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I have tried to record the result of Symonds 91 (“Symonds’ explicit Brauer induction”) in somewhat more modern form:
For G∈ FinGrp there is a linear map (homomorphism of abelian groups)
Rℂ(G)L⟶∏[H⊂G]ℤ[1dRepℂ(H)/∼]from the underlying abelian group of the representation ring to the product group of the free abelian groups that are spanned by the isomorphism classes of 1-dimensional representations over all conjugacy classes of subgroup H⊂G,
such that
L is a natural transformation of functors FinGrpop→Ab,
hence L(f*V)=f*(L(V));
L is a section of the natural transformation
∏[H⊂G]R1dℤ(H)∑ind⟶Rℂ(G)which applies induction and then sums everything up, in that the composition (∑ind)∘L is the identity:
(∑ind)∘L(V)≔∑[H⊂G]indGH[L(V)H]=VL is compatible with the total Chern classes of linear representations
Rℂ(G)c⟶∏k∈ℕH2k(BG,ℤ)via their multiplicative transfer 𝒩GH (Lemma \ref{TransferEvens}) in that
c(V)=⌣[H⊂G]𝒩GH(c(L(V)H)),hence in that the following diagram commutes:
Rℂ(G)L⟶∏[H⊂G]ℤ[1dRepℂ(H)/∼]c↓↓∏[H⊂G](c∘𝒩GH)∏k∈ℕH2k(BG,ℤ)⟵⌣[H⊂G]∏[H⊂G]a 1-dimensional representation is sent to the tuple whose component over is itself, and all whose other components vanish;
in contrast, if has no 1-dimensional direct summand, then the -compnents of is zero;
Added link to Artin’s induction theorem, a page I will now create.
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