation:(1) Department of Mathematics, University of Florida, Gainesville, FL 32611, USA
Abstract:Let G be a finite group, N a normal subgroup of G, and chi an irreducible character of G. Clifford Theory studies a whole collection of related irreducible characters of all the subgroups of G that contain N. The relationships among these characters as well as their Schur indices are controlled by the Clifford class c isin Clif(G/N, F) of chi with respect to N over some field F. This is an equivalence class of central simple G/N-algebras. Assume now that G/N is cyclic. One can obtain a new isoclinic group$$tilde G$$ and character$$tilde chi ,$$ by lsquomultiplyingrsquo each element of each coset of N in G by an appropriate power of a fixed root of unity epsi. We show that there is a simple formula to calculate the Clifford class$${tilde c}$$ of$${tilde chi }$$ in terms of c and epsi. Hence, the Clifford class c controls not only the Schur index of the characters of all the subgroups of G that contain N, it also controls the Schur indices of the characters of the corresponding characters of the isoclinic groups$$tilde G.$$When epsi is a |G/N|-th root of 1, our formula shows that then$$c = tilde c.$$ When epsi = i and |G/N| = 2, the implicit transformation on Clif(Z/2Z, F) yields a group homomorphism of the group structure introduced on the Brauer-Wall group of F to describe the Schur indices of all the irreducible characters of the double covers of the symmetric and alternating groups.Received: 17 August 2001
Keywords:20C15
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