# Stiefel-Whitney Classes of representations of some finite groups of lie type

Neha Malik  and  Steven Spallone  Indian Institute of Science Education and
Research, Pune-411008,Maharashtra,India<EMAIL_ADDRESS>Indian
Institute of Science Education and Research, Pune-411008,Maharashtra,India
<EMAIL_ADDRESS>
###### Abstract.

In this note we present the Stiefel-Whitney classes (SWCs) for orthogonal
representations of several finite groups of Lie type, namely for
$G=\operatorname{SL}(2,q),$ $\operatorname{SL}(3,q),$
$\operatorname{Sp}(4,q)$, and $\operatorname{Sp}(6,q)$, with $q$ odd. We also
describe the SWCs for $G=\operatorname{SL}(2,q)$ when $q$ is even.

###### Contents

  1. 1 Introduction
  2. 2 Notations & Preliminaries
  3. 3 Group Cohomology
  4. 4 Detection of Cohomology
  5. 5 Main Theorems

##  1\. Introduction

The theory of Stiefel-Whitney classes (SWCs) of vector bundles is an old
unifying concept in geometry. Real representations, or equivalently,
orthogonal complex representations of a group $G$ give rise to _flat_ vector
bundles over the classifying space $BG$. Via this construction one defines
SWCs of representations, which live in the group cohomology
$H^{*}(G,\mathbb{Z}/2\mathbb{Z})$. There do not seem to be many explicit
calculations of such SWCs, especially when $G$ is nonabelian. In this note we
give formulas for SWCs in terms of character values, for the groups mentioned
in the abstract.

Formulas for cyclic groups are well-known; we review this below. The second
SWCs of representations of $S_{n}$ and related groups were found in [GS20].
The case of $\operatorname{GL}(2,q)$ with $q$ odd was completed recently in
[GJ21]; their results are analogous to ours.

This note is intended only as an announcement of our calculations; the work
will appear elsewhere.

Acknowledgements. The authors would like to thank Ryan Vinroot, Dipendra
Prasad, and Rohit Joshi for helpful discussions. The research by the first
author is supported by a PhD fellowship from the Council of Scientific &
Industrial Research, India.

##  2\. Notations & Preliminaries

###  2.1. Orthogonal Representations

Let $G$ be a finite group. We say that a representation $(\pi,V)$, with $V$ a
complex finite-dimensional vector space, is _orthogonal_ , provided there
exists a non-degenerate $G$-invariant symmetric bilinear form $B:V\times
V\to\mathbb{C}$. From [BtD95, Chapter II, Section 6], we note:

######  Proposition 1.

If $(\pi,V)$ is orthogonal, then there is a $(\pi_{0},V_{0})$, with $V_{0}$ a
real vector space, so that $\pi_{0}\otimes_{\mathbb{R}}\mathbb{C}\cong\pi$.
Moreover, $\pi_{0}$ is unique up to isomorphism.

###  2.2. Stiefel-Whitney Classes

In this section we define Stiefel-Whitney classes for orthogonal complex
representations of finite groups. This is equivalent to the definition in
[Ben98] or [GKT89], given for real representations.

Recall from [MS16] that a ($d$-dimensional real) vector bundle $\mathcal{V}$
over a paracompact space $B$ is assigned a sequence

$w_{1}(\mathcal{V}),\ldots,w_{d}(\mathcal{V})$

of cohomology classes, with each $w_{i}(\mathcal{V})\in
H^{*}(B,\mathbb{Z}/2\mathbb{Z})$.

For every finite group $G$ there is a classifying space $BG$ with a principal
$G$-bundle $EG$, unique up to homotopy. From a representation $(\pi,V)$ one
can form the associated vector bundle $EG[V]$ over $BG$. Then one puts

$w_{i}^{\mathbb{R}}(\pi)=w_{i}(EG[V]);$

see for instance [Ben98] or [GKT89]. Moreover the singular cohomology
$H^{*}(BG,\mathbb{Z}/2\mathbb{Z})$ is isomorphic to the group cohomology
$H^{*}(G,\mathbb{Z}/2\mathbb{Z})$.

Hence, from a real representation $\rho$, we can associate cohomology classes
$w_{i}^{\mathbb{R}}(\rho)\in H^{i}(G,\mathbb{Z}/2\mathbb{Z})$, called Stiefel-
Whitney classes (SWCs). We prefer to work with an orthogonal complex
representation $\pi$, which by Proposition 1 comes from a unique real
representation $\pi_{0}$. We thus define

$w_{i}(\pi):=w_{i}^{\mathbb{R}}(\pi_{0}),$

for $0\leq i\leq\deg\pi$.

At this point we simply write $H^{*}(G)=H^{*}(G,\mathbb{Z}/2\mathbb{Z})$. A
group homomorphism $\varphi:G_{1}\to G_{2}$ induces a map

$\varphi^{*}:H^{*}(G_{2})\to H^{*}(G_{1}).$

The first few $w_{i}(\pi)$ have nice interpretations. Firstly,
$w_{0}(\pi)=1\in H^{0}(G)$. Moreover, the first SWC, applied to linear
characters $G\to\\{\pm 1\\}\cong\mathbb{Z}/2\mathbb{Z}$, is the well-known
isomorphism

$w_{1}:\operatorname{Hom}(G,\mathbb{Z}/2\mathbb{Z})\overset{\sim}{\to}H^{1}(G,\mathbb{Z}/2\mathbb{Z}).$

More generally, if $\pi$ is an orthogonal representation, then
$w_{1}(\pi)=w_{1}(\det\pi)$, where $\det\pi$ is simply the composition of
$\pi$ with the determinant map. When $\det\pi=1$, then $w_{2}(\pi)$ vanishes
iff $\pi$ lifts to the corresponding spin group. (See [GS20] for details.)

The SWCs are functorial in the following sense: If $\rho$ is an orthogonal
representation of $G_{2}$, then

(1) $\varphi^{*}(w(\pi))=w(\pi\circ\varphi).$

##  3\. Group Cohomology

In this section we review explicit descriptions of group cohomology for
several important groups. Later, we will describe SWCs in terms of these
descriptions. Throughout this note, we put $k=\mathbb{Z}/2\mathbb{Z}$, and
write $\Theta_{\pi}$ for the character of a representation $\pi$.

###  3.1. Cyclic Groups

Let $n$ be even, and $G=C_{n}$, the cyclic group of order $n$. Let $g$ be a
generator of $G$, and $\chi_{\bullet}$ the linear character of $G$ with
$\chi_{\bullet}(g)=e^{\frac{2\pi i}{n}}$. Write ‘sgn’ for the linear character
of order 2. It is known [Kam67] that

$H^{*}(C_{n})=\begin{cases}k[s,t]/(s^{2}),&n\equiv 0\text{ (mod }4)\\\
k[v],&n\equiv 2\text{ (mod }4)\end{cases}$

where $s=w_{1}(\operatorname{sgn}),$
$t=w_{2}(\chi_{\bullet}\oplus\chi_{\bullet}^{-1})$ for $n\equiv 0\text{ (mod
}4)$, and $v=w_{1}(\operatorname{sgn})$ when $n\equiv 2\text{ (mod }4)$.

Since the cyclic case is so easy, we can give our first example of expressing
SWCs in terms of character values.

######  Proposition 2.

Let $\pi$ be an orthogonal representation of $C_{n}$, and
$b_{\pi}=\frac{1}{4}\left(\deg\pi-\Theta_{\pi}(g^{n/2})\right)$.

  1. (1)

If $n\equiv 2\mod 4$, then

$w(\pi)=(1+v)^{2b_{\pi}}.$

  2. (2)

If $n\equiv 0\mod 4$, and $\det\pi=1$, then

$w(\pi)=(1+t)^{b_{\pi}}.$

  3. (3)

If $n\equiv 0\mod 4$, and $\det\pi\neq 1$, then

$w(\pi)=(1+s)(1+t)^{b_{\pi}}.$

Let $C_{n}^{r}$ be the $n$-fold product of $C_{n}$, with projection maps
$\operatorname{pr}_{i}:C_{n}^{r}\to C_{n}$ for $1\leq i\leq r$. By Künneth, we
have

$H^{*}(C_{n}^{r})=\begin{cases}k[s_{1},\dots,s_{r},t_{1},\dots,t_{r}]/(s_{1}^{2},\ldots,s_{r}^{2}),&n\equiv
0\text{ (mod }4)\\\ k[v_{1},\dots,v_{r}],&n\equiv 2\text{ (mod }4)\end{cases}$

where we put $s_{i}=w_{1}(\operatorname{sgn}\circ\operatorname{pr}_{i})$ and
$t_{i}=w_{2}((\chi_{\bullet}\oplus\chi_{\bullet}^{-1})\circ\operatorname{pr}_{i})$
for $n\equiv 0\text{ (mod }4)$, and
$v_{i}=w_{1}(\operatorname{sgn}\circ\operatorname{pr}_{i})$, for $n\equiv
2\text{ (mod }4)$. (Here $1\leq i\leq r$.)

###  3.2. Special Linear Groups

From [Qui72], we have for odd $q$,

$H^{*}(\operatorname{SL}(2,q))=k[e]\otimes_{k}k[b]/b^{2},$

where $\deg(b)=3$ and $e$ is the unique non-zero element in
$H^{4}(\operatorname{SL}(2,q))$.
Let $\mathcal{Z}$ be the center of $\operatorname{SL}(2,q)$, which has order
$2$. The restriction map

$\displaystyle H^{4}(\operatorname{SL}(2,q))$ $\displaystyle\to
H^{4}(\mathcal{Z})$ $\displaystyle e$ $\displaystyle\mapsto v^{4}$

is a group isomorphism.

###  3.3. Symplectic Groups

Fix the matrix $J=\begin{pmatrix}0&&0&&\ldots&&0&&1\\\ 0&&0&&\ldots&&-1&&0\\\
\vdots&&\vdots&&\iddots&&\vdots&&\vdots\\\ 0&&1&&\ldots&&0&&0\\\
-1&&0&&\ldots&&0&&0\\\ \end{pmatrix}$,

and let $G=\operatorname{Sp}(2n,q):=\\{A\in\operatorname{GL}(2n,q)\mid
A^{t}JA=J\\}$, with $q$ odd. Write $X$ for the subgroup of matrices in $G$,
all of whose nonzero entries lie either on the diagonal or the antidiagonal.
Thus

$X=\begin{pmatrix}\square&0&0&\ldots&0&0&\square\\\ 0&\ddots&&&&\iddots&0\\\
0&&\square&&\square&&0\\\ \vdots&&\vdots&\scalebox{3.0}{$\square$}_{2\times
2}&\vdots&&\vdots\\\ 0&&\square&&\square&&0\\\ 0&\iddots&&&&\ddots&0\\\
\square&0&0&\ldots&0&0&\square\\\ \end{pmatrix}.$

Note that $X$ is isomorphic to the direct product of $n$ copies of
$\operatorname{SL}(2,q)$. So there are projections

$\text{pr}_{j}:X\to\operatorname{SL}(2,q)\text{ ; }1\leq j\leq n,$

and by Künneth we have

$H^{*}(X)\cong
k[e_{1},\ldots,e_{n}]\otimes_{k}k[b_{1},\ldots,b_{n}]/(b_{1}^{2},\ldots,b_{n}^{2}),$

where $e_{j}=\text{pr}^{*}_{j}(e)$ and $b_{j}=\text{pr}^{*}_{j}(b)$ for $1\leq
j\leq n.$

##  4\. Detection of Cohomology

Let $G$ be a finite group. We say a subgroup $G^{\prime}$ _detects_ the mod-2
cohomology of $G$ when the restriction map

$H^{*}(G)\to H^{*}(G^{\prime})$

is an injection.

Suppose this is the case, and that $\pi$ is an orthogonal representation of
$G$. Then $w(\pi)$ is specified by its image in $H^{*}(G^{\prime})$, which is
actually the SWC of the restriction of $\pi$ to $G^{\prime}$. In our formulas
in the next section, when there is a detecting subgroup, we will simply again
write $w(\pi)$ for this image.

######  Lemma 1 ([AM04], Cor. 5.2, Ch. II).

A Sylow 2-subgroup of $G$ detects its mod-2 cohomology.

######  Proposition 3 ([Qui72]).

When $q$ is odd, the diagonal subgroup $A$ of $\operatorname{GL}(n,q)$ detects
its mod-2 cohomology.

From the above one can deduce:

######  Proposition 4.

When $n$ and $q$ are odd, the diagonal subgroup $A_{1}$ of
$\operatorname{SL}(n,q)$ detects its mod-2 cohomology.

Finally, there is:

######  Proposition 5 ([AM04], Lemma 6.2).

For odd $q$, the subgroup $X$ detects the mod-2 cohomology of
$\operatorname{Sp}(2n,q)$.

##  5\. Main Theorems

Now we are ready to describe the promised SWCs.

###  5.1. $G=\operatorname{SL}(2,q)$, $q$ odd

######  Theorem 1.

The total SWC of an orthogonal representation $\pi$ of
$\operatorname{SL}(2,q)$ is

(2) $w(\pi)=(1+e)^{r_{\pi}},$

where $e$ is as defined in Section 3.2, and
$r_{\pi}=\frac{1}{8}(\Theta_{\pi}(\mathbbm{1})-\Theta_{\pi}(-\mathbbm{1})).$

We can say more when $\pi$ is irreducible; write $\omega_{\pi}$ for its
central character.

######  Proposition 6 ([Gow85], Theorem 1).

If $\pi$ is irreducible orthogonal, then $\omega_{\pi}(-\mathbbm{1})=1$. If
$\pi$ is irreducible symplectic, then $\omega_{\pi}(-\mathbbm{1})=-1$.

Thus if $\pi$ is irreducible orthogonal, then
$\Theta_{\pi}(-\mathbbm{1})=\Theta_{\pi}(\mathbbm{1})$, and so $r_{\pi}=0$.
Therefore:

######  Corollary 1.1.

Let $\pi$ be an irreducible orthogonal representation of $G$. Then $w(\pi)=1$.

On the other hand, let $\rho$ be an irreducible symplectic representation of
$G$. (By this we mean that $\rho$ is an irreducible representation on a
complex vector space $V$ admitting a non-degenerate $G$-invariant
_antisymmetric_ $B:V\times V\to\mathbb{C}$.) Its double $\pi=\rho\oplus\rho$
is orthogonal. Then $\Theta_{\pi}(-\mathbbm{1})=-\Theta_{\pi}(\mathbbm{1})$,
and so

$\begin{split}r_{\pi}&=\frac{1}{8}\cdot 2\Theta_{\pi}(\mathbbm{1})\\\
&=\frac{1}{2}\deg\rho.\\\ \end{split}$

Therefore

$w(\rho\oplus\rho)=(1+e)^{\frac{1}{2}\deg\rho}.$

###  5.2. $G=\operatorname{SL}(2,q)$, $q$ even

Now let $q$ be even, say $q=2^{r}$. Let $N$ be the subgroup of upper
unitriangular matrices in $G$. Being a Sylow 2-subgroup, $N$ detects the mod-2
cohomology of $G$.
Now $N\cong(\mathbb{F}_{q},+)$ is an elementary abelian 2-group, so from
Section 3.1, we have

$H^{*}(N)\cong H^{*}(C_{2}^{r})\cong k[v_{1},\ldots,v_{r}].$

Set $n_{1}=\begin{pmatrix}1&&1\\\ 0&&1\end{pmatrix}\in N.$

Every representation $\pi$ of $G$ is orthogonal by the main result in [Vin20].

######  Theorem 2.

The total SWC of $\pi$ is

$w(\pi)=\Big{(}\prod_{v\in H^{1}(N)}(1+v)\Big{)}^{s_{\pi}}\in H^{*}(N),$

where $s_{\pi}=\frac{1}{q}(\Theta_{\pi}(1)-\Theta_{\pi}(n_{1})).$

The expansion of this product is well-known. We have

(3) $\prod_{v\in H^{1}(N)}(1+v)=1+\sum_{i=0}^{r-1}c_{r,i}\in H^{*}(N),\\\ $

with $c_{r,i}$ the famous Dickson invariants described in [Wil83]. Note that
$\deg(c_{r,i})=2^{r}-2^{i}$.

###  5.3. $G=\operatorname{SL}(3,q)$, $q$ odd

Let $q$ be odd, and put
$a_{1}=\operatorname{diag}(-1,-1,1)\in\operatorname{SL}(3,q)$. (Meaning, as
usual, the diagonal matrix with these diagonal entries.)

######  Theorem 3.

Let
$m_{\pi}=\frac{1}{8}\big{(}\Theta_{\pi}(\mathbbm{1})-\Theta_{\pi}(a_{1})\big{)}.$
Then,

$w(\pi)=\begin{cases}\big{(}(1+t_{1})(1+t_{2})(1+t_{1}+t_{2})\big{)}^{m_{\pi}}\;\;,&q\equiv
1\text{ (mod }4)\\\
\big{(}(1+v_{1})(1+v_{2})(1+v_{1}+v_{2})\big{)}^{2m_{\pi}}\;,&q\equiv 3\text{
(mod }4).\end{cases}$

######  Remark 1.

Again we encounter Dickson products as in (3), but where the vector space
$H^{1}(N)$ is replaced with $\operatorname{Span}\\{t_{1},t_{2}\\}$ or
$H^{1}(A)$.

###  5.4. $G=\operatorname{Sp}(4,q)$, $q$ odd

Let $q$ be odd, and $G=\operatorname{Sp}(4,q)$. Put
$g_{1}=\operatorname{diag}(1,-1,-1,1)\in\operatorname{Sp}(4,q).$

######  Theorem 4.

The total SWC of an orthogonal representation $\pi$ of $G$ is

$w(\pi)=((1+e_{1})(1+e_{2}))^{r_{\pi}}(1+e_{1}+e_{2})^{s_{\pi}},$

where

$\displaystyle r_{\pi}$
$\displaystyle=\frac{1}{16}\Big{(}\Theta_{\pi}(\mathbbm{1})-\Theta_{\pi}(-\mathbbm{1})\Big{)},$
$\displaystyle s_{\pi}$
$\displaystyle=\frac{1}{16}\Big{(}\Theta_{\pi}(\mathbbm{1})+\Theta_{\pi}(-\mathbbm{1})-2\Theta_{\pi}(g_{1})\Big{)}.$

Again from Theorem 1 of [Gow85], we deduce that when $\pi$ is irreducible
orthogonal, we have

$w(\pi)=(1+e_{1}+e_{2})^{s_{\pi}},$

where

$s_{\pi}=\frac{1}{8}\left(\deg\pi-\Theta_{\pi}(g_{1})\right).$

###  5.5. $G=\operatorname{Sp}(6,q)$, $q$ odd

Let

$\displaystyle g_{1}$ $\displaystyle=\text{diag}(1,1,-1,-1,1,1),$
$\displaystyle g_{2}$ $\displaystyle=\text{diag}(1,-1,-1,-1,-1,1)\in G.$

######  Theorem 5.

The total SWC of an orthogonal representation $\pi$ of $G$ is

$w(\pi)=\Big{(}\prod_{1\leq i\leq
3}(1+e_{i})\Big{)}^{r_{\pi}}\Big{(}\prod_{1\leq i<j\leq
3}(1+e_{i}+e_{j})\Big{)}^{s_{\pi}}(1+e_{1}+e_{2}+e_{3})^{t_{\pi}},$

where

$\displaystyle r_{\pi}$
$\displaystyle=\frac{1}{32}\big{(}\Theta_{\pi}(\mathbbm{1})+\Theta_{\pi}(g_{1})-\Theta_{\pi}(g_{2})-\Theta_{\pi}(-\mathbbm{1})\big{)}$
$\displaystyle s_{\pi}$
$\displaystyle=\frac{1}{32}\big{(}\Theta_{\pi}(\mathbbm{1})-\Theta_{\pi}(g_{1})-\Theta_{\pi}(g_{2})+\Theta_{\pi}(-\mathbbm{1})\big{)},\text{
and}$ $\displaystyle t_{\pi}$
$\displaystyle=\frac{1}{32}\big{(}\Theta_{\pi}(\mathbbm{1})-3\Theta_{\pi}(g_{1})+3\Theta_{\pi}(g_{2})-\Theta_{\pi}(-\mathbbm{1})\big{)}.$

Once more Theorem 1 of [Gow85] allows simplification: when $\pi$ is
irreducible orthogonal, we have

$w(\pi)=\Big{(}\prod_{1\leq i<j\leq 3}(1+e_{i}+e_{j})\Big{)}^{s_{\pi}},$

where

$s_{\pi}=\frac{1}{16}\left(\deg\pi-\Theta_{\pi}(g_{1})\right).$

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