Clifford classes for isoclinic groups

Abstract.

Let G be a finite group, N a normal subgroup of G, and χ 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 ∈ Clif(G/N, F) of χ 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 ‘multiplying’ each element of each coset of N in G by an appropriate power of a fixed root of unity ε. We show that there is a simple formula to calculate the Clifford class ${\tilde c}$ of ${\tilde \chi }$ in terms of c and ε. 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 ε is a |G/N|-th root of 1, our formula shows that then $c = \tilde c.$ When ε = 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.