Integrality of class sums

For a finite group G, the class sums form a finite basis of the center of the integral group algebra ℤ[G]. Consequently every element of this center, and in particular every class sum, is integral over ℤ.

The point is to establish integrality inside Z(ℤ[G]). Integrality of a class sum merely as an element of ℤ[G] would not retain the central algebra through which central characters factor.

The same statement over an arbitrary coefficient ring k follows by base change along ℤ[G] → k[G], which carries class sums to class sums; it cannot be proved by the argument over ℤ, because Z(k[G]) need not be a finite ℤ-module. This coefficient-general form is the one a central character Z(k[G]) →ₐ[k] k can be applied to, so it is what makes the values of a central character on the class sums algebraic integers.

Main results

• TauCeti.isIntegral_classSum: a class sum is integral over ℤ in Z(ℤ[G]).
• TauCeti.isIntegral_classSumCenter: its base change, integrality over ℤ in Z(k[G]) for an arbitrary commutative coefficient ring k.

References

• Character Theory roadmap, Layer 1, “The integral class center”.
• C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Chapters 1–3.

theorem TauCeti.isIntegral_classSum {G : Type u_1} [Group G] [Fintype G] [DecidableEq G] (C : ConjClasses G) : IsIntegral ℤ (classSumCenter C)

A class sum is integral over ℤ as an element of the center of ℤ[G].

theorem TauCeti.isIntegral_classSumCenter {G : Type u_1} [Group G] [Fintype G] [DecidableEq G] (k : Type u_2) [CommRing k] (C : ConjClasses G) : IsIntegral ℤ (classSumCenter C)

A class sum is integral over ℤ as an element of the center of k[G], for any coefficient ring k.

The center of k[G] is not a finite ℤ-module, so this does not follow by the argument over ℤ; it is transported from there along the base change ℤ[G] → k[G], which carries class sums to class sums. It is the input to the integrality of the values of a central character on the class sums.