Modules induced from a normal subgroup of prime index

###### Abstract.

Let $G$ be a finite group and $H$ a normal subgroup of prime index $p$. Let
$V$ be an irreducible $\mathbb{F}H$-module and $U$ a quotient of the induced
$\mathbb{F}G$-module $V{\uparrow}$. We describe the structure of $U$, which is
semisimple when $\textup{char}(\mathbb{F})\neq p$ and uniserial if
$\textup{char}(\mathbb{F})=p$. Furthermore, we describe the division rings
arising as endomorphism algebras of the simple components of $U$. We use
techniques from noncommutative ring theory to study
$\textup{End}_{\mathbb{F}G}(V{\uparrow})$ and relate the right ideal structure
of $\textup{End}_{\mathbb{F}G}(V{\uparrow})$ to the submodule structure of
$V{\uparrow}$.

2000 Mathematics subject classification: 20C40, 16S35

##  1\. Introduction

Throughout this paper $G$ will denote a finite group and $H$ will denote a
normal subgroup of prime index $p$. Furthermore, $V$ will denote an
irreducible (right) $\mathbb{F}H$-module, and
$V{\uparrow}=V\otimes_{\mathbb{F}H}\mathbb{F}G$ is the associated induced
$\mathbb{F}G$-module. Let $a$ be an element of $G$ not in $H$, and let
$\Delta:=\textup{End}_{\mathbb{F}H}(V)$ and
$\Gamma:=\textup{End}_{\mathbb{F}G}(V{\uparrow})$.

This paper is motivated by the following problem: “Given an irreducible
$\mathbb{F}H$-module $V$, where $\mathbb{F}$ is an arbitrary field, and a
quotient $U$ of $V{\uparrow}$, determine the submodule structure of $U$ and
the endomorphism algebras of the simple modules.” By Schur’s lemma, $\Delta$
is a division algebra over $\mathbb{F}$, so we shall need techniques from
noncommutative ring theory.

We determine the submodule structure of $U$ by explicitly realizing
$\textup{End}_{\mathbb{F}G}(U)$ as a direct sum of minimal right ideals, or as
a local ring. It suffices to solve our problem in the case when
$U=V{\uparrow}$. Henceforth $U=V{\uparrow}$.

In the case when $\mathbb{F}$ is algebraically closed of characteristic zero,
it is well known that two cases arise. Either $V$ is $G$-stable and
$V{\uparrow}$ is irreducible, or $V$ is not $G$-stable and $V{\uparrow}$ is a
direct sum of $p$ pairwise nonisomorphic irreducible submodules. In [GK96] the
structure of $V{\uparrow}$ is analyzed in the case when $\mathbb{F}$ is an
arbitrary field satisfying $\textup{char}(\mathbb{F})\neq 0$. The assumption
that $\textup{char}(\mathbb{F})\neq 0$ was made to ensure that $\Delta$ is a
field. The main theorem of [GK96] states that the structure of $V{\uparrow}$
is divided into five cases when $V$ is $G$-stable. In this paper, we drop the
hypothesis that $\textup{char}(\mathbb{F})\neq 0$, and even more cases arise
in the stable case (Theorems 5, 8 and 9). Fortunately, all these cases can be
unified by considering the factorization of a certain binomial $t^{p}-\lambda$
in a twisted polynomial ring $\Delta[t;\alpha]$, which is a (left and right)
principal ideal domain.

As we will focus on the case when $\mathbb{F}$ need not be algebraically
closed, a crucial role will be played by the endomorphism algebra
$\Delta=\textup{End}_{\mathbb{F}H}(V)$. In [GK96] the submodules of
$V{\uparrow}$ are described _up to isomorphism_. As this paper is motivated by
computational applications we will strive towards a higher standard: an
explicit description of the vectors in the submodule, and an explicit
description of the matrices in the endomorphism algebra of the submodule. This
is easily achieved in the non-stable case, which we describe for the sake of
completeness.

##  2\. The non-stable case

Let $e_{0},e_{1},\dots,e_{d-1}$ be an $\mathbb{F}$-basis for $V$ and let
$\sigma\colon H\to\textup{GL}(V)$ be the representation afforded by the
irreducible $\mathbb{F}H$-module $V$ relative to this basis. The _$g$
-conjugate_ of $\sigma$ ($g\in G$) is the representation
$g\mapsto(ghg^{-1})\sigma$, and we say that $\sigma$ is _$G$ -stable_ if for
each $g\in G$, $\sigma$ is equivalent to its $g$-conjugate. In this section we
shall assume that $\sigma$ is _not_ $G$-stable.

Let $\sigma{\uparrow}\colon G\to\textup{GL}(V{\uparrow})$ be the
representation afforded by $V{\uparrow}$ relative to the basis

$e_{0},\dots,e_{d-1},e_{0}a,\dots,e_{d-1}a,\dots,e_{0}a^{p-1},\dots,e_{d-1}a^{p-1}.$

Note that $G/H=\langle aH\rangle$ has order $p$, and we are writing
$e_{i}a^{j}$ rather than $e_{i}\otimes a^{j}$. Then

$a\sigma{\uparrow}=\begin{pmatrix}0&I&&0\\\ &&\ddots&\\\ 0&0&&I\\\
a^{p}\sigma&0&&0\end{pmatrix},\qquad
h\sigma{\uparrow}=\begin{pmatrix}h\sigma&&&\\\ &({}^{a}h)\sigma&&\\\
&&\ddots&\\\ &&&({}^{a^{p-1}}h)\sigma\end{pmatrix}$

where $h\in H$ and ${}^{a^{i}}h=a^{i}ha^{-i}$. The elements of
$\textup{End}_{\mathbb{F}G}(V{\uparrow})$ are the matrices commuting with
$G\sigma{\uparrow}$, namely the $p\times p$ block scalar matrices ${\rm
diag}(\delta,\dots,\delta)$ where $\delta\in\Delta$.

We shall henceforth assume that $V$ is $G$-stable. In particular, assume that
we know $\alpha\in\textup{Aut}_{\mathbb{F}}(V)$ satisfying

$(aha^{-1})\sigma=\alpha(h\sigma)\alpha^{-1}\qquad\text{for all $h\in H$.}$

There are ‘practical’ methods for computing $\alpha$. A crude method involves
viewing $(ah_{i}a^{-1})\sigma\alpha=\alpha(h_{i}\sigma)$, where $H=\langle
h_{1},\dots,h_{r}\rangle$, as a system of $(d/e)^{2}r$ homogeneous linear
equations over $\Delta$ in $(d/e)^{2}$ unknowns where $e=|\Delta:\mathbb{F}|$.
The solution space is $1$-dimensional if $V$ is $G$-stable, and
$0$-dimensional otherwise. A more sophisticated method, especially when
$\textup{char}(\mathbb{F})\neq 0$, involves using the meat-axe algorithm, see
[P84], [HR94], [IL00], and [NP95]. There is also a recursive method for
finding $\alpha$ which we shall not discuss here.

##  3\. The elements of $\textup{End}_{\mathbb{F}G}(V{\uparrow})$

In this section we explicitly describe the matrices in
$\Gamma=\textup{End}_{\mathbb{F}G}(V{\uparrow})$ and give an isomorphism
$\Gamma\to(\Delta,\alpha,\lambda)$ where $(\Delta,\alpha,\lambda)$ is a
general cyclic $\mathbb{F}$-algebra, see [L91, 14.5]. It is worth recalling
that if $\textup{char}(\mathbb{F})\neq 0$, then the endomorphism algebra
$\Delta$ is commutative. Even in this case, though, $\Gamma$ can be
noncommutative.

######  Lemma 1.

If $i\in\mathbb{Z}$, then $\alpha^{-i}a^{i}\colon V\to Va^{i}$ is an
$\mathbb{F}H$-isomorphism between the submodules $V$ and $Va^{i}$ of
$V{\uparrow}{\downarrow}$.

###### Proof.

It follows from Eqn (1) that

$va^{i}ha^{-i}=v\alpha^{i}h\alpha^{-i}\qquad\text{($i\in\mathbb{Z}$, $v\in
V$)}$

Replacing $v$ by $v\alpha^{-i}$ gives $v\alpha^{-i}a^{i}h=vh\alpha^{-i}a^{i}$.
Hence $\alpha^{-i}a^{i}$ is an $\mathbb{F}H$-homomorphism, and since it is
invertible, it is an $\mathbb{F}H$-isomorphism. ∎

Conjugation by $\alpha$ induces an automorphism of $\Delta$, which we also
call $\alpha$. [Proof: Conjugating the equation $h\delta=\delta h$ by $\alpha$
and using (1) shows that $\alpha^{-1}\delta\alpha\in\Delta$.] We abbreviate
$\alpha^{-1}\delta\alpha=\delta^{\alpha}$ by $\alpha(\delta)$. The reader can
determine from the context whether the symbol $\alpha$ refers to an element of
$\textup{Aut}_{\mathbb{F}}(V)$, or $\textup{Aut}_{\mathbb{F}}(\Delta)$.

It follows from Eqn (1) that

$(a^{p}ha^{-p})\sigma=\alpha^{p}h\sigma\alpha^{-p}$

for all $h\in H$. Hence $\alpha^{-p}(a^{p}\sigma)$ centralizes $H$. Therefore
$\lambda:=\alpha^{-p}(a^{p}\sigma)$ lies in $\Delta^{\times}$. Setting
$h=a^{p}$ in Eqn (1) shows $a^{p}\sigma=\alpha(a^{p}\sigma)\alpha^{-1}$. Thus

$\alpha(\lambda)=\lambda^{\alpha}=(\alpha^{-p}(a^{p}\sigma))^{\alpha}=\alpha^{-p}(a^{p}\sigma)=\lambda.$

Conjugating by $\alpha^{p}=(a^{p}\sigma)\lambda^{-1}$ induces an inner
automorphism:

$\alpha^{p}(\delta)=\delta^{\alpha^{p}}=\delta^{(a^{p}\sigma)\lambda^{-1}}=\delta^{\lambda^{-1}}=\lambda\delta\lambda^{-1}\qquad\text{($\delta\in\Delta$).}$

In summary, we have proved

######  Lemma 2.

The element $\alpha\in\textup{Aut}_{\mathbb{F}}V$ satisfying Eqn (1) induces
via conjugation an automorphism of $\Delta=\textup{End}_{\mathbb{F}H}(V)$,
also called $\alpha$. There exists $\lambda\in\Delta^{\times}$ satisfying

$\alpha^{-p}(a^{p}\sigma)=\lambda,\quad\alpha(\lambda)=\lambda,\quad\alpha^{p}(\delta)=\lambda\delta\lambda^{-1}$

for all $\delta\in\Delta$.

######  Theorem 3.

The representation $\sigma{\uparrow}\colon G\to\textup{GL}(V{\uparrow})$
afforded by $V{\uparrow}$ relative to the $\mathbb{F}$-basis

$\displaystyle e_{0},$ $\displaystyle
e_{1},\dots,e_{d-1},\dots,e_{0}\alpha^{-i}a^{i},e_{1}\alpha^{-i}a^{i},\dots,e_{d-1}\alpha^{-i}a^{i},$
(3)
$\displaystyle\dots,e_{0}\alpha^{-(p-1)}a^{p-1},e_{1}\alpha^{-(p-1)}a^{p-1},\dots,e_{d-1}\alpha^{-(p-1)}a^{p-1}$

for $V{\uparrow}$ is given by

$a\sigma{\uparrow}=\begin{pmatrix}0&\alpha&&0\\\ &&\ddots&\\\ 0&0&&\alpha\\\
\alpha\lambda&0&&0\end{pmatrix},\quad
h\sigma{\uparrow}=\begin{pmatrix}h\sigma&&&\\\ &h\sigma&&\\\ &&\ddots&\\\
&&&h\sigma\end{pmatrix}$

where $h\in H$. Moreover, there is an isomorphism from the general cyclic
algebra $(\Delta,\alpha,\lambda)$ to $\textup{End}_{\mathbb{F}G}(V{\uparrow})$

$(\Delta,\alpha,\lambda)\to\textup{End}_{{\mathbb{F}G}}(V{\uparrow})\colon\sum_{i=0}^{p-1}\delta_{i}x^{i}\mapsto\sum_{i=0}^{p-1}D(\delta_{i})X^{i}$

where

$X=\begin{pmatrix}0&I&&0\\\ &&\ddots&\\\ 0&0&&I\\\
\lambda&0&&0\end{pmatrix},\quad D(\delta)=\begin{pmatrix}\delta&&&\\\
&\alpha(\delta)&&\\\ &&\ddots&\\\ &&&\alpha^{p-1}(\delta)\end{pmatrix}$

and $\delta\in\Delta$.

###### Proof.

By Lemma 1, $\alpha^{-i}a^{i}\colon V\to Va^{i}$ is an
$\mathbb{F}H$-isomorphism. Hence $h\sigma{\uparrow}$ is the $p\times p$ block
scalar matrix given by Eqn (4b). Similarly, $a\sigma{\uparrow}$ is given by
Eqn (4a) as

$(v\alpha^{-i}a^{i})a=v\alpha\alpha^{-(i+1)}a^{i+1}\quad\text{and}\quad(v\alpha^{-(p-1)}a^{p-1})a=v\alpha(\alpha^{-p}a^{p})=v\alpha\lambda$

where the last step follows from Eqn (2a).

We follow [J96] and write $R=\Delta[t;\alpha]$ for the twisted polynomial ring
with the usual addition, and multiplication determined by
$t\delta=\alpha(\delta)t$ for $\delta\in\Delta$. The right ideal
$(t^{p}-\lambda)R$ is two-sided as

$t(t^{p}-\lambda)=(t^{p}-\lambda)t\quad\text{and}\quad\delta(t^{p}-\lambda)=(t^{p}-\lambda)\lambda^{-1}\delta\lambda$

by virtue of Eqns (2b) and (2c). The general cyclic algebra
$(\Delta,\alpha,\lambda)$ is defined to be the quotient ring
$R/(t^{p}-\lambda)R$. Since $R$ is a (left) euclidean domain, the elements of
$(\Delta,\alpha,\lambda)$ may be written uniquely as
$\sum_{i=0}^{p-1}\delta_{i}x^{i}$ where $x=t+(t^{p}-\lambda)R$, and
multiplication is determined by the rules $x^{p}=\lambda$ and
$x\delta=\alpha(\delta)x$ where $\delta\in\Delta$.

The matrices commuting with $H\sigma{\uparrow}$ are precisely the block
matrices $(\delta_{i,j})_{0\leq i,j<p}$ where $\delta_{i,j}\in\Delta$. To
compute $\textup{End}_{\mathbb{F}G}(V{\uparrow})$, we determine the matrices
$(\delta_{i,j})$ that commute with $a\sigma{\uparrow}$. If $0<i<p-1$, then
comparing row $i-1$ of both sides of
$(a\sigma{\uparrow})(\delta_{i,j})=(\delta_{i,j})(a\sigma{\uparrow})$ shows
how to express $\delta_{i,j}$ in terms of the $\delta_{i-1,k}$. Similarly, row
$p-1$ shows how to express $\delta_{p-1,j}$ in terms of $\delta_{0,k}$. It
follows that a matrix $(\delta_{i,j})$ commuting with $a\sigma{\uparrow}$ is
completely determined once we know the $0$th row $\delta_{0,j}$. We show that
the $0$th row can be arbitrary. The $0$th row of
$\sum_{i=0}^{p-1}D(\delta_{i})X^{i}$ is
$(\delta_{0},\delta_{1},\dots,\delta_{p-1})$. This element lies in $\Gamma$ as
we show that $D(\delta_{i}),X\in\Gamma$.

To see that $D(\delta)\in\Gamma$, we show that
$(a\sigma{\uparrow})D(\delta)=D(\delta)(a\sigma{\uparrow})$. The first product
equals

$(a\sigma{\uparrow})D(\delta)=\begin{pmatrix}0&\delta\alpha&&0\\\ &&\ddots&\\\
0&0&&\alpha^{-(p-2)}\delta\alpha^{p-1}\\\
\alpha\lambda\delta&0&&0\end{pmatrix}$

and the second product is identical if
$\alpha\lambda\delta=\delta^{\alpha^{p-1}}\alpha\lambda$. However, this is
true by Eqn (2c). To see that $X\in\Gamma$, write $a\sigma{\uparrow}=AX$ where
$A=\textup{diag}(\alpha,\dots,\alpha)$. It follows from Eqn (2b) that $A$ and
$X$ commute. Therefore $a\sigma{\uparrow}=AX$ and $X$ commute.

In summary, elements of $\Gamma$ may be written uniquely as
$\sum_{i=0}^{p-1}D(\delta_{i})X^{i}$ where $\delta_{i}\in\Delta$. Since
$X^{p}=\lambda I$ and $XD(\delta)=D(\alpha(\delta))X$ it follows that the map
$\sum_{i=0}^{p-1}\delta_{i}x^{i}\mapsto\sum_{i=0}^{p-1}D(\delta_{i})X^{i}$ is
an isomorphism $(\Delta,\alpha,\lambda)\to\Gamma$ as claimed. ∎

A consequence of Eqn (2c) is that $\alpha$ has order $p$ or $1$ modulo the
inner automorphisms of $\Delta$. It follows from the Skolem-Noether theorem
[CR90,3.62] that the order of $\alpha$ modulo inner automorphisms is precisely
the order of the restriction $\alpha|Z$, where $Z=Z(\Delta)$ is the centre of
$\Delta$.

##  4\. The case when $\alpha|Z$ has order $p$

In this section we determine the structure of
$\Gamma:=\textup{End}_{\mathbb{F}G}(V{\uparrow})$ in the case when $\alpha$
induces an automorphism of order $p$ on the field $Z(\Delta)$.

Of primary interest to us is Part (a) of the following classical theorem.
Although this result can be deduced from [J96, Theorem 1.1.22] and the fact
that $t^{p}-\lambda$ is a ‘two-sided maximal’ element of $\Delta[t;\alpha]$,
we prefer to give an elementary proof which generalizes [L91, Theorem 14.6].

######  Theorem 4.

Let $\Gamma$ be the general cyclic algebra $(\Delta,\alpha,\lambda)$ where
$\lambda\neq 0$ and $\alpha(\lambda)=\lambda$. Suppose that $\alpha|Z(\Delta)$
has order $p$, and fixed subfield $Z_{0}$. Then
(a) $\Gamma$ is a simple $Z_{0}$-algebra,
(b) $C_{\Gamma}(\Delta)=Z(\Delta)$,
(c) $Z(\Gamma)=Z_{0}$, and
(d) $|\Gamma:Z_{0}|=(p\textup{Deg}(\Delta))^{2}$ where
$|\Delta:Z(\Delta)|=\textup{Deg}(\Delta)^{2}$.

###### Proof.

The following proof does not assume that $p$ is prime. Let
$\gamma=\gamma_{1}x^{i_{1}}+\cdots+\gamma_{r}x^{i_{r}}$ be a nonzero element
of an ideal $I$ of $\Gamma$, where $0\leq i_{1}<\cdots<i_{r}<p$,
$\gamma_{i}\in\Delta$, and $r$ is chosen minimal. By minimality, each
$\gamma_{i}$ is nonzero. To prove Part (a) it suffices to to prove that $r=1$.
Then $I=\Gamma$ as $\gamma_{1}x^{i_{1}}\in I$ is a unit because $\gamma_{1}$
and $x$ are both units. Assume now that $r>1$. Then

$(\gamma_{1}\alpha^{i_{1}}(\delta)\gamma_{1}^{-1})\gamma-\gamma\delta=\sum_{k=2}^{r}\left(\gamma_{1}\alpha^{i_{1}}(\delta)\gamma_{1}^{-1}\gamma_{k}-\gamma_{k}\alpha^{i_{k}}(\delta)\right)x^{i_{k}}$

lies in $I$ for each $\delta\in\Delta$. By the minimality of $r$, each
coefficient of $x^{i_{k}}$ is zero. This implies that $\alpha^{i_{1}}$ equals
$\alpha^{i_{k}}$ modulo inner automorphisms for $k=2,\dots,r$. This
contradiction proves Part (a).

The proofs of Parts (b) and (c) are straightforward, so we shall omit their
proofs. Part (d) follows from $|\Gamma:\Delta|=|Z(\Delta):Z_{0}|=p$, and
$|\Delta:Z(\Delta)|=\textup{Deg}(\Delta)^{2}$ is a square. ∎

Before proceeding to Theorem 5, we define the _left-_ and _right-twisted
powers_ , $\mu^{{\sqsupset}i}$ and $\mu^{i{\sqsubset}}$, where $\mu\in\Delta$
and $i\in\mathbb{Z}$. These expressions are like norms, indeed Jacobson [J96]
uses the notation $N_{i}(\mu)$ to suggest this. These “norms”, however, are
not multiplicative in general. Consider the twisted polynomial ring
$\Delta[t;\alpha]$ and define

$(\mu
t)^{i}=\mu^{{\sqsupset}i}t^{i},\quad\textup{and}\quad(t\mu)^{i}=t^{i}\mu^{i{\sqsubset}}$

for $\mu\in\Delta$ and $i\in\mathbb{Z}$. It follows from the power laws $(\mu
t)^{i}(\mu t)^{j}=(\mu t)^{i+j}$ and $((\mu t)^{i})^{j}=(\mu t)^{ij}$ that

$\mu^{{\sqsupset}i}\alpha^{i}(\mu^{{\sqsupset}j})=\mu^{{\sqsupset}(i+j)},\quad\textup{and}\quad\mu^{{\sqsupset}i}\alpha^{i}(\mu^{{\sqsupset}i})\cdots\alpha^{i(j-1)}(\mu^{{\sqsupset}i})=\mu^{{\sqsupset}(ij)}$

for $i,j\in\mathbb{Z}$. Similar laws hold for right-twisted powers. The left-
twisted powers of nonnegative integers can be defined by the recurrence
relation

$\mu^{{\sqsupset}0}=1,\quad\textup{and}\quad\mu^{{\sqsupset}(i+1)}=\mu^{{\sqsupset}i}\alpha^{i}(\mu)=\mu\alpha(\mu^{{\sqsupset}i})\quad\textup{for
$i\geq 0$},$

and negative powers can be defined by
$\mu^{{\sqsupset}-i}=\alpha^{i}(\mu^{{\sqsupset}i})^{-1}$.

It is important in the sequel whether or not $\lambda^{-1}$ has a left-twisted
$p$th root.

######  Theorem 5.

Let $V$ be a $G$-stable irreducible $\mathbb{F}H$-module where $H\triangleleft
G$ and $|G/H|=p$ is prime. Let $\alpha$ and $\lambda$ be as in Lemma 2.
Suppose that $\alpha|Z$ has order $p$ where $Z=Z(\Delta)$ and
$\Delta=\textup{End}_{{\mathbb{F}G}}(V)$.
(a) If the equation $\mu^{{\sqsupset}p}=\lambda^{-1}$ has no solution for
$\mu\in\Delta^{\times}$, then $V{\uparrow}$ is irreducible, and
$\textup{End}_{\mathbb{F}G}(V{\uparrow})$ is isomorphic to the general cyclic
algebra $(\Delta,\alpha,\lambda)$ as per Theorem 3.
(b) If $\mu\in\Delta^{\times}$ satisfies $\mu^{{\sqsupset}p}=\lambda^{-1}$,
then $V{\uparrow}=U(\mu_{0})\dotplus\cdots\dotplus U(\mu_{p-1})$ where

$U(\mu_{j})=V\sum_{i=0}^{p-1}\mu_{j}^{{\sqsupset}i}\alpha^{-i}a^{i}\qquad\qquad(j=0,1,\dots,p-1)$

are isomorphic irreducible submodules satisfying $U(\mu_{j}){\downarrow}\cong
V$, and where $\mu_{j}^{{\sqsupset}p}=\lambda^{-1}$. Moreover, if $\rho\colon
G\to\textup{GL}(U(\mu))$ is the representation afforded by $U(\mu)$ relative
to the basis $e^{\prime}_{0},\dots,e^{\prime}_{p-1}$ where

$e^{\prime}_{j}=e_{j}\sum_{i=0}^{p-1}\mu^{{\sqsupset}i}\alpha^{-i}a^{i}\qquad(j=0,1,\dots,d-1),$

then $a\rho=\alpha\mu^{-1}$, $h\rho=h\sigma$ for $h\in H$, and

$\textup{End}_{{\mathbb{F}G}}(U(\mu))=C_{\Delta}(\alpha\mu^{-1})=\\{\delta\in\Delta\mid\delta^{\alpha}=\delta^{\mu}\\}.$

###### Proof.

By Theorem 4(a), $(\Delta,\alpha,\lambda)$ is a simple ring. In Part(a) more
is true: $(\Delta,\alpha,\lambda)$ is a division ring by [J96, Theorem
1.3.16]. By Theorem 3, $(\Delta,\alpha,\lambda)$ is isomorphic to
$\textup{End}_{\mathbb{F}G}(V{\uparrow})$ and so we have proved that
$V{\uparrow}$ is irreducible as desired.

Consider Part (b). Let $s=\mu t$ be an element of the twisted polynomial ring
$\Delta[t;\alpha]$, then $s^{i}=(\mu t)^{i}=\mu^{{\sqsupset}i}t^{i}$ and

$s\delta=\mu t\delta=(\mu\alpha(\delta)\mu^{-1})\mu
t=\mu\alpha(\delta)\mu^{-1}s.$

Therefore the map
$\Delta[t;\alpha]\to\Delta[s;\alpha\mu^{-1}]\colon\kern-2.0pt\sum_{i=0}^{p-1}\delta_{i}t^{i}\mapsto\sum_{i=0}^{p-1}\delta_{i}(\mu^{{\sqsupset}i})^{-1}s^{i}$
is an isomorphism. We are abusing notation here by identifying
$\alpha\mu^{-1}$ in $\textup{Aut}_{\mathbb{F}}(V)$ with
$\delta\mapsto\delta^{\alpha\mu^{-1}}$ in $\textup{Aut}_{\mathbb{F}}(\Delta)$.
If $y=\mu x$, then

$y^{p}=(\mu x)^{p}=\mu^{{\sqsupset}p}\lambda=\lambda^{-1}\lambda=1.$

By taking quotients we get an isomorphism
$(\Delta,\alpha,\lambda)\to(\Delta,\alpha\mu^{-1},1)$ given by

$\sum_{i=0}^{p-1}\delta_{i}x^{i}\mapsto\sum_{i=0}^{p-1}\delta_{i}(\mu^{{\sqsupset}i})^{-1}y^{i}$

where $x=t+(t^{p}-\lambda)$ and $y=s+(s^{p}-1)$.

As $y^{p}-1=(y-1)(y^{p-1}+\cdots+y+1)$, and $\Delta[s,\alpha\mu^{-1}]$ is
right euclidean it follows that $(y-1)(\Delta,\alpha\mu^{-1},1)$ is a maximal
right ideal of $(\Delta,\alpha\mu^{-1},1)$. Now $y-1$ corresponds to $\mu x-1$
which corresponds to $D(\mu)X-1$ whose kernel gives rise to the irreducible
submodule $U(\mu)$ of $V{\uparrow}$ in the statement of Part(b). We shall
reprove this, and prove a little more, using a more elementary argument.

Let $U$ be a submodule of $V{\uparrow}$ satisfying $U{\downarrow}\cong V$. Let
$\phi\colon V\to V{\uparrow}$ be an $\mathbb{F}H$-homomorphism such that
$V\phi=U{\downarrow}$. Let $\pi_{i}\colon V{\uparrow}\to Va^{i}$ be the
$\mathbb{F}H$-epimorphism given by
$(\sum_{i=0}^{p-1}v_{i}\alpha^{-i}a^{i})\pi_{i}=v_{i}\alpha^{-i}a^{i}$. Then
$\delta_{i}=\phi\pi_{i}a^{-i}\alpha^{i}$ is an $\mathbb{F}H$-homomorphism
$V\to V$, or an element of $\Delta$. Since $\pi_{0}+\pi_{1}+\cdots+\pi_{p-1}$
is the identity map $1\colon V{\uparrow}\to V{\uparrow}$, it follows that

$\phi=\phi
1=\phi(\pi_{0}+\pi_{1}+\cdots+\pi_{p-1})=\sum_{i=0}^{p-1}\delta_{i}\alpha^{-i}a^{i}.$

We now view $\phi$ as a map $V\to U$ and note that $U=Ua$. Then
$\alpha^{-1}a\colon V\to Va$, $a^{-1}\phi a\colon Va\to Ua$ and
$\phi^{-1}\colon Ua\to V$ are each ${\mathbb{F}H}$-isomorphisms. Hence their
composite, $(\alpha^{-1}a)(a^{-1}\phi a)\phi^{-1}$ is an isomorphism $V\to V$,
denoted $\mu^{-1}$ where $\mu\in\Delta^{\times}$. Rearranging gives $\phi
a=\alpha\mu^{-1}\phi$. Therefore,

$(v\phi)a=\left(v\sum_{i=0}^{p-1}\delta_{i}\alpha^{-i}a^{i}\right)a=v\alpha\mu^{-1}\sum_{i=0}^{p-1}\delta_{i}\alpha^{-i}a^{i}$

for all $v\in V$. The expression $(v\delta_{i}\alpha^{-i}a^{i})a$ equals

$v\delta_{i}\alpha\alpha^{-(i+1)}a^{i+1}=v\alpha\delta_{i}^{\alpha}\alpha^{-(i+1)}a^{i+1}=v\alpha\mu^{-1}\delta_{i+1}\alpha^{-(i+1)}a^{i+1}.$

Setting $i=p-1$ gives

$(v\delta_{p-1}\alpha^{-(p-1)}a^{p-1})a=v\alpha\delta_{p-1}^{\alpha}\alpha^{-p}a^{p}=v\alpha\delta_{p-1}^{\alpha}\lambda=v\alpha\mu^{-1}\delta_{0}.$

Therefore $\delta_{i}^{\alpha}=\mu^{-1}\delta_{i+1}$ for $i=0,\dots,p-2$ and
$\delta^{\alpha}_{p-1}\lambda=\mu^{-1}\delta_{0}$. If $\delta_{0}=0$, then
each $\delta_{i}=0$ and $\phi=0$, a contradiction. Thus $\delta_{0}\neq 0$ and
as $V\delta_{0}^{-1}\phi=U$, we may assume that $\delta_{0}=1$. It follows
from Eqn (6) that $\delta_{i}=\mu^{{\sqsupset}i}$ is the solution to the
recurrence relation: $\delta_{0}=1$ and $\mu\delta_{i}^{\alpha}=\delta_{i+1}$
for $i\geq 0$. Furthermore $\mu\delta^{\alpha}_{p-1}=\lambda^{-1}$ implies
that $\mu^{{\sqsupset}p}=\lambda^{-1}$. In summary, any submodule $U$ of
$V{\uparrow}$ satisfying $U{\downarrow}\cong V$ equals $U(\mu)$ for some $\mu$
satisfying $\mu^{{\sqsupset}p}=\lambda^{-1}$. Furthermore, by retracing the
above argument, if $\mu^{{\sqsupset}p}=\lambda^{-1}$, then $U(\mu)$ is an
irreducible submodule of $V{\uparrow}$ satisfying $U{\downarrow}\cong V$.

As $\textup{End}_{{\mathbb{F}G}}(V{\uparrow})$ is a simple ring, $V{\uparrow}$
is a direct sum of isomorphic simple submodules. Therefore,
$V{\uparrow}=U(\mu_{0})\dotplus\cdots\dotplus U(\mu_{p-1})$ as desired. It
follows from Lemma 1 that the representation $\rho\colon G\to\textup{GL}(V)$
satisfies $a\rho=\alpha\mu^{-1}$ and $h\rho=h\sigma$ for $h\in H$.
Consequently, the matrices commuting with $G\rho$ equal the elements of
$\Delta$ centralizing $a\rho$. Hence
$\textup{End}_{\mathbb{F}G}(U(\mu))=C_{\Delta}(\alpha\mu^{-1})$ as claimed. ∎

##  5\. The case when $\alpha$ is inner

In this section assume that $\alpha|Z(\Delta)$ has order $1$, or equivalently
by the Skolem-Noether theorem, that $\alpha$ is inner. Fix
$\varepsilon\in\Delta^{\times}$ such that $\alpha$ is the inner automorphism
$\alpha(\delta)=\varepsilon^{-1}\delta\varepsilon$. Clearly
$\alpha(\varepsilon)=\varepsilon$ and by Eqn (2c)
$\varepsilon^{-p}\delta\varepsilon^{p}=\alpha^{p}(\delta)=\lambda\delta\lambda^{-1}$.
Therefore, $\eta=\varepsilon^{p}\lambda\in Z(\Delta)$. If $y=\varepsilon x$,
then $y^{p}=\varepsilon^{{\sqsupset}p}x^{p}=\varepsilon^{p}\lambda=\eta$ and
$y\delta=\varepsilon
x\delta=\varepsilon\delta^{\varepsilon}x=\delta\varepsilon x=\delta y$. Hence

$(\Delta,\alpha,\lambda)\to(\Delta,1,\eta)\colon\sum_{i=0}^{p-1}\delta_{i}x^{i}\mapsto\sum_{i=0}^{p-1}\delta_{i}\varepsilon^{-i}y^{i}$

is an isomorphism. Thus we may untwist
$\textup{End}_{{\mathbb{F}G}}(V{\uparrow})$.

######  Theorem 6.

Let $V$ be a $G$-stable irreducible $\mathbb{F}H$-module where $H\triangleleft
G$ and $|G/H|=p$ is prime. Suppose that $\alpha$ induces the inner
automorphism $\alpha(\delta)=\delta^{\varepsilon}$ of the division algebra
$\Delta=\textup{End}_{\mathbb{F}H}(V)$. Then $\eta=\varepsilon^{p}\lambda\in
Z^{\times}$ where $Z=Z(\Delta)$. Suppose that $s^{p}-\eta=\nu(s)\mu(s)$ where
$\mu(s)=\sum_{i=0}^{m}\mu_{i}s^{i}$ and $\nu(s)=\sum_{i=0}^{p-m}\nu_{i}s^{i}$,
are monic polynomials in $\Delta[s]$. Then
$W_{\mu}=\sum_{i=0}^{m-1}V\sum_{j=0}^{p-m}\nu_{j}\varepsilon^{i+j}\alpha^{-(i+j)}a^{i+j}$
is a submodule of $V{\uparrow}$. Let $\rho\colon G\to\textup{GL}(W_{\mu})$ be
the representation afforded by $W_{\mu}$ relative to the basis

$e^{\prime}_{0},\dots,e^{\prime}_{d-1},\dots,e^{\prime}_{j}(\varepsilon
X)^{k},\dots,e^{\prime}_{0}(\varepsilon
X)^{m-1},\dots,e^{\prime}_{d-1}(\varepsilon X)^{m-1}$

where

$e^{\prime}_{k}=e_{k}\sum_{j=0}^{p-m}\nu_{j}\varepsilon^{j}\alpha^{-j}a^{j}=e_{k}\sum_{j=0}^{p-m}\nu_{j}(\varepsilon
X)^{j},$

and $X$ is given by Eqn (5a). Then

$a\rho=\alpha\varepsilon^{-1}\begin{pmatrix}0&1&&0\\\ &&\ddots&\\\ 0&0&&1\\\
-\mu_{0}&-\mu_{1}&&-\mu_{m-1}\end{pmatrix},$

and $h\rho=\textup{diag}(h\sigma,\dots,h\sigma)$ where $h\in H$. Moreover,

$\textup{End}_{{\mathbb{F}G}}(W_{\mu})=\left\\{\sum_{i=0}^{m-1}\delta_{i}(a\rho)^{i}\mid\delta_{i}\in\Delta\right\\}.$

If $\mu(s)\in Z[s]$, then
$\textup{End}_{{\mathbb{F}G}}(W_{\mu})\cong\Delta[s]/\mu(s)\Delta[s]\cong\Delta\otimes_{Z}\mathbb{K}$
where $\mathbb{K}=Z[s]/\mu(s)Z[s]$.

###### Proof.

Arguing as in Theorem 5, we have a series of right ideals:

$\nu(s)\Delta[s]\subseteq\Delta[s],\hskip
4.2679pt\nu(y)(\Delta,1,\eta)\subseteq(\Delta,1,\eta),\hskip
4.2679pt\nu(\varepsilon
x)(\Delta,\alpha,\lambda)\subseteq(\Delta,\alpha,\lambda),$

and $\sum_{i=0}^{n}D(\nu_{i})(\varepsilon X)^{i}\Gamma$ is a right ideal of
$\Gamma=\textup{End}_{\mathbb{F}G}(V{\uparrow})$. This right ideal corresponds
to the submodule $V{\uparrow}\sum_{i=0}^{n}D(\nu_{i})(\varepsilon
X)^{i}\Gamma$ of $V{\uparrow}$. It follows from Eqn (5a) and $(\varepsilon
X)^{p}-\eta=0$ that the minimum polynomial of $\varepsilon X$ equals
$s^{p}-\eta$.

Let $v^{\prime}=v\nu(\varepsilon X)$ where $v\in V$. Then

$v^{\prime}\mu(\varepsilon X)=v\nu(\varepsilon X)\mu(\varepsilon
X)=v((\varepsilon X)^{p}-\eta)=v\thinspace 0=0.$

This proves that (9) is a basis for

$W_{\mu}=\text{im}\;\nu(\varepsilon X)=\ker\nu(\varepsilon
X)=\sum_{i=0}^{m-1}V\sum_{j=0}^{n}\nu_{j}\varepsilon^{i+j}\alpha^{-(i+j)}a^{i+j}$

It follows from Lemma 1 that $h\rho=\textup{diag}(h\sigma,\dots,h\sigma)$ is a
block scalar matrix ($h\in H$). Since $a=\alpha X$,

$v^{\prime}(\varepsilon X)^{i}a=v^{\prime}(\varepsilon X)^{i}\alpha
X=v^{\prime}\alpha\varepsilon^{-1}(\varepsilon X)^{i+1}.$

It follows from Eqns (11) and (12) that the matrix for $a\rho$ is correct.

It is now a simple matter to show that
$\left\\{\sum_{i=0}^{m-1}\delta_{i}(a\rho)^{i}\mid\delta_{i}\in\Delta\right\\}$
is contained in $\textup{End}_{{\mathbb{F}G}}(W_{\mu})$. A familiar
calculation shows that an element of $\textup{End}_{{\mathbb{F}G}}(W_{\mu})$
is determined by the entries in its top row. As this may be arbitrary, we have
found all the elements of $\textup{End}_{{\mathbb{F}G}}(W_{\mu})$. ∎

It follows from Theorem 6 that a necessary condition for $W_{\mu}$ to be
irreducible is that $\mu(s)$ is irreducible in $\Delta[s]$. Lemma 7 describes
an important case when $\textup{End}_{\mathbb{F}G}(W_{\mu})$ is a division
ring, and hence $W_{\mu}$ is irreducible. The following proof follows Prof.
Deitmar’s suggestion [D02].

######  Lemma 7.

Let $\Delta$ be a division algebra with center $\mathbb{F}$, and let
$\mu(s)\in\mathbb{F}[s]$ be irreducible of prime degree. Suppose that no
$\delta\in\Delta$ satisfies $\mu(\delta)=0$. Then the quotient ring
$\Delta[s]/\mu(s)\Delta[s]$ is a division algebra.

###### Proof.

Let $\mathbb{K}=\mathbb{F}[s]/\mu(s)\mathbb{F}[s]$. Then $\mathbb{K}$ is a
field and $|\mathbb{K}:\mathbb{F}|=\deg\mu(s)$ is prime. Clearly
$\mu(s)\Delta[s]$ is a two-sided ideal of $\Delta[s]$, and
$\Delta[s]/\mu(s)\Delta[s]$ is isomorphic to
$\Delta_{\mathbb{K}}=\Delta\otimes_{\mathbb{F}}\mathbb{K}$. By [L91, 15.1(3)],
$\Delta_{\mathbb{K}}$ is a central simple $\mathbb{K}$-algebra, and hence is
isomorphic to $M_{n}(D)$ for some division algebra $D$ over $\mathbb{F}$. The
_degree_ of $D$ and the _Schur index_ of $\Delta_{\mathbb{K}}$ are defined as
follows

$\textup{Deg}(D)=(\dim_{\mathbb{F}}D)^{1/2}\quad\text{and}\quad\textup{Ind}(\Delta_{\mathbb{K}})=\textup{Deg}(D).$

By [P82, Prop. 13.4], $\textup{Ind}(\Delta_{\mathbb{K}})$ divides
$\textup{Ind}(\Delta)$, and $\textup{Ind}(\Delta)$ divides
$|\mathbb{K}:\mathbb{F}|\,\textup{Ind}(\Delta_{\mathbb{K}})$. Thus either

$\textup{Ind}(\Delta_{\mathbb{K}})=\textup{Ind}(\Delta)=\textup{Deg}(\Delta)=\textup{Deg}(D_{\mathbb{K}})$

and $\Delta_{\mathbb{K}}$ is a division algebra by [P82, Prop. 13.4(ii)], or
$\textup{Ind}(\Delta)$ equals
$|\mathbb{K}:\mathbb{F}|\,\textup{Ind}(\Delta_{\mathbb{K}})$. If the second
case occurred, then by [P82, Cor. 13.4], $\mathbb{K}$ is isomorphic to a
subfield of $\Delta$, and so $\mu(s)$ has a root in $\Delta$, contrary to our
hypothesis. ∎

If $\eta\not\in\Delta^{p}$, then $\eta\not\in Z^{p}$ and so $s^{p}-\eta$ is
irreducible in $Z[s]$, and it follows from Lemma 7 that
$V{\uparrow}=W_{s^{p}-\eta}$ is irreducible. Note that
$\textup{End}_{{\mathbb{F}G}}(V{\uparrow})\cong\Delta\otimes Z[\eta^{1/p}]$ is
a division algebra.

##  6\. The case when $\alpha$ is inner and $\xi^{p}=\eta$

In this section we shall assume that $\xi\in\Delta^{\times}$ satisfies
$\xi^{p}-\eta=0$. Let $y=\varepsilon x$ and $z=\xi^{-1}y=\xi^{-1}\varepsilon
x$. It is useful to consider the isomorphisms
$(\Delta,\alpha,\lambda)\to(\Delta,1,\eta)\to(\Delta,1,1)$ defined by
$x\mapsto\varepsilon^{-1}y$ and $y\mapsto\xi z$. Note $y$ and $z$ are central
in $(\Delta,1,\eta)$ and $(\Delta,1,1)$ respectively, and $y^{p}=\eta$ and
$z^{p}=1$.

######  Theorem 8.

Let $V$ be a $G$-stable irreducible $\mathbb{F}H$-module where $H\triangleleft
G$ and $|G/H|=p$ is prime. Suppose that $\alpha$ induces the inner
automorphism $\alpha(\delta)=\delta^{\varepsilon}$ of the division algebra
$\Delta=\textup{End}_{\mathbb{F}H}(V)$. Set $\eta=\varepsilon^{p}\lambda$ and
let $\xi,\omega\in\Delta$ satisfy $\xi^{p}=\eta$ and $\omega^{p}=1$. Then
$\xi\in Z=Z(\Delta)$.
(a) If $\textup{char}(\mathbb{F})\neq p$ and $\omega\neq 1$, then
$V{\uparrow}$ is an internal direct sum

$V{\uparrow}=U(\xi)\dotplus U(\xi\omega)\dotplus\cdots\dotplus
U(\xi\omega^{p-1})$

where

$U(\xi\omega^{j})=V\sum_{i=0}^{p-1}(\xi\omega^{j})^{-i}\varepsilon^{i}\alpha^{-i}a^{i}$

is irreducible, and $U(\xi\omega)\cong U(\xi\omega^{\prime})$ if and only if
$\omega$ and $\omega^{\prime}$ are conjugate in $\Delta$. If $\mu(s)$ is an
irreducible factor of $s^{p}-\eta$ in $Z[s]$, then $W_{\mu}$ defined in
Theorem 6 is a Wedderburn component of $V{\uparrow}$, and
$W_{\mu}=U(\theta_{1})\dotplus\cdots\dotplus U(\theta_{n})$ where
$\theta_{1},\dots,\theta_{n}$ are the roots of $\mu(s)$ in the field
$Z(\xi,\omega)$. In addition, the representation $\rho_{\theta}\colon
G\to\textup{GL}(U(\theta))$ afforded by $U(\theta)$ relative to the basis
$e^{\prime}_{0},\dots,e^{\prime}_{d-1}$ where

$e^{\prime}_{j}=e_{j}\sum_{i=0}^{p-1}\theta^{-i}\varepsilon^{i}\alpha^{-i}a^{i}$

satisfies

$a\rho_{\theta}=\alpha\varepsilon^{-1}\theta\qquad\text{and}\qquad
h\rho_{\theta}=h\sigma$

for $h\in H$, and
$\textup{End}_{{\mathbb{F}G}}(U(\theta))=C_{\Delta}(\theta)$.
(b) If $\textup{char}(\mathbb{F})=p$, then $\omega=1$ and $V{\uparrow}$ is
uniserial with unique composition series $\\{0\\}=W_{0}\subset
W_{1}\subset\cdots\subset W_{p}=V{\uparrow}$ where

$W_{k}=\sum_{i=1}^{k}V\sum_{j=0}^{p-i}{i+j-1\choose
j}\xi^{-j}\varepsilon^{j}\alpha^{-j}a^{j}.$

Moreover, $W_{k-1}/W_{k}\cong U(\xi)$ for $k=1,\dots,p$ and
$\textup{End}_{{\mathbb{F}G}}(U(\xi))=\Delta$.

###### Proof.

Since $z\delta=\delta z$, we see that $(\xi^{-1}\varepsilon
x)\delta=\delta(\xi^{-1}\varepsilon x)$. This implies that
$\xi^{-1}\delta=\delta\xi^{-1}$ and so $\xi\in Z$.
Case (a): Now $(\xi\omega)^{p}=\xi^{p}\omega^{p}=\eta$, hence

$y^{p}-\eta=y^{p}-(\xi\omega)^{p}=(y-\xi\omega)\left(\sum_{i=0}^{p-1}(\xi\omega)^{p-1-i}y^{i}\right).$

Therefore $V{\uparrow}\sum_{i=0}^{p-1}(\xi\omega)^{p-1-i}(\varepsilon
X)^{i}\Gamma$ is a submodule of $V{\uparrow}$ where $X$ is given by Eqn (5a).
We show directly that $U(\xi\omega)$ is a submodule of $V{\uparrow}$. This
follows from

$\displaystyle(v(\xi\omega)^{-i}\varepsilon^{i}\alpha^{-i}a^{i})a$
$\displaystyle=v\alpha(\alpha^{-1}(\xi\omega)^{-i}\varepsilon^{i}\alpha)\alpha^{-(i+1)}a^{i+1}$
(14)
$\displaystyle=v\alpha\varepsilon^{-1}\xi\omega(\xi\omega)^{-(i+1)}\varepsilon^{i+1}\alpha^{-(i+1)}a^{i+1}$

and setting $i=p-1$ in the right-hand side of Eqn (14) gives

$v\alpha\varepsilon^{-1}\xi\omega(\xi\omega)^{-p}\varepsilon^{p}\alpha^{-p}a^{p}=v\alpha\varepsilon^{-1}\xi\omega\eta^{-1}\varepsilon^{p}\lambda=v\alpha\varepsilon^{-1}\xi\omega.$

As $U(\xi\omega){\downarrow}\cong V$, we see that $U(\xi\omega)$ is an
irreducible ${\mathbb{F}G}$-submodule of $V{\uparrow}$. Setting
$\theta=\xi\omega$ establishes the truth of Eqns (12a,b).

We may calculate $\textup{Hom}(U(\xi\omega),U(\xi\omega^{\prime}))$ directly
by finding all $\delta$ in $\textup{End}_{\mathbb{F}}(V)$ that intertwine
$\rho_{\xi\omega}$ and $\rho_{\xi\omega^{\prime}}$. As $\delta$ intertwines
$h\rho_{\xi\omega}$ and $h\rho_{\xi\omega^{\prime}}$, it follows that $\delta$
commutes with $H\sigma$, and hence $\delta\in\Delta$. Also

$\delta(\alpha\varepsilon^{-1}\xi\omega)=(\alpha\varepsilon^{-1}\xi\omega^{\prime})\delta$

so $\delta^{\alpha\varepsilon^{-1}}\xi\omega=\xi\omega^{\prime}\delta$. Since
$\xi\in Z^{\times}$ and $\delta^{\alpha\varepsilon^{-1}}=\delta$, this amounts
to $\delta\omega=\omega^{\prime}\delta$. Setting $i=j$ shows that
$\textup{End}_{{\mathbb{F}G}}(U(\xi\omega))=C_{\Delta}(\omega)$.

The Galois group $\textup{Gal}(Z(\omega):Z)$ is cyclic of order dividing
$p-1$. Also $\omega$ and $\omega^{\prime}$ are conjugate in
$\textup{Gal}(Z(\omega):Z)$ if and only if they share the same minimal
polynomial over $Z$. The latter holds by Dixon’s Theorem [L91, 16.8] if and
only if $\omega$ and $\omega^{\prime}$ are conjugate in $\Delta$. Note that
$\omega$ and $\omega^{\prime}$ share the same minimal polynomial over $Z$
precisely when $\xi\omega$ and $\xi\omega^{\prime}$ share the same minimal
polynomial. This proves that $W_{\mu}$ is a Wedderburn component of
$V{\uparrow}$.

Case (b): Suppose now that $\textup{char}(\mathbb{F})=p$. Then $\omega=1$ and
Eqn (13) becomes $y^{p}-\eta=(y-\xi)^{p}=(y-\xi)(\sum_{i=0}^{p-1}{p-1\choose
i}(-\xi)^{p-1-i}y^{i})$. As

$\Gamma=\textup{End}_{\mathbb{F}G}(V{\uparrow})\cong(\Delta,\alpha,\lambda)\cong(\Delta,1,\eta)\cong(\Delta,1,1)\cong\Delta[z]/(z-1)^{p}\Delta[z]$

has a unique composition series, so too does $V{\uparrow}$. By noting that
$z=\xi^{-1}\varepsilon x$ and $D(\xi^{-1}\varepsilon)=\xi^{-1}\varepsilon$, we
see that $W_{i}=V{\uparrow}(\xi^{-1}\varepsilon X-1)^{p-i}\Gamma$ defines the
unique composition series for $V{\uparrow}$ where $X$ is given by Eqn (5a).

Let $R$ be the diagonal matrix
$\textup{diag}(1,\xi^{-1}\varepsilon,\dots,(\xi^{-1}\varepsilon)^{p-1})$, and
let $S$ be the matrix whose $(i,j)$th block is the binomial coefficient
${i\choose j}$ where $0\leq i,j<p$. A direct calculation verifies that
$R(\xi^{-1}\varepsilon X)R^{-1}=C$ and $S^{-1}CS=J$ where

$C=\begin{pmatrix}0&1&&0\\\ &&\ddots&\\\ 0&0&&1\\\
1&0&&0\end{pmatrix}\quad\text{and}\quad J=\begin{pmatrix}1&1&&\\\ &\ddots&&\\\
&&1&1\\\ &&&1\end{pmatrix}.$

Therefore $\xi^{-1}\varepsilon X-1=T^{-1}(J-1)T$ where $T=S^{-1}R$, and hence

$W_{k}=V{\uparrow}(\xi^{-1}\varepsilon
X-1)^{p-k}=V{\uparrow}\;T^{-1}(J-1)^{p-k}T=V{\uparrow}(J-1)^{p-k}T.$

It is easily seen that $\textup{im}(J-1)^{p-k}=\ker(J-1)^{k}$ is the subspace
$(0,\dots,0,V,\dots,V)$ where the first $V$ is in column $p-k$. The $(i,j)$th
entry of $T=S^{-1}R$ is $(-1)^{i+j}{i\choose j}(\xi^{-1}\varepsilon)^{j}$. The
last row of $T$ gives

$W_{1}=V\sum_{j=0}^{p-1}(-1)^{p-1+j}{p-1\choose
j}(\xi^{-1}\varepsilon)^{j}\alpha^{-j}a^{j}.$

More generally, the last $k$ rows of $T$ give

$W_{k}=\sum_{i=1}^{k}V\sum_{j=0}^{p-i}(-1)^{p-i+j}{p-i\choose
j}(\xi^{-1}\varepsilon)^{j}\alpha^{-j}a^{j}.$

Since $p-i-\ell=-(i+\ell)$ in a field (such as $\mathbb{F}$) of characteristic
$p$, we see that ${p-i\choose j}=(-1)^{j}{i+j-1\choose j}$ and the formula for
$W_{k}$ simplifies to

$W_{k}=\sum_{i=1}^{k}V\sum_{j=0}^{p-i}{i+j-1\choose
j}\xi^{-j}\varepsilon^{j}\alpha^{-j}a^{j}.$

Setting $k=1$ shows $W_{1}=U(\xi)$. A direct calculation shows that
$W_{i-1}/W_{i}\cong U(\xi)$. We showed in Part (a) that
$\textup{End}_{\mathbb{F}G}(U(\xi))$ equals $C_{\Delta}(\xi)=\Delta$. ∎

In Case (a), $C_{\Delta}(\xi\omega)$ equals $\Delta$ precisely when $\omega\in
Z$. If $\Delta$ is the rational quaternions, and $\omega$ is primitive cube
root of unity, then $C_{\Delta}(\omega)$ equals $\mathbb{Q}(\omega)$. There
are infinitely many primitive cube roots of 1 in this case, and they form a
conjugacy class of $\Delta$ by Dixon’s Theorem (as they all satisfy the
irreducible polynomial $s^{2}+s+1$ over $\mathbb{Q}$). Thus isomorphism of the
submodules $U(\xi\omega)$ is governed by conjugacy in $\Delta$, and not
conjugacy in $\textup{Gal}(\mathbb{Q}(\omega):\mathbb{Q})$.

Finally, it remains to generalize Theorem 8(a) to allow for the possibility
that $\Delta$ may not contain a primitive $p$th root of 1.

######  Theorem 9.

Let $V$ be a $G$-stable irreducible $\mathbb{F}H$-module where $H\triangleleft
G$ and $|G/H|=p$ is prime. Suppose that $\varepsilon,\xi\in\Delta$ satisfy
$\alpha(\delta)=\delta^{\varepsilon}$ ($\delta\in\Delta$) and $\xi^{p}-\eta=0$
where $\eta=\varepsilon^{p}\lambda\in Z=Z(\Delta)$. In addition, suppose that
$\textup{char}(\mathbb{F})\neq p$. Then $V{\uparrow}$ is an internal direct
sum

$V{\uparrow}=W_{\mu_{1}}\dotplus\cdots\dotplus W_{\mu_{r}}$

where $s^{p}-\eta=\mu_{1}(s)\cdots\mu_{r}(s)$ is a factorization into monic
irreducibles over $Z$, and where $W_{\mu}$ defined in Theorem 6. If $\mu(s)$
is a monic irreducible factor of $s^{p}-\eta$, and
$\mu(s)=\nu_{1}(s)\cdots\nu_{n}(s)$ where the $\nu_{i}(s)$ are monic and
irreducible in $\Delta[s]$, then $W_{\mu}$ is a Wedderburn component of
$V{\uparrow}$, and $W_{\mu}\cong W_{\nu_{n}}^{\oplus n}$ where $W_{\nu_{n}}$
is an irreducible ${\mathbb{F}G}$-module and
$\textup{End}_{{\mathbb{F}G}}(W_{\nu_{n}})$ is given in Theorem 6. In
addition,

$\textup{End}_{{\mathbb{F}G}}(W_{\nu_{n}})\cong B/\nu_{n}(s)\Delta[s]$

where
$B=\\{\delta(s)\in\Delta[s]\mid\delta(s)\nu_{n}(s)\in\nu_{n}(s)\Delta[s]\\}$
is the idealizer of the right ideal $\nu_{n}(s)\Delta[s]$.

###### Proof.

Since $\textup{char}(\mathbb{F})\neq p$, the monic polynomials
$\mu_{1}(s),\dots,\mu_{r}(s)$ are distinct and pairwise coprime in $Z[s]$.
From this it follows that $V{\uparrow}$ equals
$W_{\mu_{1}}\dotplus\cdots\dotplus W_{\mu_{r}}$. By Theorem 6,
$\textup{End}_{\mathbb{F}G}(W_{\mu})\cong\Delta_{\mathbb{K}}$ where
$\Delta_{\mathbb{K}}\cong\Delta[s]/\mu(s)\Delta[s]\cong\Delta\otimes_{Z}\mathbb{K}$,
and $\mathbb{K}$ is the field $Z[s]/\mu(s)Z[s]$. By [L91, 15.1(3)],
$\Delta_{\mathbb{K}}$ is a simple ring. Therefore $\mu(s)\Delta[s]$ is a two-
sided maximal ideal of $\Delta[s]$, and so $\mu(s)$ is called a two-sided
maximal element of $\Delta[s]$. By [J96, Theorem 1.2.19(b)],
$\Delta_{\mathbb{K}}\cong M_{n}(D)$ where $D$ is the division ring
$B/\nu_{n}(s)\Delta[s]$. Moreover, $Z(\Delta_{\mathbb{K}})\cong Z(M_{n}(D))$
so $\mathbb{K}\cong Z(D)$. Thus $W_{\mu}\cong W_{\nu_{n}}^{\oplus n}$ where
$W_{\nu_{n}}$ is an irreducible submodule of $V{\uparrow}$ and
$\textup{End}_{\mathbb{F}G}(W_{\nu_{n}})\cong D$. In addition,
$\nu_{1},\dots,\nu_{n}$ are similar [J96, Def. 1.2.7], and
$W_{\nu_{1}},\dots,W_{\nu_{n}}$ are isomorphic.

If $\mu(s),\mu^{\prime}(s)$ are distinct monic irreducible factors of
$s^{p}-\eta$ in $Z[s]$ and $\nu(s),\nu^{\prime}(s)$ in $\Delta[s]$ are monic
irreducible factors of $\mu(s)$ and $\mu^{\prime}(s)$ respectively, then it
follows from [J96, Def. 1.2.7] that $\nu(s)$ and $\nu^{\prime}(s)$ are not
similar. This means that an irreducible summand of $W_{\mu}$ is not isomorphic
to an irreducible summand of $W_{\mu^{\prime}}$. Hence the $W_{\mu}$ are
Wedderburn components as claimed. ∎

## Acknowledgment

I am very grateful to Prof. A. D. H. Deitmar for providing a proof [D02] of of
Lemma 7 in the case when $s^{p}-\eta$ has no root in $\Delta$.

## References

  * [CR90]

C.W. Curtis and I. Reiner, Methods of Representation Theory: with Applications
to Finite Groups and Orders, Vol. 1, Classic Library Edn, John Wiley and Sons,
1990.

  * [D02]

A. D. H. Deitmar, sci.math.research, September 6, 2002.

  * [GK96]

S.P. Glasby and L.G. Kovács, _Irreducible modules and normal subgroups of
prime index_ , Comm. Algebra 24 (1996), no. 4, 1529–1546. (MR 97a:20012)

  * [IL00]

G. Ivanyos and K. Lux, Treating the exceptional cases of the MeatAxe,
Experiment. Math. 9 (2000), no. 3, 373–381. (MR 2001j:16067)

  * [HR94]

D.F. Holt and S. Rees, _Testing modules for irreducibility_ , J. Austral.
Math. Soc. Ser. A 57 (1994), no. 1, 1–16. (MR 95e:20023)

  * [J96]

N. Jacobson, Finite-Dimensional Division Algebras over Fields, Springer-
Verlag, 1996.

  * [L91]

T.Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in
Mathematics 131, Springer-Verlag, 1991.

  * [NP95]

P.M. Neumann and C.E. Praeger, _Cyclic matrices over finite fields_ , J.
London Math. Soc. 52 (1995), no. 2, 263–284. (MR 96j:15017)

  * [P84]

R.A. Parker, The computer calculation of modular characters (the meat-axe),
Computational group theory (Durham, 1982), 267–274, Academic Press, London,
1984. (MR 84k:20041)

  * [P82]

R. S. Pierce, Associative Algebras, Graduate Texts in Mathematics 88,
Springer-Verlag, 1982.

S. P. Glasby
---
Department of Mathematics
Central Washington University
WA 98926-7424, USA
<EMAIL_ADDRESS>


# Can Requirements Engineering Support Explainable Artificial Intelligence?
Towards a User-Centric Approach for Explainability Requirements

Umm-E- Habiba  Institute of Software Engineering
University of Stuttgart
Stuttgart, Germany
Email<EMAIL_ADDRESS>Justus Bogner  Institute of
Software Engineering
University of Stuttgart
Stuttgart, Germany
<EMAIL_ADDRESS>Stefan Wagner  Institute of Software
Engineering
University of Stuttgart
Stuttgart, Germany
<EMAIL_ADDRESS>
###### Abstract

With the recent proliferation of artificial intelligence systems, there has
been a surge in the demand for explainability of these systems. Explanations
help to reduce system opacity, support transparency, and increase stakeholder
trust. In this position paper, we discuss synergies between requirements
engineering (RE) and Explainable AI (XAI). We highlight challenges in the
field of XAI, and propose a framework and research directions on how RE
practices can help to mitigate these challenges.

Keywords: Requirements Engineering, Explainability, XAI

##  I Introduction

In the last decade, Artificial Intelligence (AI) systems, especially based on
Machine Learning (ML), have achieved remarkable feats in many areas that were
previously computationally impossible [1]. AI is now prevalent in our
professional and personal lives. Progress has also been made in the fields of
medicine, automated vehicles, bioinformatics, and recommender systems. Despite
these advancements, trust issues and disillusionment are emerging in several
areas. One of the most common problems is the lack of transparency in these
systems, as the black-box nature of many ML models collides with the human
desire to understand the system and its output [2]. This is especially
concerning when these systems are used in contexts where they have a major
impact on human lives, such as medical diagnostic systems or financial and
legal matters.

Software systems’ growing autonomy and complexity make it harder for software
engineers and domain experts to comprehend them [3], especially if they are
composed of ML components [4]. This leads to a need for explainable systems.
Explanations help to understand system decisions, which increases confidence
in a system and improves trustworthiness. Additionally, it also justifies
actions taken, increases usability, aids in uncovering causes of mistakes, and
reduces the potential of human error because humans cannot make informed
decisions without access to the system rationale for its recommendation. An
appropriate explanation for unexpected or inaccurate system behavior might
also help to determine problems, for instance, the misinterpretation of a
requirement or a mistake in the system design.

Moreover, a lack of explainability not only causes ethical, social, and legal
issues [5, 6]. It also exacerbates distrust, reduces user acceptability [7, 8]
and stifles the adoption of innovative technology. In this regard, the Ethics
Guidelines for Trustworthy AI [9] have been proposed by the High-Level Expert
Group on Artificial Intelligence (AI HLEG). They highlighted transparency as a
key requirement for trustworthy AI, which is further comprised of
traceability, explainability, and communication. Depending on the context of
relevant stakeholders, these guidelines emphasize a proper explanation of the
decision-making process of AI systems. Suitable development techniques are
required to provide a certain level of explainability. The development process
should incorporate explainability to make it explicit. However, the concrete
scope for _explainability_ is still unclear, and many researchers are actively
exploring different courses of action to support explainability. While
approaches such as LIME [10] and Shapley values [11] are based on mathematical
methods, the Requirements Engineering (RE) community is also exploring
different approaches [12, 13] to support explainability, although it has
already been identified as a key non-functional requirement to support
transparency and reduce opacity. Current research is either inclined towards
model explainability, which particularly focuses on ML engineers, or user
experience, which addresses user needs for explainability and how users react
to certain explainability techniques. A problem arises when it comes to the
interaction between ML engineers and multiple stakeholders, as they have
different expectations and views of explainability. However, there is a lack
of studies that investigate the gap between the understanding of ML engineers
and end-users regarding explainability. We believe RE practices can help to
address this challenge.

Position and Contribution: In this paper, we provide an overview of the state-
of-the-art in Explainable AI (XAI). Our objective is to highlight current
challenges in XAI from the perspective of requirements engineering and provide
future directions on what could be options to mitigate these challenges.

##  II Explainability vs. Interpretability

Terminology related to explainability in machine learning is not consistently
used [14], with terms such as “explainable AI” [15], “interpretability” [16],
and “comprehensibility” [17]. They are all aimed at making the system more
transparent, i.e., explaining the system’s inner working and making it more
understandable.

In particular, explainability and interpretability appear frequently in the
literature, and many researchers use them interchangeably [18] as no specific
mathematical definitions for explainability or interpretability exist, nor
have they been quantified by any metric [19]. While they refer to overlapping
concepts, there is some work that attempts not only to define these two
concepts but also related notions such as comprehensibility [4, 16, 20].
Miller [21] defines interpretability as “the degree to which a human can
understand the cause of a decision”. Another definition comes from Doshi-Velez
and Kim [16], who define interpretability as “the ability to explain or
communicate in intelligible words to a human”. These definitions refer to
interpretability as a property of how input and output are associated and how
easily end-users can identify cause-and-effect relations.

In contrast, “explainability” deals with the internal logic of the system,
i.e., how the ML system has been trained and how a particular output has been
generated [22].

Doshi-Velez and Kim [16] define explanations as “the currency in which we
exchange beliefs”. Rudin [23] differentiates between interpretability and
explainability via the concepts “inherently interpretable” and “post-hoc
explanations”. In [16] and [20], the authors describe interpretability as a
broader concept and argue that explainability is essential to support
interpretability.

TABLE I: Interpretability vs Explainability in AI Literature Papers | Interpretability | Explainability
---|---|---
[4] | Interpretability addresses “how does the model work?” | Explainability addresses “what else can the model tell me?”
[16] |  | Interpretability is ”the ability to explain or to
---
present something in understandable terms to a human”
Explanations are ”the currency in which we exchange beliefs”
[20] |  | Mostly concerned with the intuitions behind the output of a model
---
”to describe the internals of a system in a way
that is understandable to humans”
| Mostly concerned with the internal logic, i.e.,
---
”models that are able to summarize the reasons for neural network
behavior, gain the trust of users, or produce insights about
the causes of their decisions”
[21] |  | “The degree to which an observer can
---
understand the cause of a decision”
“Explicitly explaining decisions to people”
[24] |  | ”An interpretation is the mapping of an abstract concept
---
into a domain that humans can make sense of”
| ”An explanation is the collection of features of the interpretable
---
domain that have contributed to a given example to produce a decision”

Table I summarizes how different authors strived to define explainability and
interpretability. However, we can see from this literature, there are no
agreed definitions of explainability or interpretability. The AI research
community is still working towards a unified position. To motivate research in
this area, researchers are trying to understand and elicit different
requirements, e.g., why these explanations and interpretations are needed,
what their context is, and who the stakeholders in these explanations are.

To address end-user needs, research must consider their perspective of
explanations. While the human-computer interaction (HCI) community is actively
exploring user-centric XAI [25, 26, 27], a more extensive contribution is
still required from the RE community to provide a systematic approach to
enable software engineers to effectively incorporate explainability
requirements in the development process. In the following section, we propose
an alliance between RE and XAI.

##  III Synergies between RE and XAI

The interactions between the two fields RE and XAI are still evolving. A few
studies discuss various interactions of RE and XAI. However, the major focus
are concrete requirements for XAI. Mark et al. [11] proposed a 5-step
methodology to understand explainable AI requirements. Their goal was to
understand explanation requirements from different perspectives. Kohl et al.
[14] proposed a process for eliciting, specifying, and verifying
explainability as a non-functional requirement. Chazette et al. [28] also
concentrated on a user perspective of explanations. Their work is more focused
on software transparency and user opinion about embedded explanations in
software. Another work [29] from the same authors explores the interaction
between explainability and other non-functional requirements. They also
describe potential trade-offs of incorporating explainability into the system.
Similarly, Liao et al. [30] interviewed 20 user experience and design
practitioners to identify gaps in current practices. They provided insights
into the design space of XAI and a question bank that can be used to create
user-centered XAI. Eiband et al. [31] improved existing user interface
guidelines by proposing a stage-based participatory process. Their approach is
aimed more towards design management and user interface design.

Most of the presented work is focused on user-centered design and interactions
of explainability with other non-functional requirement. There are studies
centered on different stakeholder perspectives, but several challenges remain.
We will outline them in the following section.

##  IV Challenges

Current research in XAI is inclined towards model explainability and that
these explanations are more focused on ML engineers rather than end-users of
the system. Most existing explanation techniques and tools are only
comprehensible by technical stakeholders with an ML background, like data
scientists and ML engineers. Approaches that pay attention to the end-user are
more rare. Although there are studies that attempted to work in the direction
of requirements engineering and try to ask stakeholders about XAI
requirements, many challenges remain.

### Absence of a mediator role

Recently, several open-source toolkits [32] emerged for XAI. They provide
explanations which are mostly comprehensible by ML engineers. Yet, these
approaches are difficult to adapt for end-users, e.g., due to the gap between
user needs and available technical capabilities to improve the effectiveness
of explainability in AI products [30]. Thus, the role of mediators arises,
whose job it is to fill the gap between stakeholders. They have the experience
to identify user needs and to convey them to the developers, which supports a
better translation of end-user needs into technical aspects. In this sense,
they serve as a bridge between users and ML engineers, keeping in mind the
demands and constraints on both sides. Thus, explainable solutions for AI
systems also need to provide explanations for the mediators, whose job is to
bridge the gaps among stakeholders.

### No coherent definition of explainability

Usually, demands for explainability and how it can be accomplished are
ambiguous [14]. Several authors presented different facets of explainability,
with no agreed on definition, which can, e.g., impact communication in AI
projects.

### Lack of stakeholder-centric approaches

Current solutions for model interpretability do not describe or specify their
intended users. Therefore, most of these solutions unwittingly wind up being
more understandable to the people who build them, i.e., ML engineers.
Alternately, the interpretable model built for the end-user has issues such as
simple visuals with general explanations that are often not useful for people
in practice [33].

### No common vocabulary for all ML stakeholders

Many terms have recently been introduced in explainable AI. However, these
terms appear interchangeably, which leads to confusion in this rapidly
expanding field. Furthermore, certification authorities, researchers, end-
user, domain experts, and ML engineers all use explainable AI with different
expectations and understanding. This leads to more problems, as each
stakeholder desires separate objectives to be satisfied by the term
explainable AI [34].

To address these challenges, we propose a general framework to provide
guidance how these issues can be minimized and how explainability can be
better supported by RE practices. For each step, we are working on more
detailed methods and guidelines to support practitioners.

##  V Proposed Framework

In this section, we i) draw a boundary between interpretability and
explainability and ii) propose a framework to mitigate the challenges stated
above.

As we have seen, these two terms have been used interchangeably, even though
some authors draw a boundary between them. As a synthesis of these
definitions, we differentiate between the two terms to make it explicit that
our framework supports explainability:

  1. 1.

Interpretability: looking at the model, i.e., the stakeholder can understand
the cause-and-effect relationship between input and output.

  2. 2.

Explainability: looking inside the model, i.e., the stakeholder can understand
the inner workings of the model and how the model has produced a particular
output.

Figure 1: Proposed Framework

Activities described in our framework will be performed by the mediating role
of the requirements engineer. This framework explicitly aims to identify
stakeholders with the requirements of explainability. Furthermore, it will
help to establish a shared understating of explainability among multiple
stakeholders. The framework comprises the following steps (Figure 1):

  1. 1.

Identifying stakeholders: In this step we aim to address the question
“explainable for whom?” We will identify relevant stakeholders for the system,
i.e., end-users, ML engineers, and domain experts. We will also determine the
stakeholders for explanations.

  2. 2.

Identifying requirements: We will elicit and document the concrete
requirements for explainability, together with rationales like why
explainability is required and what the impact of each explanation will be on
the system. We will also describe what types of explanations will be provided.
This will answer the question “why is explainability needed?”

  3. 3.

Establishing a common vocabulary: Using knowledge of end-users, domain
experts, and ML engineers, we will establish a common vocabulary. We will
introduce common terms and create a shared system understanding, which
supports discussing explainability requirements.

  4. 4.

Negotiating and validating requirements: This is an iterative process. The
identified explainability requirements will be validated in this step. This
will answer questions like “what if explainability is not possible?” or “how
much uncertainty is tolerable?” If there is any trade-off between the
requirements or some are not feasible, they will be negotiated.

  5. 5.

Classifying requirements: In this step, requirements will be classified with
respect to stakeholders. Who are the stakeholders for particular explanations,
and what do they want to achieve with these explanations? The main question to
answer is “how much explainability is enough for which stakeholder?” This
classification will also help to address each stakeholder requirements.

##  VI Conclusion and Future Work

In this position paper, we discussed differences between interpretability and
explainability, and pointed out existing challenges in the XAI space. Taking a
requirements engineering perspective, we propose a work-in-progress framework
to address the challenges, with emphasis on a user-centered approach to
explainability requirements and highlight the potential of RE to support the
explainability of AI systems.

## References

  * [1] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” _Nature_ , vol. 521, 2015\.
  * [2] L. Yang, H. Wang, and L. A. Deleris, “What does it mean to explain? a user-centered study on ai explainability,” in _International Conference on Human-Computer Interaction_. Springer, 2021.
  * [3] DARPA, “Broad agency announcement, explainable artificial intelligence (xai),” _DARPA-BAA-16-53_ , pp. 7–8, 2016.
  * [4] Z. C. Lipton, “The mythos of model interpretability: In machine learning, the concept of interpretability is both important and slippery.” _Queue_ , vol. 16, no. 3, pp. 31–57, 2018.
  * [5] B. D. Mittelstadt, P. Allo, M. Taddeo, S. Wachter, and L. Floridi, “The ethics of algorithms: Mapping the debate,” _Big Data & Society_, 2016.
  * [6] L. Floridi, J. Cowls, M. Beltrametti, R. Chatila, P. Chazerand, V. Dignum, C. Luetge, R. Madelin, U. Pagallo, F. Rossi _et al._ , “AI4People–an ethical framework for a good AI society: Opportunities, risks, principles, and recommendations,” _Minds and Machines_ , 2018.
  * [7] M. Lahijanian and M. Kwiatkowska, “Social trust: a major challenge for the future of autonomous systems,” in _2016 AAAI Fall Symposium Series_ , 2016\.
  * [8] L. R. Ye and P. E. Johnson, “The impact of explanation facilities on user acceptance of expert systems advice,” _MIS Quarterly_ , 1995.
  * [9] H. AI, “High-level expert group on artificial intelligence,” 2019.
  * [10] M. T. Ribeiro, S. Singh, and C. Guestrin, ““why should i trust you?” explaining the predictions of any classifier,” in _Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , 2016, pp. 1135–1144.
  * [11] M. Hall, D. Harborne, R. Tomsett, V. Galetic, S. Quintana-Amate, A. Nottle, and A. Preece, “A systematic method to understand requirements for explainable AI (XAI) systems,” in _Proceedings of the IJCAI Workshop on eXplainable Artificial Intelligence (XAI 2019)_ , vol. 11, 2019.
  * [12] C. T. Wolf, “Explainability scenarios: towards scenario-based xai design,” in _Proceedings of the 24th International Conference on Intelligent User Interfaces_ , 2019, pp. 252–257.
  * [13] D. Cirqueira, D. Nedbal, M. Helfert, and M. Bezbradica, “Scenario-based requirements elicitation for user-centric explainable ai,” in _International Cross-Domain Conference for Machine Learning and Knowledge Extraction_. Springer, 2020, pp. 321–341.
  * [14] M. A. Köhl, K. Baum, M. Langer, D. Oster, T. Speith, and D. Bohlender, “Explainability as a non-functional requirement,” in _2019 IEEE 27th International Requirements Engineering Conference (RE)_. IEEE, 2019.
  * [15] W. Samek, T. Wiegand, and K.-R. Müller, “Explainable artificial intelligence: Understanding, visualizing and interpreting deep learning models,” _arXiv preprint arXiv:1708.08296_ , 2017.
  * [16] F. Doshi-Velez and B. Kim, “Towards a rigorous science of interpretable machine learning,” _arXiv preprint arXiv:1702.08608_ , 2017.
  * [17] A. A. Freitas, “Comprehensible classification models: a position paper,” _ACM SIGKDD Explorations Newsletter_ , vol. 15, no. 1, pp. 1–10, 2014.
  * [18] D. V. Carvalho, E. M. Pereira, and J. S. Cardoso, “Machine learning interpretability: A survey on methods and metrics,” _Electronics_ , 2019\.
  * [19] A. Adadi and M. Berrada, “Peeking inside the black-box: a survey on explainable artificial intelligence (XAI),” _IEEE access_ , vol. 6, 2018\.
  * [20] L. H. Gilpin, D. Bau, B. Z. Yuan, A. Bajwa, M. Specter, and L. Kagal, “Explaining explanations: An overview of interpretability of machine learning,” in _2018 IEEE 5th International Conference on data science and advanced analytics (DSAA)_. IEEE, 2018, pp. 80–89.
  * [21] T. Miller, “Explanation in artificial intelligence: Insights from the social sciences,” _Artificial Intelligence_ , vol. 267, pp. 1–38, 2019.
  * [22] M. Turek, “Explainable artificial intelligence (XAI),” _Defense Advanced Research Projects Agency_ , 2018.
  * [23] C. Rudin, “Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead,” _Nature Machine Intelligence_ , vol. 1, no. 5, pp. 206–215, 2019.
  * [24] G. Montavon, W. Samek, and K.-R. Müller, “Methods for interpreting and understanding deep neural networks,” _Digital Signal Processing_ , vol. 73, 2018.
  * [25] J. J. Ferreira and M. Monteiro, “Designer-user communication for XAI: An epistemological approach to discuss XAI design,” _arXiv preprint arXiv:2105.07804_ , 2021.
  * [26] A. Abdul, J. Vermeulen, D. Wang, B. Y. Lim, and M. Kankanhalli, “Trends and trajectories for explainable, accountable and intelligible systems: An HCI research agenda,” in _Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems_ , 2018, pp. 1–18.
  * [27] D. Wang, Q. Yang, A. Abdul, and B. Y. Lim, “Designing theory-driven user-centric explainable AI,” in _Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems_ , 2019, pp. 1–15.
  * [28] L. Chazette, O. Karras, and K. Schneider, “Do end-users want explanations? analyzing the role of explainability as an emerging aspect of non-functional requirements,” in _2019 IEEE 27th International Requirements Engineering Conference (RE)_. IEEE, 2019\.
  * [29] L. Chazette and K. Schneider, “Explainability as a non-functional requirement: challenges and recommendations,” _Requirements Engineering_ , vol. 25, no. 4, 2020.
  * [30] Q. V. Liao, D. Gruen, and S. Miller, “Questioning the AI: informing design practices for explainable AI user experiences,” in _Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems_.
  * [31] M. Eiband, H. Schneider, M. Bilandzic, J. Fazekas-Con, M. Haug, and H. Hussmann, “Bringing transparency design into practice,” in _23rd International Conference on Intelligent User Interfaces_ , 2018.
  * [32] R. Marcinkevičs and J. E. Vogt, “Interpretability and explainability: A machine learning zoo mini-tour,” _arXiv:2012.01805_ , 2020.
  * [33] H. Suresh, S. R. Gomez, K. K. Nam, and A. Satyanarayan, “Beyond expertise and roles: A framework to characterize the stakeholders of interpretable machine learning and their needs,” in _Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems_ , 2021.
  * [34] A. Brennen, “What do people really want when they say they want “explainable AI?” We asked 60 stakeholders,” in _Extended Abstracts of the 2020 CHI Conference on Human Factors in Computing Systems_ , 2020.


††thanks: This work was partially supported by DARPA’s QuASAR program.
Publication of the U.S. government, not subject to U.S. copyright.

# Sub-Wavelength Imaging and Field Mapping
via EIT and Autler-Townes Splitting In Rydberg Atoms

Christopher L. Holloway<EMAIL_ADDRESS>Joshua A. Gordon  National
Institute of Standards and Technology (NIST), Electromagnetics Division, U.S.
Department of Commerce, Boulder Laboratories, Boulder, CO 80305  Andrew
Schwarzkopf  David A. Anderson  Stephanie A. Miller  Nithiwadee Thaicharoen
Georg Raithel  Department of Physics, University of Michigan, Ann Arbor, MI
48109

###### Abstract

We present a technique for measuring radio-frequency (RF) electric field
strengths with sub-wavelength resolution. We use Rydberg states of rubidium
atoms to probe the RF field. The RF field causes an energy splitting of the
Rydberg states via the Autler-Townes effect, and we detect the splitting via
electromagnetically induced transparency (EIT). We use this technique to
measure the electric field distribution inside a glass cylinder with applied
RF fields at 17.04 GHz and 104.77 GHz. We achieve a spatial resolution of
$\bf{\approx}$100 $\bf{\mu}$m, limited by the widths of the laser beams
utilized for the EIT spectroscopy. We numerically simulate the fields in the
glass cylinder and find good agreement with the measured fields. Our results
suggest that this technique could be applied to image fields on a small
spatial scale over a large range of frequencies, up into the sub-THz regime.

atom based metrology, Autler-Townes Splitting, broadband probe, electrical
field measurements and sensors, EIT, sub-wavelength imaging, Rydberg atoms

##  I Introduction

The typical probe (sensor) to measure an electric (E) field has a size on the
order of $\lambda/2$ or $\lambda/4$ (where $\lambda$ is the free-space
wavelength of the radiation one intends to measure). One example is a dipole
loaded field probe dipole . These conventional probes can only measure an
E-field strength averaged over the length of the probe. If one is interested
in measuring a field distribution in the neighborhood of a structure with
spatial features smaller than $\lambda$, the conventional probe would be
problematic. For example, consider metasurface structures apm . These
metasurfaces are typically composed of periodic arrays of inclusions or
scatterers. These scatterers are typically sub-wavelength in size (on the
order of $\lambda/10$ or smaller). Furthermore, these scatterers have even
smaller sub-structures (gaps, holes, apertures) that can be as small as
$\sim\lambda/100$. With current methods, it is virtually impossible to measure
E-field distributions on these sub-wavelength scales. The technique we
demonstrate to perform high spatial resolution mapping of RF fields could be
used in design verification and characterization of the metasurfaces and other
sub-wavelength devices and structures.

Figure 1: A four-level system.

Figure 2: Vapor cell setup for measuring EIT, with counter-propagating probe
(red) and coupling (blue) beams. The RF is applied transverse to the optical
beam propagation in the vapor cell.

To measure E-fields with sub-wavelength resolution, we take advantage of a
recently-demonstrated technique that uses atoms as field probes jim -gordon .
Here we demonstrate, for the first time, its spatial resolution capability.
The technique uses room-temperature rubidium atoms as probes, exploiting the
sensitivity of their high-lying Rydberg states to radio frequency (RF)
radiation. (The term RF is used here to cover the conventional RF, microwave,
millimeter wave, and sub-terahertz spectra.) This sensitivity reflects the
large transition matrix elements ($\wp$, on the order of $10^{3}$ to
$10^{4}ea_{0}$) for RF transitions between Rydberg states. We measure an
Autler-Townes splitting autler  of Rydberg energy levels in these atoms to
obtain the RF field strength. The energy levels are measured using
electromagnetically induced transparency (EIT) EIT ; EIT2 .

Here we describe the physical principles underlying the technique. Consider a
sample of stationary four-level atoms illuminated by a single weak (“probe”)
light field, as depicted in Fig 1. When the frequency of the light matches the
$|1\rangle$ to $|2\rangle$ atomic resonance, the atoms scatter light from the
incident beam and reduce the transmitted light intensity. If a second strong
(“coupling”) light field is applied resonant with the $|2\rangle$ to
$|3\rangle$ transition, the $|2\rangle$ and $|3\rangle$ states are mixed to
form dressed states which are close in energy. The excitation amplitudes from
$|1\rangle$ to each of these two dressed states then have opposite signs,
leading to destructive quantum interference of these excitation pathways. As
such, a transparency window is opened for the probe light: probe light
transmission is increased. This is the phenomenon known as EIT EIT . If one
applies an RF field which couples states $|3\rangle$ and $|4\rangle$, a third
dressed state is introduced between the two involved in EIT which leads to
constructive interference in the probe absorption. This splits the EIT
resonance in two, and for resonant driving fields the new transmission maxima
are split by the Rabi frequency $\Omega_{RF}$ of the $|3\rangle$ – $|4\rangle$
transition Lukin1999 ; Dutta2007 . This is known as Autler-Townes splitting of
the EIT signal tony . Therefore the frequency splitting $\Delta f_{0}$ between
the transmission maxima allows a measurement of the E-field amplitude of the
RF field via

$|E|=\frac{\hbar}{\wp}{\Omega_{RF}}=2\pi\frac{\hbar}{\wp}{\Delta f_{0}}$ (1)

where $\hbar$ is Planck’s constant and $\wp$ is the electric dipole moment of
the RF transition $|3\rangle$ to $|4\rangle$.

In order to measure the field amplitude for different RF frequencies,
different states $|3\rangle$ and $|4\rangle$ can be chosen. State $|3\rangle$
is selected by tuning the wavelength of the coupling laser. A large range of
atomic transitions can be selected, allowing measurements of microwave fields
over a correspondingly wide selection of frequencies. In essence, the atoms
act as highly-tunable, resonant RF detectors. This is a significant benefit of
using Rydberg atoms as field probes. In broadband , we use this fact to show
how the technique can be used for broadband measurements of RF E-fields,
ranging from 1 GHz to 500 GHz.

##  II Experimental Setup

The experimental setup is shown in Fig. 2. We use a cylindrical glass vapor
cell of length 75 mm and diameter 25 mm containing rubidium-85 (85Rb) atoms.
The levels $|1\rangle$, $|2\rangle$, $|3\rangle$, and $|4\rangle$ correspond
respectively to the 85Rb $5S_{1/2}$ ground state, $5P_{3/2}$ excited state,
and two Rydberg states. The probe is a 780 nm laser (“red”) which is scanned
across the $5S_{1/2}$ – $5P_{3/2}$ transition. The probe beam is focused to a
full-width at half maximum (FWHM) of 80 $\mu$m, with a power of 120 nW to keep
the intensity below the saturation intensity of the transition. Figure 3 shows
a typical transmission signal as a function of relative probe detuning
$\Delta_{p}$. The global shape of the curve is the Doppler absorption spectrum
of 85Rb at room temperature. To produce an EIT signal, we apply a counter-
propagating coupling laser (wavelength $\lambda_{c}\approx 480$ nm, “blue”)
with a power of 22 mW, focused to a FWHM of 100 $\mu$m. As an example, tuning
the coupling laser near the $5P_{3/2}$ – $50D$ Rydberg transition results in
distinct EIT transmission peaks corresponding to the transitions from
$5S_{1/2}$ to the allowed $5P_{3/2}$ hyperfine sublevels ($F=4,3,2$), which
are strongly coupled to the fine-structure-split $50D_{5/2}$ and $50D_{3/2}$
Rydberg states. The peaks for the strongest of these cascades are visible atop
the Doppler profile in Fig 3. Differential Doppler shifts between the probe
and coupling beams alter the frequency separations between EIT peaks in the
probe transmission spectrum. Splittings of $5P_{3/2}$ hyperfine states are
scaled by $1-\lambda_{c}/\lambda_{p}$, while splittings of Rydberg states are
scaled by $\lambda_{c}/\lambda_{p}$ EIT_Adams . The latter factor is relevant
to measurements of RF-induced splittings of EIT peaks.

Hereafter, for each EIT system we investigate, we focus on the strongest peak,
corresponding to the $5P_{3/2}(F=4)$ – $nD_{5/2}$ transition. We take this
peak to be at $\Delta_{p}=0$. In order to improve the signal-to-noise ratio,
we use heterodyne detection. We modulate the blue laser amplitude with a 30
kHz square wave and detect any resulting modulation of the probe transmission
with a lock-in amplifier. This removes the Doppler background and isolates the
EIT signal as shown in the black curve of Fig. 4. Here we tune the coupling
laser near the $5P_{3/2}$ – $28D_{5/2}$ transition. Application of RF to
couple states $28D_{5/2}$ and $29P_{3/2}$ splits the EIT peak as shown in the
gray curve. We measure the frequency splitting of the EIT peaks in the probe
spectrum, $\Delta f$, and determine the E-field amplitude by

$|E|=2\pi\frac{\hbar}{\wp}\frac{\lambda_{p}}{\lambda_{c}}\Delta f\quad.$ (2)

Note here the use of the Doppler scaling factor, not present in Eq. 1 for
stationary atoms.

The E-field sensing volume is determined by the overlap of the RF, probe beam,
and coupling beam within the vapor cell. Based on the geometries given above,
this volume is a cylinder of length 75 mm and diameter 80 $\mu$m. The small
optical beam diameter gives the measurement high spatial resolution in the
dimensions transverse to the optical beams, which is crucial in the
experiments presented next.

Figure 3: Probe transmission as a function of $\Delta_{p}$ for the three-level
$5S_{1/2}-5P_{3/2}-50D$ EIT system. EIT peaks are visible for transitions
corresponding to (right to left) $5P_{3/2}(F=4)$ – $50D_{5/2}$,
$5P_{3/2}(F=3)$ – $50D_{5/2}$, and $5P_{3/2}(F=4)$ – $50D_{3/2}$.

Figure 4: Black curve: EIT-signal as a function of $\Delta_{p}$ for the EIT
system $5S_{1/2}-5P_{3/2}-28D_{5/2}$. Gray curve: The $28D_{5/2}$ level is
coupled to the $29P_{3/2}$ level by a 104.77 GHz RF field.

##  III Sub-Wavelength Field Mapping

When an electromagnetic (EM) wave is incident onto a hollow dielectric
cylinder, standing waves typically develop inside the cylinder due to internal
reflections from the cylinder walls. The resulting field distribution will
vary depending on the EM frequency. Using the technique explained in the
previous section, we image the field inside our glass vapor cell (which is,
electromagnetically, a hollow dielectric cylinder) for two different RF
frequencies: 104.77 GHz and 17.04 GHz. For the 104.77 GHz measurements, we
deliver the RF with an open-ended waveguide (see Fig. 5); for 17.04 GHz we use
a horn antenna. In each of these measurements, the vapor cell is placed on a
translation stage with 12 mm of travel, see Fig. 5. The cell is then
translated in a direction perpendicular to the propagation directions of the
optical beams. This allows the imaging of RF fields inside the cell as a
function of the spatial coordinate parallel to the translation axis. The
spatial resolution is limited by the optical beam diameter (80 $\mu$m).

Figure 5: Experimental setup for field-mapping measurements with EIT. The
vapor cell is on a translation stage and is scanned with respect to the
optical beams. The waveguide in the photo is closer than that used in the
measurement.

We first perform measurements at 104.77 GHz. The blue laser is tuned to
$\approx 482.23$ nm to couple states $5{\rm P}_{3/2}$ and $28{\rm D}_{5/2}$,
and the 104.77 GHz field couples $28{\rm D}_{5/2}-29{\rm P}_{3/2}$. The open-
ended waveguide is spaced 0.14 m from the focal axis of the lasers, and is
supplied with 0.58 mW of RF power (measured with a power meter attached to the
end of the waveguide).

The cell is translated away from the source in discrete steps. At each step
position we measure the splitting of the EIT signal. We convert to an electric
field using Eq. 2, and the dipole matrix element for this transition,
$\wp=473.14ea_{0}$. Here, $e$ is the electron charge and $a_{0}$ is the Bohr
radius. The dashed line in Fig. 6 shows the measured splitting (right axis)
and the E-field amplitude (left axis) as a function of position for a step
size of 0.25 mm. The crosses show a second scan at higher resolution with a
step size of 0.10 mm, which corresponds to the larger of the two laser beam
widths. The origin in Fig. 6 corresponds to a distance of approximately 8.4 mm
between the laser beams and the inside edge of the cell that is furthest from
the source.

Figure 6: Measured EIT splitting, $\Delta f$ (right axis) and corresponding
electric field amplitude $|E|$ (left axis) as a function of position inside
the cell at 104.77 GHz.

These two sets of measurements lie on top of one another, showing the
measurement is repeatable. The results further demonstrate significant
periodic field variation inside the cell. We see up to approximately $\pm
20~{}\%$ variation in the field amplitude over the 12 mm scan. The average
period of the observed pattern is approximately half the wavelength of the RF
($\lambda_{RF}/2=1.43$ mm). The two sets of measurements yield field-mapping
resolutions of $\approx\lambda/10$ and $\approx\lambda/30$, respectively. We
have thus shown that this method enables sub-wavelength mapping of 104.77 GHz
radiation fields, and, importantly, that the achieved spatial resolution is
comparable to the laser beam spot size.

Next, we compare the measured field distribution inside the cell to the
results of a three-dimensional numerical full wave simulation performed using
a commercial finite element code. It is challenging to perform such a
numerical simulation of the actual experimental setup at high frequencies
because of computer memory requirements. This is partially due to the small RF
wavelength at 104.77 GHz, and the relatively large cell size (several RF
wavelengths at 104.77 GHz). To overcome these issues we did the following.
Instead of modeling the actual open-ended waveguide placed 0.14 m from the
cell, we performed a numerical simulation for a plane-wave impinging on the
cell with a field strength of 2.8 V/m. In order to determine this field-
strength value for the plane-wave, the open-ended waveguide source antenna was
modeled independently and the numerical simulated far-field radiation pattern
and field strength was determined. Using the measured working distance and
power mentioned above, this yields a field amplitude of 2.8 V/m at the
location of the laser beams in the experiment. The cell in the simulation has
dimensions as mentioned above for the experiment, with glass having
$\epsilon_{r}=5.5$ at RF frequencies. We assumed $\epsilon_{r}=1$ for the
region inside the cell. Figure 7 shows the numerical results for the field
inside the cell with the incident plane wave source. This contour plot shows
the expected field distribution, in which the field variation is primarily
along the propagation direction of the incident RF wave. This indicates that
our measurement method gives good spatial resolution in the only dimension
along which there is significant field strength variation. To quantitatively
compare simulation with measurement, we average the numerical results along 92
$\%$ of the length of the cell. The comparison is shown in Fig. 8, which shows
that the simulation and measurements give similar spatial variation. Both show
field distributions with a period roughly equal to half the RF wavelength. We
see good qualitative agreement between the numerical results and the data.

(a) (b)

Figure 7: Simulation of electric-field amplitude $|E|$ for a plane wave
incident onto the vapor cell from the right: (a) incident + scattered, and (b)
scattered. The region shown in the figure is a horizontal planar cut through
the center of the cell, with half of the cell shown. The E-field is on a
linear colorscale ranging from 0.7 V/m (blue) to 5 V/m (red).

Figure 8: Comparison of experimental and simulated $|E|$ as a function of
position inside the cell at 104.77 GHz.

To study field mapping in the cm-wave regime we repeat the experiment with a
17.04 GHz source. Here, the blue laser is tuned to $\approx 480.13$ nm to
couple states $5P_{3/2}$ and $50D_{5/2}$. The RF couples states $50D_{5/2}$
and $51P_{3/2}$. We use a horn antenna at a distance of 0.880 m from the laser
beams. Figure 9 shows the measured E-field as a function of position inside
the cell, where we have used $\wp=1574.83ea_{0}$. For this measurement the
origin of the position axis corresponds to the laser beams being approximately
10 mm from the inner surface of the cell that is furthest from the RF source.
The variation of the measured field is approximately $\pm 50$ $\%$ of its
average, and the observed separation between the maximum and minimum is
$\approx\lambda_{RF}/4$.

We perform numerical simulations for this case as well, using a plane wave
source with a field amplitude at the vapor cell determined from a far-field
calculation. Based on source power, cable losses, and known antenna
characteristics, the field amplitude at the location of the laser beams is
0.76 V/m. In Fig. 9, we compare the data with results obtained from the
numerical simulation. Here, agreement is good.

Figure 9: Experimental and simulated $|E|$ as a function of position inside
the cell at 17.04 GHz.

##  IV Discussion and Conclusion

In this paper we have demonstrated a technique based on Rydberg atoms, EIT,
and Autler-Townes splitting which can perform sub-wavelength imaging and field
mapping of RF radiation. We have validated the approach by comparing the
measured field inside a hollow glass cylinder to results obtained from a full-
wave numerical simulation.

While the spatial resolution of our measurements is determined by the $\approx
100\mu$m beam widths of our lasers, the spatial resolution of the method is in
principle limited by the optical diffraction limit. This is a significant
improvement over the measurement resolution achievable by conventional probes.
There are many possible applications of this technique. For example, the
sensing volume could be scanned over a printed-circuit-board (PCB) or a
metasurface in order to map their fields, as well as other applications where
E-field measurements on a small spatial resolution are desired. We aim to
demonstrate these applications in future work.

## References

  * (1) M. Kanda and L. Driver, “An isotropic electric-field probe with tapered resistive dipoles for broad-band use, 100 kHz to 18 GHz,” IEEE Trans. on Microwave Theory and Techniques, vol. 35, no. 7, pp. 124-130, Frb., 1987.
  * (2) C.L. Holloway, E.F. Kuester, J.A. Gordon, J. O’Hara, J. Booth and D.R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials”, IEEE Antennas and Propagation Magazine, vol. 54, no. 2, pp. 10-35, April 2012.
  * (3) J.A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapor cell using bright atomic resonances”, Nature Physics 8, 819, 2012.
  * (4) C.L. Holloway, J.A. Gordon, A. Schwarzkopf, D. A. Anderson, S. A. Miller, N. Thaicharoen, G. Raithel, S. Jefferts, and T. Heavner, “Broadband Rydberg Atom-Based Electric-Field Probe/Sensor: From Self-Calibrating Measurements to Sub-Wavelength Imaging,” IEEE Trans. on Antenna and Propag., 2014\.
  * (5) J.A. Sedlacek, A. Schwettmann, H. Kübler, and J. P. Shaffer, “Atom-Based Vector Microwave Electrometry Using Rubidium Rydberg Atoms in a Vapor Cell”, Phys. Rev. Lett., 111, 6, 2013
  * (6) J.A. Gordon, C.L., Holloway, S., Jefferts, and T. Heavner, “Quantum-Based SI Traceable Electric-Field Probe,”, Proc 2010 Int. Symp. on Electromagnetic Compatibility, Fort Lauderdale, FL, pp. 321-324, 25-30 July, 2010.
  * (7) J.A. Gordon, C.L. Holloway, A. Schwarzkopf, D. A. Anderson, S.A. Miller, N. Thaicharoen, and G. Raithel, “Millimeter-Wave Detection via Autler-Townes Splitting In Rubidium Rydberg Atoms,” IEEE Trans. on Antenna and Propag., 2014.
  * (8) S.H. Autler and C.H. Townes, “Stark Effect in Rapidly Varying Fields,” Physical Review, vol. 100, no. 2, pp. 703–722, 1955.
  * (9) M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in Coherent Media,” Reviews Modern Physics, vol. 77, pp. 633-673, April, 2005.
  * (10) K.J. Boller, A. Imamolu, and S.Ee Harris, “Observation of electromagnetically induced transparency,” Physical Review Letters, vol. 66, no. 20, pp. 2593-2596, May, 1991.
  * (11) M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully. “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225 (1999)
  * (12) B. K. Dutta and P. K. Mahapatra. Phys. Scr. 75 345, (2007)
  * (13) T.Y. Abi-Salloum, “Electromagnetically induced transparency and Autler-Townes splitting: Two similar but distinct phenomena in two categories of three-level atomic systems,” it Phys. Rev. A, 81, 053836, 2010.
  * (14) A.K. Mohapatra, T.R. Jackson, C.S Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett. 98, 113003 (2007).