# Stiefel-Whitney Classes of representations of $\operatorname{SL}(2,q)$

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.

We describe the Stiefel-Whitney classes (SWCs) of orthogonal representations
$\pi$ of the finite special linear groups
$G=\operatorname{SL}(2,\mathbb{F}_{q})$, in terms of character values of
$\pi$. From this calculation, we can answer interesting questions about SWCs
of $\pi$. For instance, we determine the subalgebra of
$H^{*}(G,\mathbb{Z}/2\mathbb{Z})$ generated by the SWCs of orthogonal $\pi$,
and we also determine which $\pi$ have nontrivial mod $2$ Euler class.

###### Key words and phrases:

Stiefel-Whitney classes, special linear group, quaternions

###### 1991 Mathematics Subject Classification:

Primary 20G40, Secondary 55R40

###### Contents

  1. 1 Introduction
  2. 2 Notations & Preliminaries
  3. 3 Quaternion Groups
  4. 4 $\operatorname{SL}(2,q)$ with $q$ odd
  5. 5 $\operatorname{SL}(2,q)$ with $q$ even

##  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 associates
SWCs $w_{i}(\pi)\in H^{i}(G,\mathbb{Z}/2\mathbb{Z})$ to an orthogonal
representation $\pi$ of $G$. Their sum $w(\pi)$, living in the group
cohomology $H^{*}(G,\mathbb{Z}/2\mathbb{Z})$, is known as the total Stiefel-
Whitney class of $\pi$.

There do not seem to be many explicit calculations of such SWCs, notably when
$G$ is nonabelian. (However see [8].) This paper contains formulas for SWCs in
terms of character values for $\operatorname{SL}(2,q)$, when $q$ is a prime
power, achieved via _detection results._ Formulas for SWCs of cyclic groups
are well-known; we review this for the group of order $2$ in Section 2.5. The
second SWCs of representations of $S_{n}$ and related groups were found in
[12].

The case of $\operatorname{GL}(n,q)$ with $q$ odd was completed recently by
Ganguly-Joshi in [10] and [11]. To explain how their work relates to ours, we
need to introduce “detection”.

Let $G$ be a finite group, and write $H^{*}(G,\mathbb{Z}/2\mathbb{Z})$ for the
mod $2$ group cohomology ring of $G$. Let
$H^{*}_{\operatorname{SW}}(G,\mathbb{Z}/2\mathbb{Z})$ be the subalgebra of
$H^{*}(G,\mathbb{Z}/2\mathbb{Z})$ generated by SWCs $w_{i}(\pi)$ of orthogonal
representations $\pi$ of $G$. We say a subgroup $K$ _detects_ the mod $2$
cohomology of $G$ when the restriction map

$i^{*}:H^{*}(G,\mathbb{Z}/2\mathbb{Z})\to H^{*}(K,\mathbb{Z}/2\mathbb{Z})$

is an injection, and we say $K$ detects SWCs of $G$, when this map is
injective on $H^{*}_{\operatorname{SW}}(G,\mathbb{Z}/2\mathbb{Z})$. Suppose
one of these is true, and that $\pi$ is an orthogonal representation of $G$.
Then $w(\pi)$ is specified by its image in $H^{*}(K,\mathbb{Z}/2\mathbb{Z})$,
which is actually the SWC of the restriction of $\pi$ to $K$. When there is a
detecting subgroup, one simply writes $w(\pi)$ for this image.

An essential tool in Ganguly-Joshi’s papers is a theorem of Quillen [20],
asserting that the diagonal subgroup of $\operatorname{GL}(n,q)$ detects its
mod $2$ cohomology. However, such a result is not available to us. Instead,
most of this paper is devoted to establishing the following:

######  Theorem 1.1.

Let $G=\operatorname{SL}(2,q)$ with $q$ odd. Then the center $Z$ detects SWCs
of $G$.

From this we obtain:

######  Theorem 1.2.

Let $G=\operatorname{SL}(2,q)$ with $q$ odd. Let $\pi$ be an orthogonal
representation of $G$. Then the total SWC of $\pi$ is

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

where $\mathfrak{e}$ is the non-zero element in
$H^{4}(\operatorname{SL}(2,q),\mathbb{Z}/2\mathbb{Z})$ and
$r_{\pi}=\frac{1}{8}(\chi_{\pi}(\mathbbm{1})-\chi_{\pi}(-\mathbbm{1})).$
$($Here $\mathbbm{1}$ is the identity matrix.$)$ If $\pi$ is also irreducible,
then $w(\pi)=1$.

In future work, these two theorems will be applied to determine SWCs of other
finite groups of Lie type.

The case of $q$ even is quite different. Here the upper unitriangular subgroup
$N\cong\mathbb{F}_{q}$ detects the mod $2$ cohomology of the special linear
group, and we use this to prove:

######  Theorem 1.3.

Let $(\pi,V)$ be an orthogonal representation of $G=\operatorname{SL}(2,q)$,
with $q$ even. Then the total SWC of $\pi$ is

$w(\pi)=(1+\mathcal{D})^{m_{\pi}},$

where $m_{\pi}=\dim_{\mathbb{C}}\left(V/V^{G}\right)$, where $V^{G}$ is the
$G$-fixed vectors in $V$, and $\mathcal{D}$ is the sum of the Dickson
invariants of $\mathbb{F}_{q}$, viewed as a $\mathbb{Z}/2\mathbb{Z}$-vector
space.

(Dickson invariants are reviewed in Section 5.)

There are several corollaries of Theorems 1.2 and 1.3. Let
$G=\operatorname{SL}(2,q)$. Firstly, we describe the subgroup of the complete
cohomology ring $H^{\bullet}(G,\mathbb{Z}/2\mathbb{Z})$ (see Section 2.6)
generated by $w(\pi)$, as $\pi$ varies over orthogonal representations.

######  Corollary 1.4.

This subgroup equals:

  1. (i)

$\\{(1+\mathfrak{e})^{n}:n\in\mathbb{Z}\\}$, for $q$ odd,

  2. (ii)

$\\{(1+\mathcal{D})^{n}:n\in\mathbb{Z}\\}$, for $q$ even.

Secondly, we compute $H^{*}_{\operatorname{SW}}(G,\mathbb{Z}/2\mathbb{Z})$.

######  Corollary 1.5.

The subalgebra $H^{*}_{\operatorname{SW}}(G,\mathbb{Z}/2\mathbb{Z})$ is:

  1. (i)

$\mathbb{Z}/2\mathbb{Z}[\mathfrak{e}]$, for $q$ odd,

  2. (ii)

generated by the Dickson invariants of $\mathbb{F}_{q}$, for $q$ even.

Another characteristic class of importance is the Euler class $e(\pi)\in
H^{d}(G,\mathbb{Z})$, where $d=\deg\pi$. We define
$w_{\operatorname{top}}(\pi)=w_{\deg\pi}(\pi)$; it is the “top” Stiefel-
Whitney class of $\pi$. By Property 9.5 of [19], we know that
$w_{\operatorname{top}}(\pi)$ is the reduction of $e(\pi)$ mod $2$.

######  Corollary 1.6.

Let $(\pi,V)$ be an orthogonal representation of $G$. Then
$w_{\operatorname{top}}(\pi)\neq 0$ precisely when:

  1. (i)

$\pi(-\mathbbm{1})=-1$, meaning $\pi(-\mathbbm{1})$ acts on $V$ by the scalar
$-1$, for $q$ odd,

  2. (ii)

$V^{N}=\\{0\\}$, for $q$ even. $($Equivalently, $\pi$ is cuspidal.$)$

Finally, we determine the _obstruction degree_ of an orthogonal $\pi$, meaning
the least $k>0$ with $w_{k}(\pi)\neq 0$. (If $w(\pi)$ is trivial, then this
degree is infinite.) Write $\operatorname{ord}_{2}(n)$ for the highest power
of $2$ which divides an integer $n$.

######  Corollary 1.7.

The obstruction degree of $\pi$ is:

  1. (i)

$2^{t+2}$, where $t=\operatorname{ord}_{2}(r_{\pi})$, for $q$ odd,

  2. (ii)

$2^{r+s-1}$, where $s=\operatorname{ord}_{2}(m_{\pi})$ and $q=2^{r}$.

The layout of the paper is as follows: Section 2 recalls some fundamental
notions used throughout the paper. Section 3 is dedicated to the generalized
quaternions, particularly the Quaternion group $Q$ of order 8. These groups
are featured in the detection of $\operatorname{SL}(2,q)$ when $q$ is odd.
Here we also determine $w_{4i}(\pi)$ for orthogonal representations $\pi$ of
$Q$. In Section 4, we prove Theorems 1.1 and 1.2, the corollaries for $q$ odd,
and finally compare our result to the $\operatorname{GL}(2,q)$ case. In
Section 5, we prove Theorem 1.3 and the corollaries for $q$ even.

Acknowledgements. The authors would like to thank Ryan Vinroot, Dipendra
Prasad, and Rohit Joshi for helpful discussions. This paper comes out of the
first author’s Ph.D. thesis [17] at IISER Pune, during which she was supported
by a Ph.D. fellowship from the Council of Scientific and Industrial Research,
India.

##  2\. Notations & Preliminaries

###  2.1. Representations

Let $G$ be a finite group. All our representations will be finite-dimensional.
Let $\pi$ be a representation of $G$ on a complex vector space $V$. Write
$\chi_{\pi}(g)$ for the character of $\pi$ at an element $g\in G$. If $H$ is a
subgroup of $G$, write $\operatorname{res}^{G}_{H}\pi$ for the restriction of
$\pi$ to $H$. Write $\operatorname{reg}(G)$ for the regular representation of
$G$.

We say $\pi$ is _orthogonal_ , provided there exists a non-degenerate
$G$-invariant symmetric bilinear form $B:V\times V\to\mathbb{C}$. We say it is
_symplectic_ , provided there exists a non-degenerate $G$-invariant
antisymmetric bilinear form $B:V\times V\to\mathbb{C}$.

When $\pi$ is self-dual and irreducible, it is either orthogonal or
symplectic. In this case, the Frobenius-Schur Indicator $\varepsilon(\pi)$ is
a sign defined as $1$ when $\pi$ is orthogonal, and $-1$ when $\pi$ is
symplectic.

###  2.2. Orthogonal Representations

We are especially interested in orthogonal representations.

From [4, Chapter II, Section 6], we note:

######  Proposition 2.1.

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.

Given a representation $(\pi,V)$ with dual $(\pi^{\vee},V^{\vee})$, we can
symmetrize it to get an orthogonal representation of $G$ by defining

$S(\pi):=\pi\oplus\pi^{\vee}$

on the vector space $V\oplus V^{\vee}$. We give a symmetric $G$-invariant
bilinear map $B$ on $V\oplus V^{\vee}$ as,

$B((v,\alpha),(w,\beta))=\langle\alpha,w\rangle+\langle\beta,v\rangle.$

We call $S(\pi)$ the symmetrization of $\pi.$

Every orthogonal complex representation $\Pi$ of $G$ can be decomposed as

(1) $\Pi\cong\bigoplus_{i}\pi_{i}\oplus\bigoplus_{j}S(\varphi_{j}),$

such that each $\pi_{i}$ is irreducible orthogonal and $\varphi_{j}$ are
irreducible non-orthogonal representations of $G$.

######  Definition 2.2.

A complex representation $\pi$ of $G$ is said to be an orthogonally
irreducible representation (OIR), provided $\pi$ is orthogonal, and can not be
decomposed into a direct sum of orthogonal representations.

An irreducible representation $\pi$ is orthogonally irreducible if and only if
$\pi$ is orthogonal. Moreover, for $\varphi$ irreducible and non-orthogonal,
its symmetrization $S(\varphi)$ is an OIR. In fact, the decomposition in (1)
establishes that all the OIRs of $G$ are either irreducible orthogonal $\pi$,
or of the form $S(\varphi)$ with $\varphi$ irreducible and non-orthogonal.

###  2.3. 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 [3]
or [15], given for real representations.

Recall from [19] 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 real representation $(\rho,V)$
one can form the associated vector bundle $EG[V]$ over $BG$. Then one puts

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

see for instance [3] or [15]. Moreover the singular cohomology
$H^{*}(BG,\mathbb{Z}/2\mathbb{Z})$ is isomorphic to the group cohomology
$H^{*}(G,\mathbb{Z}/2\mathbb{Z})$. At this point we simply write
$H^{*}(G)=H^{*}(G,\mathbb{Z}/2\mathbb{Z})$.

Hence, to a real representation $\rho$, we can associate cohomology classes
$w_{i}^{\mathbb{R}}(\rho)\in H^{i}(G)$, called Stiefel-Whitney classes (SWCs).
We prefer to work with an orthogonal complex representation $\pi$, which by
Proposition 2.1 comes from a unique real representation $\pi_{0}$. We thus
define

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

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

A group homomorphism $\varphi:G_{1}\to G_{2}$ induces a map
$\varphi^{*}:H^{*}(G_{2})\to H^{*}(G_{1}).$ The SWCs are functorial in the
following sense: If $\pi$ is an orthogonal representation of $G_{2}$, then

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

The SWCs are also multiplicative. This means if $\pi_{1}$ and $\pi_{2}$ are
both orthogonal, then

(3) $w(\pi_{1}\oplus\pi_{2})=w(\pi_{1})\cup w(\pi_{2}).$

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

$w_{1}:\operatorname{Hom}(G,\pm 1)\overset{\cong}{\to}H^{1}(G).$

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 [12] for details.)

To complex representations $\pi$ of $G$, are associated cohomology classes
$c_{i}(\pi)\in H^{2i}(G,\mathbb{Z})$, called Chern classes. As with SWCs,
$c_{0}(\pi)=1$ and the first Chern class, applied to linear characters, gives
the well-known isomorphism

$c_{1}:\operatorname{Hom}(G,S^{1})\xrightarrow{\cong}H^{2}(G,\mathbb{Z}).$

More generally, $c_{1}(\pi)=c_{1}(\det\pi)$ for a complex representation
$\pi$.

The SWC of a symmetrization $S(\pi)$ can be obtained from the Chern class of
$\pi$ via the formula

(4) $w(S(\pi))=\kappa(c(\pi)).$

Here $\kappa:H^{*}(G,\mathbb{Z})\to H^{*}(G,\mathbb{Z}/2\mathbb{Z})$ is the
_coefficient homomorphism_ of cohomology. (See [10, Lemma 1], based on [19,
Problem 14-B].)

From this we make an observation for later use:

######  Lemma 2.3.

If $\det\pi=1$, then $w_{i}(S(\pi))=0\text{ for }i=1,2,3$.

###### Proof.

Since Chern classes live in even degrees, all the odd SWCs of $S(\pi)$ vanish.
Moreover $c_{1}(\pi)=0$ by the determinant hypothesis. ∎

###  2.4. Steenrod Squares

For $n,i\geq 0$, there are operations on cohomology, called Steenrod Squares,

$\operatorname{Sq}^{i}:H^{n}(G)\rightarrow H^{n+i}(G).$

These are additive homomorphisms and $\operatorname{Sq}^{0}$ is the identity.
Steenrod operations are functorial, meaning for a group homomorphism
$\varphi:G\to G^{\prime}$, we have

$\varphi^{*}(\operatorname{Sq}^{i}y)=\operatorname{Sq}^{i}(\varphi^{*}(y))\text{
for all }y\in H^{n}(G^{\prime}).$

We now state the well-known Wu formula:

######  Proposition 2.4 ([18], Chapter 23, Section 6).

Let $\pi$ be an orthogonal representation of $G$. Then, the cohomology class
$\operatorname{Sq}^{i}(w_{j}(\pi))$ can be expressed as a polynomial in
$w_{1}(\pi),\ldots,w_{i+j}(\pi)$:

$\operatorname{Sq}^{i}(w_{j}(\pi))=\sum_{t=0}^{i}{j+t-i-1\choose
t}w_{i-t}(\pi)w_{j+t}(\pi).$

With $i=1$, $j=2$ for instance, we can express $w_{3}(\pi)$ in terms of
$w_{1},w_{2}$ as follows:

$w_{3}(\pi)=w_{1}(\pi)\cup w_{2}(\pi)+\operatorname{Sq}^{1}(w_{2}(\pi)).$

###  2.5. Elementary Abelian $2$-groups

Let $C_{2}=\\{\pm 1\\},$ the cyclic group of order $2$. Here we review
explicit descriptions of group cohomology for elementary abelian 2-groups
$C_{2}^{n}$ and describe SWCs in terms of these descriptions.

We write ‘sgn’ for the order $2$ linear character of $C_{2}$. It is known [16]
that

(5) $H^{*}(C_{2})\cong\mathbb{Z}/2\mathbb{Z}[v],$

where $v=w_{1}(\operatorname{sgn})$. Since this group is so simple, we can
give our first example of expressing SWCs in terms of character values. (Note
that every representation of $C_{2}$ is orthogonal.)

######  Lemma 2.5.

Let $\pi$ be a representation of $C_{2}$, and
$b_{\pi}=\frac{1}{2}\left(\deg\pi-\chi_{\pi}(-1)\right)$. Then, the total SWC
of $\pi$ is

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

###### Proof.

We can write

$\pi=a1\oplus b(\operatorname{sgn})$

for non-negative integers $a,b$. From the multiplicativity of SWCs, we obtain

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

To express $b$ in character values, consider the equations:

$\displaystyle\chi_{\pi}(1)$ $\displaystyle=a+b$ $\displaystyle\chi_{\pi}(-1)$
$\displaystyle=a-b$

giving $b=\cfrac{1}{2}(\chi_{\pi}(1)-\chi_{\pi}(-1))=b_{\pi}.$
∎

Let $C_{2}^{n}$ be the $n$-fold product of $C_{2}$, with projection maps
$\operatorname{pr}_{i}:C_{2}^{n}\to C_{2}$ for $1\leq i\leq n$. By Künneth, we
have

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

where we put $v_{i}=w_{1}(\operatorname{sgn}\circ\operatorname{pr}_{i})$.

###  2.6. SWCs for Virtual Representations

A virtual representation of $G$ can be thought as a difference
$\pi=\pi_{1}\ominus\pi_{2}$, for representations $\pi_{1},\pi_{2}$. When the
$\pi_{i}$ are orthogonal, one may define the SWC of $\pi$ as

$w(\pi)=w(\pi_{1})\cup w(\pi_{2})^{-1},$

but we must “complete” the cohomology ring so that the inversion makes sense.

More formally, let $G$ be a finite group, and let $\operatorname{RO}(G)$ be
the free abelian group generated by the isomorphism classes of OIRs (following
[4, Chapter II, Section 7]). Members of $\operatorname{RO}(G)$ are called
_virtual orthogonal representations_ of $G$.

Write

$H^{\bullet}(G)=\prod_{i}H^{i}(G),$

for the complete mod $2$ cohomology ring of $G$, consisting of all formal
infinite series $\alpha_{0}+\alpha_{1}+\cdots$, with $\alpha_{i}\in H^{i}(G)$.
(Please see [22, page 44].) Each $w(\pi)$ is invertible in this ring.

It is clear now that we may “extend” $w$ to a group homomorphism, taking
values in the unit group of $H^{\bullet}(G)$, i.e.,

(6) $w:\operatorname{RO}(G)\to H^{\bullet}(G)^{\times}.$

Later in this paper we shall determine the image of $w$ for
$G=\operatorname{SL}(2,q)$.

##  3\. Quaternion Groups

Our proof of Theorem 1.1 involves subgroups of $\operatorname{SL}(2,q)$
isomorphic to certain quaternion groups. In this section we review the group
cohomology of the quaternion group of order $8$, and the so-called
“generalized quaternions” $Q_{2^{n}}$. This will play a vital role in our
detection theorem.

###  3.1. Orthogonal Representations of $Q$

Let $Q$ be the Quaternion group of order 8. If we write $\mathbb{H}$ for the
familiar real division algebra of quaternions, then $Q$ can be quickly
described as the subgroup of $\mathbb{H}^{\times}$ generated by $i$ and $j$.

There are 4 one-dimensional representations of $Q$, which we denote by 1,
$\chi_{1},\chi_{2},\chi_{3}=\chi_{1}\otimes\chi_{2}$. Each one is orthogonal.
The group $Q$ also possesses a unique 2-dimensional irreducible representation

$\rho:Q\rightarrow\operatorname{SL}(2,\mathbb{C})$

defined by

$\rho(i)=\begin{pmatrix}i&&0\\\ 0&&-i\end{pmatrix}\text{ ,
}\;\rho(j)=\begin{pmatrix}0&&1\\\ -1&&0\end{pmatrix}.$

Of course, $\mathbb{H}$ can be described as the algebra of complex matrices of
the form

$\begin{pmatrix}s+ti&u+vi\\\ -u+vi&s-ti\end{pmatrix}\text{ where
}s,t,u,v\in\mathbb{R}.$

Since both $\rho(i)$, $\rho(j)$ are of this form, $\rho$ is an injection of
$Q$ into $\mathbb{H}^{1}$, the subgroup of norm $1$ quaternions. In
particular, $\rho$ is symplectic. Thus, $S(\rho)=\rho\oplus\rho$ is the only
OIR of $Q$, which is not irreducible.

###  3.2. The Cohomology ring $H^{*}(Q)$

The SWCs of orthogonal representations of $Q$ are certain elements in
$H^{*}(Q)$. Here we give a description of the cohomology ring $H^{*}(Q)$ with
[1, Chapter IV] as our reference.

Recall that $w_{1}$ gives an isomorphism between $\operatorname{Hom}(Q,\pm
1)=\\{1,\chi_{1},\chi_{2},\chi_{3}\\}$ and $H^{1}(Q)$. We define

$\displaystyle x$ $\displaystyle:=w_{1}(\chi_{1}),\text{ and}$ $\displaystyle
y$ $\displaystyle:=w_{1}(\chi_{2}).$

The cohomology group $H^{4}(Q)$ is one-dimensional over
$\mathbb{Z}/2\mathbb{Z}$, and we write ‘$e$’ for its nonzero element.

######  Proposition 3.1 ([1], Section V.1).

The cohomology ring of $Q$ is

$H^{*}(Q)\cong\mathbb{Z}/2\mathbb{Z}[x,y,e]/{(xy+x^{2}+y^{2},x^{2}y+xy^{2})}.$

The first few cohomology groups of $Q$ are as follows:

$\displaystyle H^{0}(Q)$ $\displaystyle=\\{0,1\\}$ $\displaystyle H^{1}(Q)$
$\displaystyle=\\{0,x,y,x+y\\}$ $\displaystyle H^{2}(Q)$
$\displaystyle=\\{0,x^{2},y^{2},x^{2}+y^{2}\\}$ $\displaystyle H^{3}(Q)$
$\displaystyle=\\{0,x^{2}y\\}$ $\displaystyle H^{4}(Q)$
$\displaystyle=\\{0,e\\}.$

Note that $x^{3}=y^{3}=0$. Moreover, one can obtain the higher cohomology
groups via the cup product with $e$ due to Proposition 3.1. This means for
$i\geq 0,$ the map

$\displaystyle H^{i}(Q)$ $\displaystyle\to H^{i+4}(Q)$ $\displaystyle z$
$\displaystyle\mapsto z\cup e$

is an isomorphism of groups.

###  3.3. SWCs of Representations of $Q$

We first compute the total SWCs of OIRs of $Q$. For the linear orthogonal
representations of $Q$, it is clear that $w(1)=1,$ $w(\chi_{1})=1+x$,
$w(\chi_{2})=1+y$, and $w(\chi_{1}\otimes\chi_{2})=1+x+y$ with $x,y\in
H^{1}(Q)$ from above.

######  Lemma 3.2.

The total SWC of $S(\rho)$ is

$w(S(\rho))=1+e,$

where $e$ is the non-trivial element of $H^{4}(Q)$.

###### Proof.

From Lemma 2.3, we have

$w_{i}(S(\rho))=0\text{ for }i=1,2,3.$

This gives $w(S(\rho))=1+w_{4}(S(\rho))$. We now prove that
$w_{4}(S(\rho))=e\in H^{4}(Q).$

Let $Z$ be the center of $Q$, which is $\\{\pm 1\\}$. Since

$\displaystyle\rho(-1)=\small\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix},$

we have the restriction
$\operatorname{res}_{Z}^{Q}\rho=\operatorname{sgn}\oplus\operatorname{sgn},$
where $\operatorname{sgn}:Z\rightarrow\mathbb{C}^{\times}$ is the linear
character of order $2$.

From (5), we have $H^{*}(Z)\cong\mathbb{Z}/2\mathbb{Z}[v]$ where
$v=w_{1}(\operatorname{sgn})$. Then, the multiplicativity of SWCs gives

$\displaystyle
w(\operatorname{res}_{Z}^{Q}S(\rho))=w(\operatorname{sgn})^{4}=(1+v)^{4}=1+v^{4},$

which implies $w_{4}(S(\rho))\neq 0$. Therefore, $w_{4}(S(\rho))=e$, since
this is the only nonzero element in $H^{4}(Q)$. ∎

We also see from the above proof that $e$ maps to $v^{4}$ under the
restriction $H^{4}(Q)\to H^{4}(Z)$. From this, we deduce that the restriction
maps in degrees divisible by $4$,

(7) $\begin{split}H^{4i}(Q)&\rightarrow H^{4i}(Z),\\\ e^{i}&\mapsto
v^{4i}\end{split}$

are isomorphisms.

Let $\pi$ be an orthogonal representation of $Q$. By (1), we can write

$\pi\cong m_{0}1\oplus m_{1}\chi_{1}\oplus m_{2}\chi_{2}\oplus
m_{2}\chi_{3}\oplus m_{4}S(\rho),$

where $m_{i}$ are non-negative integers.

######  Lemma 3.3.

Let $\pi$ be as above. Then, for $0\leq 4i\leq\deg\pi$, we have

$w_{4i}(\pi)=\binom{m_{4}}{i}e^{i},$

where $m_{4}=\frac{1}{8}(\chi_{\pi}(1)-\chi_{\pi}(-1))$.

###### Proof.

Let us consider

$S=\mathbb{Z}/2\mathbb{Z}[x,y]/{(xy+x^{2}+y^{2},x^{2}y+xy^{2})},$

which is a $6$-dimensional subalgebra of $H^{*}(Q)$. By the multiplicativity
of SWCs, we have

$\displaystyle w(\pi)$ $\displaystyle=w(\chi_{1})^{m_{1}}\cup
w(\chi_{2})^{m_{2}}\cup w(\chi_{3})^{m_{3}}\cup w(S(\rho))^{m_{4}}$
$\displaystyle=\underbrace{(1+x)^{m_{1}}(1+y)^{m_{2}}(1+x+y)^{m_{3}}}_{\quad\in\>S}(1+e)^{m_{4}}.$

Therefore, we can write $(1+x)^{m_{1}}(1+y)^{m_{2}}(1+x+y)^{m_{3}}$ as a
polynomial of the form $P(x,y)=1+Ax+By+Cx^{2}+Dy^{2}+Exy^{2}$ with
coefficients in $\mathbb{Z}/2\mathbb{Z},$ and

$\displaystyle w(\pi)$ $\displaystyle=P(x,y)(1+e)^{m_{4}}$
$\displaystyle=P(x,y)\sum_{i=0}^{m_{4}}\binom{m_{4}}{i}e^{i}$

where $e^{i}\in H^{4i}(Q)$ for each $i$. Comparing degrees gives

$w_{4i}(\pi)=\binom{m_{4}}{i}e^{i}.$

To express $m_{4}$ in terms of character values, we simply evaluate
$\chi_{\pi}$ at 1 and $-1$. Since $-1\in$Ker$(\chi_{i})$ for $i=1,2,3$ and
$\chi_{\rho}(1)=2$, $\chi_{\rho}(-1)=-2$, we have the equations:

$\displaystyle\chi_{\pi}(1)$ $\displaystyle=m_{0}+m_{1}+m_{2}+m_{3}+4m_{4},$
$\displaystyle\chi_{\pi}(-1)$ $\displaystyle=m_{0}+m_{1}+m_{2}+m_{3}-4m_{4},$

and so $m_{4}=\frac{1}{8}(\chi_{\pi}(1)-\chi_{\pi}(-1))$. ∎

###  3.4. Generalized Quaternions

The construction of the quaternion group $Q$ generalizes to give a family of
non-abelian groups which have the presentation

$Q_{2^{n}}:=\langle a,b\mid
a^{2^{n-2}}=b^{2},b^{4}=1,bab^{-1}=a^{-1}\rangle\>;\>n\geq 3.$

These groups are called generalized quaternion groups and have order $2^{n}$.
Again, there are 4 linear (orthogonal) representations of $Q_{2^{n}}$, say
$1,\psi_{1},\psi_{2},\psi_{3}.$ These can be defined on the generators of
$Q_{2^{n}}$ as:

(8) $\displaystyle\psi_{1}(a)$ $\displaystyle=-1\hskip 8.53581pt$ ,
$\displaystyle\psi_{1}(b)$ $\displaystyle=\;\;\;1$ $\displaystyle\psi_{2}(a)$
$\displaystyle=-1\hskip 8.53581pt$ , $\displaystyle\psi_{2}(b)$
$\displaystyle=-1$ $\displaystyle\psi_{3}(a)$ $\displaystyle=1\hskip
8.53581pt$ , $\displaystyle\psi_{3}(b)\>$ $\displaystyle=-1.$

Let $\zeta=e^{2\pi i/2^{n-1}}$. There is an irreducible 2-dimensional
representation $\varrho$ of $Q_{2^{n}}$ defined by:

$\varrho(a)=\small\begin{pmatrix}\zeta&&0\\\
0&&\bar{\zeta}\end{pmatrix}\quad,\quad\varrho(b)=\small\begin{pmatrix}0&&1\\\
-1&&0\end{pmatrix}.$

As earlier, $\varrho$ maps $Q_{2^{n}}$ into $\mathbb{H}^{1}$, so it is
symplectic.

Let $n>3$. We use the isomorphism

$\displaystyle w_{1}:\operatorname{Hom}(Q_{2^{n}},\pm 1)$
$\displaystyle\xrightarrow{\cong}H^{1}(Q_{2^{n}})$
$\displaystyle\\{1,\psi_{1},\psi_{2},\psi_{3}\\}$
$\displaystyle\leftrightarrow\\{0,X,Y,X+Y\\}.$

to define $X=w_{1}(\psi_{1})$ and $Y=w_{1}(\psi_{2})$.

A description of the mod $2$ cohomology ring of generalized quaternions
$Q_{2^{n}}$ for $n>3$ in [1, Chapter IV, Lemma 2.11] is as follows:

(9) $H^{*}(Q_{2^{n}})\cong\mathbb{Z}/2\mathbb{Z}[X,Y,E]/(XY,X^{3}+Y^{3}),$

where $E\in H^{4}(Q_{2^{n}})$.

######  Lemma 3.4.

We have $E=w_{4}(S(\varrho))$ in $H^{*}(Q_{2^{n}}).$

###### Proof.

This can be proved precisely as in Lemma 3.2; again $w(S(\varrho))=1+E$. ∎

######  Remark 3.5.

Let $n>3$. Let $i:Q\rightarrow Q_{2^{n}}$ be any group homomorphism. Then,

$\displaystyle i^{*}:H^{3}(Q_{2^{n}})$ $\displaystyle\rightarrow H^{3}(Q)$

is the zero map. This can be seen as follows.

From (9), we can deduce $H^{3}(Q_{2^{n}})=\\{0,X^{3}\\}$ and since $i^{*}$ is
a ring homomorphism, we have the image $i^{*}(X)\in H^{1}(Q)=\\{0,x,y,x+y\\}$.
But $x^{3}=y^{3}=(x+y)^{3}=0$ in $H^{*}(Q)$, which implies

$\displaystyle i^{*}(X^{3})=(i^{*}(X))^{3}=0.{\qed}$

Consider the subgroup $Q^{(1)}=\langle a^{2^{n-3}},b\rangle\leqslant
Q_{2^{n}}$. This subgroup is isomorphic to $Q$ by $i\leftrightarrow
a^{2^{n-3}}$ and $j\leftrightarrow b$. With this identification, it is easy to
see that

$\operatorname{res}_{Q}^{Q_{2^{n}}}\varrho=\rho.$

Write $\iota_{Q}$ for this inclusion of $Q$ into $Q_{2^{n}}$. Now

$\begin{split}\iota_{Q}^{*}(E)&=\iota_{Q}^{*}(w_{4}(\varrho))\\\
&=w_{4}(\operatorname{res}_{Q}^{Q_{2^{n}}}\varrho)\\\ &=w_{4}(\rho)\\\ &=e.\\\
\end{split}$

This means the homomorphism $\iota^{*}_{Q}:H^{*}(Q_{2^{n}})\to H^{*}(Q)$, is
an isomorphism in degree $4$. Therefore:

######  Proposition 3.6.

With notation as above, we have isomorphisms

(10) $\begin{split}{\iota_{Q}^{*}}:H^{4i}(Q_{2^{n}})&\rightarrow H^{4i}(Q)\\\
E^{i}&\mapsto e^{i}\end{split}$

for all $i\geq 0$.

##  4\. $\operatorname{SL}(2,q)$ with $q$ odd

Let $G=\operatorname{SL}(2,q)$ throughout this section, with $q$ odd. Write
‘$\mathbbm{1}$’ for the identity matrix; then the center $Z$ of $G$ is
$\\{\pm\mathbbm{1}\\}$. We begin with a description of the mod $2$ cohomology
of $G$.

######  Proposition 4.1 ([6], Chapter VI, Sec. 5).

When $q$ is odd, the mod $2$ cohomology ring of $\operatorname{SL}(2,q)$ is

$H^{*}(\operatorname{SL}(2,q))\cong\mathbb{Z}/2\mathbb{Z}[\mathfrak{e}]\otimes\mathbb{Z}/2\mathbb{Z}[\mathfrak{b}]/\langle\mathfrak{b}^{2}\rangle$

with $\deg(\mathfrak{b})=3$, $\deg(\mathfrak{e})=4$.

###  4.1. Detection

The proof of Theorem 1.1 uses the following well-known detection:

######  Lemma 4.2 ([1], Cor. 5.2, Ch. II).

Let $H$ be a finite group. A Sylow $2$-subgroup of $H$ detects its mod $2$
cohomology.

According to [13, Chapter 2, Theorem 8.3], a Sylow $2$-subgroup of $G$ is
generalized quaternion. We will therefore regard $Q_{2^{n}}$ as a subgroup of
$G$, where of course $n$ is the highest power of $2$ dividing $|G|$. We now
prove our main detection theorem.

###### Proof of Theorem 1.1.

We deduce from Proposition 4.1 that

$H^{n}(G)\cong\begin{cases}\mathbb{Z}/2\mathbb{Z},&n\equiv 0,3\;(\text{mod
}4)\\\ 0,&\text{ otherwise}.\end{cases}$

Let $\pi$ be an orthogonal representation of $G$. Let $n=4k+3$ for some $k$.
With $i=1$, $j=n-1$ such that $i+j=n$, we now apply the Wu formula from
Proposition 2.4:

$\displaystyle\operatorname{Sq}^{1}(w_{n-1}(\pi))$
$\displaystyle=w_{1}(\pi)\cup w_{n-1}(\pi)+{n-2\choose 1}w_{0}(\pi)\cup
w_{n}(\pi)$

Here, $\operatorname{Sq}^{1}(w_{n-1}(\pi))=0$ because $w_{n-1}(\pi)\in
H^{4k+2}(G)$ is zero and $\operatorname{Sq}^{1}$ is a homomorphism. Also,
$w_{1}(\pi)=0$ and $(n-2)$ is odd which implies

$w_{n}(\pi)=0\text{ when }n\equiv 3\text{ (mod }4).$

Therefore, the non-zero SWCs of representations of $G$ only occur in degrees
divisible by $4$, which means

$H^{*}_{\operatorname{SW}}(G)\subseteq\mathbb{Z}/2\mathbb{Z}[\mathfrak{e}].$

Since $Q_{2^{n}}$ is a $2$-Sylow subgroup of $G$, it detects the mod $2$
cohomology of $G$ by Lemma 4.2. Consider the Quaternion subgroup $Q\cong
Q^{(1)}\leqslant Q_{2^{n}}$. Note that the center of $Q$ is $Z$. By the
isomorphisms (7) and (10), we obtain the following sequence of inclusions for
each $i\geq 0$:

$\displaystyle H^{4i}(G)$ $\displaystyle\hookrightarrow H^{4i}(Q_{2^{n}})$
$\displaystyle\hookrightarrow H^{4i}(Q)$ $\displaystyle\hookrightarrow
H^{4i}(Z)$ $\displaystyle\mathfrak{e}^{i}\quad$ $\displaystyle\mapsto\quad
E^{i}$ $\displaystyle\mapsto\quad e^{i}\quad$ $\displaystyle\mapsto\quad
v^{4i}.$

Therefore, the subalgebra $\mathbb{Z}/2\mathbb{Z}[\mathfrak{e}]$, containing
the SWCs of $G$, injects into $H^{*}(Z)$, completing the proof.

∎

###  4.2. Formulas for SWCs

We now prove Theorem 1.2 which gives an explicit formula for the total SWCs of
orthogonal representations of $G$ in terms of character values.

###### Proof of Theorem 1.2.

To find $w(\pi)$, it is enough to work with $\operatorname{res}^{G}_{Z}\pi$
due to the detection Theorem 1.1. This restriction has been done in two steps.
We first restrict $\pi$ to the quaternion group $Q\leqslant G$, and then
further from $Q$ to $Z$.

With notations from Section 3, it looks like

$\operatorname{res}^{G}_{Q}\pi\cong m_{0}1\oplus m_{1}\chi_{1}\oplus
m_{2}\chi_{2}\oplus m_{3}\chi_{3}\oplus m_{4}(S(\rho))$

where $m_{i}$ are non-negative integers. It is easy to see that

$\begin{split}\operatorname{res}^{Q}_{Z}\chi_{i}&\cong 1\text{ for
}i=1,2,3,\text{ and }\\\
\operatorname{res}^{Q}_{Z}\rho&\cong\operatorname{sgn}\oplus\operatorname{sgn}.\end{split}$

Therefore, the further restriction of $\operatorname{res}^{G}_{Q}\pi$ to the
center $Z$ will be

$\operatorname{res}^{G}_{Z}\pi=\operatorname{res}^{Q}_{Z}\operatorname{res}^{G}_{Q}\pi\cong(m_{0}+m_{1}+m_{2}+m_{3})1\oplus
4m_{4}(\operatorname{sgn}).$

From Lemma 2.5, we then obtain

$w(\operatorname{res}^{G}_{Z}\pi)=(1+v)^{4m_{4}}=(1+v^{4})^{m_{4}},$

with $4m_{4}=\frac{1}{2}(\chi_{\pi}(\mathbbm{1})-\chi_{\pi}(-\mathbbm{1})).$
From the proof of Theorem 1.1, we have $i^{*}_{Z}(\mathfrak{e})=v^{4}$ where
$i_{Z}:Z\rightarrow G$ is the inclusion. Therefore,

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

where $r_{\pi}=m_{4}$ is the multiplicity of $S(\rho)$ in
$\operatorname{res}^{G}_{Q}\pi$.

∎

######  Example 4.3.

We have

$w(\operatorname{reg}(G))=(1+\mathfrak{e})^{\frac{1}{8}|G|}.$

###  4.3. Corollaries

We can go deeper by applying the following result due to R. Gow:

######  Theorem 4.4 ([14], Theorem 1).

Let $\pi$ be an irreducible self-dual representation of $G$ with central
character $\omega_{\pi}.$ Then, the Frobenius-Schur Indicator
$\varepsilon(\pi)$ equals $\omega_{\pi}(-\mathbbm{1})$.

We simply call this equality _Gow’s formula_.

This leads to the following:

######  Corollary 4.5.

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

###### Proof.

For $\pi$ irreducible orthogonal, the sign $\varepsilon(\pi)=1$ by definition.
Therefore

$\displaystyle r_{\pi}$
$\displaystyle=\frac{1}{8}(\chi_{\pi}(\mathbbm{1})-\chi_{\pi}(-\mathbbm{1}))$
$\displaystyle=\frac{\chi_{\pi}(\mathbbm{1})}{8}(1-\omega_{\pi}(-\mathbbm{1}))$
$\displaystyle=0.$

∎

Let $\pi$ be an irreducible, non-orthogonal representation of $G$. Then, for
the representation $S(\pi)=\pi\oplus\pi^{\vee}$, we have

$\displaystyle r_{S(\pi)}$
$\displaystyle=\frac{1}{8}\big{(}\chi_{S(\pi)}(\mathbbm{1})-\chi_{S(\pi)}(-\mathbbm{1})\big{)}$
$\displaystyle=\frac{1}{4}\big{(}\chi_{\pi}(\mathbbm{1})-\chi_{\pi}(-\mathbbm{1})\big{)}$
$\displaystyle=\frac{\chi_{\pi}(\mathbbm{1})}{4}\big{(}1-\omega_{\pi}(-\mathbbm{1})\big{)}.$

Furthermore, for symplectic $\pi$, it turns out to be

(11) $r_{S(\pi)}=\frac{\chi_{\pi}(\mathbbm{1})}{2}=\frac{\deg(\pi)}{2}$

due to Gow’s formula.

######  Corollary 4.6.

Let $G=\operatorname{SL}(2,q)$ with $q$ odd. Then the image of $w$ in (6) is

$\\{(1+\mathfrak{e})^{n}\mid n\in\mathbb{Z}\\}.$

###### Proof.

For $q=3$, there exists a unique irreducible symplectic representation
$\pi_{0}$ of $G$ with degree $2$. From Equation (11), we obtain
$r_{S(\pi_{0})}=1$, and so $w(S(\pi_{0}))=1+\mathfrak{e}.$

For $q>3$, there exist irreducible symplectic representations $\pi_{1}$ and
$\pi_{2}$ of $G$ of degrees $q+1$ and $q-1$ respectively. (See Remark 4.7 for
details.) From Equation (11), we have

$r_{S(\pi_{1})}=\frac{q+1}{2}\;,\;r_{S(\pi_{2})}=\frac{q-1}{2}$

which are co-prime. So by Bézout’s Identity, there exist integers $a,b$ such
that $a\big{(}\frac{q+1}{2}\big{)}+b\big{(}\frac{q-1}{2}\big{)}=1$.

Therefore, there is a virtual representation $\pi\in\operatorname{RO}(G)$ with
$r_{\pi}=1$ such that

$w(\pi)=1+\mathfrak{e}.$

Hence, the result follows from the multiplicativity of SWCs. ∎

######  Remark 4.7.

For completeness, we give explicit descriptions of $\pi_{1}$ and $\pi_{2}$.
This entails recalling the construction of the principal series and
irreducible cuspidal representations of $G$. The reader is referred to [9] and
[5] for more details.

Let $\widetilde{T}$ be the diagonal subgroup, isomorphic to
$\mathbb{F}_{q}^{\times}\times\mathbb{F}_{q}^{\times}$, contained in the
standard Borel subgroup $\widetilde{B}$ of upper-triangular matrices of
$\widetilde{G}=\operatorname{GL}(2,q)$. When $\alpha,\beta$ are linear
characters of $\mathbb{F}_{q}^{\times}$, we will write $\alpha\boxtimes\beta$
for the corresponding linear character of $\widetilde{T}$. Choose
$\alpha:\mathbb{F}_{q}^{\times}\to\mathbb{C}^{\times}$ satisfying

$\displaystyle\alpha(-1)$ $\displaystyle=-1$ $\displaystyle\alpha^{2}$
$\displaystyle\neq 1.$

One can inflate $\alpha\boxtimes 1$ from $\widetilde{T}$ to $\widetilde{B}$
and then consider the usual complex parabolic induction
$\pi_{\alpha}:=\operatorname{Ind}_{\widetilde{B}}^{\widetilde{G}}(\alpha\boxtimes
1)$. This is an irreducible principal series representation of $\widetilde{G}$
of degree $q+1$.

We take $\pi_{1}=\operatorname{res}^{\widetilde{G}}_{G}\pi_{\alpha}$. This
restriction is self-dual and irreducible. By Gow’s formula, $\pi_{1}$ is
symplectic.

Let $\widetilde{T}_{e}$ be an elliptic torus of $\widetilde{G}$, thus
isomorphic to $\mathbb{F}_{q^{2}}^{\times}$. Let $\widetilde{Z}$ be the center
and $N$ be the upper unitriangular subgroup of $\widetilde{G}$. Choose a
linear character $\chi$ of $\widetilde{T}_{e}$ such that

$\displaystyle\chi^{q}\neq\chi,$ $\displaystyle\;\;\chi^{2}\neq 1$
$\displaystyle\chi(-1)$ $\displaystyle=-1.$

We fix a nontrivial character $\varphi$ of $N$, and define a linear character
of $\widetilde{Z}N$ as $\chi_{\varphi}(zn)=\chi(z)\varphi(n)$. Set

$\pi_{\chi}=\operatorname{Ind}_{\widetilde{Z}N}^{\widetilde{G}}\chi_{\varphi}-\operatorname{Ind}_{\widetilde{T}_{e}}^{\widetilde{G}}\chi.$

This is an irreducible, cuspidal representation of $\widetilde{G}$ of
dimension $q-1$.

When restricted to $G$, $\pi_{\chi}$ remains irreducible and is called an
irreducible cuspidal representation of $G$. (A representation of $G$ is called
_cuspidal_ , provided each of its irreducible constituents is irreducible
cuspidal.)

Define $\pi_{2}=\operatorname{res}^{\widetilde{G}}_{G}\pi_{\chi}$. Again one
sees that $\pi_{2}$ is symplectic by Gow’s formula.

It is already known from Theorem 1.1 that
$H^{*}_{\operatorname{SW}}(G)\subseteq\mathbb{Z}/2\mathbb{Z}[\mathfrak{e}]$.
Here we make a stronger statement:

######  Corollary 4.8.

Let $G=\operatorname{SL}(2,q)$ with $q$ odd. Then,

$H^{*}_{\operatorname{SW}}(G)=\mathbb{Z}/2\mathbb{Z}[\mathfrak{e}].$

###### Proof.

For equality, we construct an orthogonal representation $\eta$ of $G$ such
that $w_{4}(\eta)=\mathfrak{e}$ as follows.

For $q>3$, consider the irreducible symplectic representations $\pi_{1}$,
$\pi_{2}$ of $G$ from Corollary 4.6. Let $\eta=S(\pi_{1})\oplus S(\pi_{2})$.
It can easily seen that $r_{\pi\oplus\pi^{\prime}}=r_{\pi}+r_{\pi}^{\prime}$
for any orthogonal $\pi,\pi^{\prime}$. Therefore, we have

$r_{\eta}=r_{S(\pi_{1})\oplus
S(\pi_{2})}=r_{S(\pi_{1})}+r_{S(\pi_{2})}=\frac{q+1}{2}+\frac{q-1}{2}=q.$

Since $q$ is odd, we have

$w(\eta)=(1+\mathfrak{e})^{q}=1+\mathfrak{e}+\cdots.$

Therefore, $w_{4}(\eta)=\mathfrak{e}.$

Moreover, we already have $w_{4}(S(\pi_{0}))=\mathfrak{e}$ for $q=3$ which
completes the proof. ∎

Next, we give a nonvanishing condition for the top SWC.

######  Corollary 4.9.

Let $G=\operatorname{SL}(2,q)$ with $q$ odd. Let $\pi$ be an orthogonal
representation of $G$. Then, the following are equivalent:

  1. (i)

$w_{\operatorname{top}}(\pi)\neq 0$.

  2. (ii)

$\pi=S(\varphi),$ where every irreducible constituent $\varphi_{i}$ of
$\varphi$ has central character satisfying
$\omega_{\varphi_{i}}(-\mathbbm{1})=-1$.

  3. (iii)

$\pi(-\mathbbm{1})=-1$, meaning $\pi(-\mathbbm{1})$ acts by the scalar $-1$.

###### Proof.

We first prove (i) and (ii) are equivalent. Assume
$\pi=S(\varphi)=\bigoplus\limits_{i=1}^{r}S(\varphi_{i})$ such that
$\omega_{\varphi_{i}}(-\mathbbm{1})=-1$ for all $1\leq i\leq r$.

Clearly $\chi_{\pi}(\mathbbm{1})=2\chi_{\varphi}(\mathbbm{1})$, and since
$\chi_{\varphi_{i}}(-\mathbbm{1})=\omega_{\varphi_{i}}(-\mathbbm{1})\chi_{\varphi_{i}}(\mathbbm{1})=-\chi_{\varphi_{i}}(\mathbbm{1})$,
we have

$\displaystyle\chi_{\pi}(-\mathbbm{1})=2\sum_{i=1}^{r}\chi_{\varphi_{i}}(-\mathbbm{1})=-2\chi_{\varphi}(\mathbbm{1}).$

This gives

$r_{\pi}=\frac{1}{2}\chi_{\varphi}(\mathbbm{1})=\frac{1}{4}\chi_{\pi}(\mathbbm{1}).$

Therefore, we have $w(\pi)=(1+\mathfrak{e})^{\frac{1}{4}\deg\pi}$, implying

$w_{\operatorname{top}}(\pi)=\mathfrak{e}^{\frac{1}{4}\deg\pi}\neq 0.$

For the converse, suppose $w_{\operatorname{top}}(\pi)\neq 0$. Being
orthogonal, we can write $\pi$ as

$\pi=\bigoplus_{i}\rho_{i}\oplus\bigoplus_{j}S(\phi_{j})\oplus\bigoplus_{k}S(\varphi_{k})$

such that each $\rho_{i}$ is irreducible orthogonal, whereas
$\phi_{j},\varphi_{k}$ are irreducible non-orthogonal with
$\omega_{\phi_{j}}(-\mathbbm{1})=1$ for each $j$, and
$\omega_{\varphi_{k}}(-\mathbbm{1})=-1$ for each $k$.
From Theorem 1.2 and Corollary 4.5, we obtain

$\displaystyle w(\pi)$
$\displaystyle=\prod_{k}(1+\mathfrak{e})^{\frac{\deg\varphi_{k}}{2}}$
$\displaystyle=(1+\mathfrak{e})^{\frac{\deg\varphi}{2}},\text{ where
}\varphi=\bigoplus_{k}\varphi_{k}.$

Now, the condition $w_{\operatorname{top}}(\pi)\neq 0$ implies
$\deg\pi=4\cdot\frac{\deg\varphi}{2}$. Therefore,

$\displaystyle\sum\limits_{i}\deg\rho_{i}+2\sum\limits_{j}\deg\phi_{j}+2\sum\limits_{k}\deg\varphi_{k}$
$\displaystyle=2\deg\varphi$
$\displaystyle\sum\limits_{i}\deg\rho_{i}+2\sum\limits_{j}\deg\phi_{j}$
$\displaystyle=0,$

which means $\rho_{i}$ and $S(\phi_{j})$ don’t appear in $\pi$. Hence
$\pi=\bigoplus\limits_{i}S(\varphi_{i})$, with each $\varphi_{i}$ irreducible
and $\omega_{\varphi_{i}}(-\mathbbm{1})=-1$.

Next, suppose that $\pi_{1},\ldots,\pi_{r}$ are representations of $G$, and
$\pi$ is their direct sum. Then $\pi$ satisfies condition (iii) iff each
$\pi_{i}$ satisfies condition (iii). (Because a block matrix equals the
negative of the identity matrix iff each block is the negative of the identity
matrix.) In particular, this shows that (ii) implies (iii). By Gow’s formula,
no irreducible orthogonal representation satisfies condition (iii), and so by
decomposing $\pi$ we see that (iii) implies (ii).

∎

For $\pi$ orthogonal, let $k_{0}$ be the least $k>0$ such that $w_{k}(\pi)\neq
0$. Then $w_{k_{0}}(\pi)$ is known as the obstruction class of $\pi$,
following [10].

######  Corollary 4.10.

Let $\pi$ be an orthogonal representation of $G$. Put
$t=\operatorname{ord}_{2}(r_{\pi})$. Then the obstruction class of $\pi$ is
$w_{2^{t+2}}(\pi)=\mathfrak{e}^{2^{t}}$.

###### Proof.

By Theorem 1.2,

$w_{k}(\pi)=\begin{cases}{r_{\pi}\choose i}\mathfrak{e}^{i},\;&\;0\leq
k=4i\leq\deg\pi\\\ 0,\;&\text{ otherwise}.\end{cases}$

Therefore the problem amounts to finding the smallest positive $k=4i$, so that
the binomial coefficient ${r_{\pi}\choose i}$ is odd. This is an elementary
application of Lucas’s Theorem [7]; proceeding as in [10, Proposition 3], we
find that the smallest such $k$ is $k=2^{t+2}$, giving the corollary.

∎

###  4.4. Relation with $\operatorname{GL}(2,q)$

Ganguly and Joshi compute SWCs for orthogonal representations $\pi$ of
$\widetilde{G}=\operatorname{GL}(2,q)$ in [10], with $q$ odd. Let us review
their result here, and see that it is harmonious with ours.

Their SWCs are identified with members of $H^{*}(\widetilde{T})$ for the
bicyclic group $\widetilde{T}$, isomorphic to $C_{q-1}\times C_{q-1}$. When
$q\equiv 3\>(\operatorname{mod}\>4)$, the cohomology is generated by elements
$v_{1},v_{2}$ of degree $1$, by Künneth. When $q\equiv
1\>(\operatorname{mod}\>4)$, it is generated by elements
$s_{1},s_{2},t_{1},t_{2}$, where $\deg s_{i}=1$ and $\deg t_{i}=2$. Put
$\delta=0$ if $\det\pi=1$ and $\delta=1$ otherwise.

In these terms, the total SWC of $\pi$ is

(12)
$w(\pi)=\begin{cases}(1+\delta(s_{1}+s_{2}))(1+t_{1}+t_{2}+t_{1}t_{2})^{r_{\pi}}(1+t_{1}+t_{2})^{s_{\pi}/2},&q\equiv
1\;(\operatorname{mod}\>4)\\\\[2.84526pt]
(1+v_{1}^{2}+v_{2}^{2}+v_{1}^{2}v_{2}^{2})^{r_{\pi}}(1+v_{1}+v_{2})^{s_{\pi}},&q\equiv
3\;(\operatorname{mod}\>4),\end{cases}$

for a certain integer $s_{\pi}$, and
$r_{\pi}=\dfrac{1}{8}(\chi_{\pi}(\mathbbm{1})-\chi_{\pi}(-\mathbbm{1}))$ as
above.

Our two papers are compatible via the following diagram, formed from the
restriction maps on cohomology:

${H^{*}(\widetilde{G})}$${H^{*}(G)}$${H^{*}(\widetilde{T})}$${H^{*}(Z)}$

The RHS of (12) lies in $H^{*}(\widetilde{T})$. The left map is injective by
Quillen’s work, and the right map is injective on SWCs by Theorem 1.1. The
lower map $H^{*}(\widetilde{T})\to H^{*}(Z)$ maps $v_{i}$ to $v$ for $q\equiv
3\>(\operatorname{mod}\>4)$, and when $q\equiv 1\>(\operatorname{mod}\>4)$, it
maps $s_{i}$ to $0$ and $t_{i}$ to $v^{2}$. This gives:

$w(\pi|_{Z})=(1+v^{4})^{r_{\pi}}$

in all cases. Thus, for orthogonal representations of $G$ which extend to
orthogonal representations of $\widetilde{G}$, our Theorem 1.2 agrees with the
main result of [10].

######  Remark 4.11.

Not every orthogonal representation of $G$ extends to $\widetilde{G}$. For
instance when $q\equiv 1\>(\operatorname{mod}\>4)$, the representation
$\pi=\operatorname{Ind}_{\widetilde{B}}^{\widetilde{G}}(\operatorname{sgn}\boxtimes
1)$ (with $\widetilde{B}$ the upper triangular subgroup of $\widetilde{G}$),
when restricted to $G$ is the sum of two irreducible orthogonal
representations, neither of which extend to $\widetilde{G}$.

##  5\. $\operatorname{SL}(2,q)$ with $q$ even

Let $G=\operatorname{SL}(2,q)$ with $q=2^{r}$ throughout this section. Recall
that we write $N$ for the subgroup of upper unitriangular matrices. It is
well-known that this is a Sylow $2$-subgroup of $G$, and therefore $N$ detects
the mod $2$ cohomology of $G$ by Lemma 4.2. All representations of $G$ are
orthogonal by the main result in [21].

###  5.1. Formulas for SWCs

The subgroup $N$ is an elementary abelian $2$-group of rank $r$, so as in
Section 2.5, the mod $2$ cohomology ring of $N\cong C_{2}^{r}$ may be written
as

$H^{*}(N)\cong
H^{*}(C^{r}_{2})\cong\mathbb{Z}/2\mathbb{Z}[v_{1},v_{2},\ldots,v_{r}].$

Let $T$ be the subgroup of diagonal matrices in $G$. That is

$T=\Bigg{\\{}\small\begin{pmatrix}a&&0\\\
0&&a^{-1}\end{pmatrix}:a\in\mathbb{F}_{q}^{\times}\Bigg{\\}}.$

The conjugation action of $T$ on $N$ is equivalent to the action of
$\mathbb{F}_{q}^{\times}$ on $\mathbb{F}_{q}$ through multiplication by
squares. Since $\mathbb{F}_{q}^{\times}$ has odd order, this action is
transitive on $\mathbb{F}_{q}-\\{0\\}$. Therefore:

######  Lemma 5.1.

$T$ acts transitively on the non-trivial linear characters of $N$.

Let $\widehat{N}=\operatorname{Hom}(N,\mathbb{C}^{\times})$, the character
group of $N$. From the above lemma, the $T$-orbits of $\widehat{N}$ are:

$\\{1\\}\;,\;\\{\chi:\chi\neq 1\\}.$

Let us define

$\operatorname{reg}(N)^{\prime}=\operatorname{reg}(N)\ominus
1=\bigoplus_{\chi\neq 1}\chi.$

Let $\pi$ be a representation of $G$. Since $\operatorname{res}^{G}_{N}\pi$ is
$T$-invariant, it is of the form

(13) $\begin{split}\operatorname{res}^{G}_{N}\pi&\cong\ell_{\pi}1\oplus
m_{\pi}\operatorname{reg}(N)^{\prime},\end{split}$

where $\ell_{\pi},m_{\pi}$ are non-negative integers.

######  Lemma 5.2.

Let $\pi$ be a non-trivial irreducible representation of $G$. Then,
$m_{\pi}=1$.

###### Proof.

For $r\geq 2$, it is known that $G$ has no non-trivial normal subgroups. For
non-trivial $\pi$, we must have $m_{\pi}\geq 1$ because ker$(\pi)=\mathbbm{1}$
and $N\not\leqslant\ker(\pi)$. Now from (13), we have

$\deg\pi=\ell_{\pi}+m_{\pi}(q-1).$

If $m_{\pi}>1$, then the sum $\ell_{\pi}+m_{\pi}(q-1)$ would be greater than
the highest possible degree for an irreducible representation of $G$, which is
$(q+1)$. So actually $m_{\pi}=1$.

For $r=1$, we have $\operatorname{SL}(2,2)\cong S_{3}$ with only two non-
trivial irreducible representations. That is $\pi$ is either the ‘sgn’
representation or the 2-dimensional standard representation of $S_{3}.$ Here
we can see from direct calculations that $m_{\pi}=1$.

∎

It follows that $m_{\pi}=\dim_{\mathbb{C}}(V/V^{G})$, where $V$ denotes the
representation space of $\pi$. It is also the number of non-trivial
irreducible constituents of $\pi$. We can get a character formula for
$m_{\pi}$ by taking character values at $n_{0}=\begin{pmatrix}1&&1\\\
0&&1\end{pmatrix}\in N.$ In fact,

$\displaystyle\chi_{\pi}(\mathbbm{1})$
$\displaystyle=\ell_{\pi}+(q-1)m_{\pi}\text{ and }$
$\displaystyle\chi_{\pi}(n_{0})$ $\displaystyle=\ell_{\pi}-m_{\pi},$

so that $m_{\pi}=\frac{1}{q}(\chi_{\pi}(\mathbbm{1})-\chi_{\pi}(n_{0}))$.

We are now ready to describe the SWCs of representations of $G$. Since $N$ is
a detecting subgroup, we may and will identify $w(\pi)$ with its image in
$H^{*}(N)$.

######  Theorem 5.3.

Let $q=2^{r}.$ Let $\pi$ be a representation of $\operatorname{SL}(2,q)$.
Then, the total SWC of $\pi$ is

$w(\pi)=\big{(}\prod_{v\in H^{1}(N)}(1+v)\big{)}^{m_{\pi}}.$

###### Proof.

The restriction $\operatorname{res}^{G}_{N}\pi$, from (13), is of the form

$\operatorname{res}^{G}_{N}\pi\cong(\ell_{\pi}-m_{\pi})1\oplus
m_{\pi}\operatorname{reg}(N).$

Since $\operatorname{reg}(N)=\sum\limits_{\chi\in\widehat{N}}\chi$, we use the
isomorphism between $\widehat{N}$ and $H^{1}(N)$ and the multiplicativity of
SWCs to obtain

$\displaystyle w(\operatorname{res}^{G}_{N}\pi)$
$\displaystyle=w(\operatorname{reg}(N))^{m_{\pi}}$
$\displaystyle=\big{(}\prod_{v\in H^{1}(N)}(1+v)\big{)}^{m_{\pi}}.$

∎

######  Remark 5.4.

The formula in Theorem 5.3 above is also valid for
$\widetilde{G}=\operatorname{GL}(2,q)$ with $q$ even, since $N$ is also a
Sylow $2$-subgroup of $\widetilde{G}$. Thus this result in some sense
completes [10].

###  5.2. Dickson Invariants

The expansion of the product above is well-known. We have

(14) $\prod_{v\in H^{1}(N)}(1+v)=1+\sum_{i=1}^{r}d_{i}\in H^{*}(N),\\\ $

where $d_{i}$ are _Dickson invariants_ in the algebra $H^{*}(N)$.

We digress for a moment to recount the theory of Dickson invariants for mod
$2$ spaces. Let $E$ be an $\mathbb{F}_{2}$-vector space. The ring of
polynomials from $E$ to $\mathbb{F}_{2}$ can be identified with the symmetric
algebra $S[E^{\vee}]$ on the dual space $E^{\vee}$. The linear group
$\operatorname{GL}(E)$ acts on $S[E^{\vee}]$ via the contragradient map. It is
natural to look for the invariants under this action.

######  Theorem 5.5 ([23]).

Suppose $\dim(E)=r$. Then the ring of invariants
$S[E^{\vee}]^{\operatorname{GL}(E)}$ is a polynomial algebra generated by
elements $d_{i}$ called Dickson invariants, for $1\leq i\leq r$. The
polynomial $d_{i}$ has degree $2^{r}-2^{r-i}$. Moreover,

$\prod_{v\in E^{\vee}}(1+v)=1+\sum_{i=1}^{r}{d_{i}}\in S[E^{\vee}].$

These invariants are described explicitly in terms of certain determinants.
(See [23], or [1, Chapter III] for instance.)

Let us illustrate with some examples. If $E=\mathbb{F}_{2}$, then
$S[E^{\vee}]$ is a polynomial algebra with only one generator $v$, that is
$S[E^{\vee}]\cong\mathbb{Z}/2\mathbb{Z}[v]$. Here $d_{1}=v$ giving the ring of
invariants

$S[E^{\vee}]^{\operatorname{GL}(V)}\cong\mathbb{Z}/2\mathbb{Z}[v]^{\operatorname{GL}(1,2)}=\mathbb{Z}/2\mathbb{Z}[v].$

Also Theorem 5.3 for $G=\operatorname{SL}(2,2)$ and $\pi\neq 1$ irreducible
says that

$w(\pi)=1+v=1+d_{1}.$

Next, suppose $\dim E=2$. Then $S[E^{\vee}]$ is a polynomial algebra with two
generators, say $v_{1},v_{2}$ such that
$S[E^{\vee}]\cong\mathbb{Z}/2\mathbb{Z}[v_{1},v_{2}]$. Here the ring of
invariants
$S[E^{\vee}]^{\operatorname{GL}(E)}\cong\mathbb{Z}/2\mathbb{Z}[v_{1},v_{2}]^{\operatorname{GL}(2,2)}$
is generated by

$\displaystyle d_{1}$ $\displaystyle=v_{1}^{2}+v_{2}^{2}+v_{1}v_{2},$
$\displaystyle d_{2}$ $\displaystyle=v_{1}v_{2}(v_{1}+v_{2}).$

Theorem 5.3 gives: For $G=\operatorname{SL}(2,4)$, the total SWC of a non-
trivial irreducible representation $\pi$ of $G$ is

$w(\pi)=1+d_{1}+d_{2}.$

Now let $\mathcal{D}=\sum\limits_{i=1}^{r}d_{i}$ be the sum of the Dickson
invariants, so that we can succinctly write

$w(\pi)=(1+\mathcal{D})^{m_{\pi}},$

which is Theorem 1.3 from the Introduction.

######  Example 5.6.

We have

$w(\operatorname{reg}(G))=(1+\mathcal{D})^{q^{2}-1}.$

###  5.3. Corollaries

Now we prove the corollaries mentioned in the Introduction.

######  Corollary 5.7.

Let $G=\operatorname{SL}(2,q)$ with even $q$. Then the image of $w$ in (6) is

$\\{(1+\mathcal{D})^{n}\mid n\in\mathbb{Z}\\}.$

###### Proof.

Let $\pi$ be a non-trivial irreducible representation of $G$. Then, its total
SWC is

$w(\pi)=1+\mathcal{D},$

since $m_{\pi}=1$ by Lemma 5.2. With the virtual representation $n\pi$, we
obtain
$(1+\mathcal{D})^{n}$ in the image of $w$ for each $n\in\mathbb{Z}$. ∎

The above proof also gives:

######  Corollary 5.8.

Let $G=\operatorname{SL}(2,q)$ with $q=2^{r}$. Then,

$H^{*}_{\operatorname{SW}}(G)=\mathbb{Z}/2\mathbb{Z}[d_{1},\ldots,d_{r}].$

######  Corollary 5.9.

Let $G=\operatorname{SL}(2,q)$ with $q$ even. Let $(\pi,V)$ be a
representation of $G$. Then, $w_{\operatorname{top}}(\pi)\neq 0$ if and only
if $V^{N}=\\{0\\}$.

###### Proof.

From the preceding, we have

$w_{\operatorname{top}}(\operatorname{reg}(N)^{\prime})=d_{r},$

since this invariant has degree $q-1$. The highest degree term of $w(\pi)$ is
therefore

$d_{r}^{m_{\pi}},$

which has degree $m_{\pi}(q-1)$; this is the degree of $\pi$ iff
$\ell_{\pi}=0$ iff $V^{N}=\\{0\\}$. ∎

######  Remark 5.10.

A representation $\pi$ of $\operatorname{SL}(2,q)$ is cuspidal iff
$V^{N}=\\{0\\}$. (See [2, page 410].)

######  Corollary 5.11.

Let $G=\operatorname{SL}(2,q)$ with $q=2^{r}$. Let $\pi$ be a representation
of $G$. Put $s=\operatorname{ord}_{2}(m_{\pi})$. Then, the obstruction class
of $\pi$ is equal to $w_{2^{r+s-1}}(\pi)=d_{1}^{2^{s}}.$

###### Proof.

From Theorem 1.3, we have

$w(\pi)=\sum_{i=0}^{m_{\pi}}{m_{\pi}\choose i}\mathcal{D}^{i}.$

As in Corollary 4.10, we can say that $m_{\pi}\choose 2^{s}$ is the first odd
binomial coefficient appearing in the above sum. By expanding the term

${m_{\pi}\choose 2^{s}}\mathcal{D}^{2^{s}}={m_{\pi}\choose
2^{s}}(d_{1}^{2^{s}}+\ldots+d_{r-1}^{2^{s}}+d_{r}^{2^{s}}),$

we can deduce that ${m_{\pi}\choose 2^{s}}d_{1}^{2^{s}}$ has the least degree,
which is $(2^{r-1}\cdot 2^{s})$. Therefore, the least $k>0$ such that
$w_{k}(\pi)\neq 0$ is $2^{r+s-1}$ as claimed. ∎

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††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Comparing tunneling spectroscopy and charge sensing of
Andreev bound states in a semiconductor-superconductor hybrid nanowire
structure

Deividas Sabonis  Center for Quantum Devices, Niels Bohr Institute, University
of Copenhagen, 2100 Copenhagen, Denmark  Microsoft Quantum Lab–Copenhagen,
Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
David van Zanten  Center for Quantum Devices, Niels Bohr Institute, University
of Copenhagen, 2100 Copenhagen, Denmark  Microsoft Quantum Lab–Copenhagen,
Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
Judith Suter  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, 2100 Copenhagen, Denmark  Microsoft Quantum Lab–Copenhagen, Niels
Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark  Torsten
Karzig  Microsoft Quantum, Station Q, University of California, Santa Barbara,
California 93106-6105, USA  Dmitry I. Pikulin  Microsoft Quantum, Station Q,
University of California, Santa Barbara, California 93106-6105, USA  Microsoft
Quantum, Redmond, Washington 98052, USA  Jukka I. Väyrynen  Microsoft Quantum,
Station Q, University of California, Santa Barbara, California 93106-6105, USA
Department of Physics and Astronomy, Purdue University, West Lafayette,
Indiana 47907, USA  Eoin O’Farrell  Center for Quantum Devices, Niels Bohr
Institute, University of Copenhagen, 2100 Copenhagen, Denmark  Microsoft
Quantum Lab–Copenhagen, Niels Bohr Institute, University of Copenhagen, 2100
Copenhagen, Denmark  Davydas Razmadze  Center for Quantum Devices, Niels Bohr
Institute, University of Copenhagen, 2100 Copenhagen, Denmark  Microsoft
Quantum Lab–Copenhagen, Niels Bohr Institute, University of Copenhagen, 2100
Copenhagen, Denmark  Peter Krogstrup  Center for Quantum Devices, Niels Bohr
Institute, University of Copenhagen, 2100 Copenhagen, Denmark  Microsoft
Quantum Materials Lab–Copenhagen, 2800 Lyngby, Denmark  Charles M. Marcus
Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen,
2100 Copenhagen, Denmark  Microsoft Quantum Lab–Copenhagen, Niels Bohr
Institute, University of Copenhagen, 2100 Copenhagen, Denmark

###### Abstract

Transport studies of Andreev bound states (ABSs) are complicated by the
interplay of charging effects and superconductivity. Here, we compare
transport approaches to ABS spectroscopy in a semiconductor-superconductor
island to a charge-sensing approach based on an integrated radio-frequency
single-electron transistor. Consistency of the methods demonstrates that fast,
non-invasive charge sensing allows accurate quantitative measurement of ABSs
while eluding some complexities of transport.

††preprint: APS/123-QED

Experiments with superconductor-semiconductor hybrid structures have
demonstrated the presence of Andreev bound states (ABSs) in proximitized
semiconductors by a variety of methods, including tunneling spectroscopy in
NIS junctions (N is normal, S is superconductor, I is an insulating tunnel
barrier), Coulomb blockade spectroscopy in superconducting islands with normal
leads (NISIN) Higginbotham  _et al._ (2015); Albrecht  _et al._ (2016); Shen
_et al._ (2018); Deng  _et al._ (2016); Vaitiekėnas  _et al._ (2020), and SNS
spectroscopy of Josephson junctions Laroche  _et al._ (2019); van Zanten  _et
al._ (2020); Sabonis  _et al._ (2020); Kringhøj  _et al._ (2021); Tosi  _et
al._ (2019); Hays  _et al._ (2020, 2018); van Woerkom  _et al._ (2017). These
methods reveal distinct spectroscopic features and raise different technical
challenges. With growing interest in detecting and controlling Majorana zero
modes (MZMs) across Josephson junctions Ginossar and Grosfeld (2014); Hassler
_et al._ (2011); Hyart  _et al._ (2013) and distinguishing them from ABSs, it
is important to compare methods in systems that can be probed several ways.

In this Article, we report measurements of an electrostatically gated multi-
segment hybrid InAs/Al nanowire (NW) device [see Fig. 1(a)] that is
reconfigurable in situ, allowing a comparison of spectroscopic signatures of
the same ABS by three methods: a Josephson junction (SIS), a Coulomb island
(SISIN), and a radio-frequency (RF) charge sensor. We observed consistency
between methods, noting that RF charge detection was especially fast, and,
unlike transport, does not alter charge occupancy during measurement.

Figure 1: (a) False coloured scanning electron micrograph of the device
consisting of a InAs nanowire with segments of epitaxial Al (scale bar 1
$\mu$m). The main segment of the nanowire defines the Coulomb blockaded
superconducting island (S) that is separated with the gate-tuneable barriers
(I) from the superconducting segment on the left (S) and normal segment (N) on
the right. Voltages applied to plunger gates P1, P2 and P3 tune the carrier
density in the proximitized regions whereas gates C1 and C2 are used for
controlling the tunnel barrier height. The island is capacitively coupled to a
single-electron transistor charge sensor at the right end of the nanowire that
is a part of the resonant circuit connected to a right lead (green). (b)
Schematics of the superconductor-insulator-superconductor system (SIS). The
induced superconducting gap ($\Delta$) and subgap state energy ($\delta$) on
the island side are indicated together with bias voltage $V_{B}$. (c)
Schematics of the superconducting island system confined by barriers (I) from
superconducting contact on the left (S) and normal contact on the right (N).
(d) SIS tunneling conductance $g$ as a function of axial magnetic field $B$
and source-drain bias $V_{B}$. Starting at $B$ = 0.6 T (red marker) the
feature in conductance at $V_{B}$ = 220 $\mu$V becomes independent on the
further increase in magnetic field.

The device was fabricated from an InAs NW grown by molecular beam epitaxy,
with Al grown on three facets of the hexagonal core. Using conventional
lithographic processing, Al was removed by wet etching in $\sim$100 nm
segments adjacent to cutter gates C1 and C2, providing gate-controllable
barriers. Blanket atomic layer deposited ${\rm HfO}_{2}$ insulated all gates
from the NW. Depleting the InAs with C1 and C2 yielded an SISIN device with an
S island separated by tunnel barriers (I) from an S lead on the left and N
lead on the right. Voltages applied to plunger gates P1-P3 tuned carrier
density in adjacent proximitized semiconductor regions as well as the charge
offset on the S island. At the far right, a bare InAs segment (again, created
by etching of Al), served as RF single-electron transistor (RF-SET) that was
capacitively coupled to the main S island by a floating gate [see Fig. 1(a)].
Data acquisition followed Ref. van Zanten  _et al._ (2020) and charge sensing
followed Ref. Razmadze  _et al._ (2019). Measurements were performed in a
dilution refrigerator with a 6-1-1 T vector magnet.

To generate ABSs in the nanowire, we applied a moderate magnetic field, $B$ =
600 mT, along the nanowire axis and set gate C1 to the tunneling regime
(conductance $g\ll e^{2}/h$), junction C2 to the open regime ($g\sim
e^{2}/h$), and gates P1-P3 to $-4$ V to reduce density in the NW. In this
configuration, SIS spectroscopy [Figs. 1b)] was used to locate a single subgap
state by tuning P3. Figure 1(d) shows differential conductance $g$ as a
function of bias $V_{B}$ and $B$ in this SIS configuration. Near $B=0$ a
precursor of the finite-field ABS is seen at $V_{B}~{}\sim~{}\pm 330~{}\mu$V.
Increasing $B$ leads to a single ABS moving down with an effective $g$ factor
of $\sim 6$. Starting at $B\sim 0.6$ T [red marker in Fig. 1(d)] the feature
at $V_{B}$ = 220 $\mu$V becomes independent of the further increase $B$. This
presumably results from competition between a reduction of the induced
superconducting gap and a zero-energy-crossing ABS Shen  _et al._ (2018).
Unless otherwise noted, measurements described below are at $B$ = 0.6 T.

The location of the ABS within the NW could be inferred from its dependence on
P1 and P2. Specifically, spectroscopy showed little response to $V_{P1}$ and a
strong response to $V_{P2}$, as shown in Figs. 2(a,b), indicating that the ABS
was located on the right side of cutter gate C1, as represented in Fig. 1(b).
The energy of the ABS approached the gap edge for $V_{P2}\sim-1.6$ V and
$\sim-1.35$ V with a minimum at around $V_{P2}\sim-1.5$ V. Differential
conductance as a function of bias, $V_{B}$, and gate P1 [Fig. 2(c)] were
carried out at $V_{P2}=-1.49$ V.

Figure 2: (a) Two dimensional map of superconductor-insulator-superconductor
tunneling differential conductance $g$, as a function of source-drain bias
$V_{B}$ and gate voltage $V_{P2}$ at $B$ = 0.6 T. ABS shows a response to the
voltage on plunger P2. (b) Differential conductance $g$ as a function of
$V_{B}$ and voltage $V_{P1}$ measured at $V_{P2}$ = $-$1.490 V. A weak
response of the ABS with respect to gate P1 indicates the location of the
state as being on the right side of the barrier (superconducting island). (c)
Line-cuts at red and black marker positions in (a), at the minimum ABS energy
(red) and with the ABS energy merging with the continuum (black).

Following SIS spectroscopy, gates C1 and C2 were set to the tunneling regime
to create a superconducting island, providing SISIN spectroscopy. Figure 3(a)
shows the resulting two-dimensional conductance map as a function of bias and
island plunger voltage $V_{P3}$. In the SISIN configuration, transport showed
1$e$-periodic resonances at high bias, $V_{B}~{}\gtrsim~{}400~{}\mu$V, and
2$e$-periodic features at low bias. The black dashed diagonal line in Fig.
3(a) indicates the alignment of the chemical potential of the left lead and
the island. Four features along this diagonal are highlighted: Near $V_{B}=0$,
faint 2$e$-periodic features that become stronger along the diagonal at
$V_{B}\sim 65~{}\mu$V ($\bullet$), accompanied by negative differential
conductance (NDC); Further increasing bias yielded stronger 1$e$-periodic
features starting at $V_{B}\sim 160~{}\mu$V ($\blacksquare$) along with
stronger NDC; Around $V_{B}\sim 240~{}\mu$V ($\blacklozenge$) the conductance
increase was largely independent of $V_{P3}$; Finally, at
$V_{B}~{}\sim~{}300~{}\mu$V the strong 1$e$-periodic conductance resonances
($\bigstar$) were observed.

Figure 3: (a) Coulomb blockade measured in differential conductance $g$ as a
function of the island plunger voltage $V_{P3}$ and a source-drain bias
$V_{B}$ at $V_{P2}$ = $-$1.456 V and axial magnetic field $B$ = 0.6 T. Pink
marker indicates the bias $V_{B}$ value at which the 2$e$-periodic negative
differential conductance is observed whereas green marker indicates the
position of 1$e$ negative differential conductance. The diagonal dashed line
indicates the $V_{B}$ \- $V_{P3}$ configuration for which the superconducting
island and the left superconducting lead stay at the same potential. Four
markers along the diagonal trace indicated the threshold values for the bias
that enable the transport processes discussed in the main text. (b) Line-cuts
from (a) at the pink and the green marker positions in (a) indicating 2$e$ and
1$e$-periodic features together with the associated negative differential
conductance.

These marked features in Fig. 3(a) are consistent with a model of an SISIN
device with a single ABS at energy $\delta$ on the S island with electrostatic
energy $E_{C}(n-n_{g})^{2}$, where $E_{C}$ is the charging energy, $n$ is the
number of electrons on the island, and $n_{g}$ is the dimensionless gate
voltage. In particular, the 2$e$-periodic features at zero bias arise from
resonant Cooper pair tunneling between S lead and S island at odd integer
values of $n_{g}$. At increased bias, the resonance occurs at
$2V_{B}=4E_{C}n_{g}$ [dashed line in Fig. 3(a)], with the S island on
resonance with the left S lead while the energy difference from the Fermi
level in the N lead increases. For the case $\delta<E_{C}$, the 2$e$-periodic
features are expected to be suppressed around zero bias, as Cooper pair
resonances are between excited states above the odd parity ground state of the
occupied ABS close to odd integer values of $n_{g}$. The 2$e$-periodic
features around zero bias in Fig. 3(a) suggest that indeed $\delta<E_{C}$. At
$V_{B}=E_{C}-\delta$ [$\bullet$ in Fig. 3(a)], the ABS can empty into the
right normal lead, enabling resonant Cooper pair tunneling, brightening the
2$e$-periodic feature. At yet larger bias with fixed $n_{g}$, the resonance
condition for Cooper pair tunneling from left is exceeded, resulting in NDC.
At $V_{B}~{}=~{}\Delta_{L}+\delta-E_{C}$ [$\blacksquare$ in Fig. 3(a)] above-
gap quasiparticles in the left S lead can excite the ABS thus allowing single-
electron transport that no longer requires resonant Cooper pair processes,
resulting in 1$e$-periodic resonances in $n_{g}$. Here, NDC presumably results
when the coherence peak in the density of states the S lead surpasses the
bound state energy. At $V_{B}~{}=~{}\Delta_{L}$ [$\blacklozenge$ in Fig. 3(a)]
quasiparticles from the S lead can transfer to the N lead via elastic
cotunneling through the ABS or other subgap states, which are only weakly
dependent on $n_{g}$. Finally, at $V_{B}=\Delta_{L}+\Delta_{I}-E_{C}$
[$\bigstar$ in Fig. 3(a)] single-electron transport between continuum states
of the left S lead and the S island become energetically allowed. Figure 3(b)
shows line cuts from Fig. 3(a), indicating both 2$e$ (pink) and 1$e$ (green)
features along with the associated NDC. Further details of the four features
are discussed in Appendix .2.

We extract a charging energy $E_{C}$ $\sim$ 85 $\mu$eV from independent
transport data of Coulomb diamonds (not shown). Using this value, the above
model is consistent with an ABS energy $\delta=20~{}\mu$eV and superconducting
gaps $\Delta_{L}\sim 220~{}\mu$eV and $\Delta_{I}\sim 160~{}\mu$eV, in the
left S lead and S island, both consistent with values seen in similar hybrid
nanowires Chang  _et al._ (2015). The condition $E_{C}<\Delta_{I}$ is
consistent with the 2$e$ features at low bias in Fig. 3(a).

Transport features observed in SISIN spectroscopy are rather complicated, even
with the relatively simple situation of a single ABS on the island. Previous
work has showed that RF charge readout offers some advantages compared to
transport, particularly speed and simplicity of the data Razmadze  _et al._
(2019); van Veen  _et al._ (2019); de Jong  _et al._ (2019). To perform charge
readout on the present device, we use the integrated charge sensor and examine
the same ABS investigated above via transport.

Figure 4(a) shows transport-based Coulomb blockade spectroscopy in the SISIN
configuration performed over a larger range of P2 gate voltages. The effect of
the ABS can be seen as an envelope modulation of Coulomb blockade that follows
the shape of the ABS that was previously shown in the Fig. 2(a). To emphasize
that the modulation is connected to the ABS, red markers in Fig. 4(a) indicate
the tips of Coulomb diamonds. Charge sensing data from the same gate regime as
Fig. 4(a) is shown in Fig. 6. Charge sensing shows explicitly the transition
in periodicity of superconducting island occupancy whenever the bound state
energy $\delta$ is lowered with respect to charging energy $E_{C}$ of the
island. This change in occupancy was implied by the model, but not explicitly
visible in transport data. Figure 4(b) shows the effect of the 2$e$ to 1$e$
transition as a function of plunger voltage $V_{P2}$ measured with the the
integrated RF-SET. The small overshoot of the even level (blue) at $V_{P2}$ =
$-$1.51 V is associated to ABS that is oscillating around zero energy as a
function of $V_{P2}$. Gate P3 in the Fig. 6 is also responsible for tuning the
ABS however with a smaller lever arm compared to P2 compatible with the ABS
being localized closer to the barrier C1 side. We note that the suppression of
the 2$e$-periodic signal around zero bias in Fig. 4(a) is correlated with the
onset of 1$e$-periodicity in Fig. 4(b). This indicates that the exchange of
Cooper pairs between the island and superconducting left lead, despite being
aligned in chemical potential, is indeed suppressed when $\delta<E_{C}$. The
latter requirement is supported by the observation of re-entrant resonant
Cooper pair processes at $V_{B}~{}=~{}E_{C}-\delta$.

Figure 4: (a) Coulomb blockade spectroscopy measured in conductance $g$
through the island in the extended plunger P2 voltage range. The effect of the
ABS can be indicated as an envelope modulation of Coulomb blockade features
following the shape of the ABS from the Fig. 2(a) (red markers around the tip
of Coulomb diamonds). (b) Normalised peak spacing of even (blue) and odd
(green) Coulomb valleys as a function of $V_{P2}$ extracted from charging
sensing data (see Fig. 6). The 2$e$ to 1$e$ transition in periodicity of
Coulomb blockade results from the ABS energy becoming smaller than the
charging energy, controlled by $V_{P2}$. (c) Bound state profile and energy
extracted by three different methods: SIS transport spectroscopy (black),
SISIN Coulomb blockade spectroscopy (red) and RF charge sensing - in a gate
voltage regime where the charge sensing data was acquired (green). Green and
red data have been shifted horizontally by 30 mV to compensate capacitive
coupling to $V_{C2}$.

The three measurements methods of the same ABS are compared in Fig. 4(c).
Black data points show the ABS profile extracted using SIS spectroscopy [Fig.
2(a)], red - Coulomb blockade measurements where the ABS energy $\delta$ is
determined using the onset of Coulomb blockade features [Fig. 4(a)] and green
- RF-SET measurements [Fig. 6]. For SIS and SISIN transport measurements both
positive and negative values of $V_{B}$ were used. An offset in $V_{C2}$ of
$\sim$ 30 mV between SIS and SISIN measurements, presumably due to cross-
capacitance to $V_{C2}$ has been removed. Correspondence between the ABS
energy profiles measured using transport (SIS and SISIN) and charge sensing
shows that the charge periodicity of the island is indeed affected by the ABS
energy. It also shows that RF measurements with increased acquisition rate
Razmadze  _et al._ (2019); de Jong  _et al._ (2021) is a useful alternative to
transport.

## Appendices

###  .1 Transport processes

Figure 5 shows schematics of the transport processes corresponding to the
markers in Fig. 3(a), within our model. In Fig. 5(a) the 2$e$-periodic process
at $V_{B}$ = $E_{C}$ $-$ $\delta$ ($\bullet$) is visualized. Typically Cooper
pair transfer is resonant around odd integer $n_{g}$ values. However, in the
case of a subgap state energy $\delta<E_{C}$ the 2$e$ transport is blocked. At
$V_{B}>E_{C}$ $-$ $\delta$ the transport can get unblocked by emptying the
subgap state into the right normal lead. Panel (b) depicts 1$e$-periodic
process at bias $V_{B}$ = $\Delta_{L}$ $-$ ($E_{C}$ $-$ $\delta$)
($\blacksquare$). The island can relax its energy when going from the even to
the odd parity state around odd integer $n_{g}$ values or from odd parity to
even parity at even $n_{g}$ values. The relaxation reduces the bias $V_{B}$
threshold for single-electron transfers from the left lead. In Fig. 5(c) a
single-electron process at $V_{B}$ = $\Delta_{L}$ ($\blacklozenge$) can be
associated with elastic cotunneling through the bound state or other subgap
states. In Fig. 5(d), again, a 1$e$-periodic process at bias $V_{B}$ =
$\Delta_{L}$ \+ $\Delta_{I}$ $-$ $E_{C}$ ($\bigstar$) happens. This process is
similar to (b). The system can save an energy $E_{C}$ for the threshold when
going from even parity state to odd parity state around odd integer $n_{g}$ or
from odd to even parity at even integer values of $n_{g}$. Both (b) and (d)
require poisoning or some other way of bringing the island out of its ground
state. Since both of these are at substantial bias $V_{B}$ $\sim$ $\Delta_{L}$
it becomes possible to create the corresponding excited quasiparticle states,
for example by splitting a Cooper pair from the left lead (thus gaining energy
$2V_{B}$) to create two quasiparticles on the island.

Figure 5: Model of transport processes through the superconducting island for
different bias configurations. (a) 2$e$-periodic process at $V_{B}$ = $E_{C}$
$-$ $\delta$. (b) 1$e$-periodic process at bias $V_{B}$ = $\Delta_{L}$ $-$
($E_{C}$ $-$ $\delta$). (c) 1$e$-periodic process constant in $n_{g}$ at
$V_{B}$ = $\Delta_{L}$ (elastic cotunneling through the bound state or other
subgap states). (d) 1$e$-periodic process at bias $V_{B}$ = $\Delta_{L}$ \+
$\Delta_{I}$ $-$ $E_{C}$. This process is similar to (b). The system can save
an energy $E_{C}$ for the threshold when going from even to odd parity state
around odd integer $n_{g}$ values or from odd to even parity at even integer
$n_{g}$.

###  .2 Charge sensing of bound state

The charge sensor was capacitively coupled to the superconducting island via
floating coupler. Fig. 6 shows a two dimensional gate map recorded in
demodulated voltage $V_{\rm RF}$ from the charge sensor as a function of gate
voltages $V_{P2}$ and $V_{P3}$ (line average along $V_{P3}$ axis subtracted).
Measurements are performed at $V_{B}$ = 0. Constant charge positions appear as
bright plateaus interrupted by transitions between the two charge states
(dark). The jump at $V_{P2}$ = $-$1.485 V is likely due to electrostatic
background charges in the nanowire environment or in the dielectric covering
the nanowire. The effect of the ABS can be seen as a change of gate space
periodicity along the $V_{P2}$ axis. It starts from 2$e$-periodic behaviour at
more negative values of $V_{P2}$. Then as the ABS energy $\delta$ is lowered
starting at $V_{P2}$ = $-$1.490 V it transitions through the even-odd and
eventually 1$e$ regime recovering back to 2$e$ for more positive values of
$V_{P2}$. Gate P3 is also responsible for tuning the ABS however with a
smaller lever arm compared to P2 compatible with the ABS being localised
closer to the barrier C1 side.

Figure 6: Two dimensional gate map measured in charge sensor response $V_{\rm
RF}$ as a function of plunger voltages $V_{P3}$ and $V_{P2}$. The map shows
that the island changes its periodicity from the 2$e$ at lower end of $V_{P2}$
voltages, to even-odd and eventually 1$e$ whenever the energy of the bound
state $\delta$ is lowered below the charging energy $E_{C}$ of the
superconducting island. The system recovers 2$e$ periodicity for more positive
values of $V_{P2}$.

###  .3 Fabrication and measurement

The 100 nm diameter nanowire is grown using the vapor-liquid-solid technique
in a molecular beam epitaxy system with the InAs [111] substrate crystal
orientation. Following the growth, Al is deposited epitaxially in situ on
three facets of the nanowire with an average thickness of 10 nm. The nanowire
is then positioned by hand on the substrate using a micro-manipulator tool,
which gives few-micrometer placement precision. Using standard lithography,
the nanowire is patterned and Al is etched away where tunneling barrier
controlling gates are defined. Ti/Au contacts and gates are then patterned and
thermally evaporated.

Signal demodulation was carried out using a Zurich Instruments Ultra High
Frequency Lock-in (ZIUHFLI) access: https://www.zhinst.com/products/uhfli
which uses digital processing but the principle of operation is similar to
analog demodulation.

## Acknowledgments

We thank Shivendra Upadhyay for help with fabrication. Research was supported
by Microsoft, the Danish National Research Foundation, and the European
Research Council under grant 716655. Judith Suter acknowledges financial
support from the Werner Siemens Foundation Switzerland.

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