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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 GG \in FinGrp there is a linear map (homomorphism of abelian groups)

    R (G)L[HG][1dRep (H) /] R_{\mathbb{C}}\big( G \big) \overset {L} {\longrightarrow} \underset{ [H \subset G] }{\prod} \mathbb{Z} \left[ 1dRep_{\mathbb{C}}\big( H \big)_{/\sim} \right]

    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 HGH \subset G,

    such that

    1. LL is a natural transformation of functors FinGrp opAbFinGrp^{op} \to Ab,

      hence L(f *V)=f *(L(V))L\big( f^\ast V\big) = f^\ast( L(V) );

    2. LL is a section of the natural transformation

      [HG]R 1d(H)indR (G) \underset{ [H \subset G] }{\prod} R^{1d}_{\mathbb{Z}}\big( H\big) \overset {\sum ind} {\longrightarrow} R_{\mathbb{C}}\big( G \big)

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

      (ind)L(V)[HG]ind H G[L(V) H]=V \big( \sum ind \big) \circ L(V) \coloneqq \underset{ [H \subset G] }{ \sum } ind_H^G\left[ L(V)_H \right] \;=\; V
    3. LL is compatible with the total Chern classes of linear representations

      R (G)ckH 2k(BG,) R_{\mathbb{C}}\big( G \big) \overset{c}{\longrightarrow} \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big( B G, \mathbb{Z} \big)

      via their multiplicative transfer 𝒩 H G\mathcal{N}_H^G (Lemma \ref{TransferEvens}) in that

      c(V)=[HG]𝒩 H G(c(L(V) H)), c \big( V \big) \;=\; \underset{ [H \subset G] }{\smile} \mathcal{N}_H^G \Big( c \big( L(V)_H \big) \Big) \,,

      hence in that the following diagram commutes:

      R (G) L [HG][1dRep (H) /] c [HG](c𝒩 H G) kH 2k(BG,) [HG] [HG]ProdkH 2k(BG,) \array{ R_{\mathbb{C}}\big( G\big) &\overset{L}{\longrightarrow}& \underset{ [H \subset G] }{\prod} \mathbb{Z} \left[ 1dRep_{\mathbb{C}}\big( H \big)_{/\sim} \right] \\ {}^{\mathllap{c}}\Big\downarrow && \Big\downarrow {}^{ \underset{ [H \subset G] }{\prod} \left( c \circ \mathcal{N}_H^G \right) } \\ \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big( B G, \mathbb{Z} \big) & \underset{ \underset{ [H \subset G] }{\smile} }{\longleftarrow} & \underset{ [H \subset G] }{\prod} \underset{ k \in \mathbb{Z} } {\Prod} H^{2k}\big( B G, \mathbb{Z}\big) }
    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