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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 5th 2013
• (edited Aug 13th 2014)

added to equivariant K-theory comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.

(Also finally added references to Green and Julg at Green-Julg theorem).

This all deserves to be prettified further, but I have to quit now.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 2nd 2018

slightly expanded the paragraph “Relation to representation theor” (here), adding mentioning also of $KO^0_G(\ast) \simeq R_{\mathbb{R}}(G)$

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 2nd 2018
• (edited Oct 2nd 2018)

I have added statement of the remarkable fact that equivariant KO-theory of the point subsumes the representations rings over $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$:

$KO_G^n(\ast) \;\simeq\; \left\{ \array{ 0 &\vert& n = 7 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{R}}(G) &\vert& n = 6 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 5 \\ R_{\mathbb{H}}(G) \phantom{/ R_{\mathbb{R}}(G) } &\vert& n = 4 \\ 0 &\vert& n = 3 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{H}}(G) &\vert& n = 2 \\ R_{\mathbb{R}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 1 \\ R_{\mathbb{R}}(G) \phantom{/ R_{\mathbb{R}}(G)} &\vert& n =0 } \right.$
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 9th 2018
• (edited Oct 9th 2018)

added (towards the end of this subsection) the expression for $c_1$ of a complex representation regarded as a vector bundle over $B G$ (from the appendix of Atiyah 61)

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeOct 9th 2018

Is this what was meant

$c_1(V) = c_1(\wedge^n V)$?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 10th 2018

Yes, that’s indeed what is meant (but we may want to add superscripts to make it seem less surprising):

First we define $c_1$ for line bundles/1d reps, then we define $c_1$ on any vector bundle/rep by saying that it’s the previously defined $c_1$ of the determinant line bundle/top exterior power.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 10th 2018
• (edited Oct 10th 2018)

Have expanded the respective paragraph to now read like so:

For 1-dimensional representations $V$ the first Chern class of $\widehat{V}$ is just the canonical isomorphism of 1-dimensional characters with group cohomology of $G$ and then with ordinary cohomology of the classifying space $B G$

$c_1\left(\widehat{(-)}\right) \;\colon\; Hom(G, U(1)) \overset{\simeq}{\longrightarrow} H^2_{grp}(G,\mathbb{Z}) \simeq H^2(B G, \mathbb{Z}) \,,$

while for any $n$-dimensional representation $V$ the first Chern class is this isomorphism applied to the $n$th exterior power $\wedge^n V$ of $V$ (which is a 1-dimensional representation, namely the “determinant line bundle” of $widehat{V}$, to which the previous definition of $c_1$ applies):

$c_1(V) = c_1(\wedge^n V) \,.$
• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeOct 10th 2018

I think I’m struggling with the grammar. So first Chern class is an equivalence, so a map? Then “the first Chern class of $X$” is the image of this map applied to $X$? But above you’re saying this image for $\widehat{V}$ is an isomorphism.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeOct 10th 2018
• (edited Oct 10th 2018)

For 1-dimensional representations $V$ their first Chern class $c_1(\widehat{V}) \in H^2(B G, \mathbb{Z})$ is their image under the canonical isomorphism from 1-dimensional characters in $Hom_{Grp}(G,U(1))$ to the group cohomology $H^2_{grp}(G, \mathbb{Z})$ and further to the ordinary cohomology $H^2(B G, \mathbb{Z})$ of the classifying space $B G$:

$c_1\left(\widehat{(-)}\right) \;\colon\; Hom_{Grp}(G, U(1)) \overset{\simeq}{\longrightarrow} H^2_{grp}(G,\mathbb{Z}) \overset{\simeq}{\longrightarrow} H^2(B G, \mathbb{Z}) \,.$

More generally, for $n$-dimensional representations $V$ their first Chern class $c_1(\widehat V)$ is the previously defined first Chern-class of the line bundle $\widehat{\wedge^n V}$ corresponding to the $n$-th exterior power $\wedge^n V$ of $V$. The latter is a 1-dimensional representation, corresponding to the determinant line bundle $det(\widehat{V}) = \widehat{\wedge^n V}$:

$c_1(\widehat{V}) \;=\; c_1(det(\widehat{V})) \;=\; c_1( \widehat{\wedge^n V} ) \,.$
• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeOct 10th 2018

Much clearer!

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTime1 day ago

appended to the previous discussion the explicit formula for $c_1$ of an $n$-dimensional representation $V$ as a polynomial in its character values (here):

$c_1(V) = \chi_{\left(V^{\wedge^n}\right)} \;\colon\; g \;\mapsto\; \underset{ { k_1,\cdots, k_n \in \mathbb{n} } \atop { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n } }{\sum} \underoverset{ l = 1 }{ n }{\prod} \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l ! } \left(\chi_V(g^l)\right)^{k_l}$