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Conjugacy class sum

From Wikipedia, the free encyclopedia

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

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

The functions are the class sums of G.

In the group algebra

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

.

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

.

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

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  • 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.