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A canonical Brauer induction formula. (English) Zbl 0718.20005
Représentations linéaires des groupes finis, Proc. Colloq., Luminy/Fr. 1988, Astérisque 181-182, 31-59 (1990).
[For the entire collection see Zbl 0699.00023.]
Let G be a finite group, R(G) the character ring of G and \(R_+(G)\) the free abelian group on the set of conjugacy classes of pairs (H,\(\phi\)) where H is a subgroup of G and \(\phi\) is a 1-dimensional representation of H. The linear map \(b_ G: R_+(G)\to R(G)\) assigning to the conjugacy class of (H,\(\phi\)) the induced character \(ind^ G_ H(\phi)\) is surjective, by Brauer’s theorem. The author shows that there are unique maps \(a_ G: R(G)\to R_+(G)\) which commute with restriction and are trivial on 1-dimensional representations. Among many other nice functorial properties the maps \(a_ G\) satisfy \(b_ Ga_ G=id_{R(G)}\), so \(a_ G\) gives an induction formula. This result is related to a paper by V. Snaith [Invent. Math. 94, 455-478 (1988; Zbl 0704.20009)]. However, the map \(a_ G\) is different from Snaith’s construction, and the method of proof is different. The author’s approach is purely algebraic whereas Snaith uses tools from topology. This interesting paper also discusses inductions from special classes of subgroups and the Artin exponent of a finite group.

MSC:
20C15 Ordinary representations and characters
19A22 Frobenius induction, Burnside and representation rings