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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 26th 2018
   • (edited Sep 26th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am getting the feeling that the theory of Schur indices for passing between rational representations and complex representation of a finite group is secretly the same as the statement of the Adams conjecture for equivariant cohomology. Could that be? Does this resonate with anyone? So for $G$ a finite group, the way to turn an irreducible complex representation $V \in R_{\mathbb{C}}(G) \simeq KU_G(\ast)$ into a rational representation in $R_{\mathbb{Q}}(G)$ is to 1. "average out the Adams operation" by passing to $$ V + \Psi V + \Psi^2 V + \cdots + \Psi^{\vert G\vert} V $$ 1. multiply the result with the Schur index $n$ of $V$. Then there is an irreducible rational rep $W \in R_{\mathbb{Q}}(G)$ such that $$ W \otimes_{\mathbb{Q}} \mathbb{C} \;\simeq\; n \big( V + \Psi V + \Psi^2 V + \cdots + \Psi^{\vert G\vert} V \big) $$ (The formulation of this classical fact with the perspective of Adams operations made explicit is highlighted for instance in tom Dieck's lecture notes on p. 95 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf)).) But in many cases the map $\beta$ in $$ \mathbb{S}_G(\ast) \simeq A(G) \overset{\beta}{\longrightarrow} R_{\mathbb{Q}}(G) \overset{}{\longrightarrow} R_{\mathbb{C}}(G) $$ is surjective (for instance for $G$ a cyclic group, [here](https://ncatlab.org/nlab/show/permutation+representation#WhenAllVirtualLinearRepsAreVirtualPermutationReps)) and, moreover, admits a canonical splitting. Hence in these cases we may think of the abve construction of a rational representation from a complex representation as really being the construction of a class in equivariant stable cohomotopy $\mathbb{S}_G(\ast)$ from a class in equivariant K-theory. With that in mind, it becomes manifest how something at least closely analogous to the [[Adams conjecture]] is going on. There we have a map from plain (i.e. not equivariant) $K$-theory to plain stable cohomotopy $$ J \;\colon\; KU^0(X) \longrightarrow \mathbb{S}^1(X) $$ (this complex version of $J$ and the Adams conjecture-theorem is for instance on p. 5 [here](http://www-math.mit.edu/~hrm/papers/miller-priddy-g-stable.pdf)) The statement of the Adams-conjecture theorem is that there is a natural number $n$ such that $J$ annihilates $n ( \Psi^k - id )$. But this just says that $n J$ acts by averaging out Adams operations followed by multiplying the result by some $n$... ...just as above for the passage from complex reps to rational reps, thought of as the passage from equivariant complex K-theory of the point to equivariant stable cohomotopy of the point. Is this more than a close analogy? Looking for "equivariant Adams conjecture" I find an announcement of such a thing in Waner's "Equivariant classifying spaces and fibrations" ([pdf](https://www.jstor.org/stable/pdf/1998063.pdf)), with reference > [Wa1] , S. Waner, "Cyclic groups actions and the adams conjecture" (to appear) but maybe that article never actually appeared? I don't seem to find any incarnation of it.

   I am getting the feeling that the theory of Schur indices for passing between rational representations and complex representation of a finite group is secretly the same as the statement of the Adams conjecture for equivariant cohomology.

   Could that be? Does this resonate with anyone?

   So for $G$ a finite group, the way to turn an irreducible complex representation $V \in R_{\mathbb{C}}(G) \simeq KU_G(\ast)$ into a rational representation in $R_{\mathbb{Q}}(G)$ is to

   1. “average out the Adams operation” by passing to

      $$V + \Psi V + \Psi^2 V + \cdots + \Psi^{\vert G\vert} V$$

   2. multiply the result with the Schur index $n$ of $V$.

   Then there is an irreducible rational rep $W \in R_{\mathbb{Q}}(G)$ such that

   $$W \otimes_{\mathbb{Q}} \mathbb{C} \;\simeq\; n \big( V + \Psi V + \Psi^2 V + \cdots + \Psi^{\vert G\vert} V \big)$$

   (The formulation of this classical fact with the perspective of Adams operations made explicit is highlighted for instance in tom Dieck’s lecture notes on p. 95 (pdf).)

   But in many cases the map $\beta$ in

   $$\mathbb{S}_G(\ast) \simeq A(G) \overset{\beta}{\longrightarrow} R_{\mathbb{Q}}(G) \overset{}{\longrightarrow} R_{\mathbb{C}}(G)$$

   is surjective (for instance for $G$ a cyclic group, here) and, moreover, admits a canonical splitting. Hence in these cases we may think of the abve construction of a rational representation from a complex representation as really being the construction of a class in equivariant stable cohomotopy $\mathbb{S}_G(\ast)$ from a class in equivariant K-theory.

   With that in mind, it becomes manifest how something at least closely analogous to the Adams conjecture is going on. There we have a map from plain (i.e. not equivariant) $K$-theory to plain stable cohomotopy

   $$J \;\colon\; KU^0(X) \longrightarrow \mathbb{S}^1(X)$$

   (this complex version of $J$ and the Adams conjecture-theorem is for instance on p. 5 here)

   The statement of the Adams-conjecture theorem is that there is a natural number $n$ such that $J$ annihilates $n ( \Psi^k - id )$. But this just says that $n J$ acts by averaging out Adams operations followed by multiplying the result by some $n$…

   …just as above for the passage from complex reps to rational reps, thought of as the passage from equivariant complex K-theory of the point to equivariant stable cohomotopy of the point.

   Is this more than a close analogy?

   Looking for “equivariant Adams conjecture” I find an announcement of such a thing in Waner’s “Equivariant classifying spaces and fibrations” (pdf), with reference

   [Wa1] , S. Waner, “Cyclic groups actions and the adams conjecture” (to appear)

   but maybe that article never actually appeared? I don’t seem to find any incarnation of it.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 26th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAre the reasonably numerous hits for the phrase inadequate, e.g., [this](http://math.uchicago.edu/~may/PAPERS/40.pdf), [this](https://projecteuclid.org/euclid.ijm/1256065410) and [that](https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=29&iss=3&rank=5)?

   Are the reasonably numerous hits for the phrase inadequate, e.g., this, this and that?

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 26th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexGoogle Scholar doesn't find anything for "equivariant Adams conjecture" after [this](https://msp.org/agt/2009/9-4/agt-v9-n4-p01-p.pdf) in 2009. Nice motivating introduction.

   Google Scholar doesn’t find anything for “equivariant Adams conjecture” after this in 2009. Nice motivating introduction.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 26th 2018
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexPeople on the alg-top mailing list would know about that Waner preprint, I guess.

   People on the alg-top mailing list would know about that Waner preprint, I guess.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 26th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for digging out references! Am collecting them [here](https://ncatlab.org/nlab/show/Adams+conjecture#ReferencesInEquivariantCohomology).

   Thanks for digging out references! Am collecting them here.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeSep 27th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have now brought into _[[Schur index]]_ more details on the question in [#1](https://nforum.ncatlab.org/discussion/9045/equivariant-adams-conjecture-and-schur-indices/?Focus=72128#Comment_72128), see [this remark](https://ncatlab.org/nlab/show/Schur%20index#EquivariantJHomomorphism).

   I have now brought into Schur index more details on the question in #1, see this remark.