Conjugacy class sum

In abstract algebra, a conjugacy class sum, or simply class sum, is a function defined for each conjugacy class of a finite group G as the sum of the elements in that conjugacy class. The class sums of a group form a basis for the center of the associated group algebra.

Definition

Let G be a finite group, and let C₁,...,C_k be the distinct conjugacy classes of G. For 1 ≤ i ≤ k, define

[math]\displaystyle{ \overline{C_i}=\sum_{g\in C_i}g. }[/math]

The functions [math]\displaystyle{ \overline{C_1},\ldots,\overline{C_k} }[/math] are the class sums of G.

In the group algebra

Let CG be the complex group algebra over G. Then the center of CG, denoted Z(CG), is defined by

[math]\displaystyle{ \operatorname{Z}(\mathbf{C}G) = \{f \in \mathbf{C}G \mid \forall g\in \mathbf{C}G, fg = gf \} }[/math].

This is equal to the set of all class functions (functions which are constant on conjugacy classes). To see this, note that f is central if and only if f(yx) = f(xy) for all x,y in G. Replacing y by yx⁻¹, this condition becomes

[math]\displaystyle{ f(xyx^{-1})=f(y) \text{ for } x,y \in G }[/math].

The class sums are a basis for the set of all class functions, and thus they are a basis for the center of the algebra.

In particular, this shows that the dimension of Z(CG) is equal to the number of class sums of G.

References

• Goodman, Roe; and Wallach, Nolan (2009). Symmetry, Representations, and Invariants. Springer. ISBN 978-0-387-79851-6. See chapter 4, especially 4.3.
• James, Gordon; and Liebeck, Martin (2001). Representations and Characters of Groups (2nd ed.). Cambridge University Press. ISBN 0-521-00392-X. See chapter 12.