# Total Stiefel Whitney classes for real representations of
$\operatorname{GL}_{n}$ over $\mathbb{F}_{q},\mathbb{R}$ and $\mathbb{C}$

Jyotirmoy Ganguly, Rohit Joshi  The Institute of Mathematical Sciences, IV
Cross Road, CIT Campus, Taramani, Chennai-600113, Tamil Nadu, India
<EMAIL_ADDRESS>IISER Pune, Dr. Homi Bhabha Road,Pune-411008,
Maharashtra, India<EMAIL_ADDRESS>
###### Abstract.

We compute the total Stiefel Whitney class for a real representation $\pi$ of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$, where $q$ is odd in terms of
character values of $\pi$ on order $2$ diagonal elements. We also compute the
total Stiefel Whitney class of $\operatorname{GL}_{n}(\mathbb{R})$ and
$\operatorname{GL}_{n}(\mathbb{C})$.

###### Contents

  1. 1 Introduction
  2. 2 Notation and Preliminaries
    1. 2.1 Irreducible Representations of $C_{m}^{n}$
    2. 2.2 Cohomology of $GL_{n}(\mathbb{F}_{q})$, for $q\equiv 1\pmod{4}$
  3. 3 SWC of $S_{n}$ Invariant Representations of $C_{2}^{n}$
  4. 4 proof of main theorem
  5. 5 Detection Results
  6. 6 Calculations of Some SWC
  7. 7 Some Applications
    1. 7.1 Lifting Criteria for Real Representations of $\operatorname{GL}_{n}(\mathbb{F}_{q})$
    2. 7.2 Non-Detection by An-isotropic Torus
  8. 8 The Case of $\operatorname{GL}_{n}(\mathbb{C})$ and $\operatorname{GL}_{n}(\mathbb{R})$
  9. 9 Examples
    1. 9.1 Principal Series Representation
    2. 9.2 Cuspidal Representations
      1. 9.2.1 Computation of $w_{2}(\pi)$
      2. 9.2.2 Computation of $w_{4}(\pi)$
    3. 9.3 Steinberg Representation
      1. 9.3.1 Computation of $w_{2}(\pi)$
      2. 9.3.2 Computation of $w_{4}(\pi)$

##  1\. Introduction

Let $\pi$ be a real representation of a finite group $G$. One can define
Stiefel Whitney classes (SWC) $w_{i}(\pi)$ for $0\leq i\leq\deg\pi$ as members
of the cohomology groups $H^{i}(G)=H^{i}(G,\mathbb{Z}/2\mathbb{Z})$. The total
Stiefel Whitney class of $\pi$ in the cohomology ring $H^{*}(G)$ is defined as

$w(\pi)=w_{0}(\pi)+w_{1}(\pi)+\cdots+w_{d}(\pi),$

where $d=\deg\pi$. (see [Ben91, Section 2.6, page no. 50] for details).

In this article, we compute the total SWC for a real representation $\pi$ of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$, where $q$ is odd, in terms of the
character values on diagonal elements of order $2$. The computations for the
particular case of $n=2$ can be found in [GJ22]. We also compute the total SWC
of an orthogonal representation of $\operatorname{GL}_{n}(\mathbb{C})$ and
$\operatorname{GL}_{n}(\mathbb{R})$. The computation of second SWC for
representations of $S_{n}$ and related groups can be found in [GS20]. Explicit
expressions for the first and second SWC of the representations of
$\mathrm{O}_{n}(\mathbb{R})$ were found in [GJ21] in terms of highest weights.

We will now describe our results in some more detail. Let $\mathfrak{X}$
denote the set of binary strings of length $n$. For $a=(a_{1},a_{2},\ldots
a_{n})\in\mathfrak{X}$, define
$\nu(a)=\sum\limits_{i=1}^{n}a_{i}\in\mathbb{Z}$. Put
$\mathfrak{X}_{i}=\\{a\in\mathfrak{X}\mid\nu(a)=i\\}$. Let $C_{m}$ denote the
additive cyclic group of order $m$.

Consider the linear character $\chi^{j}$ of $C_{m}$ given by
$\chi^{j}(a)=\zeta_{m}^{j}$ where $a$ is the generator of $C_{m}$ and
$\zeta_{m}$ is the primitive $m$th root of unity. For $m$ even write
‘$\operatorname{sgn}$’ for $\chi^{m/2}$. For a binary string
$a\in\mathfrak{X}$ consider the representation $\operatorname{sgn}_{a}$ of
$C_{m}^{n}$, where $m$ is even, defined as
$\operatorname{sgn}_{a}=\bigotimes\limits_{j=1}^{n}(\operatorname{sgn})^{a_{j}},$
where

$(\operatorname{sgn})^{\epsilon}=\begin{cases}\mathbb{1},\quad\text{if
$\epsilon=0$},\\\ \operatorname{sgn},\quad\text{if $\epsilon=1$}.\end{cases}$

Here we use some terminologies from supercharacter theory. For more details we
refer the reader to [DI08]. Consider the action of $S_{n}$ on $C_{2}^{n}$ by
permuting the copies of $C_{2}$. If we identify $C_{2}^{n}$ with set of binary
strings $\mathfrak{X}$, then the superclasses are $\mathfrak{X}_{i}$ which are
orbits of $S_{n}$ action.

We will identify $C_{2}^{n}$ with the subgroup

$\text{diag}(\pm 1,\ldots,\pm 1)\subset\operatorname{GL}_{n}(F),$

where $k=\mathbb{F}_{q},\mathbb{R}$ or $\mathbb{C}$ depending upon the
context. We will write

$h_{i}=\operatorname{diag}(\underbrace{-1,\ldots,-1}_{i\,\text{times}},1,\ldots,1)\in\operatorname{GL}_{n}(F).$

Then $h_{i}$ represents the superclass $\mathfrak{X}_{i}$. Consider the
representations

$\sigma_{i}=\bigoplus\limits_{l\in\mathfrak{X}_{i}}\operatorname{sgn}_{l}$ (1)

of $C_{m}^{n}$ for $1\leq i\leq n$. Then clearly the supercharacters are
$\sigma_{i}$.

For $l\in\mathfrak{X}$, consider the linear character $\chi(l)$ of $C_{m}^{n}$
given by $\chi(l)=\bigotimes\limits_{k=1}^{n}\chi^{l_{k}}$. For $1\leq i\leq
n$, take $e_{i}\in\mathfrak{X}$ such that $e_{ij}=1$ if $i=j$ and $e_{ij}=0$
otherwise. If $4\mid m$ then

$H^{*}(C_{m}^{n})=\dfrac{\mathbb{Z}/2\mathbb{Z}[s_{1},\ldots,s_{n},t_{1},\ldots,t_{n}]}{(s_{1}^{2},\ldots,s_{n}^{2})},$

where $s_{i}=w_{1}(\operatorname{sgn}_{e_{i}})$ and
$t_{i}=w_{2}(\chi(e_{i})_{\mathbb{R}})$. On the other hand

$H^{*}(C_{2}^{n})=\mathbb{Z}/2\mathbb{Z}[v_{1},\ldots,v_{n}],$

where $v_{i}=w_{1}(\operatorname{sgn}_{e_{i}})$. Let $v=(v_{1},\ldots,v_{n})$
and $t=(t_{1},\ldots,t_{n})$ and $b_{1}=\sum\limits_{i=1}^{n}s_{i}$. For
$l\in\mathfrak{X}_{i}$ we write $l\cdot v$ (resp. $l\cdot t$) for
$\sum\limits_{i=1}^{n}l_{i}v_{i}$ (resp. $\sum\limits_{i=1}^{n}l_{i}t_{i}$).

For a real representation $\pi$ of $\operatorname{GL}_{n}(F)$ where
$F=\mathbb{F}_{q}$ or $\mathbb{R}$ or $\mathbb{C}$, consider the term

$c_{k}=\frac{1}{2^{n}}\sum\limits_{i=0}^{n}\chi_{\sigma_{i}}(h_{k})\chi_{\pi}(h_{i}),$

where $M=(\chi_{\sigma_{i}}(h_{k}))$ is the matrix which is exactly the
supercharacter table. Thus we explicitly compute the table by proving that
$\chi_{\sigma_{i}}(h_{k})$ is the coefficient of $y^{i}$ in the expression
$(1-y)^{k}(1+y)^{n-k}$(see Proposition 2).

For a real representation $\pi$ of $\operatorname{GL}_{n}(\mathbb{F}_{q})$ we
define

$\delta=\begin{cases}0,&\text{if $\det\pi=1$},\\\
1,&\text{otherwise}.\end{cases}$

For a subgroup $H$ of $G$ we denote by $w^{H}(\pi)$ (resp. $w_{i}^{H}(\pi)$)
the total Stiefel Whitney class
$w(\pi\mid_{H})=\operatorname{res}^{G}_{H}(w(\pi))$(resp.
$w_{i}(\pi\mid_{H})$). Let $D$ denote the diagonal torus of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$. We have
$\mathrm{res}:H^{*}(\operatorname{GL}_{n}(\mathbb{F}_{q}))\hookrightarrow
H^{*}(D)$ for $q\equiv 1\pmod{4}$ and
$\mathrm{res}:H^{*}(\operatorname{GL}_{n}(\mathbb{F}_{q}))\hookrightarrow
H^{*}(C_{2}^{n})$ for $q\equiv 3\pmod{4}$ are injections. Our main theorem,
proved in Section 4, says that

######  Theorem 1 (Main Theorem).

Let $\pi$ be a real representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$
where $q$ is odd.

  1. (1)

If $q\equiv 1\pmod{4}$ then

$w^{D}(\pi)=\left(1+\delta
b_{1}\right)\prod_{i=1}^{n}\left(\prod_{p\in\mathfrak{X}_{i}}(1+p\cdot
t)\right)^{c_{i}/2}.$

  2. (2)

If $q\equiv 3\pmod{4}$ then

$w^{C_{2}^{n}}(\pi)=\prod_{i=1}^{n}\left(\prod_{l\in\mathfrak{X}_{i}}(1+l\cdot
v)\right)^{c_{i}}.$

We use the same approach to compute the total Stiefel Whitney class of real
representations of $\operatorname{GL}_{n}(\mathbb{C})$ and
$\operatorname{GL}_{n}(\mathbb{R})$. We have
$\operatorname{res}:H^{*}(\operatorname{GL}_{n}(k))\hookrightarrow
H^{*}(C_{2}^{n})$ is injective.

######  Theorem 2.

Let $\pi$ be a real representation of $\operatorname{GL}_{n}(\mathbb{C})$ or
$\operatorname{GL}_{n}(\mathbb{R})$. Then the total Stiefel Whitney class of
$\pi$ is given by

$w^{C_{2}^{n}}(\pi)=\prod_{i=1}^{n}\left(\prod_{l\in\mathfrak{X}_{i}}(1+l\cdot
v)\right)^{c_{i}}.$ (2)

In [Nak60, Corollary $6.7$] Nakaoka proved that the cohomology of the
symmetric group stabilizes. In particular, the result says that if $i<n/2$,
then the map $\operatorname{res}:H^{i}(S_{n})\to H^{i}(S_{2i})$ is an
isomorphism. We prove a similar result for
$\operatorname{GL}_{n}(\mathbb{F}_{q})$.

######  Theorem 3.

The map $\operatorname{res}:H^{i}(\operatorname{GL}_{m}(\mathbb{F}_{q}))\to
H^{i}(\operatorname{GL}_{n-1}(\mathbb{F}_{q}))$, where $m\geq n$ and $q\equiv
1\pmod{4}$ is injective for $i<2n-1$.

We have a similar result for the case when $q\equiv 3\pmod{4}$ (see Theorem
8).

The theorem 12 recovers the main result in [JS21, Theorem $1$] for real
representations of $\operatorname{GL}_{n}(\mathbb{F}_{q})$ where $q$ is odd.

Theorem 3 allows further simplification to calculate the second and fourth
Stiefel Whitney classes. Let $\pi$ be a real representation of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$, where $q$ is odd. Then we have

$\begin{split}w_{2}^{D}(\pi)&=\dfrac{m_{\pi}}{2}\sum_{i=1}^{n}t_{i},\text{if}\,\,q\equiv
1\pmod{4},\\\
w_{2}^{C_{2}^{n}}(\pi)&=\dbinom{m_{\pi}}{2}\left(\sum\limits_{i=1}^{n}v_{i}^{2}\right),\text{if}\,\,q\equiv
3\pmod{4},\end{split}$

where $m_{\pi}=\frac{\dim\pi-\chi_{\pi}(h_{1})}{2}$. In particular, if
$q\equiv 1\pmod{4}$, then

$w_{4}^{D}(\pi)=\binom{m_{\pi}/2}{2}\sum_{i=1}^{n}t_{i}^{2}+\frac{\dim\pi-\chi_{\pi}(h_{2})}{8}\sum_{1\leq
i<j\leq n}t_{i}t_{j}.$

As an application of the result we obtain the following:

######  Theorem 4.

Let $\pi$ be a real principal series representation for
$\operatorname{GL}_{n}(\mathbb{F}_{q})$ where $q\equiv 1\pmod{4}$. Then
$w_{4}(\pi)=0$ for $n\geq 5$.

We also compute the second SWC for $q$ odd and the fourth SWC for $q\equiv
1\pmod{4}$.

We now describe the layout of the paper. In Section 2 recall some basic facts
on cohomology of the groups $\operatorname{GL}_{n}(\mathbb{F}_{q})$ and
$C_{m}^{n}$. In Section 3 we compute the SWC for $S_{n}$ invariant real
representations of $C_{2}^{n}$. We prove the main theorem in Section 4. The
theorems 3 and 8 are proved in Section 5. Using the detection results we
compute the second and the fourth SWC in Section 6. Section 7 shows some
applications of the theory. Section 8 deals with the cases of
$\operatorname{GL}_{n}(\mathbb{C})$ and $\operatorname{GL}_{n}(\mathbb{R})$.
We work out some examples in Section 9.

##  2\. Notation and Preliminaries

A complex representation $(\pi,V)$ of a finite group $G$ is called orthogonal
if it preserves a non degenerate symmetric bilinear form. We call $\pi$ real
if the representation space $V$ is real. For a complex representation $\pi$
one can consider its realisation $(\pi_{\mathbb{R}},V_{\mathbb{R}})$ by
forgetting the complex structure of $V$ and regarding it as a real vector
space.

###  2.1. Irreducible Representations of $C_{m}^{n}$

Take the set of ordered pairs

$\mathcal{P}=\\{\tau=(\tau_{1},\ldots,\tau_{n})\mid\tau_{j}\in\mathbb{Z}/m\mathbb{Z},1\leq
j\leq n\\}.$

Consider the subset $\mathcal{P}^{\prime}$ of $\mathcal{P}$ given by

$\mathcal{P}^{\prime}=\\{\tau=(\tau_{1},\ldots,\tau_{n})\mid\tau_{j}\in\\{0,m/2\\},1\leq
j\leq n\\}.$

Consider the linear character
$\chi_{\tau}=(\chi^{\tau_{1}}\otimes\chi^{\tau_{2}}\otimes\cdots\otimes\chi^{\tau_{n}})$
of $C_{m}^{n}$ (See Section 2)). We have $\chi_{-\tau}=\chi_{\tau}^{\vee}$.
Note that $\chi_{\tau}$ is real if $\tau\in\mathcal{P}^{\prime}$. Define an
equivalence relation on $\mathcal{P}$ as

$\tau\sim(-\tau_{1},-\tau_{2},\ldots,-\tau_{n}).$

Put $\widehat{\mathcal{P}}=(\mathcal{P}-\mathcal{P}^{\prime})/\sim$. Therefore
a real irreducible representations of $C_{m}^{n}$ is either of the form
$(\chi_{\tau})_{\mathbb{R}}$ for $\tau\in\widehat{\mathcal{P}}$ or of the form
$\chi_{\tau}$ where $\tau\in\mathcal{P}^{\prime}$.

###  2.2. Cohomology of $GL_{n}(\mathbb{F}_{q})$, for $q\equiv 1\pmod{4}$

The cohomology ring of $\operatorname{GL}_{n}(\mathbb{F}_{q})$, where $q\equiv
1\pmod{4}$ is given by

$H^{*}(\operatorname{GL}_{n}(\mathbb{F}_{q}),\mathbb{Z}/2\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}[a_{2},a_{4},\ldots,a_{2n}]\otimes
E[b_{1},b_{3},\ldots,b_{2n-1}],$ (3)

where

$a_{2l}=\sum_{i_{1}<i_{2}<\ldots<i_{l}}t_{i_{1}}\cdots t_{i_{l}},\quad
b_{2l-1}=\sum_{i_{1}<i_{2}<\ldots<i_{l},1\leq j\leq
l}t_{i_{1}}\cdots\hat{t}_{i_{j}}\cdots t_{i_{l}}s_{i_{j}},\quad 1\leq l\leq
n,$

and $E[b_{1},\ldots,b_{2n-1}]$ denotes the alternating algebra. For details we
refer the reader to [FP06, Theorem $8.1$, page $288$].

We know that the symmetric group $S_{n}$ is the Weyl group of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$ with respect to the diagonal torus
$D$. It acts on $D$ by permuting the diagonal entries.

######  Proposition 1.

Let $\pi$ be a representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$. Then
$\pi\mid_{D}$ is $S_{n}$ invariant.

###### Proof.

We leave the proof to the reader. ∎

This enables us to work with $S_{n}$ invariant representations of $D$.

######  Definition 1.

Let $G^{\prime}$ be a subgroup of a finite group $G$. We say that $G^{\prime}$
_detects_ the (mod $2$) cohomology of $G$, provided that the restriction map

$\operatorname{res}:H^{*}(G,\mathbb{Z}/2\mathbb{Z})\to
H^{*}(G^{\prime},\mathbb{Z}/2\mathbb{Z})$ (4)

is injective.

The action of $S_{n}$ on $D$ induces an action of $S_{n}$ on $H^{*}(D)$. We
write $H^{*}(D)^{S_{n}}$ to denote the subalgebra fixed under the action.

######  Theorem 5 (Corollary, page $571$ Quillen).

The diagonal subgroup $D$ detects the cohomology of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$. In fact, the homomorphism

$H^{*}(\operatorname{GL}_{n}(\mathbb{F}_{q}))\to H^{*}(D)^{S_{n}}$

induced from the restriction of $\operatorname{GL}_{n}(\mathbb{F}_{q})$ to $D$
is injective.

##  3\. SWC of $S_{n}$ Invariant Representations of $C_{2}^{n}$

An $S_{n}$ invariant representation of $C_{2}^{n}$ will be of the form

$\pi=\bigoplus_{i=0}^{n}c_{i}\sigma_{i},$ (5)

where $\sigma_{i}$ is as in (1).

######  Proposition 2.

The character value $\chi_{\sigma_{i}}(h_{k})$ is equal to the coefficient of
$y^{i}$ in the expression $(1-y)^{k}(1+y)^{n-k}$.

###### Proof.

Fix two integers $i,k$ such that $0\leq i,k\leq n$. Let $R_{l}$ denote the set
of elements $a\in\mathfrak{X}_{i}$ such that the first $k$ components of $a$
contains $l$ number of $1$’s. Then for $a\in R_{l}$ we have
$\chi_{\operatorname{sgn}_{a}}(h_{k})=(-1)^{l}.$ Take
$\tau_{l}=\bigoplus\limits_{a\in R_{l}}\operatorname{sgn}_{a}$. Then

$\chi_{\tau_{l}}(h_{k})=\binom{k}{l}\binom{n-k}{i-l}(-1)^{l}.$

Observe that $\bigoplus\limits_{l=0}^{k}\tau_{l}=\sigma_{i}$. Thus

$\chi_{\sigma_{i}}(h_{k})=\sum_{l=0}^{k}\binom{k}{l}\cdot\binom{n-k}{i-l}(-1)^{l}.$
(6)

Consider the product

$\left(\sum_{l=0}^{k}(-1)^{l}\binom{k}{l}y^{l}\right)\cdot\left(\sum_{l=0}^{i}\binom{n-k}{i-l}y^{-l}\right).$

Then $\chi_{\sigma_{i}}(h_{k})$ is the constant term in the above product.
Observe that it is also constant term in the product

$\left(\sum_{l=0}^{x}(-1)^{l}\binom{k}{l}y^{l}\right)\cdot\left(\sum_{l=i-(n-k)}^{i}\binom{n-k}{i-l}y^{-l}\right),$

We make a change of variable $z=i-l$. The above expression is

$\displaystyle
y^{-i}\left(\sum_{l=0}^{k}(-1)^{l}\binom{k}{l}y^{l}\right)\cdot\left(\sum_{i-l=(n-k)}^{0}\binom{n-k}{i-l}y^{i-l}\right)$
$\displaystyle=y^{-i}(1-y)^{k}\left(\sum_{x=0}^{n-k}\binom{n-k}{z}y^{z}\right)$
$\displaystyle=y^{-i}(1-y)^{k}(1+y)^{n-k}.$

This gives the desired result.

∎

From (5) we have

$\chi_{\pi}(h_{i})=\sum_{j=0}^{n}c_{j}\cdot\chi_{\sigma_{j}}(h_{i})$ (7)

Define matrix $M=(m_{ij})$ where $0\leq i,j\leq n$ such that

$m_{ij}=\chi_{\sigma_{j}}(h_{i})$ (8)

. Take
$a={}^{t}\left(\chi_{\pi}(h_{0}),\ldots,\chi_{\pi}(h_{k})\right)\quad\text{and}\quad
m={}^{t}\left(c_{0},\ldots,c_{k}\right).$ Observe that $a=M\cdot m$.

######  Theorem 6.

We have

$M^{2}=2^{n}I_{n+1},$

where $I_{n+1}$ denotes the identity matrix of rank $n+1$.

###### Proof.

Let

$R_{i}=(m_{i0},\ldots,m_{in}),$

be the $i$-th row of $M$. Note that from Proposition 2 the element $m_{il}$,
$0\leq l\leq n$, is equal to the coefficient of $y^{l}$ in
$(1-y)^{i}(1+y)^{n-i}$.

We calculate the $x$-th row in the matrix $M^{2}$.

$\displaystyle\sum m_{xi}R_{i}$
$\displaystyle=\sum_{i=0}^{n}\chi_{\sigma_{i}}(h_{x})(1-y)^{i}(1+y)^{n-i}$
$\displaystyle=\sum_{i=0}^{n}\left(\sum_{l=0}^{x}\binom{x}{l}\cdot\binom{n-x}{i-l}(-1)^{l}\right)(1-y)^{i}(1+y)^{n-i}$
$\displaystyle=\sum_{l=0}^{x}\binom{x}{l}(-1)^{l}\left(\sum_{i=0}^{n}\binom{n-x}{i-l}(1-y)^{i}(1+y)^{n-i}\right)$
$\displaystyle=\sum_{l=0}^{x}\binom{x}{l}(-1)^{l}\left(\sum_{t=0}^{n-x}\binom{n-x}{t}(1-y)^{t+l}(1+y)^{n-t-l}\right)$
$\displaystyle=\sum_{l=0}^{x}\binom{x}{l}(-1)^{l}(1-y)^{l}\left(\sum_{t=0}^{n-x}\binom{n-x}{t}(1-y)^{t}(1+y)^{n-t-
l-x+x}\right)$
$\displaystyle=\sum_{l=0}^{x}\binom{x}{l}(-1)^{l}(1-y)^{l}(1+y)^{x-l}\left(\sum_{t=0}^{n-x}\binom{n-x}{t}(1-y)^{t}(1+y)^{n-x-t}\right)$
$\displaystyle=\sum_{l=0}^{x}\binom{x}{l}(y-1)^{l}(y+1)^{x-l}(1-y+1+y)^{n-x}$
$\displaystyle=(2y)^{x}2^{n-x}$ $\displaystyle=2^{n}y^{x}.$

∎

Since $M$ is invertible, we have

$m=M^{-1}\cdot a.$ (9)

######  Proposition 3.

Let $\pi$ be the representation of $C_{2}^{n}$ as in equation (5). Then we
have

$c_{k}=\frac{1}{2^{n}}\sum_{i=0}^{n}\chi_{\sigma_{i}}(h_{k})\chi_{\pi}(h_{i}).$

###### Proof.

The proof follows from Equations (8) (9) and Theorem 6. ∎

######  Theorem 7.

Let $\pi$ be a $S_{n}$ invariant representation of $C_{2}^{n}$ of the form
$\pi=\bigoplus\limits_{i=1}^{n}c_{i}\sigma_{i}$.Then we have

$w(\pi)=\prod_{i=1}^{n}\left(\prod_{l\in\mathfrak{X}_{i}}(1+l\cdot
v)\right)^{c_{i}}$ (10)

where
$c_{i}=\frac{1}{2^{n}}\sum\limits_{k=0}^{n}\chi_{\sigma_{k}}(h_{i})\chi_{\pi}(h_{k})$.

###### Proof.

We have

$w(\sigma_{i})=\prod_{l\in\mathfrak{X}_{i}}(1+l\cdot v)$

and

$w(\pi)=\prod_{i=1}^{n}(w(\sigma_{i}))^{m_{i}}.$

Now from Theorem 3 we have

$c_{i}=\frac{1}{2^{n}}\sum_{k=0}^{n}\chi_{\sigma_{k}}(h_{i})\chi_{\pi}(h_{k}).$

Hence the result follows.

∎

##  4\. proof of main theorem

For a subset $\text{I}=\\{i_{1},i_{2},\ldots,i_{l}\\}$ of $\\{1,2,\ldots,n\\}$
and for a multiset $\text{J}=\\{j_{1},j_{2},\ldots,j_{k}\\}$ where
$j_{i}\in\\{1,2,\ldots,n\\}$ we define

$s^{\text{I}}=s_{i_{1}}\cdots s_{i_{j}}\text{ and
}t^{\text{J}}=t_{j_{1}}t_{j_{2}}\cdots t_{j_{k}}.$

Observe that
$H^{*}(C_{4m}^{n})=\dfrac{\mathbb{Z}/2\mathbb{Z}[s_{1},\ldots,s_{n},t_{1},\ldots,t_{n}]}{(s_{i}^{2})}$
is a $\mathbb{Z}/2\mathbb{Z}$ vector space with basis
$s^{\text{I}}t^{\text{J}}$, where $\text{I}(\text{ resp. }\text{J})$ varies
over all subsets ( resp. multi-sets) over $\\{1,2,\ldots,n\\}$.

######  Proposition 4.

Let $q\equiv 1\pmod{4}$. If $\pi$ is a real representation of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$ and if
$w(\pi)=\sum\limits_{\text{I},\text{J}}a_{\text{I},\text{J}}s^{\text{I}}t^{\text{J}}$,
then elements of the form $s^{\text{I}}$ have coefficient $0$ i.e.
$a_{\text{I},\emptyset}=0$ if $|\text{I}|\geq 2$.

###### Proof.

Consider the projection map

$\mathrm{pr}:\dfrac{\mathbb{Z}/2\mathbb{Z}[s_{1},\ldots,s_{n},t_{1},\ldots,t_{n}]}{\langle
s_{i}^{2}\rangle}\to\dfrac{\mathbb{Z}/2\mathbb{Z}[s_{1},\ldots,s_{n}]}{\langle
s_{i}^{2}\rangle}$

defined by $\mathrm{pr}(s_{i})=s_{i}$ and $\mathrm{pr}(t_{i})=0$ for all $i$.
This gives $\operatorname{pr}(a_{2i})=0$ for all $i$ and
$\operatorname{pr}(b_{2i-1})=0$ for $i>1$. Then from Equation (3) we have

$\mathrm{pr}\left(H^{*}(\operatorname{GL}_{n}(\mathbb{F}_{q}))\right)=\mathbb{Z}/2\mathbb{Z}[b_{1}].$
(11)

Now if $s^{I}$ appears as a term in $w(\pi)$, where $|I|\geq 2$, then it
survives in $\operatorname{pr}(w(\pi))$. But this contradicts Equation (11).

∎

###### Proof of Main Theorem 1.

Let $\widehat{P}$ be as defined in Section 2. Let $\pi$ be a real
representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$. Recall that for a
subgroup $H$ of $\operatorname{GL}_{n}(\mathbb{F}_{q})$ we write
$w_{i}^{H}(\pi)$ and $w^{H}(\pi)$ to denote $w_{i}(\pi\mid_{H})$ and
$w(\pi\mid_{H})$ respectively.

Since $\pi\mid_{D}$ is an $S_{n}$ invariant representation of $D$ we write

$\pi\mid_{D}=\mathbbm{1}^{\oplus m_{0}}\bigoplus_{i=1}^{n}\sigma_{i}^{\oplus
m_{i}}\bigoplus_{\tau\in\widehat{P}}((\chi_{\tau})_{\mathbb{R}})^{\oplus
m_{\tau}}.$ (12)

Therefore we may write

$\displaystyle w^{D}(\pi)=$ $\displaystyle w(\mathbbm{1}^{\oplus
m_{0}}\bigoplus_{i=1}^{n}\sigma_{i}^{\oplus m_{i}})\cdot
w(\bigoplus_{\tau\in\widehat{P}}((\chi_{\tau})_{\mathbb{R}})^{\oplus
m_{\tau}})$ $\displaystyle=$
$\displaystyle(1+Q(s_{1},\ldots,s_{n}))(1+P(t_{1},\ldots,t_{n}))\left(\text{
where }P,Q\text{ do not have constant term}\right),$ $\displaystyle=$
$\displaystyle(1+\delta b_{1})(1+P(t_{1},\ldots t_{n}))\left(\text{ follows
from Proposition}\ref{polysi}\right),$

This calculation shows that $w_{1}^{D}(\pi)=\delta b_{1}$. From this we
conclude that

$\delta=\begin{cases}0,\text{ if }\det\pi=1,\\\ 1,\text{ otherwise.
}\end{cases}$

Consider the representation $\psi=(\chi^{1})_{\mathbb{R}}$ of $C_{q-1}$. Let
$C_{2}<C_{q-1}$ be the subgroup of order $2$. Then the restriction map
$res:H^{*}(C_{q-1})\to H^{*}(C_{2})$ maps $t=w_{2}(\psi)$ to
$w_{2}^{C_{2}}(\psi)=w_{2}(\operatorname{sgn}\oplus\operatorname{sgn})=v^{2}$.
Thus the restriction map $res_{n}:H^{*}(C_{q-1}^{n})\to H^{*}(C_{2}^{n})$ maps
$t_{i}$ to $v_{i}^{2}$ and $s_{i}$ to $0$.

$\displaystyle w^{C_{2}^{n}}(\pi)$
$\displaystyle=1+P(v_{1}^{2},\ldots,v_{n}^{2})$ (13)
$\displaystyle=(1+P(v_{1},\ldots,v_{n}))^{2}.$ (14)

On the other hand $\pi\mid_{C_{2}^{n}}$ is an $S_{n}$ invariant
representation, therefore by Theorem 7 we have

$w^{C_{2}^{n}}(\pi)=\prod_{i=1}^{n}(\prod_{l\in\mathfrak{X}_{i}}(1+\sum_{j}l_{j}v_{j}))^{c_{i}},$

where
$c_{i}=(\dfrac{1}{2^{n}})\sum\limits_{k=0}^{n}\chi_{\sigma_{k}}(h_{i})\chi_{\pi}(h_{k}).$
Now since $H^{*}(C_{2}^{n})=\mathbb{Z}/2\mathbb{Z}[v_{1},\ldots,v_{n}]$ is a
Unique Factorization domain, we have
$1+P(v_{1},\ldots,v_{n})=\prod_{i=1}^{n}(\prod_{l\in\mathfrak{X}_{i}}(1+\sum_{j}l_{j}v_{j}))^{c_{i}/2}.$

Thus we obtain

$w^{D}(\pi)=(1+\delta
b_{1})\prod_{i=1}^{n}(\prod_{l\in\mathfrak{X}_{i}}(1+l\cdot t))^{c_{i}/2},$

where $l\cdot t=\sum_{j}l_{j}t_{j}$.

In the case $q\equiv 3\pmod{4}$ we have

$\mathrm{res^{D}_{C_{2}^{n}}}:H^{*}(D)\to H^{*}(C_{2}^{n})$

is an isomorphism as $C_{2}^{n}$ is the $2$ Sylow subgroup of $D$. (see [GJ22,
Propositions $5$ and $6$]). Now we apply Theorem 7 to obtain

$w^{C_{2}^{n}}(\pi)=\prod_{i=1}^{n}(\prod_{l\in\mathfrak{X}_{i}}(1+\sum_{j}l_{j}v_{j}))^{c_{i}},$

where again
$c_{i}=(\dfrac{1}{2^{n}})\sum_{k}\chi_{\sigma_{k}}(h_{i})\chi_{\pi}(h_{k}).$
This completes the proof of the main theorem. ∎

##  5\. Detection Results

We write

$E[x_{1},\ldots,x_{n}]=\dfrac{\mathbb{Z}/2\mathbb{Z}[x_{1},\ldots,x_{n}]}{(x_{i}^{2})}.$

Let $D_{k}$ denote the diagonal torus of
$\operatorname{GL}_{k}(\mathbb{F}_{q})$. Take $m\geq n$. We have

$H^{*}(D_{m})=\dfrac{\mathbb{Z}/2\mathbb{Z}[t_{1},\ldots,t_{m},s_{1},\ldots,s_{m}]}{(s_{i}^{2})},$

and

$H^{*}(D_{n-1})=\dfrac{\mathbb{Z}/2\mathbb{Z}[t_{1}^{\prime},\ldots,t_{n-1}^{\prime},s_{1}^{\prime},\ldots,s_{n-1}^{\prime}]}{(s_{i}^{2})}.$

Consider the inclusion map
$j_{n}:\operatorname{GL}_{n-1}(\mathbb{F}_{q})\to\operatorname{GL}_{m}(\mathbb{F}_{q})$
given by

$j_{n}(A)=\begin{pmatrix}A&0\\\ 0&I_{m-n+1}\end{pmatrix},$

where $A\in\operatorname{GL}_{n-1}(\mathbb{F}_{q})$. It will induce a map on
the diagonal subgroups namely $j_{n}:D_{n-1}\to D_{m}$ (we use the same
notation for the induced map). We have

$j_{n}^{*}(t_{l})=\begin{cases}t_{l}^{\prime},&\text{if $l<n$}\\\ 0,&\text{if
$l\geq n$}\end{cases}$

Similarly

$j_{n}^{*}(s_{l})=\begin{cases}s_{l}^{\prime},&\text{if $l<n$}\\\ 0,&\text{if
$l\geq n$}\end{cases}$

It will induce a map

$j_{n}^{*}:\mathbb{Z}/2\mathbb{Z}[a_{2},\ldots,a_{2m}]\otimes
E[b_{1},b_{3},\ldots,b_{2m-1}]\to\mathbb{Z}/2\mathbb{Z}[a_{2}^{\prime},\ldots,a_{2n-2}^{\prime}]\otimes
E[b_{1}^{\prime},\ldots,b_{2n-3}^{\prime}],$

where

$j_{n}^{*}(a_{2l})=a_{2l}^{\prime}=\sum\limits_{1\leq
i_{1}<i_{2}<\ldots<i_{l}\leq n-1}t_{i_{1}}^{\prime}\cdots t_{i_{l}}^{\prime},$

for $l<n$ and

$j_{n}^{*}(b_{2k-1})=b_{2k-1}^{\prime}=\sum\limits_{1\leq j\leq
k}\sum\limits_{1\leq i_{1}<i_{2}<\ldots<i_{k}\leq
n-1}t_{i_{1}}^{\prime}\cdots\hat{t^{\prime}}_{i_{j}}\cdots
t_{i_{k}}^{\prime}s_{i_{j}}^{\prime},$

for $k<n$. Therefore $j_{n}^{*}(a_{2l})=0$ when $l\geq n$ and
$j_{n}^{*}(b_{2k-1})=0$ for $k\geq n$.

###### Proof of Theorem 3.

Put $R_{m}=\mathbb{Z}/2\mathbb{Z}[a_{2},\ldots,a_{2m}]$,
$E_{m}=E[b_{1},b_{3},\ldots,b_{2m-1}],R^{\prime}_{n-1}=\mathbb{Z}/2\mathbb{Z}[a_{2}^{\prime},\ldots,a^{\prime}_{2n-2}]$
and
$E_{n-1}^{\prime}=E[b_{1}^{\prime},b_{3}^{\prime},\ldots,b^{\prime}_{2n-3}]$.
Since the elements $a_{2},\ldots,a_{2n-2}$ are algebraically independent, we
have

$\ker j_{n}^{*}\mid_{R_{m}}=\langle a_{2n},a_{2n+2},\ldots,a_{2m}\rangle.$

Take $R_{m}=\bigoplus_{i}A_{i}$ and
$R_{n-1}^{\prime}=\bigoplus_{i}A_{i}^{\prime}$, where $A_{i}$ and
$A_{i}^{\prime}$ are the $i$-th degree components. Then comparing degrees we
have $A_{i}\cap\ker j_{n}^{*}=0$, for $i<2n$. So the map $j_{n}^{*}:A_{i}\to
A_{i}^{\prime}$ is injective for $i<2n-1$.

Note that $E[b_{1},b_{3},\ldots,b_{2m-1}]$ is a $\mathbb{Z}/2\mathbb{Z}$
vector space with basis elements

$b_{2i_{1}-1}\cdots b_{2i_{k}-1},$

where $1\leq i_{1}<i_{2}<\cdots<i_{k}\leq m$ and $1\leq k\leq m$. In each
basis element a particular $b_{j}$ can have exponent $1$ or $0$. Therefore
$\dim E[b_{1},b_{3},\ldots,b_{2m-1}]=2^{m}$. Similarly $\dim
E[b_{1}^{\prime},b_{3}^{\prime},\ldots,b_{2n-3}^{\prime}]=2^{n-1}$. Since
$j_{n}^{*}$ is surjective we have $\dim\ker
j_{n}^{*}\mid_{E_{m}}=2^{m}-2^{n-1}$.

Consider the ideal $\mathfrak{I}=\langle
b_{2n-1},b_{2n+1},\ldots,b_{2m-1}\rangle$ of $E_{m}$. As a
$\mathbb{Z}/2\mathbb{Z}$ vector space $\dim\mathfrak{I}=2^{m}-2^{n-1}$. Also
for $x\in\mathfrak{I}$ one has $j_{n}^{*}(x)=0$. Therefore we obtain

$\ker j_{n}^{*}\mid_{E_{m}}=\langle b_{2n-1},b_{2n+1},\ldots,b_{2m-1}\rangle.$

Let $E_{m}=\bigoplus_{i}B_{i}$ and
$E_{n-1}^{\prime}=\bigoplus_{j}B_{j}^{\prime}$. Comparing the degrees we get
$\ker j_{n}^{*}\cap B_{i}=0$ for $i<2n-1$. So the map $j_{n}^{*}:B_{i}\to
B_{i}^{\prime}$ is injective for $i<2n-1$.

The $i$th graded part of $R_{m}\otimes E_{m}$ is of the form
$\bigoplus\limits_{k+l=i}A_{k}\otimes B_{l}$. Similarly, the $i$th graded part
of $R_{(n-1)}^{\prime}\otimes E_{(n-1)}^{\prime}$ is of the form
$\bigoplus\limits_{k+l=i}A_{k}^{\prime}\otimes B_{l}^{\prime}$. To show the
injectivity of the map $j_{n}^{*}$ it is enough to show that the map
$j_{n}^{*}:A_{k}\otimes B_{i-k}\to A_{k}^{\prime}\otimes B_{i-k}^{\prime}$ is
injective for every $0\leq k\leq i<2n-1$. It follows from the fact that the
maps $j_{n}^{*}:A_{k}\to A_{k}^{\prime}$ and $j_{n}^{*}:B_{k}\to
B_{k}^{\prime}$ are injective for $0\leq i<2n-1$. ∎

######  Theorem 8.

The map $\operatorname{res}:H^{i}(\operatorname{GL}_{m}(\mathbb{F}_{q}))\to
H^{i}(\operatorname{GL}_{n-1}(\mathbb{F}_{q}))$, where $m\geq n$ and $q\equiv
3\pmod{4}$ is injective for $i<n$.

###### Proof.

From Theorem 5 we obtain that the map
$\operatorname{res}_{k}:H^{i}(\operatorname{GL}_{k}(\mathbb{F}_{q}))\to
H^{i}(D_{k})^{S_{k}}$ is injective. Let
$\alpha:H^{i}(\operatorname{GL}_{m}(\mathbb{F}_{q}))\to
H^{i}(\operatorname{GL}_{n-1}(\mathbb{F}_{q}))$ denote the restriction map.

We have
$H^{*}(D_{k})^{S_{k}}=\mathbb{Z}/2\mathbb{Z}[v_{1},\ldots,v_{k}]^{S_{k}}$.
Consider the map

$\beta_{n}^{*}:\mathbb{Z}/2\mathbb{Z}[v_{1},\ldots,v_{m}]^{S_{m}}\to\mathbb{Z}/2\mathbb{Z}[v_{1}^{\prime},\ldots,v_{n-1}^{\prime}]^{S_{n-1}},$

given by

$\beta_{n}^{*}(v_{i})=\begin{cases}v_{i}^{\prime},&\text{if $i<n$},\\\
0,&\text{if $i\geq n$}.\end{cases}$

Note that $\beta_{n}^{*}$ induces a map

$\beta_{ni}^{*}:H^{i}(D_{m})^{S_{m}}\to H^{i}(D_{n-1})^{S_{n-1}}.$

Consider the following commutative diagram.

$H^{i}(\operatorname{GL}_{m}(\mathbb{F}_{q}))$$H^{i}(\operatorname{GL}_{n-1}(\mathbb{F}_{q}))$$H^{i}(D_{m})^{S_{m}}$$H^{i}(D_{n-1})^{S_{n-1}}$$\alpha$$\operatorname{res}_{m}$$\beta_{ni}^{*}$$\operatorname{res}_{n-1}$

Since the maps $\operatorname{res}_{m}$ and $\operatorname{res}_{n-1}$ are
injective, it is enough to prove that the map $\beta_{ni}^{*}$ is injective.
Let $f_{i}^{k}$ denote the elementary symmetric polynomial of degree $i$ in
$k$ variables $v_{1},\ldots,v_{k}$, where $k\geq i$. Note that
$\beta_{ni}^{*}(f_{i}^{k})\neq 0$ for $1\leq i\leq n-1\leq k$. We know that
the set of elementary symmetric functions form a basis for the algebra
$\mathbb{Z}/2\mathbb{Z}[v_{1},\ldots,v_{m}]^{S_{m}}$. An element in
$\mathbb{Z}/2\mathbb{Z}[v_{1},\ldots,v_{m}]^{S_{m}}$ of degree $i\leq n-1$ is
of the form $P(f_{1}^{m},\ldots,f_{i}^{m})$, where $P$ is a polynomial (see
[Lan02, Theorem $6.1$, page $191$]). We have
$\beta_{ni}^{*}(P(f_{1}^{m},\ldots,f_{i}^{m}))=P(f_{1}^{n-1},\ldots,f_{i}^{n-1})$.
Therefore $\beta_{n}^{*}(P(f_{1}^{m},\ldots,f_{i}^{m}))\neq 0$ as the elements
$f_{1}^{n-1},\ldots,f_{i}^{n-1}$ are algebraically independent. So
$\beta_{ni}^{*}$ is injective for $i\leq n-1$. ∎

##  6\. Calculations of Some SWC

Let
$i_{k}:\operatorname{GL}_{k}(\mathbb{F}_{q})\hookrightarrow\operatorname{GL}_{n}(\mathbb{F}_{q})$,
where $k<n$, denote the injection of $\operatorname{GL}_{k}(\mathbb{F}_{q})$
inside $\operatorname{GL}_{n}(\mathbb{F}_{q})$ at the top left corner with
rest diagonal entries $1$. When $q\equiv 1\pmod{4}$ we have
$i_{1}^{*}(a_{2})=t_{1}$ and when $q\equiv 3\pmod{4}$ we have
$i_{2}^{*}(\sum_{i=1}^{n}v_{i}^{2})=v_{1}^{2}+v_{2}^{2}$.

For a real representation $\pi$ of $\operatorname{GL}_{1}(\mathbb{F}_{q})$,
Theorem 1 gives

$w(\pi)=\begin{cases}(1+\delta s_{1})(1+t_{1})^{c_{1}/2},&\text{ if }q\equiv
1\pmod{4},\\\ (1+v_{1})^{c_{1}},&\text{ if }q\equiv 3\pmod{4},\end{cases}$

For a real representation $\pi$ of $\operatorname{GL}_{2}(\mathbb{F}_{q})$,
Theorem 1 says

$w(\pi)=\begin{cases}(1+\delta(s_{1}+s_{2}))((1+t_{1})(1+t_{2}))^{c_{1}^{\prime}/2}(1+t_{1}+t_{2})^{c_{2}^{\prime}/2},&\text{
if }q\equiv 1\pmod{4},\\\
(1+v_{1})^{c_{1}^{\prime}}(1+v_{2})^{c_{1}^{\prime}}(1+v_{1}+v_{2})^{c_{2}^{\prime}},&\text{
if }q\equiv 3\pmod{4}.\end{cases}$

For the case of $\operatorname{GL}_{1}(\mathbb{F}_{q})$ we have

$\begin{pmatrix}c_{0}\\\ c_{1}\\\ \end{pmatrix}=(1/2)\begin{pmatrix}1&1\\\
1&-1\\\ \end{pmatrix}\cdot\begin{pmatrix}\chi_{\pi}(h_{0})\\\
\chi_{\pi}(h_{1})\\\ \end{pmatrix},$

while for the case of $\operatorname{GL}_{2}(\mathbb{F}_{q})$ we have

$\begin{pmatrix}c_{0}^{\prime}\\\ c_{1}^{\prime}\\\ c_{2}^{\prime}\\\
\end{pmatrix}=(1/4)\begin{pmatrix}1&2&1\\\ 1&0&-1\\\ 1&-2&1\\\
\end{pmatrix}\cdot\begin{pmatrix}\chi_{\pi}(h_{0})\\\ \chi_{\pi}(h_{1})\\\
\chi_{\pi}(h_{2})\\\ \end{pmatrix}.$

For $q\equiv 3\pmod{4}$ we compute $w_{1}(\pi)$ in terms of character values.

######  Theorem 9.

Let $\pi$ be a real representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$,
where $q\equiv 3\pmod{4}$. Then we have

$w_{1}(\pi)=m_{\pi}\left(\sum_{i=1}^{n}v_{i}\right).$

###### Proof.

Since $i=1$ in Theorem 8, we have $n=2$. So for the representation $\pi$ we
have

$w_{1}(\pi)=(c_{1}^{\prime}+c_{2}^{\prime})\cdot(v_{1}+v_{2}).$

Since $c_{1}^{\prime}+c_{2}^{\prime}=m_{\pi}$, we have the result.

∎

Next we compute the second SWC.

######  Theorem 10.

Let $\pi$ be a real representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$,
where $q$ is odd. Then we have

$w_{2}(\pi)=\begin{cases}\dfrac{m_{\pi}}{2}a_{2},&\text{if}\,\,q\equiv
1\pmod{4}\\\ \\\
\dbinom{m_{\pi}}{2}\left(\sum\limits_{i=1}^{n}v_{i}^{2}\right),&\text{if}\,\,q\equiv
3\pmod{4}.\end{cases}$

###### Proof.

For a real representation $\pi$ of $\operatorname{GL}_{1}(\mathbb{F}_{q})$,
where $q\equiv 1\pmod{4}$, we have

$w_{2}(\pi)=\frac{c_{1}}{2}t_{1}.$

We have $c_{1}=(1/2)(\chi_{\pi}(h_{0})-\chi_{\pi}(h_{1}))=m_{\pi}.$ Thus by
Theorem 3 we have

$w_{2}(\pi)=\frac{m_{\pi}}{2}a_{2}.$

For a real representation $\pi$ of $\operatorname{GL}_{2}(\mathbb{F}_{q})$ and
$q\equiv 3\pmod{4}$ we have

$w_{2}(\pi)=\binom{c_{1}^{\prime}+c_{2}^{\prime}}{2}(v_{1}^{2}+v_{2}^{2})+c_{1}^{\prime
2}v_{1}v_{2}.$

We know from [GJ22, Lemma $4.4$]) that $c_{1}^{\prime}$ is even. Moreover we
have
$c_{1}^{\prime}+c_{2}^{\prime}=(1/2)(\chi_{\pi}(h_{0})-\chi_{\pi}(h_{1}))=m_{\pi}$.
Hence the Theorem follows from Theorem 8. ∎

Using the detection result we compute the fourth SWC in terms of character
values when $q\equiv 1\pmod{4}$.

######  Theorem 11.

Let $\pi$ be a real representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$,
where $q\equiv 1\pmod{4}$. Then

$w_{4}(\pi)=\binom{m_{\pi}/2}{2}\sum_{i=1}^{n}t_{i}^{2}+\frac{\dim\pi-\chi_{\pi}(h_{2})}{8}\sum_{1\leq
i<j\leq n}t_{i}t_{j}.$

###### Proof.

From Theorem 3 we know that the map
$\operatorname{res}:H^{4}(\operatorname{GL}_{n}(\mathbb{F}_{q}))\to
H^{4}(\operatorname{GL}_{2}(\mathbb{F}_{q}))$ is injective. We have

$w^{\operatorname{GL}_{2}(\mathbb{F}_{q})}(\pi)=(1+\delta(s_{1}+s_{2}))\left((1+t_{1})(1+t_{2})\right)^{c_{1}^{\prime}/2}(1+t_{1}+t_{2})^{c_{2}^{\prime}/2}.$

Then we calculate

$\displaystyle\operatorname{res}(w_{4}(\pi))$
$\displaystyle=\left(\binom{c_{1}^{\prime}/2}{2}+\binom{c_{2}^{\prime}/2}{2}+\frac{c_{1}^{\prime}c_{2}^{\prime}}{4}\right)(t_{1}^{2}+t_{2}^{2})+\frac{c_{1}^{\prime}}{2}t_{1}t_{2}$
$\displaystyle=\binom{\frac{c_{1}^{\prime}}{2}+\frac{c_{2}^{\prime}}{2}}{2}+\frac{c_{1}^{\prime}}{2}t_{1}t_{2}$
$\displaystyle=\binom{m_{\pi}/2}{2}(t_{1}^{2}+t_{2}^{2})+\frac{\dim\pi-\chi_{\pi}(h_{2})}{8}t_{1}t_{2}.$

So we have

$w_{4}(\pi)=\binom{m_{\pi}/2}{2}\sum_{i=1}^{n}t_{i}^{2}+\frac{\dim\pi-\chi_{\pi}(h_{2})}{8}\sum_{1\leq
i<j\leq n}t_{i}t_{j}.$

∎

##  7\. Some Applications

###  7.1. Lifting Criteria for Real Representations of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$

A real representation $\pi$ of a finite group $G$ is called spinorial provided
it lifts to the $\mathrm{Pin}$ group, which is a double cover of the
orthogonal group. One knows that $\pi$ is spinorial iff
$w_{2}(\pi)+w_{1}^{2}(\pi)=0$ (see [GKT89, page $328$]).

The main result in [JS21, Theorem $1$] provides a criterion for spinoriality
of a real representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$ in terms
of character values except for some particular pairs $(n,q)$. We recover the
result here for $q$ odd.

######  Theorem 12.

Let $\pi$ be a real representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$,
where $q$ is odd. Then

  1. (1)

for $q\equiv 1\pmod{4}$, we have $\pi$ is spinorial iff $m_{\pi}\equiv
0\pmod{4}$,

  2. (2)

for $q\equiv 3\pmod{4}$, we have $\pi$ is spinorial iff $m_{\pi}\equiv 0$ or
$3\pmod{4}$.

###### Proof.

We know that $\pi$ is spinorial iff $w_{2}(\pi)+w_{1}^{2}(\pi)=0$. From
Theorems 3 and 8 we deduce that $\pi$ is spinorial iff
$res(w_{2}(\pi)+w_{1}^{2}(\pi))=0$, where
$\operatorname{res}:H^{*}(\operatorname{GL}_{n}(\mathbb{F}_{q}),\mathbb{Z}/2\mathbb{Z})\to
H^{*}(\operatorname{GL}_{k}(\mathbb{F}_{q}),\mathbb{Z}/2\mathbb{Z})$ denotes
the restriction map, where $k=1$ for $q\equiv 1\pmod{4}$ and $k=2$ for
$q\equiv 3\pmod{4}$.

From Theorems 9 and 10, it follows that for a real representation $\pi$ of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$, we have

$\operatorname{res}(w_{2}(\pi)+w_{1}^{2}(\pi))=\begin{cases}\dfrac{m_{\pi}}{2}t_{1},&\text{
if }q\equiv 1\pmod{4},\\\
\dfrac{m_{\pi}(m_{\pi}+1)}{2}(v_{1}^{2}+v_{2}^{2}),&\text{ if }q\equiv
3\pmod{4}.\end{cases}$

∎

###  7.2. Non-Detection by An-isotropic Torus

Let $a\in\mathbb{F}_{q^{n}}^{\times}$ and
$G=\operatorname{GL}_{n}(\mathbb{F}_{q})$, where $q$ is odd. We fix a basis of
$\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$. Then the map $x\to a\cdot x$ gives
a linear map $M_{a}\in G$. Then
$M=\\{M_{a},a\in\mathbb{F}_{q^{n}}^{\times}\\}$ is a cyclic group of order
$q^{n}-1$ and is called an anisotropic torus.

######  Theorem 13.

The anisotropic torus $M$ does not detect the mod-2 cohomology of $G$.

###### Proof.

Consider the representation $\pi=(\chi)_{\mathbb{R}}$, where
$\chi=\chi^{j}\circ\det$ and $j$ is odd. One has

$\displaystyle\chi_{\pi}(h_{1})$ $\displaystyle=2\chi_{j}(\det(h_{1}))$
$\displaystyle=2\chi_{j}(-1)$ $\displaystyle=2(-1)^{j}.$

Since $j$ is odd we have $\chi_{\pi}(h_{1})=-2$. This gives

$m_{\pi}=\frac{1}{2}(2+2)=2.$

Then from Theorem 10 one obtains $w_{2}(\pi)\neq 0.$

Let $\hat{g}$ be a generator of $M$. Now we have

$w_{2}(\pi\mid_{M})=\kappa(c_{1}(\chi^{j}\circ\det\mid_{M})).$

We have

$\chi(\hat{g})=\chi^{j}\circ\det(\hat{g})=\zeta_{q-1}^{j}=\zeta_{q^{2}-1}^{(q+1)j}.$

Take $u=c_{1}(\dot{\chi})\in H^{2}(M,\mathbb{Z})$ where
$\dot{\chi}\in\mathrm{Hom}(M,\mathbb{C}^{\times})$ such that
$\dot{\chi}(\hat{g})=\zeta_{q^{2}-1}$. As a result we get

$\displaystyle\kappa(c_{1}(\chi^{j}\circ\det\mid_{M}))$
$\displaystyle=\kappa((q+1)ju)$ $\displaystyle=(q+1)j\kappa(u)$
$\displaystyle=0.$

This shows that $w_{2}(\pi\mid_{M})=\operatorname{res}^{G}_{M}(w_{2}(\pi))=0$
whereas $w_{2}(\pi)\neq 0$. ∎

##  8\. The Case of $\operatorname{GL}_{n}(\mathbb{C})$ and
$\operatorname{GL}_{n}(\mathbb{R})$

For a Lie group $G$ we write $H^{i}(G)$ to denote the (singular) cohomology
$H^{i}(BG)$ of a classifying space $BG$ of $G$. Consider the subgroup

$\Gamma=\operatorname{diag}(\pm 1,\pm 1,\ldots,\pm 1)\cong C_{2}^{n}$

of $\operatorname{GL}_{n}(k)$, where $k=\mathbb{R}\text{ or }\mathbb{C}$. Take
$T=(S^{1})^{n}$ to be the diagonal torus of the unitary group $U_{n}$. Now we
establish some useful detection results.

######  Proposition 5.

The mod $2$ cohomology of $\operatorname{GL}_{n}(\mathbb{C})$ is detected by
the group $\Gamma$.

###### Proof.

From [T+87, Theorem $2.1$] we obtain that the maps
$i_{1}^{*}:H^{*}(\operatorname{GL}_{n}(\mathbb{C}))\to H^{*}(U_{n})$ and
$i_{2}^{*}:H^{*}(U_{n})\to H^{*}(T)$ are injective. As a result we conclude
that the mod $2$ cohomology of $\operatorname{GL}_{n}(\mathbb{C})$ is detected
by the diagonal torus $T$. First we prove that the mod $2$ cohomology of
$S^{1}$ is detected by $C_{2}$. Note that the circle group $S^{1}$ is
isomorphic to $\mathrm{SO}_{2}(\mathbb{R})$. We know that
$H^{*}(\mathrm{SO}_{2}(\mathbb{R}))=\mathbb{Z}/2\mathbb{Z}[w_{2}]$, where
$w_{2}$ denotes the second Stiefel Whitney class of standard representation of
$\mathrm{SO}_{2}(\mathbb{R})$. The standard representation of
$\mathrm{SO}_{2}(\mathbb{R})$ can be thought of as the representation
$\pi=\rho\oplus\rho^{\vee}$ of $S^{1}$ where $\rho(z)=z$ for $z\in S^{1}$. So
$w_{2}=w_{2}(\pi)$. One can consider $C_{2}=\\{\pm 1\\}$ as a subgroup of
$S^{1}$ to obtain a map
$\operatorname{res}:H^{*}(\mathrm{SO}_{2}(\mathbb{R}))\to
H^{*}(C_{2},\mathbb{Z}/2\mathbb{Z})$. We have
$H^{*}(C_{2},\mathbb{Z}/2\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}[v]$, where $\deg
v=1$. Note that $\pi\mid_{C_{2}}=\operatorname{sgn}\oplus\operatorname{sgn}$.
Thus we have

$\operatorname{res}w_{2}(\pi)=w_{2}(\operatorname{sgn}\oplus\operatorname{sgn})=w_{1}^{2}(\operatorname{sgn})\neq
0.$

So $\operatorname{res}(w_{2})=v^{2}$. This implies that the map
‘$\operatorname{res}$’ is injective. Now the result follows from Kunneth
formula.

∎

######  Proposition 6.

The mod $2$ cohomology of $GL_{n}(\mathbb{R})$ is detected by $\Gamma$.

###### Proof.

From [Mit01, Theorem $7.7$] we know that $B\operatorname{GL}_{n}(\mathbb{R})$
is homotopy equivalent to $BO_{n}(\mathbb{R})$. Thus the restriction map
$\operatorname{res}_{O}:H^{*}(\operatorname{GL}_{n}(\mathbb{R}))\to
H^{*}(O_{n}(\mathbb{R}))$ is an isomorphism. From [T+87, Theorem 2.2] we
obtain that $\operatorname{res}_{\Gamma}:H^{*}(O_{n}(\mathbb{R}))\to
H^{*}(\Gamma)$ is an injection. Thus the $\mathbb{Z}/2\mathbb{Z}$ cohomology
of $\operatorname{GL}_{n}(\mathbb{R})$ is detected by the subgroup
$\Gamma\cong C_{2}^{n}$. ∎

###### Proof of Theorem 2.

Given a real representation $\pi$ of $\operatorname{GL}_{n}(\mathbb{R})(\text{
or }\operatorname{GL}_{n}(\mathbb{C}))$, we calculate the total Stiefel
Whitney class of $\psi=\pi\mid_{\Gamma}$. Moreover $\psi$ is a $S_{n}$
invariant representation as it is a restriction of representation of
$\operatorname{GL}_{n}(\mathbb{R})(\text{ or
}\operatorname{GL}_{n}(\mathbb{C}))$. Now the Theorem 2 follows from Theorem
7. ∎

######  Remark 1.

Let $\pi$ be a representation of $\mathrm{O}_{n}(\mathbb{R})$. Then the proof
of Proposition 6 gives that the total SWC of $\pi$ is same as the expression
in Equation (2).

##  9\. Examples

We have

$w_{2}(\pi)=\begin{cases}\dfrac{m_{\pi}}{2}a_{2},&\text{ if }q\equiv 1\mod
4,\\\ \\\ \dbinom{m_{\pi}}{2}(\sum v_{i}^{2}),&\text{ if }q\equiv 3\mod
4,\end{cases}$

where $m_{\pi}=(1/2)(\dim\pi-\chi_{\pi}(h_{1}))$. Also

$w_{4}(\pi)=\binom{m_{\pi}/2}{2}\sum_{i}t_{i}^{2}+\frac{n_{\pi}}{4}\sum_{i<j}t_{i}t_{j},$

where $n_{\pi}=\dfrac{\dim\pi-\chi_{\pi}(h_{2})}{2}$.

###  9.1. Principal Series Representation

Let $\chi_{1},\chi_{2},\ldots,\chi_{n}$ are linear characters of
$\mathbb{F}_{q}^{\times}$ and let $B$ be the Borel subgroup of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$ . Put

$\pi=\mathrm{Ind}_{B}^{G}(\chi_{1}\boxtimes\chi_{2}\boxtimes\cdots\boxtimes\chi_{n}).$

Such representations are called principal series representations of $G$. Such
a representation $\pi$ is orthogonal iff

$\\{\chi_{1},\ldots,\chi_{n}\\}=\\{\chi_{1}^{-1},\ldots,\chi_{n}^{-1}\\},$

as multisets. We introduce the notation

$\displaystyle[n]_{q}!$ $\displaystyle=\prod_{i=1}^{n}\dfrac{q^{i}-1}{q-1}$
$\displaystyle=(q+1)(q^{2}+q+1)\cdots(q^{n-1}+q^{n-2}+\cdots+q+1).$

From [Car85, Proposition $7.5.3$, page no. $233$] we obtain the following
result.

$\chi_{\pi}(h_{k})=[k]_{q}![n-k]_{q}!\sum_{1\leq i_{1}<i_{2}<\cdots<i_{k}\leq
n}\chi_{i_{1}}(-1)\chi_{i_{2}}(-1)\cdots\chi_{i_{k}}(-1).$ (15)

Put

$T_{i}=(1+q+q^{2}+\cdots+q^{i}).$

So we calculate

$\displaystyle m_{\pi}$ $\displaystyle=\frac{1}{2}(\dim\pi-\chi_{\pi}(h_{1}))$
$\displaystyle=\frac{1}{2}\left([n]_{q}!-[n-1]_{q}!\left(\sum_{i=1}^{n}\chi_{i}(-1)\right)\right)$
$\displaystyle=\frac{1}{2}[n-1]_{q}!\left(T_{n-1}-\sum_{i=1}^{n}\chi_{i}(-1)\right).$

Similarly one has

$\displaystyle n_{\pi}$ $\displaystyle=\frac{1}{2}(\dim\pi-\chi_{\pi}(h_{2}))$
$\displaystyle=\frac{1}{2}[n-2]_{q}!\left(T_{n-2}T_{n-1}-(1+q)\sum_{i<j}\chi_{i}(-1)\chi_{j}(-1)\right).$

######  Theorem 14.

Let $\pi$ be a real principal series representation for
$\operatorname{GL}_{n}(\mathbb{F}_{q})$ where $q$ is odd. Then $w_{2}(\pi)=0$
for $n\geq 3$.

###### Proof.

From [JS21, Thorem $5$] we know that if $n\geq 3$, then all the real principal
series representations of $\operatorname{GL}_{n}(\mathbb{F}_{q})$ are
spinorial. Now $\pi$ is spinorial iff $w_{2}(\pi)=w_{1}^{2}(\pi)$ (see [GS20,
Proposition $6.1$]). If $q\equiv 1\pmod{4}$,then $w_{1}^{2}(\pi)=0$. Therefore
$w_{2}(\pi)=0$.

Now consider the case for $q\equiv 3\pmod{4}$. We have

$m_{\pi}=\frac{1}{2}[n-1]_{q}!\left(T_{n-1}-\sum_{i=1}^{n}\chi_{i}(-1)\right).$

Also

$T_{n-1}\equiv\sum_{i=1}^{n}\chi_{i}(-1)\equiv n\pmod{2}.$

This gives $T_{n-1}-\sum\limits_{i=1}^{n}\chi_{i}(-1)\equiv 0\pmod{2}$. We
have $[m]_{q}!\equiv 0\pmod{4}$ for $m\geq 2$. For $n\geq 3$, we have
$\upsilon_{2}(m_{\pi})\geq 2$. Therefore $\dbinom{m_{\pi}}{2}\equiv
0\pmod{2}$. Therefore $w_{2}(\pi)=0$ for $n\geq 3$.

∎

###### Proof of Theorem 4.

We have

$\displaystyle n_{\pi}$
$\displaystyle=\frac{1}{2}[n-2]_{q}!\left(T_{n-2}T_{n-1}-(1+q)\sum_{i<j}\chi_{i}(-1)\chi_{j}(-1)\right).$

Observe that $T_{n-2}T_{n-1}-(1+q)\sum_{i<j}\chi_{i}(-1)\chi_{j}(-1)$ is
always even. Also $\upsilon_{2}([n-2]_{q}!)\geq 3$ for $n\geq 6$. Therefore
$n_{\pi}/4$ remains even for $n\geq 6$.

We have

$m_{\pi}=\frac{1}{2}[n-1]_{q}!\left(T_{n-1}-\sum_{i=1}^{n}\chi_{i}(-1)\right).$

Also

$T_{n-1}\equiv\sum_{i=1}^{n}\chi_{i}(-1)\equiv n\pmod{2}.$

This gives $T_{n-1}-\sum\limits_{i=1}^{n}\chi_{i}(-1)\equiv 0\pmod{2}$. For
$n\geq 6$, the term $m_{\pi}/2$ remains even. So we have
$\dbinom{m_{\pi}/2}{2}\equiv\dfrac{m_{\pi}}{4}\pmod{2}$. Now
$\upsilon_{2}(m_{\pi})\geq 4$ for $n\geq 6$. Therefore $w_{4}(\pi)=0$ for
$n\geq 6$.

For $n=5$ the two possible real Principal series representations are
$\pi_{1}=\\{\mathbb{1},\chi_{i},\chi_{i}^{-1},\chi_{j},\chi_{j}^{-1}\\}$ and
$\pi_{2}=\\{\operatorname{sgn},\chi_{i},\chi_{i}^{-1},\chi_{j},\chi_{j}^{-1}\\}$.

We calculate

$\displaystyle m_{\pi_{1}}$
$\displaystyle=\frac{1}{2}\left([5]_{q}!-[4]_{q}!(1+2(-1)^{i}+2(-1)^{j})\right)$
$\displaystyle=\frac{1}{2}(q+1)^{2}(q^{2}+1)(q^{2}+q+1)\left(q(q+1)(q^{2}+1)-2(-1)^{i}-2(-1)^{j}\right).$

This shows $\upsilon_{2}(m_{\pi_{1}})\geq 4$. Similarly one obtains
$\upsilon_{2}(m_{\pi_{2}})\geq 4$.

We have

$\displaystyle n_{\pi_{1}}$
$\displaystyle=\frac{1}{2}\left([5]_{q}!-[2]_{q}![3]_{q}!(2+2(-1)^{i}+2(-1)^{j}+4(-1)^{i+j})\right)$
$\displaystyle=\frac{1}{2}(q+1)^{2}(q^{2}+q+1)\left((q^{2}+1)(q^{4}+q^{3}+q^{2}+q+1)-(2+2(-1)^{i}+2(-1)^{j}+4(-1)^{i+j})\right)$
$\displaystyle=\frac{1}{2}(q+1)^{2}(q^{2}+q+1)\left(\\{q^{6}+q^{5}+2(q^{4}+q^{3}+q^{2})\\}+\\{q-1\\}-\\{2(-1)^{i}+2(-1)^{j})+4(-1)^{i+j}\\}\right).$

Note that the all the terms in the curly brackets and $(q+1)^{2}$ are
divisible by $4$. Therefore $\upsilon_{2}(n_{\pi_{1}})\geq 3$. Similar
calculation shows that $\upsilon_{2}(n_{\pi_{2}})\geq 3$. Now the result
follows from Theorem 11. ∎

Here we list down the value of $w_{4}(\pi)$, where $\pi$ is an orthogonal
principal series representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$ for
$q\equiv 1\pmod{4}$.

  1. (1)

Take $n=3$ and $\pi=\\{\mathbb{1},\chi_{j},\chi_{j}^{-1}\\}$ or
$\\{\operatorname{sgn},\chi_{j},\chi_{j}^{-1}\\}$ Then

$w_{4}(\pi)=\sum_{i=1}^{3}t_{i}^{2}+\sum_{1\leq i<j\leq 3}t_{i}t_{j}$

if $q\equiv 1,9\pmod{16}$ and $j$ is odd or $q\equiv 5,13\pmod{16}$ and $j$ is
even. Otherwise $w_{4}(\pi)=0$.

  2. (2)

Take $n=4$ and
$\pi=\\{\mathbb{1},\operatorname{sgn},\chi_{i},\chi_{i}^{-1}\\}$. Then

$w_{4}(\pi)=\begin{cases}\sum\limits_{i=1}^{4}t_{i}^{2},&\text{if $i$ is
odd},\\\ 0,&\text{if $i$ is even}.\end{cases}$

  3. (3)

Take $n=4$ and $\pi=\\{\chi_{j},\chi_{j}^{-1},\chi_{k},\chi_{k}^{-1}\\}$.

$w_{4}(\pi)=\begin{cases}\sum\limits_{i=1}^{4}t_{i}^{2},&\text{if $j$ and $k$
have different parity},\\\ 0,&\text{otherwise}.\end{cases}$

###  9.2. Cuspidal Representations

Let $\theta$ be a regular character of full anisotropic torus $T$ of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$. Then there is a cuspidal
representation $\pi_{\theta}$ associated to it. The representation
$\pi_{\theta}$ is real iff there exists a Weyl group element $w$ such that
$w(\theta)=\theta^{-1}$.

Let $\pi$ be a real Cuspidal representation of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$.

####  9.2.1. Computation of $w_{2}(\pi)$

We know that $\chi_{\pi}(h_{1})=0$ and $\dim\pi=\psi_{n-1}(q)$ where

$\psi_{m}(q)=\prod_{i=1}^{m}(q^{i}-1).$

Thus $m_{\pi}=\psi_{n-1}(q)/2$.

######  Theorem 15.

Let $\pi$ be a real Cuspidal representation of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$ such that $n\geq 3$. Then
$w_{2}(\pi)=0$.

###### Proof.

For $n\geq 3$, the term $\dfrac{(q-1)(q^{2}-1)}{2}$ divides $m_{\pi}$, Thus
$m_{\pi}\equiv 0\mod 4$, and therefore $w_{2}(\pi)=0$. ∎

For $n=2$ and $q\equiv 1\mod{4}$, we have $m_{\pi}=(q-1)/2$ and
$w_{2}=\frac{q-1}{4}a_{2}$.

$w_{2}(\pi)=\begin{cases}0,&\text{ if }q\equiv 1\mod 8,\\\ a_{2},&\text{ if
}q\equiv 5\mod 8.\\\ \end{cases}$

For $n=2$ and $q\equiv 3\mod 4$, we have $m_{\pi}=(q-1)/2$ and
$w_{2}=\binom{m_{\pi}}{2}\cdot(v_{1}^{2}+v_{2}^{2})$. Therefore

$w_{2}(\pi)=\begin{cases}0,&\text{ if }q\equiv 3\mod 8,\\\
v_{1}^{2}+v_{2}^{2},&\text{ if }q\equiv 7\mod 8.\\\ \end{cases}$

####  9.2.2. Computation of $w_{4}(\pi)$

We summerize the computations in the following table.

Table 1. Computation of $w_{4}(\pi)$ where $\pi$ is the Cuspidal representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$ $q\mod 16$ |  $\theta(-1)$ |  $\binom{m_{\pi}/2}{2}\pmod{2}$ |  $n_{\pi}/4\pmod{2}$
---|---|---|---
$1$ |  1 |  0 |  0
$1$ |  $-1$ |  0 |  0
$5$ |  $1$ |  0 |  0
$5$ |  $-1$ |  0 |  1
$9$ |  $1$ |  1 |  0
$9$ |  $-1$ |  1 |  0
$13$ |  $1$ |  1 |  0
$13$ |  $-1$ |  1 |  1

###  9.3. Steinberg Representation

Let $\pi$ be the Steinberg representation of
$\operatorname{GL}_{n}(\mathbb{F}_{q})$.

####  9.3.1. Computation of $w_{2}(\pi)$

We have $\chi_{\pi}(h_{1})=p^{k}$ where $p^{k}$ is the highest power of $p$
dividing $|Z_{G}(h_{1})|$ (see [Car85, Theorem $6.4.7$, page $195$]). We have
$Z_{G}(h_{1})\cong\operatorname{GL}_{1}(\mathbb{F}_{q})\times\operatorname{GL}_{n-1}(\mathbb{F}_{q})$.
Thus $\chi_{\pi}(h_{1})=q^{(1/2)(n-1)(n-2)}$. Therefore

$\displaystyle m_{\pi}$
$\displaystyle=\dfrac{q^{(1/2)(n(n-1))}-q^{(1/2)(n-1)(n-2)}}{2}$
$\displaystyle=(1/2)\cdot q^{(1/2)(n-1)(n-2)}(q^{n-1}-1)$
$\displaystyle=\dfrac{q-1}{2}\cdot
q^{(1/2)(n-1)(n-2)}\cdot(1+q+q^{2}+\cdots+q^{n-2}).$

For $q\equiv 1\pmod{4}$ we have

$w_{2}(\pi)=\begin{cases}a_{2},&\text{ if }q\equiv 5\mod 8\text{ and for
}n\text{ even },\\\ 0,&\text{ otherwise }.\end{cases}$

For $q\equiv 3\pmod{4}$ we have

$w_{2}(\pi)=\begin{cases}\sum\limits_{i=1}^{n}v_{i}^{2},&\text{ if }q\equiv
3\mod 8\text{ and for }n\equiv 0\pmod{4},\\\
\sum\limits_{i=1}^{n}v_{i}^{2},&\text{ if }q\equiv 7\mod 8\text{ and for
}n\equiv 2\pmod{4},\\\ 0,&\text{ otherwise }.\end{cases}$

####  9.3.2. Computation of $w_{4}(\pi)$

We summerize the computations in the following table.

Table 2.  Computation of $w_{4}(\pi)$ where $\pi$ is the Steinberg representation of $\operatorname{GL}_{n}(\mathbb{F}_{q})$ $q\mod 8$ |  $n\mod 4$ |  $\binom{m_{\pi}/2}{2}\mod 2$ |  $n_{\pi}/4\mod 2$
---|---|---|---
$1$ |  $0,1,2,3$ |  0 |  0
$5$ |  $0$ |  1 |  0
$5$ |  $1$ |  0 |  1
$5$ |  $2$ |  0 |  0
$5$ |  $3$ |  1 |  1
$9$ |  $1,3$ |  0 |  0
$9$ |  $0,2$ |  1 |  0
$13$ |  $0$ |  0 |  0
$13$ |  $1$ |  0 |  1
$13$ |  $2$ |  1 |  0
$13$ |  $3$ |  1 |  1

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