# nLab characteristic class of a linear representation

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

cohomology

# Contents

## Idea

Given a linear representation of a finite group $G$ on a finite-dimensional vector space $V$, this induces an associated vector bundle over the classifying space $B G$. The characteristic classes of this vector bundle, notably it Chern classes or Pontryagin, are hence entirely determined by the linear representation, and may be associated with it.

## Definition

Under the identification of the representation ring with the equivariant K-theory of the point (see there) and the Atiyah-Segal completion map

$R_{\mathbb{C}}(G) \simeq KU_G^0(\ast) \overset{ \widehat{(-)} }{\longrightarrow} KU(BG)$

one may ask for Chern classes of the K-theory class $\widehat{V} \in KU(B G)$ expressed in terms of the actual character of the representation $V$.

(…)

## Examples

### First Chern class

There is a closed formula at least for the first Chern class (Atiyah 61, appendix):

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 linear 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 $det(\widehat{V}) = \widehat{\wedge^n V}$:

$c_1(\widehat{V}) \;=\; c_1(det(\widehat{V})) \;=\; c_1( \widehat{\wedge^n V} ) \,.$

More explicitly, via the formula for the determinant as a polynomial in traces of powers (see there) this means that the first Chern class of the $n$-dimensional representation $V$ is expressed in terms of its character $\chi_V$ as

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

For example, for a representation of dimension $n = 2$ this reduces to

$c_1(V) = \chi_{V \wedge V} \;\colon\; g \;\mapsto\; \frac{1}{2} \left( \left( \chi_V(g)\right)^2 - \chi_V(g^2) \right)$

###### Example

Let $G =\mathbb{Z}_{2n+1}$ be a finite cyclic group of odd order and let $k[\mathbb{Z}_{2n+1}]$ be its regular representation. Then the first Chern class vanishes:

$c_1\big( k[\mathbb{Z}_{2n+1}]\big) \;=\; 0$
###### Proof

The underlying set of $\mathbb{Z}_{2n+1}$ constitutes the canonical linear basis of $k[\mathbb{Z}_{2n+1}]$. Moreover, this carries a canonical linear order $(e, g_1, g_2, \cdots, g_{2n+1})$. With respect to this ordering, the action of each group element $g \in \mathbb{Z}_n$ is by a cyclic permutation. Since for odd number of elements the signature of a cyclic permutation is $+1$, it follows that for every group element

$g(e \wedge g_1 \wedge \cdots \wedge g_{2n+1}) = + e \wedge g_1 \wedge \cdots \wedge g_{2n+1} \,.$

This shows that the character of $\wedge^{2n+1}k[\mathbb{Z}_{2n+1}]$ equals that of the trivial representation $\mathbf{1}$

## Properties

### Splitting principle

Let $G$ be a finite group, let the ground field to be the complex numbers.

Then by the Brauer induction theorem every virtual representation

$[V] \in R_\mathbb{C}(G)$

has a presentation as a virtual combination of induced representations of 1-dimensional representations:

$[V] \;=\; \underset{ \mathclap{ {H_i \hookrightarrow G} \atop { { W_i \in Rep(H_i) \,, } \atop { dim(W_i) =1 } } } }{\sum} n_i \left[ ind_{H_i}^G W_i \right]$

Of course this expansion is not unique.

According to Symonds 91, p. 4 & Prop. 2.4, there is a natural choice for this expansion, and for this there holds a splitting principle for the corresponding Chern classes summarized in the total Chern class (formal sum of all Chern classes)

$c(V) \;\coloneqq\; 1 + c_2(V) + c_2(V) + \cdots \;\in\; \underset{k}{\prod} H^{2 k}\big( B G , \mathbb{Z}\big)$

as follows:

(2)$ch \left( V \right) \;=\; \underset{ \mathclap{ {H_i \hookrightarrow G} \atop { { W_i \in Rep(H_i)\,, } \atop { dim(W_i) = 1 } } } }{\prod} \mathcal{N}_{H_i}^G \Big( \overset{ \mathclap{ = (1 + c_1(W_i)) } }{ \overbrace{ ch\left(W_i\right) } } {}^{\alpha(W_i)} \Big) \;\; \in \underset{ k \in \mathbb{N} }{\prod} H^{2k}\big(B G, \mathbb{Z} \big)$

where

1. the transfer maps

(3)$\mathcal{N}_H^G \;\colon\; H^\bullet(B H, \mathbb{Z}) \longrightarrow H^\bullet( B G, \mathbb{Z} )$

are from Evens 63, bottom of p. 7,

2. the $\alpha(W_i)$-s are the Euler characteristics of certain CW-complexes, described in Symonds 91, p. 3.

Here over the brace we used that the $W_i$ are 1-dimensional, so that at most their first Chern class may be non-vanishing.

Notice that the transfer maps (3) are multiplicative under cup product (Evens 63, prop. 4), whence Symonds 91 refers to them as the “mutliplicative transfer”.

## References

• Michael Atiyah, Appendix of Characters and cohomology of finite groups, Publications Mathématiques de l’IHÉS, Volume 9 (1961) , p. 23-64 (numdam)

• Leonard Evens, A Generalization of the Transfer Map in the Cohomology of Groups, Transactions of the American Mathematical Society Vol. 108, No. 1 (Jul., 1963), pp. 54-65 (doi:10.1090/S0002-9947-1963-0153725-1, jstor:1993825)

• Leonard Evens, On the Chern Classes of Representations of Finite Groups, Transactions of the American Mathematical Society, Vol. 115 (Mar., 1965), pp. 180-193 (doi:10.2307/1994264)

• F. Kamber, Philippe Tondeur, Flat Bundles and Characteristic Classes of Group-Representations, American Journal of Mathematics, Vol. 89, No. 4 (Oct., 1967), pp. 857-886 (doi:10.2307/2373408)

• Ove Kroll, An Algebraic Characterisation of Chern Classes of Finite Group Representations, Bulletin of the LMS, Volume19, Issue3 May 1987 Pages 245-248 (doi:10.1112/blms/19.3.245)

• J. Gunarwardena, B. Kahn, C. Thomas, Stiefel-Whitney classes of real representations of finite groups, Journal of Algebra Volume 126, Issue 2, 1 November 1989, Pages 327-347 (doi:10.1016/0021-8693(89)90309-8)

• Arnaud Beauville, Chern classes for representations of reductive groups (arXiv:math/0104031)

Last revised on May 15, 2019 at 10:09:49. See the history of this page for a list of all contributions to it.