# The performance of a new Kuraray wavelength shifting fiber YS-2

I. Alekseev,11footnotetext: Corresponding author. M. Danilov  V. Rusinov  E.
Samigullin  D. Svirida  E. Tarkovsy

###### Abstract

Wavelength shifting fibers are widely used for light collection from
scintillation counters, which allow connection of various scintillation planes
to relatively small photocathodes of photodetectors and especially tiny
photocathodes of silicon photo-multipliers. In October 2020 Kuraray announced
production of a new branch of faster fibers. We performed a comparison of the
new fiber YS-2 to a mature Y-11. The fiber YS-2 demonstrated decay time nearly
two times shorter than that of Y-11: $\approx$4.0 ns versus $\approx$7.4 ns.
At the same time its light yield and attenuation length are as good as of
Y-11, which makes YS-2 a good choice for timing scintillation detectors.

##  1 Introduction

One of the typical applications of wavelength shifting (WLS) fibers is a light
collection from scintillation detectors. In usual design a WLS fiber is
surrounded by a scintillator by placing the fiber in a groove or a hole in the
scintillator medium or attached to the edge of the scintillator plane. In most
cases this technique is not good for timing measurements. The first problem is
in geometry, producing large variations of light trajectories. But the other
problem is in the time distribution of photon emission by the WLS fiber.
Nevertheless there are many cases when even some moderate time resolution is
important. Examples are scintillation counters of CHOD hodoscope of the NA62
experiment [1], where, in the presence of a very intense beam, the time
resolution is important to suppress overlapping events, a new CMD-3 Time-of-
Flight system [2], or new scintillation detectors for the DANSS experiment
upgrade [3], where the time difference between signals from both ends of the
scintillation strip allows reconstructing the coordinate. Kuraray announcement
of the new line of WLS fibers with shorter decay time looks very promising
[4]. We report here the results of the comparison of 1 mm Kuraray fibers:
YS-2MJ Multi (YS-2) and Y-11(200)M Multi (Y-11).

##  2 Decay time measurement

A single photo-electron mode was used to measure the decay time. After an
instant excitation of the WLS fiber at $t=0$ the probability to emit a photon
at some time $t$ will follow an exponential law:

$N(t)=\left\\{{\begin{array}[]{ll}0&\;,\;\mathrm{when}\;t<0\\\
N_{0}e^{-t/\tau}&\;,\;\mathrm{when}\;t\geq 0\\\ \end{array}}\right.\;,$

where $N_{0}$ is the initial emission intensity and $\tau$ is the decay time.
In case of some time measurement uncertainty, which follows Gaussian
distribution, the probability will be given by a convolution of an exponential
decay and this Gaussian function. The resulting distribution is given by an
equation:

$N(t)=C\left(1+\mathrm{erf}(\frac{t-t_{0}-\sigma^{2}/\tau}{\sqrt{2}})\right)e^{-(t-t_{0})/\tau}\;,$
(2.1)

where $C$ is the normalization coefficient, $t_{0}$ is the start time,
including light in the fiber propagation delay, $\sigma$ is the system time
resolution, $\tau$ is the decay time and $\mathrm{erf}()$ is the Gauss error
function [5]. We use short laser pulses (less than 85 ps FWHM) and select
events with only a single photon observed. To extract the decay time the time
distribution of these events was fit by a sum of formula (2.1) and time-
independent SiPM noise.

A schematic layout of the decay time measurement is shown in figure 1. A 1.5 m
length piece of tested fiber was polished on one end and connected to
HAMAMATSU S12825-050C silicon photo-multiplier (SiPM). The other end of the
fiber was painted black to avoid reflection. An open end of a fiber-optic
light guide was placed a few millimeters from the tested fiber at three
different distances 5, 10, and 20 cm from its SiPM end. The light guide
delivers very short pulses from UV picosecond laser VisUV-266-355-532 by
PicoQuant GmbH. An output with a 355 nm wavelength was used. The laser was
triggered by a pulser with a 3 kHz repetition rate. The signal from SiPM was
amplified and connected to one input of UVFD64 waveform digitizer module [6].
Another input of the module was used for laser output synchronization pulse,
providing the trigger for DAQ and the time reference. UWFD64 uses 12 bit 125
MSPS flash ADC to digitize each of its 64 inputs, two of which were used in
this work. An example of a digitized single pixel pulse is shown in figure 3.
The laser power was adjusted so that only one or no photo-electron was
produced in most of the events. The distribution of events over pulse
amplitude is shown in figure 3. Peaks corresponding up to five pixels fired
are seen. Zero pixel peak is not shown. For the cleanest selection of single
pixel events, an elliptical cut over two-dimensional distribution charge
(integral) versus amplitude is used (figure 5). This allows rejecting events
with more than one single pixel pulses, corresponding to several photo-
electrons in one event separated by some time.

Figure 1:  A schematic layout of decay time measurement.

Figure 2: Digitized single pixel waveform.

Figure 3: Signal amplitude distribution.

Figure 4: Distribution of the pulse charge (integral) over its amplitude. The
ellipse shows the cut, selecting single pixel events.

Figure 5: Single pixel time distribution for direct SiPM illumination. The
curve represents a fit by Gaussian function and uniform background.

The system time resolution was measured in a dedicated experiment, where a
laser light guide was directed to the SiPM without a WLS fiber. The time
distribution of single pixel events was fitted by a sum of Gaussian and
uniform background (see figure 5). The measured $\sigma=0.8019\pm 0.0007$ ns
was fixed for further fits of the fiber measurements. The decay time of both
YS-2 and Y-11 WLS fibers was measured at 3 distances from the laser beam spot
to the SiPM. No dependence on the distance was observed. Time distributions
for 10 cm distance are shown in figure 6. The results of the measurements are
summarized in table 1.

Figure 6: Single pixel time distribution for fibers YS-2 (left) and Y-11 (right). The curve represents a fit by the function (2.1) and uniform background. Table 1: The measured decay time $\tau$ in nanoseconds. Only statistical errors are given in 5, 10 and 20 cm columns. The last column presents the values provided by the manufacturer [4]. Fiber | 5 cm | 10 cm | 20 cm | Average $\pm$ RMS | Kuraray datasheet
---|---|---|---|---|---
YS-2 | $3.949\pm 0.006$ | $4.005\pm 0.006$ | $4.003\pm 0.005$ | $3.99\pm 0.03$ | 3.2
Y-11 | $7.293\pm 0.010$ | $7.295\pm 0.010$ | $7.522\pm 0.010$ | $7.37\pm 0.11$ | 6.9

We collected about 2 million events for each configuration out of which
0.8-0.9 million satisfy single pixel criteria, so the statistics in each point
are very high and statistical errors are small. Values of the decay time
$\tau$ are stable to the fit range and Gaussian width $\sigma$ variation. An
estimation of the systematic error could be estimated from the result
variation at different distances. RMS of the values at 3 distances was
calculated and indicated as an error in the last but one column of the table
1.

##  3 Light output measurement with cosmic rays

An important question for WLS fiber is the light output in the real detector
geometry. For the comparison of YS-2 and Y-11 a sample scintillation counter
with 4 grooves was manufactured from polystyrene (see figure 7). Fibers were
cut at 23 cm and polished on both ends. One end of a fiber was connected to
SiPM and the other end was left open. Four fibers of one type were put into
the grooves without any glue and exposed to cosmic rays. Then fibers were
exchanged keeping SiPM and grooves correspondence and another exposition
taken. The results are presented in table 2. Individual values have a
statistical error of about 2% not shown in the table. All 4 fibers are in the
same conditions, so their variation is used for error estimation, which is
shown in the last column together with average values. This simple test
demonstrated that the light yield of YS-2 is at least as good as of Y-11.

Figure 7: Layout of the test counter with 4 grooves. Grooves numbers are also shown. Table 2: Median light output for cosmic particles crossing the test scintillator counter. Values are given for each fiber. Groove | 1 | 2 | 3 | 4 | Average $\pm$ RMS
---|---|---|---|---|---
YS-2, ph.e. | 30.7 | 24.6 | 28.2 | 25.5 | $27.3\pm 2.4$
Y-11, ph.e. | 24.8 | 25.5 | 28.6 | 22.7 | $25.4\pm 2.1$

##  4 Longitudinal attenuation measurement

Measurement of light attenuation was performed with the help of 90Sr beta
source placed above a rectangular piece of the scintillator ($30\times
30\times 3$ mm3) with a hole drilled along its 30 mm side. A scintillation
counter was placed below the scintillator with the hole and allowed to trig on
the electrons passing through it. The set of the source and scintillators may
slide along the tested fiber for measurements at various distances. 1.2 m
length pieces were used, polished on both ends with SiPM attached to one end.
Median light output in photo-electrons is shown in figure 8 as a function of
the distance from the readout end. Longitudinal attenuation of YS-2 looks to
be as small as of Y-11. Our YS-2 samples were relatively short, so we cannot
compare our attenuation length value for YS-2 ($1.45\pm 0.07$ m), calculated
using the data between 0.3 and 1.2 m, with manufacture’s value of 3.5 m,
corresponding to distances $1.0-3.0$ m [7].

Figure 8: Signal longitudinal attenuation in YS-2 (full circles) and Y-11
fibers (open squares).

##  5 Conclusions

Timing response of scintillation counters with WLS fibers is already studied
relatively well [8, 9, 10] and many more. The decay time measured for Y-11
fiber is in an agreement with the previous measurement [9]. In that work a fit
of large (many photons) signal shape was performed and the result was $8.8\pm
1.5$ ns for Y-11(M200) and $7.3\pm 1.1$ ns for Y-11(MS250). This time is
essentially larger than that of BCF-92 by Saint-Gobain [11] ($2.4\pm 0.4$ ns
[9]). But BCF-92 demonstrates significantly lower light yield – the average
ratio for various scintillation materials and the same detector geometry
between Y-11 and BCF-92 is $1.85\pm 0.07$ [10]. For the new Kuraray WLS YS-2
the light output and attenuation length are as good as of Y-11. The decay time
of YS-2 is nearly two times shorter. This makes YS-2 a good choice for timing
measurements, leaving well behind Y-11 with its comparatively large decay time
and BCF-92 with its comparatively low light yield.

## Acknowledgments

This work is supported by the Ministry of Science and Higher Education of the
Russian Federation under Contract No. 075-15-2020-778.

## References

  * [1] V. Duk et al., JINST 11 (2016) P06001
  * [2] A. Amirkhanov et al., Nucl. Instrum. Meth. in Physics Research A 936 (2019) 598
  * [3] Svirida D J. Phys. Conf. Ser. 1690 (1) (2020) 012179
  * [4] Short Decay time Wavelength Shifter (YS-series). http://kuraraypsf.jp/pdf/YSSeries_201007.pdf
  * [5] M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1972
  * [6] I.G. Alekseev et al., Instruments and Experimental Techniques 61 (3) (2018) 349
  * [7] Plastic Scintillating Fibers. https://www.kuraray.com/uploads/5a717515df6f5/PR0150_psf01.pdf
  * [8] R. Wojcik et al., Nuclear Instruments and Methods in Physics Research A 342 (1994) 416
  * [9] V. Brekhovskih et al., The WLS Fiber Time Properties Study, Preprint CERN-LHCb-2000-039
  * [10] G.I. Britvich et al., Instruments and Experimental Techniques 58 (2) (2015) 211
  * [11] Organic Scintillation Materials and Assemblies. https://www.crystals.saint-gobain.com/sites/imdf.crystals.com/files/documents/organics-plastic-scintillators.pdf


In this section, we will discuss three key factors of  ():
numerically stable estimation, alignment shaping, and streaming finetuning.
It is noteworthy that the monotonic alignment estimation, denoted as $\alpha$, discussed in this section is based on a single attention head.
Following the same design as [Ma et al., 2019], the same estimation of $\alpha$ is applied to every attention head in  as integrated into the Transformer [Vaswani et al., 2017] model.
Notably, only the infinite lookback [Arivazhagan et al., 2019] variant of monotonic attention is applied.

§.§ Numerically Stable Estimation

Similar to [Raffel et al., 2017],
the objective of the monotonic estimation is to calculate the expected alignment $\alpha_{i, j}$,
from stepwise action probability $p_{i, j}$.
Nevertheless, the numerical instability arises from the denominator in Equation <ref>,
particularly when dealing with the multiplication of several small probabilities.
To address this issue, we introduce a innovative numerically stable approach for estimation of monotonic attention.

In accordance with the approach proposed by [Raffel et al., 2017],
the monotonic alignment between the $i$-th target item and the $j$-th source item can be represented as:

\begin{equation}
\alpha_{i,j} = p_{i,j} \sum_{k=1}^j \alpha_{i-1, k} \prod_{l=k}^{j-1}(1 - p_{i,l})
\end{equation}

This expression can be reformulated in a matrix multiplication format:

\begin{equation}
\alpha_{i,:} = p_{i,:} \odot \alpha_{i-1, :} \mathbf{T}(i)
\label{eq:alpha_matrix_multi}
\end{equation}

where $\mathbf{T}(i)$ represents a transition matrix, with each of its elements defined as:

\begin{equation}
\mathbf{T}(i)_{m,n} =
\begin{cases}
\prod_{l=m}^{n-1} (1 - p_{i,l}) & m < n\\
1 & m = n\\
0 & m > n \\
\end{cases}
\end{equation}

$\mathbf{T}(i)_{m,n}$ indicates the probability of the policy at the $(i-1)$-th target step consecutively skipping from the $m$-th to the $n$-th inputs.
Moreover, the transition matrix can be further expressed as

\begin{equation}
\mathbf{T}(i) = \texttt{triu}_0\left(\texttt{cumprod}_2(1 - \texttt{triu}_1\left( J_{|X| \times 1} \texttt{roll}_1(p_{i,:}) \right))\right)
\end{equation}

The operations within the equation can be efficiently executed in parallel on GPUs,
as elaborated in <Ref>.
Reframing Equation <ref>, we arrive at the following expression:

\begin{equation}
\alpha_{i,:} = p_{i,:} \odot \alpha_{i-1, :} \texttt{triu}_0\left(\texttt{cumprod}_2(1 - \texttt{triu}_1\left( J_{|X| \times 1} \texttt{roll}_1(p_{i,:}) \right))\right)
\label{eq:emma}
\end{equation}

It is noteworthy that this estimation process is also closed-form, with the desirable properties of being numerically stable and unbiased,
as it does not require a denominator as the product of probabilities within the equation.
A comprehensive derivation of this closed-form estimation is provided in <Ref>.

§.§ Alignment Shaping

When training the infinite lookback variant of monotonic attention,
it is necessary to add latency regularization in order to prevent the model from learning a trivial policy.
Without latency regularization,
the optimal policy to minimize cross-entropy loss is to the entire the sequence before starting the translation.
Therefore, we applied latency and variance regularizations to control the tradeoff between translation quality and latency of the learned simultaneous translation policy.

The latency of the alignment describes how much partial input information is needed by the model to generate part of the translation.
The reduction of latency is commonly achieved by introducing a regularization term derived from the estimated alignment.
Consistent with prior work,
such as [Arivazhagan et al., 2019, Ma et al., 2019],
the expected delays $\bar{\mathbf{D}}=\bar{d_1},...,\bar{d}_{|\mathbf{Y}|}$ are estimated from the expected alignment $\alpha$ during training time.
The expected delay of target token $y_i$, denoted as $\bar{d}_i$, is computed as

\begin{equation}
\bar{d}_i = \texttt{E}[j | i] = \sum_{k=1}^{|\mathbf{X}|} k \alpha_{i,k}
\end{equation}

Given a latency metric $\mathcal{C}$, the loss term is then computed as

\begin{equation}
\mathcal{L}_{\text{latency}} = \mathcal{C}(\bar{\mathbf{D}})
\end{equation}

The variance of the alignment characterizes the certainty of an estimation.
It is noteworthy that an alignment estimation can be low latency but high variance.
For instance, a random walk policy, which yields a monotonic alignment of linear latency,
has a huge variance on the estimation.
[Arivazhagan et al., 2019] proposed a method to reduce the uncertainty by introducing a Gaussian noise to the input of stepwise probability network.
Nevertheless, empirical results show that this method is not efficient, especially when applied to speech translation models.
Therefore, we propose an alternative regularization-based strategy.

Denote the $\mathcal{V} = \bar{v_1}, ..., \bar{v}_{|\mathbf{Y}|}$ as the expected variances of the monotonic alignment.
The expected variance of target token $y_i$, denoted as $\bar{v}_i$, can be expressed as

\begin{equation}
\bar{v}_i = \texttt{E}[(j - \texttt{E}[j | i])^2 | i] = \texttt{E}[j^2 | i] -  \texttt{E}[j | i]^2 = \sum_{k=1}^{|\mathbf{X}|} k^2 \alpha_{i,k} - \left(\sum_{k=1}^{|\mathbf{X}|} k \alpha_{i,k}\right)^2
\end{equation}

We then introduce the alignment variance loss as the following:

\begin{equation}
\mathcal{L}_{\text{variance}} = \sum_{i=1}^{|\mathbf{Y}|}  \bar{v}_i \
\end{equation}

To further reduce the alignment variance, we proposed an enhanced stepwise probability network as

\begin{equation}
p_{i, j} = \texttt{Sigmoid}\left(\frac{ \texttt{FFN}_s(s_{i-1})^T \texttt{FFN}_h(h_j) + b }{\tau} \right)
\end{equation}

$\texttt{FFN}_s$ and $\texttt{FFN}_s$ serve as energy projections, constructed using multi-layer feedforward networks,
which increase the expressive capability of stepwise probability network over linear projection adopted by prior monotonic work.
$b$ is a learnable bias, initialized by a negative value.
Its purpose is to schedule an easier the policy optimization process from the offline policy.
$\tau$ is the temperature factor, to encourage polarized output from stepwise probability network.

Finally, we optimize the model with the following objective

\begin{equation}
\mathcal{L}(\theta) = -\text{log}(\mathbf{Y} | \mathbf{X}) + \lambda_{\text{latency}} \mathcal{L}_{\text{latency}} +  \lambda_{\text{variance}} \mathcal{L}_{\text{variance}}
\end{equation}

where $\lambda_{\text{latency}}$ and $\lambda_{\text{variance}}$ are the loss weights.

§.§ Simultaneous Fine-tuning

In most prior work on simultaneous translation,
the model is usually trained from scratch.
However, this approach often requires substantial resources when dealing with extensive or multilingual scenarios.
For instance, it can be a significant challenge to retrain a simultaneous model with the configuration from recent large-scale multilingual models,
such as Whisper or SeamlessM4T.
To leverage the recent advancements achieved with large foundational translation models and enhance the adaptability of the simultaneous translation model, we introduce a method for Simultaneous Fine-tuning.

Denote the an arbitrary offline encoder-decoder translation model as $\mathcal{M}(\theta^o_e, \theta^o_d)$, with $\theta_e$ representing the encoder parameters and $\theta_d$ representing the decoder parameters.
The simultaneous model is denoted as $\mathcal{M}(\theta_e, \theta_d, \theta_p)$, where $\theta_p$ denotes the policy network.
Simultaneous fine-tuning involves initializing $\theta_e$ with $\theta^o_e$ and $\theta_d$ with $\theta^o_d$.
During the training process, the encoder parameters $\theta_e$ remain fixed, and optimization is only performed on $\theta_d$ and $\theta_p$.
This design is motivated by the assumption that the generative components of the model,
namely $\theta_e$ and $\theta_d$,
should closely resemble those of the offline model.
In simultaneous setting, they are adapted to partial contextual information.

§.§.§ Streaming Inference

We used  [Ma et al., 2020] to build the inference pipeine.
The overall inference algorithm is illustrated in <Ref>.
For streaming speech input, we update the whole encoder every time a new speech chunk is received by the model.
Then, we run the decoder to generate a partial text translation based on the policy.

[1]
$\mathbf{X}$: Input streaming speech.
$\mathbf{Y}$: Output text.
$t_{\text{\emma{}}}$ : Decision threshold for
$i \gets 1$, $j \gets 0$, $k \gets 0$
$s_0 \gets \texttt{TextDecoder}(y_{0})$
$y_{i-1} \neq \texttt{EndOfSequence}$
$j \gets j + 1$
$h_{\leq j} \gets \texttt{SpeechEncoder}(\mathbf{X}_{\leq j})$
$y_{i-1} \neq \texttt{EndOfSequence}$
$p \gets 1$
$\texttt{StepwiseProbabilty}$ in all attention head
$p \gets \texttt{min}(p, \texttt{StepwiseProbabilty}(h_j, s_{i-1}))$
$p < t_{\text{\emma{}}}$
$y_i, s_i \gets \texttt{TextDecoder}(s_{<i}, h_{\leq j})$
$k \gets k + 1$
$i \gets i + 1$

§ RESULTS

§.§ Quality-Latency Trade-off

We present the quality-latency trade-off on the bilingual setting.
<Ref> shows the BLEU score under different latency settings.
We can see that the  model significantly outperforms the Wait-k model on all the latency regions in both directions.

Spanish to English

English to Spanish

The quality-latency trade-off for bilingual model.

We present the quality-latency trade-off on the multilingual setting, in <Ref>.
SeamlessM4T- can achieve decent translation quality in a much shorter training time compared with training from scratch.

$t_{\text{\emma{}}}$     BLEU     AL

SeamlessM4T         28.8     -
0.5     26.0     1.75

0.7     26.4     1.88

0.4     25.9     1.68

0.6     26.2     1.81

Average translation quality and latency under multilingual setting with different latency decision thresholds $t_{\text{\emma{}}}$.

Visualization of monotonic alignment. Left side is stepwise probability $p$ and right side is estimated alignment $\alpha$.

§.§ Visualization

We also visualize the policy learned from  as <Ref>.
We can see that with the guidance of latency and variance regularization, the model can learn a monotonic policy aligning input speech and output text.

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# Extreme events generated in microcavity lasers and their predictions by
reservoir computing

T<EMAIL_ADDRESS>H. X. Zhou1, Q. Fang2, Y. N. Han1, X. X.
Guo1, Y. H. Zhang1, C. Qian3, H. S. Chen3, S. Barland4, S. Y.
<EMAIL_ADDRESS>G. L. Lippi4,333Gian-Luca.LIPPI@univ-
cotedazur.fr 1 State Key Laboratory of Integrated Service Networks, Xidian
University, Xi’an 710071, China  2 School of Electronics and Information,
Hangzhou Dianzi University, Hangzhou 310018, China  3 Interdisciplinary Center
for Quantum Information, College of Information Science and Electronic
Engineering,Zhejiang University, Hangzhou 310027, China  4 Université de la
Côte d’Azur, Institute de Physique de Nice, UMR 7010 CNRS, France

###### Abstract

Extreme events generated by complex systems have been intensively studied in
many fields due to their great impact on scientific research and our daily
lives. However, their prediction is still a challenge in spite of the
tremendous progress that model-free machine learning has brought to the field.
We experimentally generate, and theoretically model, extreme events in a
current-modulated, single-mode microcavity laser operating on orthogonal
polarizations, where their strongly differing thresholds – due to cavity
birefringence – give rise to giant light pulses initiated by spontaneous
emission. Applying reservoir-computing techniques, we identify in advance the
emergence of an extreme event from a time series, in spite of coarse sampling
and limited sample length. Performance is optimized through new hybrid
configurations that we introduce in this paper. Advance warning times can
reach 5ns, i.e. approximately ten times the rise time of the individual
extreme event.

##  I Introduction

In recent years, Extreme Events (EEs: very rare and large amplitude
fluctuations) have been a topic of great interest in various fields due to
their significant impact on society Cavalcante2013 . While oceanic rogue
waves, recognized by their height and steepness well in excess of their
average Dysthe2008 ; Pelinovsky2008 ; Osborne2010 , are the best-known
examples, EEs are known to exist, beyond oceanography Onorato2001 , in
astrophysics Hudson1991 , in the atmosphere Stenﬂo2009 , in geology
Bogachev2008 , and even in optics Metayer2014 ; Baronio2012 . Regardless of
their different physical origins, these EEs can be seen as emergent phenomena,
where information coming from one physical system can benefit the others.

Optics offers many advantages in this respect, allowing well-controlled
laboratory conditions, good detection techniques, as well as extremely fast
timescales that allow the collection of large data samples in a limited time.
Powerful instrumentation detects and stores long datasets which may easily
contain an isolated EE in the midst of a large number of lower-level
oscillations, as shown from experiments and models in nonlinear fibers
Solli2007 ; Dudley2008 ), lasers subject to external injection Bonatto2011 ;
Pisarchik2011 , to self-injection through feedback Uy2017 ; Tlidi2017 ;
Lee2016 , to the presence of an intracavity saturable absorber Herink2017 ;
Coulibaly2017 ; Rimoldi2017 , to soliton interactions Walczak2017 , or
predicted in resonant parameteric oscillators Oppo2013 .

Paralleling the interest in fundamental understanding of EEs generation in
different systems, their prediction is of paramount practical importance,
independenly of the physical details. While model-free machine learning
techniques have made significant progress, practical restrictions often limit
the amount of information available, increasing the relevance of model-free
detection, based on Reservoir Computing (RC), capable of operating with
incomplete data with limited accuracy Pammi2023 : a very realistic real-life
scenario.

In this paper, we use EEs which appear in the emission of one polarization
channel of a low-frequency-pump-modulated semiconductor microcavity laser.
Their observation is corroborated by numerical modelling, based on a standard
spin-flip model, which confirm the presence of EEs whose number can be tuned
by a suitable choice of modulation parameters. We show that RC is capable of
predicting the emergence of EEs with good accuracy. Given the broad variety of
RC implementations, in order to identify its most successful forms for this
task we analyse different schemes and propose several hybrid configurations.
Our work provides a new platform for studying EEs in a microcavity system and
beyond, thus opening up a new field of investigation for EEs prediction.

##  II Experimental design

The experimental setup is shown in Fig. 1. The physical system is a single-
mode semiconductor microcavity laser diode (M-LD, Ulm Photonics, single mode,
$\lambda=(980\pm 3){\rm nm}$) where the $\beta$ factor, which represents the
fraction of spontaneous emission coupled into the lasing mode, is estimated at
$\approx 10^{-3}$ Wang2015 . The laser pump is sinusoidally modulated at
$\approx 600{\rm MHz}$ by a function generator (FG, Agilent E4421B), coupled
to the stabilized current supply with a bias-tee. In the following, we will
choose a below-threshold bias Wang2016 ; Lippi2022  for the weak field
polarization. The output laser beam is then split in a 10:90 ratio after
passing though a beam splitter (BS), where the weak beam is detected by a
power meter while the other one is analyzed by two channels separated by a
Polarized Beam Splitter (PBS), preceded by a Half-Way-Plate (HW) to align the
laser’s polarization axes with the PBS. The two orthogonal polarization
components of the beam are then detected by two fiber-coupled fast
photodetectors (Thorlabs PDA8GS). Finally, the temporal signals of the two
polarization states are recorded by a digital oscilloscope (LeCroy Wave Master
8600A, sampling rate 10Gs/s, analog bandwidth 6 GHz).

Figure 1:  Experimental design: M-LD, microcavity laser diode; OI, Faraday
optical isolator; BS, Beam splitter; PD, photodiode; PM, power meter; HW,
half-wave plate; PBS, polarization beam splitter; XP, X-polarization; YP,
Y-polarization; OSC, oscilloscope; FG, Function generator.

##  III Characterization of temporal EEs

Fig. 2 shows the measured input-output function curves of the TE (blue) and TM
(orange) modes under free-running conditions; the relative pump axis (Pump
$(P/P_{th})$) is obtained by dividing the current supplied to the laser by the
threshold current of the TE mode ($P^{TE}_{th}\approx$ 0.20 mA). The laser
behavior curves show that there is a significant difference in the threshold
between TM and TE modes. Specifically, the threshold for the TM mode
($P^{TM}_{th}\approx$ 1.60 mA) is approximately 8 times larger than the one
for the TE mode. We remark that in the broad interval of single-polarization
emission, the latter is stable, devoid of the switching behaviour often
observed Sondermann2004  in Vertical Cavity Surface Emitting Lasers (VCSELs,
the schematic representation is shown in the inset of Fig. 2). Thus, in our
device the TE polarization entirely controls the microcavity emission, with
output which increases linearly with pump. In other words, no deterministic
interpolarization dynamics is present in the measured signal.

Figure 2:  Input-output lasing behaviour curves for TE (blue) and TM (orange)
modes of the free running laser: the measured threshold of TE mode is 0.20 mA,
and TM mode is around 1.60 mA. Inset: schematic cross-sectional view of
vertical cavity laser structure.

When the laser is modulated at $I_{dc}=5.5P^{TE}_{th}$ by a 600-MHz sinusoidal
signal with a moderate modulation amplitude, such that the TM threshold is not
attained, the temporal response of the TE mode exhibits periodic behavior, as
shown in Fig. 3 (bottom panel). In contrast, the TM mode dynamics remains
dominated by spontaneous emission (Fig. 3, top panel), hardly affected by the
small pump modulation below threshold (noise from the detection chain mixes
in, masking possible residual oscillations).

Figure 3:  Temporal dynamics within 100 ns of TE (blue) and TM (orange) modes
of the modulated laser.

Increasing the modulation amplitude to levels which bring the laser, in
transient, close to or beyond the TM mode threshold, one observes large
amplitude fluctuations in the TM channel. Fig. 4 shows the histogram of the TM
mode emission under these conditions, where the long tail at large amplitudes
reveals the presence of non-gaussian statistics and the green part of the
distribution represents the EEs. The latter are determined using the standard
EE discrimination criterion based on $I_{EE}>\langle I\rangle+8\sigma_{I}$
Bonatto2011 , where $\langle I\rangle$ stands for the average measured
intensity and $\sigma_{I}$ for its standard deviation. The inset of Fig. 4
shows a typical rare event in the time domain: the very large amplitude pulse
is surrounded by a background and by smaller spikes which correspond to a
regular transient excitation of the TM mode.

Figure 4:  Experimental histograms with EEs, the tail marked by green color
represents the events whose intensity is larger than the threshold; inset:
typical EE with very large amplitude.

##  IV Numerical confirmation and analysis

The experimental observations can be interpreted with the help of the Spin
Flip Model (SFM) Regalado1997 ; Valle2008  for a current-modulated
semiconductor laser:

$\displaystyle\frac{dE_{\pm}}{dt}=\kappa(1+i\alpha)[(N\pm
n)-1]E_{\pm}-(\gamma_{a}+i\gamma_{p})E_{\mp}$ $\displaystyle+\sqrt{\beta(N\pm
n)}\xi\pm\,,$ (1)
$\displaystyle\frac{dn}{dt}=-\gamma_{s}n-\gamma[(N+n)|E_{+}|^{2}-(N-n)|E_{-}|^{2}]\,,$
(2)
$\displaystyle\frac{dN}{dt}=-\gamma[N-I+(N+n)|E_{+}|^{2}+(N-n)|E_{-}|^{2}]\,,$
(3)

where $E_{\pm}$ are the left and right circularly polarized components of the
slowly varying optical field. $N$ describes the evolution of the total carrier
number, in excess of its value at transparency, normalized to the same
quantity evaluated at the lasing threshold. The variable $n$ represents
instead the difference between the population carrier numbers which interact
with the two different polarizations, normalized in the same way as
Regalado1997 . The decay rate of the field in the cavity is denoted by
$\kappa$; $\alpha$ stands for the electric field’s phase-amplitude coupling.
The carrier relaxation rate is $\gamma$; $\gamma_{s}$ is the coupling rate
between the two circularly polarized radiation channels, coming from the
different microscopic relaxation mechanisms that equilibrate the carriers’
spin. $\gamma_{p}$ ($\gamma_{a}$) represents the degree of linear
birefringence (dichroism) per intracavity round-trip time. Essentially, these
parameters reflect the degree to which the intracavity radiation is split into
two different polarization states. The spontaneous emission factor $\beta$
matches the one already introduced. The noise terms $\xi_{\pm}$ are zero–mean
Gaussian white noise terms used to model random fluctuations in the system.

The normalized injection current $I$ is modulated by a sinusoidal function,
which reproduces the experiment, with a constant component $I_{dc}$, a
modulation amplitude $\Delta I$ and a modulation frequency $f_{m}$:
$I(t)=I_{dc}+\Delta I\cos(2\pi f_{m}t)$. The two orthogonal polarization
components (TE and TM) have strongly separate thresholds due to cavity
birefringence, modelled by the parameter $\gamma_{p}$. Their detection is
reconstructed through the projection onto the $X$ and $Y$ axes, respectively,
computed as:

$\displaystyle I_{x}=|E_{x}|^{2}=\frac{1}{2}(E_{+}+E_{-})^{2}\,,$ (4)
$\displaystyle I_{y}=|E_{y}|^{2}=\frac{1}{2}(E_{+}-E_{-})^{2}\,,$ (5)

where $I_{j}$ is proportional to the measured intensity in each polarization
channel.

To investigate the parameter region where the extreme events can be found, we
perform intensive numerical simulations using the SFM model. After determining
the optimal bias position ($I_{dc}=6.03P^{TE}_{th}$), we proceed to modulate
the laser at this specific point footnote . Fig. 5 shows a $10\mu s$ long
snapshot of the TM intensity evolution for $\Delta I=5.2P_{th}$ and
$f_{m}=600MHz$. The temporal dynamics show the isolated intensity pulses which
overcome the threshold for EEs (red line, placed at $I\approx 0.7316$). For
comparison, the inset diplays an experimental sequence. The satisfactory
agreement justifies the use of numerical simulations as a complement in
testing RC recognition expnum . We remark that EEs strongly depend on $\Delta
I$ and $f_{m}$ in the simulations and appear only in a narrow parameter
interval.

Figure 5:  Simulated temporal dynamics of the TM mode in a time interval of 10
$\mu$s: the red, dashed line indicates the threshold of EEs. Inset:
experimental temporal dynamics of the TM mode in a $0.4ms$ time interval.

The simplicity of the model, which clearly separates into a deterministic and
a stochastic part, lends credibility to the action of the spontaneous emission
as source of the EEs Bonatto2011 ; Zamora-Munt2013 , since the complexity of
an attractor collision is not present in our system in the conditions of the
experiment Metayer2014 .

##  V Prediction of EEs by RC

RC is a novel type of recurrent neural network (RNN) Tanaka2019  that has
demonstrated exceptional performance in nonlinear channel equalization
Masaad2022 , time series prediction, and speech recognition Hasegawa2023 ;
Jaeger2004 ; Tanaka2019 . Its unique architecture allows RC to efficiently
process complex and high-dimensional data, making it a promising tool for
various prediction and classification tasks. Here, our goal is to obtain an
accurate statistical prediction for the temporal EEs.

We numerically implement three different RC configurations (Fig. 6) as basic
architectures for EE prediction. The first is a single RC (panel a). Its
implementation could also be based on a semiconductor laser (SL) with a
feedback loop (FL) Brunner2013 ; Wang2023  as a replacement for the nonlinear
nodes (yellow box). The input layer takes an input function signal $u(t)$,
which is then multiplied by a mask consisting of randomly assigned integers
(-1 and +1) to obtain the signal $S(t)$. The information from the FL is
sampled at $\theta$ intervals to collect the high-dimensional information
state: $state(n)$. As a result of the sampling in the FL, the $N$ virtual
nodes are generated. The output layer remains the same as in the Echo State
Network (ESN).

The second approach, called parallel reservoir configuration, could be
implemented with multiple SLs with FL set up in parallel (Fig. 6b). Here,
$u(t)$ is multiplied by different mask signals $mask_{1}$ and $mask_{2}$ to
obtain $S_{1}(t)$ and $S_{2}(t)$, respectively, which are then injected into
different SLs. In the output layer, the virtual node states are used to
generate the output signal by calculating a weighted linear sum of the virtual
node states. Thus, the total number of virtual nodes is given as
$N_{total}=kN$, where $k$ is the number of SLs and $N$ is the number of
virtual nodes for each SL. In this scheme, we set $k=2$ for simplicity. The
advantages of a parallel configuration are an increase in flow rate and
redundancy, which ensure uniterrupted operation also in case of failure of one
(or more) nodes. Therefore, this configuration is preferred for applications
where high flow rates and increased reliability are essential.

As a third method, we propose a dual-training RC configuration Gu2022  (Fig.
6c). It consists of an input layer, a storage layer, and an output layer and
requires two training processes. In the first, the input $u_{1}(t)$ is
processed in the same way as the single RC. For the second, the input signal
$u_{2}(t)$ is constructed from the difference between the target values and
the predicted values of the first training process. Both $u_{1}(t)$ and
$u_{2}(t)$ are converted into different mask signals. Finally, the output of
the system is the sum of $Y_{1}$ and $Y_{2}$.

Figure 6:  Different prediction configurations: (a) single RC; (b) parallel
RC; (c) dual-training RC.

The performance evaluation of different RCs is based on a data set of 300,000
data points for training and an additional 100,000 points for testing. The
virtual nodes of the single, parallel, and dual training RCs were all set to
$N_{total}=100$, while the period of the mask signal was set to a specific
value: $T=2ps$. In the current work, the feedback delay time was set to
$\tau=1ns$, the time between the output signal and the input signal, while the
sampling interval was set to $\theta=10ps$, the time between successive
samples.

Figure 6a’-c’ displays the results of the EE prediction, after training, for
each RC. The original (target) signal is in black, while the red curve shows
the predicted signal. There is a close resemblance in all cases, albeit some
differences in amplitudes. Overall, RCs appear to be a promising tool in
forecasting EEs. A closer analysis provides a better insight into RC
performance.

Fig. 7a-c shows the calculated prediction accuracy of EEs over time for the
three different RCs. Single and parallel RCs have comparable performance.
Their prediction accuracy is above 0.9 for shorter warning times, but starts
to rapidly decrease after $\approx 1{\rm ns}$. A short plateau appears,
followed by further decline which reaches 50% accuracy when the warning time
reaches $\approx 3.25{\rm ns}$. The dual-training RC, instead, proves to be
much more efficient in terms of overall performance. In fact, its accuracy
remains abot 50% even when the warning time is pushed to the $5{\rm ns}$.

Finally, to evaluate the accuracy of the different RC configurations in
predicting the time of appearance of EEs, we calculated the errors between the
original and predicted time positions and plotted the distributions for each
error value. As shown in Fig. 7d-e, the horizontal axis represents the
deviation values in units of time, measured in ns. The vertical axis shows the
number of EEs corresponding to each error value, thus offering a quantitative
and visual measure of the timing errors in predicting an EE. In addition, a
higher concentration of data points at or near zero time deviation signals a
higher accuracy level, while a scattered distribution of data points indicates
a poorer one.

For the single RC, even though the highest column is at position
$\Delta\tau=0$, there are still several errors, with $\Delta\tau$ reaching
values as large as $0.338{\rm ns}$ (Fig. 7d). This suggests that while single
RC may be accurate for some aspects of the task, there is still significant
room for improvement. For the parallel RC, $|\Delta\tau|$ further regroups
around $0$, indicating a better temporal prediction with a maximum deviation
of $0.3{\rm ns}$ (Fig. 7e); a marked improvement over the single RC. Fig. 7f
shows that the majority of errors in the dual-training RC occur in the range
of $|\Delta\tau|<0.1ns$, indicating a higher level of accuracy than the other
two RCs, and a maximum deviation $0.2ns$. Thus, the dual-training RC offers
superior prediction capabilities and is more effective for accurate EEs
prediction.

Figure 7: Calculated forecasting accuracy (a-c) and histograms of error
distributions (d-f) with different RCs configurations: (a) and (d), single RC;
(b) and (e), parallel RC; (c) and (f), Dual-training RC.

##  VI Discussion and conclusions

The analysis has yielded EE predictions that, in our experimental and
numerical data, exceed 50% accuracy for the dual-training RC network at times
as long as 5 ns. To put this in context and to evaluate the performance more
generally, it is important to compare it to the intrinsic timescale of the EE,
i.e., its rise-time. Estimating the rise time as the time taken for the signal
to reach its maximum, starting from 10% of its height (to eliminate the
influence of the background), we find a typical rise time of $\tau_{rt}\approx
0.5ns$. In relative terms, this means that the dual training network is able
to predict an EE with $\tau_{f,0.7}\approx$ 70% accuracy with a warning time
of $\tau_{f}\approx 5\tau_{rt}$ (Fig. 7f). This time increases to
$\tau_{f,0.5}>10\tau_{rt}$ when the accepted accuracy is 50%. For events whose
main characteristic is the intensity of the phenomenon (e.g. water waves for
oil rigs or ships) such an advance would amount to a few minutes warning.

Figure 8: Computed second-order, time-delayed autocorrelation on a temporal
time sequence containing EEs (solid, blue line). Best fit for the
autocorrelation (red, dashed line), with slope $a=-6.321\times 10^{9}s^{-1}$.

Another way to quantify the amount of time warning is to compare the advance
time to the autocorrelation time of the signal. Fig. 8 shows the fit, on a
semi-logarithmic scale, of the time-delayed second-order autocorrelation
$\left(g^{(2)}(\tau)=\frac{\langle s(t)s(t-\tau)\rangle}{\langle
s(t)\rangle^{2}}.\right)$. The slope of the line corresponds to a correlation
time of $\tau_{c}\approx 0.16ns$. The autocorrelation (solid line, Fig. 8)
shows a correlation recovery at $\tau\approx 0.1ns$ and a much smaller one at
$\tau\approx 0.25ns$. Comparing with the 70% accuracy of the warning time, we
find $\tau_{f,0.7}\approx 14\times\tau_{c}$; $\tau_{f,0.7}$ is still about ten
times longer than the second revival in the correlation. Reading from Fig. 7f
the 90% accuracy time, we can estimate $\tau_{f,0.9}\approx 5\times\tau_{c}$.
These numbers are important as a relative measure of the prediction
performance of the dual training RC. In fact, prediction times up to 5 times
the correlation time are a very valuable tool for predicting extreme phenomena
Jacques-Dumas2022 .

In conclusion, we have experimentally generated EEs from a vertical
microcavity semiconductor laser emitting in a dominant polarization mode (TE)
with the pump biased below the threshold of the weak polarization mode (TM),
and modulated in time. The EE arise from the rare strong amplification of a
fluctuation (spontaneous emission) that occurs close to the pump maximum,
favoring the dominance of the TM polarization. Using the spin-flip model for
semiconductor lasers, we have reproduced high-resolution data for the EE, with
features consistent with those of the experiment. The EE prediction has been
obtained with the help of Reservoir Computing (RC), for which we have
numerically implemented three schemes. The dual-training RC proves to be the
best in terms of both advance warning time and error distribution, and works
well on both experimental and numerical data. The advance warning exceeds five
characteristic times and can reach $\approx$ 15 times, depending on the
criteria adopted, as explained above.

The growing importance of EEs and early warning, not only in basic research
Wolde1997 ; Chandler2005 ; Kohn2005 , but also in the prediction of natural
phenomena and disasters (extreme heat waves Jacques-Dumas2022 , extreme
precipitation Zhang2023 , tropical cyclones or hurricanes Bi2023 ), gives
great relevance to the investigation of techniques for their optimization.
Optical systems, such as microcavity lasers, offer the advantage of
flexibility and ease of measurement, the potential to accumulate large amounts
of data in a short time, and laboratory controlled conditions.

## Acknowledgment

This work is partially supported by several sources: National Key Research and
Development Program of China (2021YFB2801900, 2021YFB2801901, 2021YFB2801902,
2021YFB2801904), National Natural Science Foundation of China (Grant No.
61804036), Zhejiang Province Commonweal Project (Grant No. LGJ20A040001).

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# On the degrees of irreducible characters fixed by some field automorphism

Nicola Grittini

###### Abstract

It is known that, if all the irreducible real valued characters of a finite
group are of odd degree, then the group has a normal Sylow 2-subgroup. In this
paper, we prove and analogous result for solvable groups, by taking into
account the degree of irreducible characters fixed by some field isomorphism
of prime order $p$.

We prove it as a consequence of a more general result.

Keywords— character degrees, prime divisors, characters field of values,
normal $p$-subgroups

##  1 Introduction

The search for results linking the normal structure of a finite group and the
degrees of its irreducible characters is a classical problem in character
theory. The best known result linking the two properties is the celebrated
Ito-Michler theorem, and many variants of the theorem have already been found.

In [1] a variant of Ito-Michler theorem is studied, involving real-valued
characters. In fact, [1, Theorem A] states that, if the prime 2 does not
divide the degree of any real-valued irreducible character of a finite group
$G$, then $G$ has a normal Sylow 2-subgroup.

If we see the real-valued characters as the characters fixed by the complex
conjugation, it is possible to generalize this theorem for any prime $p$, at
least for solvable groups.

######  Theorem A.

Let $G$ be a finite solvable group, let $p$ be a prime number, and let
$\sigma\in\operatorname{Gal}(\mathbb{Q}_{\left|G\right|}\mid\mathbb{Q})$ of
order $p$. Suppose that $p$ does not divide the degree of any
$\sigma$-invariant irreducible character of $G$. Then, $G$ has a normal Sylow
$p$-subgroup.

Theorem A generalizes [1, Theorem A] for solvable groups, since [1, Theorem A]
follows directly from Theorem A if we take the complex conjugation as
$\sigma$.

If we assume the group $G$ not to be solvable, Theorem A is no longer valid,
as we can see if we consider $\operatorname{PSL}(2,8)$ and $p=3$, or
$\operatorname{Sz}(32)$ and $p=5$. However, in this situation, a weaker
version of the theorem is still valid.

######  Theorem B.

Let $G$ be a finite group, let $p$ be a prime number, let
$\sigma\in\operatorname{Gal}(\mathbb{Q}_{\left|G\right|}\mid\mathbb{Q})$ of
order $p$ and let $P$ be a Sylow $p$-subgroup of $G$. Suppose that $p$ does
not divide the degree of any $\sigma$-invariant irreducible character of $G$.
Then, $\operatorname{N}_{P}(G)$ intersects every chief factor of $G$ non-
trivially.

We mention that our proof of Theorem B relies on the Classification of Finite
Simple Groups.

It is not hard to see that Theorem A is actually a corollary of Theorem B, and
we will prove it as such.

##  2 Preliminary results

We first prove some preliminary results. We begin with some results about
$p$-groups acting on finite vector spaces.

######  Lemma 2.1.

Let $p$ be a prime and let $P>1$ be a $p$-group. Let $V$ be a finite vector
space which is also an irreducible $P$-module, and suppose that
$\operatorname{C}_{V}(P)=0$. Let $\sigma\in\operatorname{Aut}(V)$ of order $p$
such that, for every $v\in V$ and every $x\in P$,
$(v^{x})^{\sigma}=(v^{\sigma})^{x}$. Then, there exists $g\in P$ such that
$v^{\sigma}=v^{g}$ for every $v\in V$.

###### Proof.

Notice that, since $\operatorname{C}_{V}(P)=0$, then
$\operatorname{C}_{P}(V)<P$ and, thus, if we replace $P$ with
$\bar{P}=P/\operatorname{C}_{P}(V)$ we may assume $\operatorname{C}_{P}(V)=1$.
Moreover, since $V$ is an irreducible $P$-module, then $V=\langle
v^{P}\rangle$ for every $0\neq v\in V$.

Let $z\in Z=\operatorname{Z}(P)$ and suppose $z\in\operatorname{C}_{P}(v)$ for
some non-trivial $v\in V$; then, for every $y\in P$,
$(v^{y})^{z}=(v^{z})^{y}=v^{y}$. Since by hypothesis $V=\langle v^{P}\rangle$,
it follows that $z$ centralizes all the generators of $V$ and, thus,
$z\in\operatorname{C}_{P}(V)=1$. Therefore, $Z\cap\operatorname{C}_{P}(v)=1$
for every $0\neq v\in V$.

Let now $H=\langle\sigma\rangle\times Z$ and consider the group
$\Gamma=H\ltimes V$. Since $H$ is an abelian $p$-group and it is not cyclic,
it follows from [3, Corollary 6.10] that $\Gamma$ is not a Frobenius group. As
a consequence of [3, Theorem 6.4], there exists $w\in V\setminus\\{0\\}$ such
that $D=\operatorname{C}_{H}(w)\neq 1$. However, we know from the previous
paragraph that $Z\cap D=1$; thus, there exists $z\times\sigma^{-m}\in D$ for
some $m$ coprime with $p$ and some $z\in Z$. Now, let $a\in\mathbb{Z}$ such
that $am=bp+1$ for some $b\in\mathbb{Z}$, then
$(z\times\sigma^{-m})^{a}=z^{a}\times\sigma^{-1}\in D$ and, thus,
$w^{z^{a}}=w^{\sigma}$. We may take $g=z^{a}\in\operatorname{Z}(P)$.

Let $v\in V$. Since, by hypothesis, $V=\langle w^{P}\rangle$, we can write
$v=\sum_{i=1}^{k}n_{i}w^{p_{i}}$ for some $p_{i}\in P$ and some
$n_{i}\in\mathbb{Z}$. Now we have that

$v^{\sigma}=\sum_{i=1}^{k}(n_{i}w^{p_{i}})^{\sigma}=\sum_{i=1}^{k}(n_{i}w^{\sigma})^{p_{i}}=\sum_{i=1}^{k}(n_{i}w^{g})^{p_{i}}=\sum_{i=1}^{k}(n_{i}w^{p_{i}})^{g}=v^{g}$

and the thesis follows. ∎

######  Lemma 2.2.

Let $p$ be a prime and let $P>1$ be a $p$-group. Let $V$ be a finite vector
space which is also a $P$-module, and suppose that
$\operatorname{C}_{V}(P)=0$. Let $\sigma\in\operatorname{Aut}{V}$ of order $p$
such that, for every $v\in V$ and every $x\in P$, $v^{\sigma}\in\langle
v\rangle$ and $(v^{x})^{\sigma}=(v^{\sigma})^{x}$. Then, there exists $g\in P$
and $0\neq v\in V$ such that $v^{\sigma}=v^{g}$ and
$\left|P:\operatorname{C}_{P}(v)\right|=\min\\{\left|P:\operatorname{C}_{P}(w)\right|\mid
0\neq w\in V\\}$.

###### Proof.

At first, we recall that, since $V$ is a finite vector space, then it is a
vector space over a field of finite characteristic $q$. From the fact that
$\operatorname{C}_{V}(P)=0$, it follows that $q\neq p$, since otherwise
$\Gamma=P\ltimes V$ is a $p$-group and, thus,
$\operatorname{C}_{V}(P)=\operatorname{Z}(\Gamma)\cap V>0$.

Thus, by Maschke theorem, we can write $V$ as a sum of irreducible $P$-modules
$V=W_{1}\oplus...\oplus W_{n}$. Let $v\in V$ such that
$\left|P:\operatorname{C}_{P}(v)\right|=\min\\{\left|P:\operatorname{C}_{P}(w)\right|\mid
0\neq w\in V\\}$, we claim that $v$ can be chosen such that $v\in W_{i}$ for
some $i\in\\{1,...,n\\}$. In fact, let $v=w_{1}+...+w_{n}$ for some uniquely
determined $w_{i}\in W_{i}$, for each $i\in\\{1,...,n\\}$. Then,
$\operatorname{C}_{P}(v)=\bigcap_{i=1}^{n}\operatorname{C}_{P}(w_{i})$ and, by
the minimality of $\left|P:\operatorname{C}_{P}(v)\right|$, it follows that,
for each $i\in\\{1,...,n\\}$, either $w_{i}=0$ or
$\operatorname{C}_{P}(w_{i})=\operatorname{C}_{P}(v)$. Since $v\neq 0$, then
there exists $i\in\\{1,...,n\\}$ such that $w_{i}\neq 0$ and, if we replace
$v$ with $w_{i}$, we prove the claim.

Thus, without loss of generality, we can claim that there exists $v\in W_{1}$
such that
$\left|P:\operatorname{C}_{P}(v)\right|=\min\\{\left|P:\operatorname{C}_{P}(w)\right|\mid
0\neq w\in V\\}$. Moreover, notice that ${W_{1}}^{\sigma}\leq W_{1}$ and,
thus, $\sigma_{\mid_{W_{1}}}\in\operatorname{Aut}(W_{1})$ and either
$\sigma_{\mid_{W_{1}}}=1$ or $o(\sigma_{\mid_{W_{1}}})=p$. Since
$\operatorname{C}_{W_{1}}(P)\leq\operatorname{C}_{V}(P)=0$ and $W_{1}$ is an
irreducible $P$-module, it follows from Lemma 2.1 that there exists $g\in P$
such that $v^{\sigma}=v^{g}$. ∎

We now need some preliminary results concerning character theory.

######  Proposition 2.3.

Let $G$ be a finite group, let $p$ be a prime and let
$\sigma\in\operatorname{Gal}(\mathbb{Q}_{\left|G\right|}\mid\mathbb{Q})$ such
that $o(\sigma)$ is a power of $p$. Suppose $H\leq G$ such that
$p\nmid\left|G:H\right|$ and suppose, for some $\psi\in\operatorname{Irr}(H)$,
that $\psi(1)$ divides the degree of every irreducible constituent of
$\psi^{G}$. If $\psi$ is fixed by $\sigma$, then $\sigma$ also fixes some
irreducible constituents of $\psi^{G}$.

Notice that the hypothesis of $\psi(1)$ dividing the degree of every
irreducible constituent of $\psi^{G}$ is automatically verified if $H$ is
normal in $G$.

###### Proof.

Since $\psi^{\sigma}=\psi$, then also $(\psi^{G})^{\sigma}=\psi^{G}$ and,
thus, $\sigma$ acts on the irreducible constituents of $\psi^{G}$. Let
$C_{1},...,C_{k}$ be the orbits of this action, then we have

$\left|G:H\right|\psi(1)=\psi^{G}(1)=\sum_{i=1}^{k}\left|C_{i}\right|n_{i}\chi_{i}(1),$

where $\chi_{i}$ is a representative of $C_{i}$ for each $i\in\\{1,...,k\\}$,
and $n_{i}$ is the multiplicity of $\chi_{i}$ in $\psi^{G}$. Notice that,
since $o(\sigma)=p^{a}$ for some $a\in\mathbb{N}$, then the size of each orbit
$\left|C_{i}\right|$ is a power of $p$.

It follows that

$\left|G:H\right|=\sum_{i=1}^{k}\left|C_{i}\right|n_{i}\frac{\chi_{i}(1)}{\psi(1)}$

and, since $p\nmid\left|G:H\right|$ and the quotients
$\frac{\chi_{i}(1)}{\psi(1)}$ are all integers, we have that
$\left|C_{i}\right|=1$ for some $i\in\\{1,...,k\\}$ and, thus, there exists
some $\chi=\chi_{i}\in\operatorname{Irr}(G\mid\psi)$ such that
$\chi^{\sigma}=\chi$. ∎

######  Lemma 2.4.

Let $G$ be a finite group and let $N\lhd G$ such that
$(\left|G/N\right|,\left|N\right|)=1$. Let
$\sigma\in\operatorname{Gal}(\mathbb{Q}_{\left|G\right|}\mid\mathbb{Q})$ and
suppose that, for some $\varphi\in\operatorname{Irr}(N)$, there exists $g\in
G$ such that $\varphi^{\sigma}=\varphi^{g}$. Then, there exists
$\chi\in\operatorname{Irr}(G\mid\varphi)$ such that $\chi^{\sigma}=\chi$.
Moreover, $\chi$ can be chosen such that
$\chi(1)=\varphi(1)\left|G:\operatorname{I}_{G}(\varphi)\right|$.

###### Proof.

Let $T=\operatorname{I}_{G}(\varphi)$ and notice that
$T^{g}=\operatorname{I}_{G}(\varphi^{g})=\operatorname{I}_{G}(\varphi^{\sigma})=T$.
Let $n=o(\varphi)$, then by [2, Corollary 6.27] there exists a unique
character $\hat{\varphi}\in\operatorname{Irr}(T\mid\varphi)$ extending
$\varphi$ such that $o(\hat{\varphi})=n$. Moreover, $\hat{\varphi}^{\sigma}$
lies over $\varphi^{\sigma}$ and, analogously, it is the unique character in
$\operatorname{Irr}(T\mid\varphi^{\sigma})$ such that
$o(\hat{\varphi}^{\sigma})=o(\varphi^{\sigma})=n$. Let
$\chi=\hat{\varphi}^{G}\in\operatorname{Irr}(G)$ and notice that
$(\hat{\varphi}^{\sigma})^{G}=\chi^{\sigma}$.

Since $\hat{\varphi}^{g}$ lies over $\varphi^{g}=\varphi^{\sigma}$, and
$o(\hat{\varphi}^{g})=o(\hat{\varphi})=n$, it follows by uniqueness that
$\hat{\varphi}^{g}=\hat{\varphi}^{\sigma}$ and, thus,

$\chi=(\hat{\varphi}^{g})^{G}=(\hat{\varphi}^{\sigma})^{G}=\chi^{\sigma}.$

Finally, $\chi(1)=\left|G:T\right|\varphi(1)$ and the lemma is proved. ∎

Finally, we need a preliminary result to deal with nonabelian simple groups.
It is known that a group has no non-trivial rational characters if and only if
it is a group of odd order (the proof, however, requires the Classification of
Finite Simple Groups). We can see a proof of this fact in [4, Theorem 8.2].

######  Theorem 2.5.

_[4, Theorem 8.2]_ Let $G$ be a finite group. If $G$ has exactly one
irreducible rational character, then $G$ has odd order.

######  Corollary 2.6.

Let $S$ be a nonabelian simple group, then $S$ has a non-trivial rational
character.

##  3 Proofs of Theorem A and Theorem B

We are now ready to prove Theorem A and Theorem B.

We begin with the proof of Theorem B, since Theorem A will follow as a
corollary.

###### Proof of Theorem B.

We prove the theorem by induction on $\left|G\right|$. Let
$N=\operatorname{N}_{G}(P)$ and let $M/K$ be a chief factor of $G$, by
induction we can assume that $K=1$ and, thus, that $M$ is a minimal normal
subgroup of $G$. If $p\mid\left|M\right|$, then $1<P\cap M\leq N\cap M$ and
the theorem is proved in this case. Thus, we can assume $M$ to be a
$p^{\prime}$-group.

Notice that, in this situation, $N\cap M=\operatorname{C}_{M}(P)$. Therefore,
we need to prove that $C=\operatorname{C}_{M}(P)>1$. We prove it both when $M$
is a non-abelian minimal normal subgroup and when $M$ is abelian.

Case 1. $M$ is non-abelian.

Suppose $M$ is a non-abelian $p^{\prime}$-subgroup. Then, it is a direct
product of non-abelian simple groups which are transitively permuted by $G$.
Write $M=S_{1}\times...\times S_{n}$, where each $S_{i}$ is simple non-
abelian, and write $S=S_{1}$. By Corollary 2.6, there exists
$1_{S}\neq\delta\in\operatorname{Irr}(S)$ which is rational and, therefore,
$\sigma$-invariant. Let

$\psi=\delta\otimes 1_{S_{2}}\otimes...\otimes
1_{S_{n}}\in\operatorname{Irr}(M)$

and notice that $\psi^{\sigma}=\psi$. Let $T=\operatorname{I}_{G}(\psi)$ and
let $P_{0}=P\cap T$; by eventually replacing $\psi$ with some $G$-conjugate,
we can assume that $P_{0}$ is a Sylow $p$-subgroup for $T$.

Now, by [2, Corollary 6.27], there exists a unique character
$\hat{\psi}\in\operatorname{Irr}(P_{0}M\mid\psi)$ extending $\psi$ such that
$o(\hat{\psi})=o(\psi)$. Since also
$\hat{\psi}^{\sigma}\in\operatorname{Irr}(P_{0}M\mid\psi)$ and
$o(\hat{\psi}^{\sigma})=o(\hat{\psi})$, $\hat{\psi}$ is $\sigma$-invariant.
Moreover, $\hat{\psi}(1)=\psi(1)$ divides the degree of every irreducible
constituent of $\hat{\psi}^{T}$. Since $p\nmid\left|T:P_{0}M\right|$, it
follows from Proposition 2.3 that there exists
$\theta\in\operatorname{Irr}(T)$ lying over $\hat{\psi}$, and thus lying also
over $\psi$, such that $\theta^{\sigma}=\theta$. If
$\chi=\theta^{G}\in\operatorname{Irr}(G)$, then

$\chi^{\sigma}=(\theta^{G})^{\sigma}=(\theta^{\sigma})^{G}=\theta^{G}=\chi$

and, by hypothesis, $p\nmid\chi(1)=\left|G:T\right|\theta(1)$. It follows that
$p\nmid\left|G:T\right|$ and, thus, $P\leq T$ and
$\psi\in\operatorname{Irr}_{P}(M)$, with $\operatorname{Irr}_{P}(M)$ being the
set of the $P$-invariant irreducible characters of $M$. By Glauberman
correspondence, see [2, Theorem 13.1],
$\left|\operatorname{Irr}(C)\right|=\left|\operatorname{Irr}_{P}(M)\right|>1$
and it follows that also $C>1$.

Case 2. $M$ is abelian.

Assume $M$ to be abelian, then it is an elementary abelian $q$-group for some
$q\neq p$. Let $V=\operatorname{Irr}(M)$, $V$ has the structure of a finite
vector space over a field of characteristic $q$ and it is also a $P$-module,
with $P$ acting on $V$ by conjugation. If we assume
$C=\operatorname{C}_{M}(P)=1$, then also $\operatorname{C}_{V}(P)=1_{M}$.
Moreover, the action of $\sigma$ on $V$ commutes with the action of $P$ and,
for each $\xi\in V$, $\ker(\xi)=\ker(\xi^{\sigma})$ and, therefore,
$\xi^{\sigma}\in\langle\xi\rangle$. By Lemma 2.2, there exist $g\in P$ and
$\lambda\in V$ such that $\lambda^{\sigma}=\lambda^{g}$. Moreover, we can
choose $\lambda$ such that
$\left|P:\operatorname{I}_{P}(\lambda)\right|=\min\\{\left|P:\operatorname{I}_{P}(\xi)\right|\mid
1_{M}\neq\xi\in\operatorname{Irr}(M)\\}$ and, in particular, it follows that
$\left|P:\operatorname{I}_{P}(\lambda)\right|\mid\theta(1)$ for each
$\theta\in\operatorname{Irr}(PM)$ such that $M\nleq\ker(\theta)$.

Now, by Lemma 2.4, there exists $\eta\in\operatorname{Irr}(PM)$ lying over
$\lambda$ such that $\eta^{\sigma}=\eta$ and
$\eta(1)=\left|P:\operatorname{I}_{P}(\lambda)\right|$. Let $\psi$ be an
irreducible constituent of $\eta^{G}$, then
$\psi_{PM}=n_{1}\theta_{1}+...+n_{r}\theta_{r}$ for some $n_{i}\in\mathbb{N}$
and some $\theta_{i}\in\operatorname{Irr}(PM)$ such that, for each
$i\in\\{1,...,r\\}$, $M\nleq\ker(\theta_{i})$, since $M\nleq\ker(\psi)$ and
$M$ is normal in $G$. It follows from the previous paragraph that $\eta(1)$
divides the degree of every $\theta_{i}$, for each $i\in\\{1,...,r\\}$, and
thus $\eta(1)\mid\psi(1)$.

Thus, $\eta(1)$ divides the degree of every irreducible constituent of
$\eta^{G}$ and, since $p\nmid\left|G:PM\right|$, it follows from Proposition
2.3 that $\sigma$ fixes some irreducible constituent $\chi$ of $\eta^{G}$. By
hypothesis, $p\nmid\chi(1)$ and, thus,
$p\nmid\eta(1)=\left|P:\operatorname{I}_{P}(\lambda)\right|$. It follows that
$1_{M}\neq\lambda\in\operatorname{Irr}_{P}(M)$ and, again by Glauberman
correspondence,
$\left|\operatorname{Irr}(C)\right|=\left|\operatorname{Irr}_{P}(M)\right|>1$
and, thus, $C>1$. ∎

######  Corollary 3.1 (Theorem A).

Let $G$ be a finite solvable group, let $p$ be a prime number, and let
$\sigma\in\operatorname{Gal}(\mathbb{Q}_{\left|G\right|}\mid\mathbb{Q})$ of
order $p$. Suppose that $p$ does not divide the degree of any
$\sigma$-invariant irreducible character of $G$. Then, $G$ has a normal Sylow
$p$-subgroup.

###### Proof.

We prove the theorem by induction on $\left|G\right|$. Let $M$ be a minimal
normal subgroup of $G$ and let $P\in\operatorname{Syl}_{p}\left(G\right)$; if
$M$ is a $p$-group, then $M\leq P$ and, by induction, $P/M\lhd G/M$. Thus, it
follows that in this case $P\lhd G$.

We can now assume that $\operatorname{O}_{p}(G)=1$ and that $M$ is an abelian
$q$-group for some $q\neq p$. By induction, we have that $PM\lhd G$. Moreover,
by Theorem B, we have that

$1<\operatorname{N}_{P}(G)\cap
M=\operatorname{C}_{M}(P)=M\cap\operatorname{Z}(PM).$

It follows that $M\cap\operatorname{Z}(PM)>1$ and, since
$\operatorname{Z}(PM)\lhd G$ and $M$ is a normal minimal subgroup of $G$, we
have that $M\leq\operatorname{Z}(PM)$ and $PM=P\times M$.

Therefore, $P\leq\operatorname{O}_{p}(PM)\leq\operatorname{O}_{p}(G)=1$ and
the thesis follows also in this case. ∎

## References

  * [1] S. Dolfi, G. Navarro, P. H. Tiep : “Primes dividing the degrees of the real characters”. _Math. Z._ 259 (2008), 755–-774.
  * [2] I. M. Isaacs: _Character theory of finite groups_. New York: Dover Publications, Inc., 1994.
  * [3] I. M. Isaacs: _Finite group theory_. Vol. 92 Graduate Studies in Mathematics. American Mathematical Society, 2008.
  * [4] G. Navarro, P. H. Tiep: “Rational irreducible characters and rational conjugacy classes in finite groups”. _Trans. Amer. Math. Soc._ 360 (2008), 2443-–2465.