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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 28th 2019

    am starting something here

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 29th 2019
    • (edited Jan 29th 2019)

    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)

    R(G)L[HG][1dRep(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 HG,

    such that

    1. L is a natural transformation of functors FinGrpopAb,

      hence L(f*V)=f*(L(V));

    2. L is a section of the natural transformation

      [HG]R1d(H)indR(G)

      which applies induction and then sums everything up, in that the composition (ind)L is the identity:

      (ind)L(V)[HG]indGH[L(V)H]=V
    3. L is compatible with the total Chern classes of linear representations

      R(G)ckH2k(BG,)

      via their multiplicative transfer 𝒩GH (Lemma \ref{TransferEvens}) in that

      c(V)=[HG]𝒩GH(c(L(V)H)),

      hence in that the following diagram commutes:

      R(G)L[HG][1dRep(H)/]c[HG](c𝒩GH)kH2k(BG,)[HG][HG]
    4. a 1-dimensional representation W1dRep(G) /R (G)W \in 1dRep\big(G\big)_{/\sim} \subset R_{\mathbb{C}}\big(G\big) is sent to the tuple L(W)=(W,0,0,)L(W) = (W,0,0, \cdots) whose component over GGG \subset G is VV itself, and all whose other components vanish;

    5. in contrast, if VRep (G) /V \in Rep_{\mathbb{C}}\big( G \big)_{/\sim} has no 1-dimensional direct summand, then the GG-compnents of L(V)L(V) is zero;

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorJohn Baez
    • CommentTimeMar 11th 2025

    Added link to Artin’s induction theorem, a page I will now create.

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeMar 12th 2025

    Added info comparing Brauer’s and Artin’s theorems.

    diff, v5, current