nLab characteristic class of a linear representation

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

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

(Atiyah 61, appendix, item (7))

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

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

(see also e.g. tom Dieck 09, p. 45)

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

see at Symonds’ explicit Brauer induction

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)

splitting principle:

• Peter Symonds, A splitting principle for group representations, Comment. Math. Helv. (1991) 66: 169 (dml:140229, doi:10.1007/BF02566643)