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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 27th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexam starting something. Just saving for the moment, not done yet <a href="https://ncatlab.org/nlab/revision/Schur+index/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Schur+index">current</a>

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

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 27th 2018
   • (edited Sep 27th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexokay, I have recorded the basic definition/statement I tried to bring out the nice picture as in [tom Dieck 09, 6.2](https://ncatlab.org/nlab/show/Schur+index#tomDieck09), 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 $\widehat k$-irrep with all its distinct Galois translates its _Galois group averaging_ ([here](https://ncatlab.org/nlab/show/Schur+index#GaloisGroupAveraging)) 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 $\Psi^n - id$ for $n$ coprime to ${\vert G\vert}$. Because this makes the relation to the [[Adams conjecture]] manifest. Hope I got this right. But representation theorists please feel invited to criticize. <a href="https://ncatlab.org/nlab/revision/diff/Schur+index/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Schur+index/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Schur+index">current</a>

   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 $\widehat k$-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 $\Psi^n - id$ for $n$ coprime to ${\vert G\vert}$. 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

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 27th 2018
   • (edited Sep 27th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have also added a remark ([here](https://ncatlab.org/nlab/show/Schur+index#EquivariantJHomomorphism)) 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](https://nforum.ncatlab.org/discussion/9045/equivariant-adams-conjecture-and-schur-indices/?Focus=72128#Comment_72128)) 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 $G$ for which $A(G) \overset{\beta}{\to} R_{\mathbb{Q}}(G)$ is surjective -- because it looks directly analogous. This may be straightforward to check by chasing through Segal's proof of $\mathbb{S}^0_G(\ast) \simeq A(G)$. 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.

   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 $G$ for which $A(G) \overset{\beta}{\to} R_{\mathbb{Q}}(G)$ is surjective – because it looks directly analogous.

   This may be straightforward to check by chasing through Segal’s proof of $\mathbb{S}^0_G(\ast) \simeq A(G)$. 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.