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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 29th 2022
   • (edited Jan 29th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am trying to understand in full detail the "splitting of the rationalized representation ring functor" (namely of $Orb(G)^{op} \to Rng \;:\; G/H \mapsto \mathbb{Q} \otimes R_{\mathbb{C}}(H)$) that is described on pages 231-232 & 237-238 of * Wolfgang Lück, Bob Oliver, *Chern characters for the equivariant K-theory of proper G-CW-complexes*, pages 249-262 in: Jaume Aguadé, Carles Broto, Carles Casacuberta (eds.), *Cohomological Methods in Homotopy Theory*, Barcelona Conference on Algebraic Topology, Bellaterra, Spain, June 4–10, 1998, Springer 2001 ([doi:10.1007/978-3-0348-8312-2](https://link.springer.com/book/10.1007/978-3-0348-8312-2), [p. 231-232 & p. 237-238 pdf](https://ncatlab.org/schreiber/files/LueckOliver_p231-232_p237-238.pdf)) with reference to p. 102-103 in * Jean-Pierre Serre, *Linear Representations of Finite Groups*, Springer 1977 ([doi:10.1007/978-1-4684-9458-7](https://doi.org/10.1007/978-1-4684-9458-7), [p.102-103 pdf](https://ncatlab.org/schreiber/files/SerreRepTheory_p102-103.pdf)) and reviewed in a tad more detail on p. 22-24 in: * Guido Mislin, Alain Valette, *Proper Group Actions and the Baum-Connes Conjecture*, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 ([doi](https://link.springer.com/book/10.1007/978-3-0348-8089-3), [p. 22-24 pdf](https://ncatlab.org/schreiber/files/MislinValette_p22-24.pdf)) $\,$ It's supposedly elementary representation theory, occupies just the handful of pages extracted in the above pdf-s, and I get the gist; yet I remain unclear about some details. $\,$ So for $G$ a finite group, we consider the system of complex-representation rings (character rings) $R(H) \,\coloneqq\, R_{\mathbb{C}}(H)$ of its subgroups $H \subset G$ as a functor on the opposite of its [[orbit category]]: $$ \array{ Orb(G)^{op} &\xrightarrow{\;\;}& Rng \\ G/H &\mapsto& R(H) \,; } $$ and we consider the class functions $e^{[C]}_{\vert H}$ on $H$ which for a conjugacy class $[C]$ of a cyclic subgroup $C \subset G$ are the "characteristic functions" that answer to whether an element $h \in H$ generates a subgroup conjugate to $C$: $$ \array{ e^{[C]}_{\vert H} &:& H &\longrightarrow& \mathbb{C} \\ && h &\mapsto& \left\{ \begin{array}{lcl} 1 &\vert& [\langle h\rangle] \,=\, [C] \;\in\; Sub(G)_{/\sim_{\conj}} \\ 0 &\vert& otherwise. \end{array} \right. } $$ Now the first claim is that these class functions are in fact rational combinations of characters (in fact of characters of rational representations, but we don't need this) hence are in fact elements of the rationalized representation rings $$ e^{[C]}_{\vert H} \;\in\; \mathbb{Q} \otimes R(H) \xhookrightarrow{\;} Cl(H) \,. $$ More concretely, from the first displayed equation in the proof of Prop. 4.1 in Lueck & Oliver ([p. 232](https://ncatlab.org/schreiber/files/LueckOliver_p231-232_p237-238.pdf#page=2)), the claim seems to be that $ord(G) \cdot e^{[C]}_{\vert H}$ are characters, and that's what Serre's Thm. 25, Cor. 2 (?) ([p. 102-103](https://ncatlab.org/schreiber/files/SerreRepTheory_p102-103.pdf)) is referenced for. This sounds like an elementary fact, but I still need to think more about it. But once we know that the $e^{[C]}_{\vert H}$ are rational characters, the splitting of the rationalized representation ring that we are after follows readily: Multiplication by the $e^{[C]}_{\vert H}$ clearly projects onto a complete set of subrings and this construction is clearly functorial in $H$ (regarded as $G/H \in Orb(G)$), so that $$ \mathbb{Q} \otimes R(-) \;\; \simeq \;\; \underset{ [C] }{\prod} \, e^{[C]}_{\vert(-)} \cdot \big( \mathbb{Q} \otimes R(-) \big) \;\;\;\; \in \;\; Func\big( Orb(G)^{op},\, Rng \big) \,. $$ Now it's about fully understanding the factors in this product of functors. First there is a claim about identifying the value of the functor $e^{[C]}_{\vert H} \cdot \big( \mathbb{Q} \otimes R(H) \big) $ with some fixed locus inside the cyclotomic field $\mathbb{Q}(\zeta_{ord(C)})$. I can see how the cyclotomic field here is the subring of the rational character ring generated by a suitable irrep of $C$, but I am still unsure how exactly to state and see the claimed identification. But maybe I'll leave that and further issues to followup comments.

   I am trying to understand in full detail the “splitting of the rationalized representation ring functor”

   (namely of $Orb(G)^{op} \to Rng \;:\; G/H \mapsto \mathbb{Q} \otimes R_{\mathbb{C}}(H)$)

   that is described on

   pages 231-232 & 237-238 of

   • Wolfgang Lück, Bob Oliver, Chern characters for the equivariant K-theory of proper G-CW-complexes, pages 249-262 in: Jaume Aguadé, Carles Broto, Carles Casacuberta (eds.), Cohomological Methods in Homotopy Theory, Barcelona Conference on Algebraic Topology, Bellaterra, Spain, June 4–10, 1998, Springer 2001 (doi:10.1007/978-3-0348-8312-2, p. 231-232 & p. 237-238 pdf)

   with reference to p. 102-103 in

   • Jean-Pierre Serre, Linear Representations of Finite Groups, Springer 1977 (doi:10.1007/978-1-4684-9458-7, p.102-103 pdf)

   and reviewed in a tad more detail on p. 22-24 in:

   • Guido Mislin, Alain Valette, Proper Group Actions and the Baum-Connes Conjecture, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 (doi, p. 22-24 pdf)

   $\,$

   It’s supposedly elementary representation theory, occupies just the handful of pages extracted in the above pdf-s, and I get the gist; yet I remain unclear about some details.

   $\,$

   So for $G$ a finite group, we consider the system of complex-representation rings (character rings) $R(H) \,\coloneqq\, R_{\mathbb{C}}(H)$ of its subgroups $H \subset G$ as a functor on the opposite of its orbit category:

   $$\array{ Orb(G)^{op} &\xrightarrow{\;\;}& Rng \\ G/H &\mapsto& R(H) \,; }$$

   and we consider the class functions $e^{[C]}_{\vert H}$ on $H$ which for a conjugacy class $[C]$ of a cyclic subgroup $C \subset G$ are the “characteristic functions” that answer to whether an element $h \in H$ generates a subgroup conjugate to $C$:

   $$\array{ e^{[C]}_{\vert H} &:& H &\longrightarrow& \mathbb{C} \\ && h &\mapsto& \left\{ \begin{array}{lcl} 1 &\vert& [\langle h\rangle] \,=\, [C] \;\in\; Sub(G)_{/\sim_{\conj}} \\ 0 &\vert& otherwise. \end{array} \right. }$$

   Now the first claim is that these class functions are in fact rational combinations of characters (in fact of characters of rational representations, but we don’t need this) hence are in fact elements of the rationalized representation rings

   $$e^{[C]}_{\vert H} \;\in\; \mathbb{Q} \otimes R(H) \xhookrightarrow{\;} Cl(H) \,.$$

   More concretely, from the first displayed equation in the proof of Prop. 4.1 in Lueck & Oliver (p. 232), the claim seems to be that $ord(G) \cdot e^{[C]}_{\vert H}$ are characters, and that’s what Serre’s Thm. 25, Cor. 2 (?) (p. 102-103) is referenced for. This sounds like an elementary fact, but I still need to think more about it.

   But once we know that the $e^{[C]}_{\vert H}$ are rational characters, the splitting of the rationalized representation ring that we are after follows readily: Multiplication by the $e^{[C]}_{\vert H}$ clearly projects onto a complete set of subrings and this construction is clearly functorial in $H$ (regarded as $G/H \in Orb(G)$), so that

   $$\mathbb{Q} \otimes R(-) \;\; \simeq \;\; \underset{ [C] }{\prod} \, e^{[C]}_{\vert(-)} \cdot \big( \mathbb{Q} \otimes R(-) \big) \;\;\;\; \in \;\; Func\big( Orb(G)^{op},\, Rng \big) \,.$$

   Now it’s about fully understanding the factors in this product of functors.

   First there is a claim about identifying the value of the functor $e^{[C]}_{\vert H} \cdot \big( \mathbb{Q} \otimes R(H) \big)$ with some fixed locus inside the cyclotomic field $\mathbb{Q}(\zeta_{ord(C)})$. I can see how the cyclotomic field here is the subring of the rational character ring generated by a suitable irrep of $C$, but I am still unsure how exactly to state and see the claimed identification.

   But maybe I’ll leave that and further issues to followup comments.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 30th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexBut maybe I don't need any of this after all, as we should have a more elegant way to deduce the "delocalized" form of the twist of equivariant K-theory. Namely, the topological classifying stack for twisted equivariant K-theory is simply: $$ \array{ Fred \sslash PU_{{}_{\omega}} \\ \big\downarrow \\ \mathbf{B} PU_{{}_{\omega}} } \;\;\;\; \in \; \big(SmthGrpd_{\infty}\big)_{\mathbf{B} PU_{{}_{\omega}}} $$ in that $G$-equivariant K-theory cocycles are stacky sections ([p. 202](https://ncatlab.org/schreiber/files/EquivariantInfinityBundles_220230b.pdf#page=202)) $$ \array{ && Fred \sslash PU_{{}_{\omega}} \\ & \nearrow & \big\downarrow \\ X \sslash G &\longrightarrow& \mathbf{B} PU_{{}_{\omega}} } $$ (where the horizontal map is "stable"). Now applying $Map(\mathbf{B} \mathbb{Z} -)$ to this diagram and using the analysis of intertwiners of stable projective representations [p. 185](https://ncatlab.org/schreiber/files/EquivariantInfinityBundles_220230b.pdf#page=185) this tuns into sections of the following pullback bundle: $$ \array{ \Big( \underset{g \in G}{\coprod} Fred^{\langle g \rangle} \Big) \sslash G &\longrightarrow& Map\big(\mathbf{B}\mathbb{Z},\, Fred \sslash PU_{{}_{\omega}} \big)^{stbl} \\ \big\downarrow & & \big\downarrow \\ \Big( \underset{g \in G}{\coprod} X^g \Big) \sslash_{{}_{ad}} G &\longrightarrow& \underset{g \in G}{\coprod} \mathbf{B} Hom\big( \langle g\rangle,\, \mathrm{U}(1) \big) \times \mathbf{B} PU_{{}_{\omega}} } $$ Here the bottom map encodes now, aside from the equivariant $PU_{{}_{}\omega}$-bundle, a $C_G(g)$-equivariant complex line bundle on each fixed locus $X^g$ with structure group reduced to $\mathbb{Z}/ord(g) \,\simeq\, Hom\big( \langle g \rangle,\, \mathrm{U}(1) \big)$ (hence a "local system") which twists vector bundles of $\langle g \rangle$-representations by tensoring these reps with these group characters, hence which, after decomposing these actions into irreps, twists vector bundles in all possible ways that they can be twisted by a $\mathbb{Z}/n$-local system. That's the form of the local system twist of equivariant K-theory as seen in the literature, e.g. (3.5) in FHT06 ([p. 8](https://arxiv.org/pdf/math/0206257.pdf#page=8)).

   But maybe I don’t need any of this after all, as we should have a more elegant way to deduce the “delocalized” form of the twist of equivariant K-theory.

   Namely, the topological classifying stack for twisted equivariant K-theory is simply:

   $$\array{ Fred \sslash PU_{{}_{\omega}} \\ \big\downarrow \\ \mathbf{B} PU_{{}_{\omega}} } \;\;\;\; \in \; \big(SmthGrpd_{\infty}\big)_{\mathbf{B} PU_{{}_{\omega}}}$$

   in that $G$-equivariant K-theory cocycles are stacky sections (p. 202)

   $$\array{ && Fred \sslash PU_{{}_{\omega}} \\ & \nearrow & \big\downarrow \\ X \sslash G &\longrightarrow& \mathbf{B} PU_{{}_{\omega}} }$$

   (where the horizontal map is “stable”). Now applying $Map(\mathbf{B} \mathbb{Z} -)$ to this diagram and using the analysis of intertwiners of stable projective representations p. 185 this tuns into sections of the following pullback bundle:

   $$\array{ \Big( \underset{g \in G}{\coprod} Fred^{\langle g \rangle} \Big) \sslash G &\longrightarrow& Map\big(\mathbf{B}\mathbb{Z},\, Fred \sslash PU_{{}_{\omega}} \big)^{stbl} \\ \big\downarrow & & \big\downarrow \\ \Big( \underset{g \in G}{\coprod} X^g \Big) \sslash_{{}_{ad}} G &\longrightarrow& \underset{g \in G}{\coprod} \mathbf{B} Hom\big( \langle g\rangle,\, \mathrm{U}(1) \big) \times \mathbf{B} PU_{{}_{\omega}} }$$

   Here the bottom map encodes now, aside from the equivariant $PU_{{}_{}\omega}$-bundle, a $C_G(g)$-equivariant complex line bundle on each fixed locus $X^g$ with structure group reduced to $\mathbb{Z}/ord(g) \,\simeq\, Hom\big( \langle g \rangle,\, \mathrm{U}(1) \big)$ (hence a “local system”) which twists vector bundles of $\langle g \rangle$-representations by tensoring these reps with these group characters, hence which, after decomposing these actions into irreps, twists vector bundles in all possible ways that they can be twisted by a $\mathbb{Z}/n$-local system.

   That’s the form of the local system twist of equivariant K-theory as seen in the literature, e.g. (3.5) in FHT06 (p. 8).