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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2018

    am starting something. Just saving for the moment, not done yet

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2018
    • (edited Sep 27th 2018)

    okay, I have recorded the basic definition/statement

    I tried to bring out the nice picture as in tom Dieck 09, 6.2, in particular I focused, for the moment, on field extensions from \mathbb{Q}.

    I have taken the liberty of calling the direct sum of a k^\widehat k-irrep with all its distinct Galois translates its Galois group averaging (here) since I find that saying this helps to see at a glance what’s really going on here.

    I have also taken the liberty of giving an alternative characterization of this “Galois group averaged representation”, namely as the smallest rep that is in the kernel of Ψ nid\Psi^n - id for nn coprime to |G|{\vert G\vert}. Because this makes the relation to the Adams conjecture manifest. Hope I got this right.

    But representation theorists please feel invited to criticize.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2018
    • (edited Sep 27th 2018)

    I have also added a remark (here) that highlights the close similarity of the construction to the J-homomorphism and the Adams conjecture.

    So I highlighted in particular the (open?) question (raised in another thread here) of whether indeed the Schur index construction is the incarnation of the equivariant J-homomorphism over the point, and if the “Galois group averaging” involved is the equivariant Adams conjecture-statement on the point – at least for those groups GG for which A(G)βR (G)A(G) \overset{\beta}{\to} R_{\mathbb{Q}}(G) is surjective – because it looks directly analogous.

    This may be straightforward to check by chasing through Segal’s proof of 𝕊 G 0(*)A(G)\mathbb{S}^0_G(\ast) \simeq A(G). If this question turns out to be really open, I’ll sit down and do that. But I’d much rather just cite it from somewhere. If it is indeed true.

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