# A positive feedback method based on F-measure value for Salient Object
Detection

Ailing Pan 1 Chao Dai 1 Chen Pan 1 Corresponding author. Email:
<EMAIL_ADDRESS>Dongping Zhang 1 This work was supported by Key Research and
Development Projects in Zhejiang Province of China (NO.2021C03192,
2023C01032). Yunchao Xu 1
1Department of Computer Engineering, China JiLiang University, Hangzhou,
Zhejiang, 310018, China

###### Abstract

The majority of current salient object detection (SOD) models are focused on
designing a series of decoders based on fully convolutional networks (FCNs) or
Transformer architectures and integrating them in a skillful manner. These
models have achieved remarkable high performance and made significant
contributions to the development of SOD. Their primary research objective is
to develop novel algorithms that can outperform state-of-the-art models, a
task that is extremely difficult and time-consuming. In contrast, this paper
proposes a positive feedback method based on F-measure value for SOD, aiming
to improve the accuracy of saliency prediction using existing methods.
Specifically, our proposed method takes an image to be detected and inputs it
into several existing models to obtain their respective prediction maps. These
prediction maps are then fed into our positive feedback method to generate the
final prediction result, without the need for careful decoder design or model
training. Through applying the positive feedback method for decision fusion of
multi-model perception results. Moreover, our method is adaptive and can be
implemented based on existing models without any restrictions. Experimental
results on five publicly available datasets show that our proposed positive
feedback method outperforms the latest 12 methods in five evaluation metrics
for saliency map prediction. Additionally, we conducted a robustness
experiment, which shows that when at least one good prediction result exists
in the selected existing model, our proposed approach can ensure that the
prediction result is not worse. Our approach achieves a prediction speed of 20
frames per second (FPS) when evaluated on a low configuration host and after
removing the prediction time overhead of inserted models. These results
highlight the effectiveness, efficiency, and robustness of our proposed
approach for salient object detection. Code and Saliency maps will be
available.

##  1 Introduction

Salient object detection (SOD) aims to mimic the human visual perception
system to capture the most prominent regions in given images or videos. It can
serve as a pre-processing step for other realated computer vision tasks,
including object tracking[1], action recognition[2], video segmentation[3],
and image captioning[4].

Existing SOD models can be mainly divided into traditional algorithm-based and
deep learning-based approaches. Traditional SOD methods rely on handcrafted
features and use these features to predict saliency maps in a bottom-up
manner. Common handcrafted features include center prior[5] and distance
transform[6], which only contain low-level clues and perform poorly in complex
scenes. With the emergence of deep learning techniques, SOD has made
remarkable progress. Recent SOD models are mainly implemented based on fully
convolutional networks (FCNs) and Transformer architectures, among which FCN
is still the mainstream SOD architecture. SOD FCNs-based models mainly design
a series of decoders around feature extraction, refinement or enhancement, and
multi-level feature fusion and set specific loss functions to assist model
training[7][8][9][10][11][12][13][14][15][16]. Inspired by the impressive
performance of the Transformer in the field of natural language processing
(NLP), researchers have also carried out some innovative work using the
Transformer architecture in the SOD research field and achieved impressive
performance improvements. These models mainly rely on the powerful long-
distance feature correlation capture ability of the Transformer architecture
and help the model generate more complete predictions by effectively
extracting global contextual clues[17][18]. In addition, a hybrid architecture
of FCN and Transformer has been proposed by integrating the advantages of both
models[19][20].

The main research content of the above work is to design different network
models carefully based on the characteristics of the SOD task and demonstrate
the performance beyond the previous proposed models, playing an extremely
important role in the rapid development process of the SOD research field.
However, the process of carrying out these works is usually challenging, time-
consuming, and requires difficultly obtaining higher performance than the
latest methods. Therefore, in contrast to the above work, this paper proposes
a positive feedback approach based on F-measure value for SOD, which is based
on existing methods to obtain more accurate prediction results. The approach
allows any existing method to be inserted, such as traditional algorithms and
various deep learning methods, and can achieve performance beyond the inserted
methods.

In summary, the main contributions of this paper are:

  * •

Under the current SOD research background, we have explored how to utilize
existing methods rarely. This paper proposes a positive feedback approach
based on F-measure value for SOD, which allows any method to be inserted and
performs better than the inserted method in terms of performance. The
prediction process of the approach is training-free, adaptive weights, and
contains only one tunable hyperparameter.

  * •

Based on the F-measure value, a positive feedback process is designed, which
does not require human involvement during the calculation process, and is a
completely self-updating process.

##  2 Related Work

Existing SOD models can be mainly divided into traditional algorithm-based and
deep learning-based approaches, with the latter being the current mainstream
SOD method.

###  2.1 FCNs-based network

FCNs-based SOD models mainly design a series of decoders around feature
extraction, refinement or enhancement, and multi-level feature fusion and set
specific loss functions to assist model training.

The successful performance of SOD models depends on the effective integration
of multi-level features. Several studies have highlighted the significant
differences between the low-level and high-level features extracted from
backbone networks. To address this issue, Chen et al.[21] proposed a
progressive context aware feature aggregation module, while Dai et al.[10]
developed a middle layer feature extraction module to achieve better feature
fusion results. Moreover, the integration of multiscale features has been
shown to improve SOD detection performance in various studies. For instance,
Zhang et al.[15] developed a neural structure based search unit to
automatically determine the multiscale features that need to be aggregated.
Fang et al.[9] proposed a densely nested-based network framework to utilize
multiscale high-level feature maps effectively. Meanwhile, Zhuge et al.[8]
used various multiscale feature extraction methods to obtain diverse
multiscale features and designed an integrity channel enhancement module to
highlight salient objects. Base on work[22], Wu et al. [14] proposed a dynamic
pyramid convolution to extract multiscale features instead of fixed size
convolutions. Effective supervision strategy have also been employed to help
models predict salient objects more accurately. For instance, Wei et al.[22]
proposed pixel position aware losss to guide networks to pay more attention to
local details, while Yang et al.[23] proposed progressive self-guided loss to
guide network learning to more complete salient regions. Additionally, Xu et
al.[16] designed a knowledge review network to roughly locate salient regions
first and then finely segment salient objects. Wu et al. [7] proposed a
decomposition and completion network to predict the saliency, edge, and
skeleton maps respectively, and then filled the saliency map using edge and
skeleton maps.

###  2.2 Transformer-based network

In the field of SOD, the incorporation of global context information is
extremely crucial for accurately and completely predicting salient regions.
Transformer-based network architectures, renowned for their ability to capture
long-range dependencies, have been successfully introduced in this domain.

To this end, Liu et al.[17] proposed a unified transformer-based model for
SOD, which facilitates the propagation of global context information among
image patches. Meanwhile, Zhang et al.[18] developed a generative vision
transformer network that generates a pixel-level uncertainty map, effectively
representing the significance confidence of salient objects. Additionally, Ren
et al.[24] proposed a simple yet effective deeply-transformer network that
preserves more unifying global-local representations to gradually restore
spatial details.

###  2.3 Hybrid framework-based network

To enhance the accuracy and completeness of prediction results in the field of
SOD, it is important to effectively extract and utilize both global context
from deep features and local context from shallow features. However, achieving
this in network frameworks can be challenging. Therefore, some hybrid network
frameworks have emerged as the times require.

Zhu et al.[25] proposed a deep supervised fusion transformer network,
extending the applicability of FCN to the transformer architecture for the
first time. They employed a transformer encoder to extract multiscale features
and designed a multiscale aggregation module to aggregate these features in a
coarse-to-fine manner. Similarly, Yun et al.[19] proposed a self-refined
transformer network that leverages the transformer encoder to capture long-
distance dependencies and designed a context refinement module. The module is
employed to integrate global context with decoder features and refine and
locate local details automatically. In contrast, Wang et al.[26] still used
FCN as an encoder and utilized a transformer module for multi-level feature
fusion to address the limited receptive field of FCN.

##  3 Method

###  3.1 Overview of the proposed Method

Figure 1: The overall pipeline of the proposed method.

Figure 1 shows structure of the proposed method, which includes a multi-branch
model structure and a positive feedback prediction structure. The former is
used to place existing SOD model sequences, including traditional algorithm
models, deep-learning supervised or unsupervised models. The positive feedback
prediction structure will iteratively calculate the respective weights
according to the outputs of all branches to obtain the final saliency maps,
where the weight calculation process is completely adaptive. In order to
verify our method more conveniently, we use the precious models in advance to
generate saliency maps, which are marked as $Smp_{i}(i=1,2,3,4)$.

###  3.2 Positive feedback prediction algorithm

The positive feedback prediction algorithm is summarized in Table 1, and its
conceptual principle is illustrated in Figure 2. Hereinafter, $smp_{n},N$
refers to the saliency maps generated by the multi-branch models and the
number of branch. $\varepsilon$ is a threshold, which is set to 0.95. The
$imbinarize$($\cdot$) is a binarization function and the $mat2grey$ is a
graying function. The $computerFmeasure$($\cdot$) is the function for
calculating $F$-$measure$[27].

The key steps of the algorithm:
Step1. Input: outputs ($smp_{n}$) of the multi-branch model structure are used
as inputs of the positive feedback prediction structure, and perform the
binarization function to get binary maps ($b\\_smp_{n}$).
Step2. Initialization: the above inputs are fused in the way of pixel-level
addition, and perform the binarization function to get binary maps
($B\\_Smp_{0}$). At this time, the weights of each branch are consistent.
Step3. Iteration: calculate $F$-$measure$ with the input ($b\\_smp_{n}$) of
each brach and the latest fusion result ($B\\_Smp_{i-1}$). Update the weights
of each branch, recalculate the fusion result ($Smp_{i}$) from the new weights
and perform the binarization function to get binary maps ($B\\_Smp_{i}$).
Step4. Judgement: calculate $F$-$measure$ with the latest and last fusion
result and compare it with $\varepsilon$. If it is greater than the threshold,
the two successive resluts are considered to be simmilar enough, and the
latest fusion result is taken as the final output; Otherwise, execute Step3
again.

Table 1: Positive feedback prediction algorithm. Input: |
---|---
|  $smp_{n},N$;
Output: |
|  $Smp_{i}$;
1: |  $B\\_Smp_{i-1}=imbinarize(mat2grey(\sum_{n=1}^{N}smp_{n})),i=1$;
2: |  $b\\_smp_{n}=imbinarize(smp_{n})$;
3: |  while True do
4: |  $F_{n}=computerFmeasure(b\\_smp_{n},B\\_Smp_{i-1})$;
5: |  $F_{sum}=\sum_{n=1}^{N}F_{n}$; $\alpha_{n}={F_{n}}\setminus{F_{sum}}$;
6: |  $Smp_{i}=\sum_{n=1}^{N}{\alpha_{n}}\cdot smp_{n};B\\_Smp_{i}=imbinarize(Smp_{i})$;
7: |  $F=computerFmeasure(B\\_Smp_{i},B\\_Smp_{i-1})$;
8: |  if $F\geq\varepsilon$ then
9: |  $break$;
10: |  else
11: |  $i=i+1$;
12: |  end if
13: | end while
14: |  $Smp_{i}=mat2grey(Smp_{i})$;
15: |  return $Smp_{i}$.
Figure 2: The conceptual principle of positive feedback prediction algorithm.

##  4 Experimental results and analysis

###  4.1 Datasets and Evaluation Metrics

To validate the efficacy of the proposed method in this paper, we performed a
series of experiments on five publicly available datasets. The datasets
utilized in this study are briefly described as follows: DUTS[28] comprises a
total of 10,000 images for training and 5,000 images for testing, with only
the test set being used in this study. DUT-OMRON[29] consists of 5,019 images
with complex structures and backgrounds. HKU-IS[30] contains 4,447 maps with
multiple salient objects. PASCAL-S[31] contains 850 natural images. ECSSD[32]
comprises a collection of 1,000 images obtained from the internet.

To quantitatively evaluate the proposed method, we adopt six evaluation
metrics as the performance measures, including mean absolute error
($MAE$)[33], maximum F-measure ($mF$) score, and S-measure ($Sm$) score[34],
precision-recall ($PR$) curves.

###  4.2 Implementation Details

To facilitate the testing of the proposed positive feedback method, we
implemented the following steps. Firstly, we generated prediction maps of a
multi-branch model using the model disclosed by the author and Python tools.
Secondly, we evaluated the positive feedback method using Matlab tools.

To evaluate the effectiveness and characteristics of our proposed method, we
conducted comparative, ablation, and robustness experiments. In the
comparative experiment, we compared the performance with other state-of-the
arts models. Results showed that the positive feedback mechanism improved the
model’s performance on multiple metrics. In the ablation experiment, we
compared the performance of the positive feedback mechanism with pixel-level
addition fusion on five datasets. In the robustness experiment, we manually
selected good and bad renderings from each model and input them into the
positive feedback mechanism to observe the visual comparison results.

In total, we conducted two sets of experiments. In the first set, we used four
branches, namely MSFNet (2021)[15], PAKRNet (2021)[16], ICONet (2022)[8], and
SelReformer (2022)[19]. The performance of the 2021 methods are weaker than
that of the 2022 methods. In the second set, we used two branches, namely
DPNet (2022)[14] and SelReformer. The performance of these two methods is
relatively close. For detailed experimental procedures and analysis, please
refer to each subsection.

Table 2: Quantitative evaluation. The mean absolute error (MAE, smaller is
better), maximum F-measure (mF, larger is better) and S-measure (Sm, larger is
better). The results for each saliency detection method are reported on five
different datasets, with the top three performing methods highlighted in red,
green, and blue. It is important to note that ”Our-SSSS” and ”Our-SS” denote
the results of the four-branch and two-branch models, respectively.

Methods | DUTS | ECSSD | HKU-IS | PASCAL-S | DUT-OMRON
---|---|---|---|---|---
mF | MAE | Sm | mF | MAE | Sm | mF | MAE | Sm | mF | MAE | Sm | mF | MAE | Sm
DCNet[7] | .894 | .035 | .895 | .952 | .032 | .928 | .939 | .027 | .922 | .872 | .062 | .861 | .823 | .051 | .845
BiconNet[35] | .888 | .038 | .890 | .949 | .034 | .927 | .939 | .029 | .923 | .877 | .063 | .863 | .817 | .053 | .842
DNTD[9] | .892 | .033 | .891 | .946 | .034 | .922 | .938 | .028 | .920 | .878 | .064 | .857 | .803 | .051 | .828
EDNet[12] | .895 | .035 | .892 | .951 | .032 | .927 | .941 | .026 | .924 | .886 | .062 | .865 | .828 | .049 | .850
EFNet[10] | .898 | .034 | .895 | .948 | .034 | .925 | .939 | .027 | .922 | .875 | .063 | .864 | .822 | .054 | .843
MRINet[11] | .899 | .035 | .894 | .950 | .032 | .927 | .941 | .027 | .922 | .877 | .060 | .864 | .829 | .054 | .848
RCSBNet[13] | .899 | .035 | .881 | .944 | .034 | .922 | .938 | .027 | .919 | .882 | .059 | .860 | .810 | .049 | .835
MSFNet[15] | .878 | .034 | .877 | .941 | .033 | .915 | .927 | .027 | .908 | .863 | .061 | .852 | .799 | .050 | .832
PAKRNet[16] | .907 | .033 | .900 | .953 | .032 | .928 | .943 | .027 | .924 | .873 | .067 | .858 | .834 | .050 | .853
ICONet[8] | .893 | .037 | .892 | .951 | .031 | .931 | .942 | .027 | .925 | .884 | .060 | .870 | .830 | .059 | .846
DPNet[14] | .917 | .028 | .912 | .954 | .031 | .931 | .950 | .023 | .934 | .894 | .054 | .877 | .834 | .049 | .853
SelReformer[19] | .916 | .027 | .911 | .958 | .027 | .936 | .947 | .024 | .931 | .894 | .051 | .881 | .837 | .043 | .861
Our-SSSS | .922 | .032 | .909 | .960 | .029 | .941 | .950 | .025 | .935 | .898 | .055 | .884 | .863 | .043 | .877
Our-SS | .931 | .028 | .919 | .962 | .028 | .943 | .954 | .023 | .939 | .899 | .052 | .889 | .859 | .046 | .869

###  4.3 Comparison with the State-of-the-arts

We compare the proposed method with 12 state-of-the-art methods, including
DCNet[7], BiconNet[35], DNTD[9], EDNet[12], EFNet[10], MRINet[11],
RCSBNet[13], MSFNet[15], PAKRNet[16], ICONet[8], DPNet[14], SelReformer[19].
To ensure a fair and objective comparison, we will utilize the saliency maps
provided by the authors and employ the same standardized evaluation function
to calculate each metric.

In Table 2, we present the quantitative comparison results based on MAE,
maximum F-measure, and S-measure. Our methods demonstrate the most
comprehensive performance across all five datasets, as evidenced by our
superior results across all three metrics. Notably, our method, Our-SS,
consistently outperforms the second-best result in terms of mF and Sm by 1.5%,
0.4%, 0.4%, 0.5%, and 0.9%, 0.8%, 0.6%, 0.9% on DUTS, ECSSD, HKU-IS, and
PASCAL-S datasets, respectively. Additionally, our method, Our-SSSS, performs
exceptionally well on ECSSD, HKU-IS, and PASCAL-S datasets, particularly on
the DUT-OMRON dataset, where it consistently outperforms SelReformer[19], the
strongest model among the four-branch structures. Moreover, in Figure 3, we
provide precision-recall curves (PR) for all five datasets. Our curves
demonstrate exceptional performance across most thresholds, particularly on
the DUT-OMRON and DUTS datasets.

Figure 3: Performance comparison with 15 state-of-the-art methods on five
saliency datasets. The red solid line and the blue solid line denote our
models.

###  4.4 Ablation experiment

Within this subsection, we have performed ablation experiments on two datasets
to demonstrate the efficacy and distinctive attributes of the proposed
approach. Specifically, we have replaced the proposed positive feedback fusion
with pixel-level addition and have presented the experimental outcomes in
Table 3. The results reveal that our proposed method yields superior overall
performance compared to the direct addition way. Notably, in the ablation
experiment pertaining to the four-branch architecture, our proposed approach
has demonstrated a considerably more substantial advantage, especially for the
mean absolute error (MAE) metric, which has reduced by 7.2%and 7.7%,
correspondingly.

Table 3: Ablation evaluation on two datasets.

Methods | PASCAL-S | DUT-OMRON
---|---|---
mF | MAE | Sm | mF | MAE | Sm
ablation-4S | .891 | .060 | .882 | .863 | .047 | .874
Our-SSSS | .898 | .055 | .884 | .863 | .043 | .877
ablation-2S | .897 | .053 | .888 | .858 | .046 | .869
Our-SS | .899 | .052 | .889 | .859 | .046 | .869

Based on the results of the two sets of comparative experiments on ablation,
we can derive several professional conclusions. Firstly, in cases where there
is a significant difference in performance between the selected methods and
their placement in a multi-branch structure, the proposed method exhibits
greater advantages over direct fusion. Conversely, when the performance
difference between the selected methods is minor, the proposed method can only
exhibit limited superiority over direct fusion. Additionally, it is important
to note that the performance synthesis level achieved through positive
feedback fusion is consistently superior to that of each individual model
within a multi-branch structure.

###  4.5 Robustness experiment

To assess the proposed method’s robustness against interference, we conducted
adversarial experiments on a four-branch structure. Despite the outstanding
detection performance of many existing methods, their capabilities are not
consistently strong, as depicted in Figure 4. To better showcase the
superiority of our approach, we manually selected several sets of input
predictions with both desirable and undesirable effects for the proposed
method, and presented visual comparisons in Figure 4. This not only explains
why the proposed method is superior to methods introduced into a multi-branch
structure, but also superior to direct fusion, as it allows for automatic
weight updating to achieve satisfactory results, instead of treating all
branches equally. Furthermore, it should be noted that even in the absence of
good predictions in the four-branch structural model, a favorable outcome may
still be yielded, as demonstrated in the seventh row.

Figure 4: Visual comparisons from the four-branch structure.

## Conclusion

Distinguished from most existing efforts in salient object detection (SOD),
this study focuses on enhancing the accuracy of previously established
algorithms. Specifically, we introduce a positive feedback approach based
F-measure value for SOD that comprises a multi-branch model structure and a
positive feedback prediction structure. The method involves feeding input
images into the multi-branch model structure to generate their corresponding
saliency maps, and then processing these maps through the positive feedback
prediction structure for obtaining the final result via positive feedback
calculation. Notably, our method requires no model training, entails minimal
hyperparameter tuning, and features automatic calculation. By integrating
multiple existing models, our method surpasses the performance of individual
models. Our proposed method achieves a prediction speed of 20 frames per
second (FPS) even on a low-end host, while removing the prediction time
overhead of inserted models. Our findings suggest that higher performance of
the inserted models leads to better overall results, and the initialization
stage is crucial due to the positive feedback mechanism. Selecting models with
insignificant performance differences weakly showcases the advantages of our
approach. We believe that our study is not only of significant research value
but also has practical applications, and thus, merits the attention and
scrutiny of future researchers.

## Conflict of interests

The authors declare that they have no conflict of interests.

## Acknowledgments

This work was supported by Key Research and Development Projects in Zhejiang
Province of China (NO.2021C03192, 2023C01032).

## Author contributions

Ailing Pan wrote the paper, conceived and designed the experiments;
Chao Dai assisted in revising paper, collected the data, performed the
experiments and recorded experimental results;
Chen Pan proposed the core idea of the paper and will act as the corresponding
author;
Dongping Zhang and Yunchao Xu are participants in the fund projects.
All authors agree to this submission.

## Code and data availability

Code and Saliency maps will be available at:
https://github.com/dc3234/PF/tree/main.

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# Quantifying the rise and fall of scientific fields

Chakresh Singh  Université de Paris, INSERM U1284, Center for Research and
Interdisciplinarity (CRI), F-75006 Paris, France  Emma Barme  Université de
Paris, INSERM U1284, Center for Research and Interdisciplinarity (CRI),
F-75006 Paris, France  Robert Ward  School of Public Policy, Georgia Institute
of Technology, Atlanta, GA 30332  Liubov Tupikina  Université de Paris, INSERM
U1284, Center for Research and Interdisciplinarity (CRI), F-75006 Paris,
France  Nokia Bell labs, France  Marc Santolini  Université de Paris, INSERM
U1284, Center for Research and Interdisciplinarity (CRI), F-75006 Paris,
France  Corresponding author<EMAIL_ADDRESS>
###### Abstract

Science advances by pushing the boundaries of the adjacent possible. While the
global scientific enterprise grows at an exponential pace, at the mesoscopic
level the exploration and exploitation of research ideas is reflected through
the rise and fall of research fields. The empirical literature has largely
studied such dynamics on a case-by-case basis, with a focus on explaining how
and why communities of knowledge production evolve. Although fields rise and
fall on different temporal and population scales, they are generally argued to
pass through a common set of evolutionary stages.To understand the social
processes that drive these stages beyond case studies, we need a way to
quantify and compare different fields on the same terms. In this paper we
develop techniques for identifying scale-invariant patterns in the evolution
of scientific fields, and demonstrate their usefulness using 1.5 million
preprints from the arXiv repository covering $175$ research fields spanning
Physics, Mathematics, Computer Science, Quantitative Biology and Quantitative
Finance. We show that fields consistently follows a rise and fall pattern
captured by a two parameters right-tailed Gumbel temporal distribution. We
introduce a field-specific rescaled time and explore the generic properties
shared by articles and authors at the creation, adoption, peak, and decay
evolutionary phases. We find that the early phase of a field is characterized
by the mixing of cognitively distant fields by small teams of
interdisciplinary authors, while late phases exhibit the role of specialized,
large teams building on the previous works in the field. This method provides
foundations to quantitatively explore the generic patterns underlying the
evolution of research fields in science, with general implications in
innovation studies.

##  1 Introduction

Quantifying the dynamics of scientific fields can help us understand the past
and design the future of scientific knowledge production. Several studies have
investigated the emergence and evolution of scientific fields, from the
discovery of new concepts to their adaptation and modification by the
scientific community [1, 2, 3, 4]. In particular, methods ranging from
bibliometric studies [5, 6] to network analyses [7, 8, 4] and natural language
processing [9, 10] have been implemented on large publication corpora to
monitor the propagation of concepts across articles [11, 12] and the social
interactions between researchers that are producing them [8, 13, 14].

The definition of research fields often relies on data-driven strategies using
self-reported keywords or content analysis. For example, the use of granular
author self-reported topic annotations from well-defined classification
schemes such as PACS (Physics and Astronomy Classification Scheme), MESH terms
and keywords has allowed to construct topic co-occurrence networks and extract
clusters corresponding to potential research fields [7, 9]. Beyond self-
reported annotations, other methods have exploited the citation network
between research articles to group articles by relatedness and map the
knowledge flow within and across research fields [12, 4], and inferential
methods have leveraged Natural Language Processing techniques to automatically
identify key topics of research [11, 10, 15] and their relations. These
various methods provide clusters of closely related topics corresponding to
putative research fields, allowing to monitor how the changing relations
between topics and ideas underlie the dynamic evolution and mutual
interactions between fields. Beyond topic-centric approaches, other methods
have leveraged the interactions between researchers to define research
communities with shared research interests. For example, the co-authorship
network between researchers has been shown to undergo a topological transition
during the emergence of a new field [13, 14]. Co-authorship relations also
influence the individual evolution of research interests and foster the
emergence of a consensus in a research community [16, 17, 18].

While fields rise and fall on different temporal and population scales, they
are generally argued to pass through a common set of evolutionary stages [19,
1]. These stages delineate how diverse actors and behaviors are involved in
successive phases. To study these temporal patterns, dynamical models were
introduced to characterize the evolution of research fields [20, 5] and the
spread of innovation [21, 22, 23]. Yet, we are still lacking a unified
framework to delineate stereotyped stages in the evolution of scientific
fields that can be validated over a large number of well-annotated research
fields.

Here, we address this gap by developing techniques for identifying scale-
invariant patterns in the evolution of fields. We demonstrate their usefulness
using a large corpus of 1.45 million articles from the arXiv repository with
self-reported field tags spanning 175 research fields in Physics, Computer
Science, Mathematics, Finance, and Biology. We show that the evolution of
fields follows a right-tailed distribution with two parameters characterizing
peak location and distribution width. This allows us to collapse the temporal
distributions onto a single rise-and-fall curve and delineate different
evolutionary stages of the fields: creation, adoption, peak, early decay, and
late decay. We then describe the characteristics of articles and authors
across these stages. We finish by discussing these results and their
implications for further work in science and innovation.

Figure 1: a Example of an article in arXiv, highlighting the metadata
extracted using the arXiv API. b The daily number of articles submitted to
arXiv since 1986 shows an exponential growth over time, with a doubling period
of 6 years. The data also shows strong seasonality with 10 times fewer
articles over the weekends. c Heatmap representing the share of articles in
each field (rows) over time (columns). Field are identified using the subject
tags within articles. The heatmap is row normalized for comparison across
fields. Rows are ordered in chronological order of their peak time. The right
side panel shows the total number of articles published in each field as
horizontal bars.

##  2 Results

###  2.1 Description of the data

Since its launch in $1991$, the arXiv repository has become a major venue for
community research, gaining considerable importance across the fields of
Physics, Mathematics and Computer Science. As an open and free contribution
platform, it provides an equal opportunity for publication to researchers
globally, and plays a dominant role in the diffusion of knowledge [24] and the
evolution of new ideas [25].

When submitting a contribution, authors declare the research fields that the
article is contributing to by selecting from a list of subject tags. Here we
collected information about authors, date of publication, and research fields
of 1,456,403 arXiv articles until 2018 (see Methods section and Fig 1a). The
number of articles and authors exhibit an exponential growth over time with a
doubling period of 6 years (see Fig 1b and S1). To control for this effect,
here we focus for each field $i$ on the yearly share of articles
$f_{i,y}=n_{i,y}/N_{y}$, where $n_{i,y}$ is the number of articles published
in the considered field at year $y$ and $N_{y}$ is the total number of
articles in arXiv in the same year. We represent the temporal distributions of
all fields in Fig 1c by chronological peak time. Over the past 30 years, the
research interests have shifted from high-energy physics to computer science
and more recently economy.

###  2.2 Quantifying the rise and fall of scientific fields

Despite differences in overall number of articles and eventual duration, we
observe a general rise-and-fall pattern across research fields (Fig 1c),
prompting us to explore if a simple model can capture their temporal
variation. Extreme value theory [26] predicts that under a broad range of
circumstances, temporal processes displaying periods of incubation (such as
incubation of ideas) or processes with multiple choice (such as the choice of
ideas or research fields) follow skewed right-tailed extreme value
distributions. Examples of such processes can be found in diverse areas, for
example when modeling the evolution of scientific citations [27] or disease
incubation periods [28, 29]. Following these insights, here we use the Gumbel
distribution (Eq. 1) as an ansatz to model the observed field temporal
distributions. Belonging to the general class of extreme value distributions
[30, 26], it provides interpretable parameters for the peak location $\alpha$
and distribution width $\beta$ (Fig. S3). Denoting by $t$ the time since the
first article was published in the field, the share of articles $G(t)$ follows
Eq. 1:

$G(t)=\frac{1}{\beta}e^{\frac{-(t-\alpha)}{\beta}}e^{-e^{\frac{-(t-\alpha)}{\beta}}}$
(1)

where $\alpha$ is the location parameter and $\beta$ the scale parameter.

Figure 2:  a. Conditions for a field to be included in the analysis. b-d
Gumbel fits for the fields with the largest numbers of articles in Physics,
Mathematics and Computer Science: Mathematics - Combinatorics (b), Material
Science (c) and Computational Complexity (d). e. Evolution of the 72 studied
fields after temporal re-scaling from Eq. 2. The blue curve represents the
Gumbel fit, and red dots correspond to the empirical average over equal-sized
bins. Error bars indicate standard error.

In order to estimate the model fit, we consider fields that satisfy three
conditions: (i) longevity – having at least 10 years of activity to ensure a
sufficient observation period, (ii) unimodality – we exclude multimodal
distributions as it would require introducing a mixture model going beyond the
scope of this study and (iii) completeness – we require the peak of the
distribution to be at least 3 years away from the beginning and the end of the
collection period to ensure that we capture sufficient data on both sides of
the distribution. This reduces the number of fields to $72$, which we consider
in our analyses below.

Using a least-squares optimization fitting procedure (see Methods), we show
that 66 out of 72 fields ($91.6\%$) exhibit a significant goodness of fit
($k<0.3$ and $p>0.05$ under KS-test, see Fig S5). We show in Fig 2b-d the
temporal distributions and Gumbel fits for the fields with the largest total
numbers of articles in Physics, Mathematics and Computer Science. After
obtaining the location $\alpha$ and scale $\beta$ parameters from the fitting
procedure, we compute for each field the re-scaled time:

$t^{{}^{\prime}}=\frac{t-\alpha}{\beta}$ (2)

By re-normalizing fields with this standardized time, we observe that the
various temporal distributions align on a single curve, highlighting the
shared patterns of rise and fall across the fields studied (Fig 2e). In
particular, the Gumble distribution provides a more stringent fit of the
tails, as can be observed when comparing to a symmetric, Gaussian fit in a log
scale (Fig S5).

###  2.3 Characterizing the stages of research field evolution

Using the rescaled time from Eq. 2, we next explore the characteristics of
articles and researchers at different stages of a research field evolution. We
adopt hereafter the standard delineations of stages from the innovation
diffusion literature [21] and define 5 periods of research field evolution
(creation, adoption, peak, early decay, and late decay) delineated at the re-
scaled times corresponding respectively to the $2.5\%$, $16\%$, $50\%$ and
$84\%$ quantiles of the Gumbel distribution in Fig2e (blue curve). We then
group articles within these categories for each field and examine the
variation of their characteristics when averaging across all fields.

We consider characteristics of the articles submitted at various field stages,
and of the authors who submit them. For articles, we focus on the number of
fields reported (article multidisciplinarity), the number of authors (team
size), the number of references made to other arXiv articles, and the number
of citations received within arXiv (article impact). For authors, we consider
their career stage at the time of submitting the article (seniority), the
total number of articles submitted to arXiv (longevity), the number of fields
their articles span during their career and the number of fields per article
(author multidisciplinarity). We average these characteristics over the
article coauthors for which we have a unique identifier (ORCID). In the case
of career stage $s$, we use Eq. 3, where $N_{art}$ is the chronological rank
of the current article across the author’s publications and $N_{tot}$ is the
total number of articles:

$s=\frac{N_{art}-1}{N_{tot}-1}$ (3)

We show in Fig 3 the average values of these features for each stage across
the $72$ fields along with random expectations (see Methods 4.4). In the
context of article metrics (Fig 3a), we find that the early stages of research
fields are characterized by interdisciplinary articles ($2.36$ fields, vs.
$2.05$ for late decay) co-authored by small teams (2 authors vs 4.5). As
fields evolve, we observe a steady growth in the number of references to
earlier arXiv articles, indicating that the community builds on earlier works
in arXiv (Fig. 3 and Fig. S6a when restricting to the same field). Finally, we
find that article impact, measured by the number of citations within arXiv, is
maximal at the Adoption phase before the field has reached its peak. The
citation count observes a similar trend in the case of total citations within
arXiv shown in Fig. 3a as well as citations within arXiv received in the first
five years (Fig. S6b). For author metrics (Fig. 3b), we find that the early
stages of research fields are characterized by multidisciplinary authors (16.9
fields in career for creation vs 7.9 in career for late decay, and 2.13 fields
per article vs 1.91 fields per article), who tend to be in their early career
(8% of total duration vs 60%) with the longest longevity (55 papers vs 27).

Figure 3: Characteristics of articles and authors at different evolutionary
stages. The observed values are averaged over all fields (red bars). Gray bars
correspond to the average field-specific random expectation (see Methods 4.4).
Bottom plots represent the relative difference between observed and random
values. Error bars denote standard errors for observed values (red) and
standard deviation for random values (gray). a Article-centric features:
number of fields reported in the article (multidisciplinarity), number of
authors (team size), number of references made to other arXiv articles, and
number of citations received within arXiv (impact). b Author centric features:
career stage at the time of submitting the article (seniority), total number
of articles submitted to arXiv (longevity), number of fields their articles
span during their career and average number of fields per article
(multidisciplinarity).

###  2.4 Cognitive distance and early innovation

Figure 4: a Co-occurrence network of arXiv field tags. Nodes are colored based
on the major research area they belong to (Physics, Computer Science,
Mathematics, Statistics, Quantiative Finance, Quantitative Biology). Barplots
in b,c follow the same method than in Fig 3. b Average cognitive distance
across the field tags of articles. c Average cognitive distance across all the
field tags used by authors throughout their career.

The previous results show that works submitted in early phases of research
fields tend to mix a larger number of field tags. However, this measure does
not take into consideration the various levels of similarity between fields.
For example, publishing an article within sub-fields of physics is different
than publishing an article mixing quantitative biology, computer science, and
physics. This is rendered apparent when examining the co-occurrence network of
fields across arXiv articles (Fig. 4a). In the co-occurrence network, nodes
represent field tags, and edges represent their co-occurrence across articles.
To define edge weights, we first compute the number of co-occurrences between
two fields across the whole period. We then compute a hypergeometric p-value
that the two fields would have this number of co-occurrences given the number
of times they each have occurred. Lower p-values indicate stronger similarity.
We define the weight $W_{ij}$ between fields $i$ and $j$ as
$-log_{10}(p_{ij})$, where $p_{ij}$ is the hypergeometric p-value. Edges with
$p>0.01$ are finally filtered out. The network represents the landscape of
fields in the arXiv, with closely related fields clustering together into
communities corresponding to 6 broader categories: Physics (purple),
Quantitative biology (gray), Computer Science (green), Mathematics (blue),
Statistics (pink) and Quantitative Finance (orange).

Using this network embedding, we define the Cognitive distance $C_{i,j}$
between field tags $i$ and $j$ as the weighted shortest path
$C_{i,j}=\sum_{e}\frac{1}{W_{e}}$, where $e$ are the edges on the shortest
path between the two tags $i$ and $j$ and $W_{e}$ are their weights in the co-
occurrence network. This cognitive distance allows us to provide a weighted
proxy for interdisciplinarity. In particular, it allows to quantify the
distance between disconnected fields: an example of this is shown in Fig. S8a
where q-fin.ec (Economics) connects to hep-ph (High Energy Physics) by a path
length of $4$.

We use this measure to compute for each article with at least two field tags
the maximum cognitive distance between any pair of tags. We find that articles
published in the early stages of a research field have a significantly larger
cognitive distance, while the measure decays to the random level by peak stage
(Fig 4b). Similarly, for authors we find that in earlier stages authors
publish in cognitively distant fields, which narrows down to similar fields in
later stages (Fig. 4c). The relative difference with random at the creation
stage is more stringent that the previous measure using number of tags
(articles: 0.3 vs 0.1, authors: 0.8 vs 0.1), strengthening our previous
observation.

##  3 Discussion

In this study, we leverage the field annotation of 1.5M articles from the
arXiv preprint repository to explore the scale-invariant patterns in the
evolution of scientific fields and highlight the attributes of articles and
researchers across different evolutionary stages. We show that research fields
follow a right-tailed Gumbel temporal distribution, allowing to rescale their
evolution over a single curve. We demonstrate the usefulness of this approach
by highlighting characteristics shared by articles and authors across the
various stages of a field evolution. We observe that early stages are
characterized by articles written by small teams of early career,
interdisciplinary authors, while late stages exhibit the role of large, more
specialized teams. This supports the general finding that small teams disrupt
while large teams develop science and technology [31, 32]. We find that
maximum impact, measured by citations, is reached before the peak of the field
evolution during the Adoption stage. This may reflect foundational works
underlying the subsequent attractivity of the field and moving it to the
‘peak’ phase. In addition, we observe a steady increase in the within-field
references to earlier work as fields evolve. This suggests a consolidation of
the community over the particular body of work produced in the field, though
further work on the citation and collaboration networks would be needed to
investigate this aspect.

The main contribution of our work is to provide a method to rescale fields and
associate research patterns to standardized evolutionary stages. However, this
study has limitations. First, to capture sufficient data on the rise and fall
patterns of research fields, we limited ourselves to a subset of 72 fields out
of the 175 available. In particular, the choice of keeping only unimodal
fields could be overcome by implementing a simple extension of our approach by
using a mixture model, thereby capturing different “waves” of interest within
a research field. In addition, to avoid ambiguity in author names we focused
only on authors for which we could extract ORCID IDs, limiting the study to a
small and potentially biased subset of authors. Future work should extend such
analyses to larger databases with disambiguated authors and topic annotation
to gain in generality. Finally, while being an open repository, authors
submitting to the arXiv need to be invited by another existing member from the
main field of interest. These create social ‘chaperoning’ constraints [33]
that might influence the type of authors observed at various stages.

Overall, this study contributes to the Science of Science literature by
proposing a simple method to investigate the generic temporal properties of
research fields, and highlighting its use in the context of arXiv. Future work
should be conducted to provide mechanistic models recapitulating the observed
patterns, and extending these analyses to larger datasets. We expect these
insights to be helpful for researchers and policymakers interested in the
emergence and development of research fields and more broadly in the dynamics
of innovation [34].

##  4 Methods

###  4.1 Dataset extraction

We extracted the publication metadata from the arXiv website using the arXiv
API. The data spans years 1986 to 2018, with a total of 1,456,404 articles.
For each article we retrieved the following characteristics: a) the unique
article ID, b) the timestamp of article submission, c) the list of subjects
categories (field tags), d) the citations received within arXiv, e) the
references to other arXiv articles, and f) the list of last names of authors.
We show an example article in Fig. 1a. Furthermore, we extracted when possible
the ORCID IDs of the authors that declared it in arXiv. The number of unique
ORCID IDs was 50,402, allowing to disambiguate these authors’ names.

###  4.2 Fitting procedure

Uni-modality Test: For filtering multi-modal fields we use the $diptest$ R
library to compute the dip unimodality test. We remove fields that fail the
test ($p<0.05$).

Least square optimization: For the selected fields, we strip years before the
first publication to only consider years since first article. We then
constrain the mode of the fitted distribution to coincide with the empirical
one, and we fit the location and scale parameters using least-square
optimization.

###  4.3 Assigning articles and authors to evolutionary stages

We first collect for each field all articles containing the field tag. We
associate each article to the evolutionary stage corresponding to the re-
scaled time obtained for that particular field. We then assign the authors of
each article with an ORCID ID to the corresponding evolutionary stage. Note
that articles with multiple field tags can be assigned to different stages of
evolution corresponding to the re-scaled times of the different tags.

###  4.4 Randomization

The observed features in Fig 3 are compared with random expectation by
shuffling for each field the re-scaled times across articles. This procedure
is repeated 50 times for each field and we compute the average for each stage.
Finally, we compute the average and standard deviation across fields.

##  5 Acknowledgements

Thanks to the Bettencourt Schueller Foundation long term partnership, this
work was partly supported by the CRI Research Fellowship to Marc Santolini.

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##  Appendix A Supplementary Information

###  A.1 arXiv as a dataset

Paul Ginsparg created arXiv in 1991. It was initially designed for sharing
preprint articles with friends and colleagues [35]. The reasons why
researchers favor uploading their articles on arXiv are diverse. With a low
threshold in the review phase and a minimal time between submission and online
appearance, it provides a fast way for researchers to share their results with
the scientific community. This in turn helps them in getting feedback from the
larger ecosystem and gain intellectual precedence for their claims. The
management team of arXiv follows a strict and systematic procedure ensuring
accurate classification of an article to its subject domain (see Field tags
management). Though arXiv is lenient in its quality control as compared to a
stricter “peer-reviewed” system, an earlier study reports that $\sim 64\%$
arXiv articles end up publishing in WOS (Web Of Science) indexed journals and
many journals also have started accepting arXiv preprint for submissions [24],
supporting the credibility of arXiv articles.

Field tags management \- Users can choose appropriate field tags for their
articles from the existing ones. They, however, cannot create their tags. The
tags assigned by users are then reviewed by moderators of different subject
domains and changed if deemed necessary. New field tags can only be introduced
by the arXiv administration. They do consider proposals from researchers for
introducing new tags and only after considering multiple factors such as the
size of the research community, frequency of articles appearing in the field,
or its impact on arXiv. A recent example of this was the introduction of two
new tags in $2018$: econ.TH and econ.GN, corresponding to Economics Theory and
Economics General. This happened after a community of economists proposed it
to arXiv. However, most of the field tags appeared in the initial years (see
Fig. S2).

Growth rate \- To calculate the growth rate of the arXiv dataset, we consider
the growth function as defined in Eq4, with growth rate $r$:

$N(t)=N_{0}e^{rt}$ (4)

We then fit the cumulative number of articles and number of authors in the
dataset over time as shown in Fig. S1.

Figure S1: a Cumulative number of articles submitted to arXiv in time. b
Cumulative number of (unique) authors. Both number of articles and of authors
grow exponentially with a doubling period of $\sim$ 6 years.

The growth rates $r$ for articles and authors are respectively 0.117 and
0.119. Hence the doubling period i.e $\frac{ln2}{r}$ for articles and authors
is resp. 5.9 and 5.8 years.

Figure S2: Cumulative number of field tags across time, among the $72$
studied. After an initial growth in the early years, the number of unique tags
stays constant.

###  A.2 Example of Gumbel distribution

To get more insights on the role of the $\alpha$ and $\beta$ parameters, we
show in Fig. S3 some examples of Gumbel distributions with varying parameters.
The location parameter $\alpha$ corresponds to the peak location, while the
scale parameter $\beta$ corresponds to the distribution width. Fields with a
low $\beta$ have a rapid rise followed by a rapid decay with a long tail.
These could be the fields promoted by sudden advances in science and
technologies or economics, for example, Pricing of Securities in Quantitative
finance (q-fin.pr) (Fig. S4a). On the other hand fields with a large $\beta$
have a gradual rise and fall with a long tail in the decay phase – for example
Condensed Matter Material Sciences (cond-mat.mtrl-sci) (Fig. S4b).

Figure S3: Examples of Gumbel distributions for different location and shape
parameter values. Figure S4: Empirical distribution and Gumbel fits for the
fields of a Quantitative Finance (q-fin.pr) and b Condensed matter material
sciences (cond-mat.mtrl-sci).

###  A.3 Fitting the empirical data

Normalizing the Gumbel Distribution Function

Since for each field we only observe a finite sampling period of the full
distribution, we need to normalize the Gumbel distribution between times
$t_{1}$ and $t_{2}$ to improve the fit. Given the Gumbel function $G(x)$, we
find the normalizing constant such that:

$\displaystyle C\int_{t_{1}}^{t_{2}}G(x,\alpha,\beta)dx$ $\displaystyle=1$ (5)
$\displaystyle
C\int_{t_{1}}^{t_{2}}\frac{1}{\beta}e^{\frac{-(x-\alpha)}{\beta}}e^{-e^{\frac{-(x-\alpha)}{\beta}}}dx$
$\displaystyle=1$ (6)

Let $y=e^{-\frac{(x-\alpha)}{\beta}}$ $\implies$
$dy=-\frac{1}{\beta}e^{-\frac{(x-\alpha)}{\beta}}dx$. Replacing above in Eq.6
and adjusting limits we get:

$\displaystyle
C\int_{e^{-\frac{(t_{1}-\alpha)}{\beta}}}^{e^{-\frac{(t_{2}-\alpha)}{\beta}}}-e^{-y}dy$
$\displaystyle=1$ (7) $\displaystyle
Ce^{-y}\Bigr{|}_{e^{-\frac{(t_{1}-\alpha)}{\beta}}}^{e^{-\frac{(t_{2}-\alpha)}{\beta}}}$
$\displaystyle=1$ (8) $\displaystyle
C\left[e^{-{e^{-\frac{(t_{2}-\alpha)}{\beta}}}}-e^{-{e^{-\frac{(t_{1}-\alpha)}{\beta}}}}\right]$
$\displaystyle=1$ (9) $\displaystyle
C=\frac{1}{\left[e^{-{e^{-\frac{(t_{2}-\alpha)}{\beta}}}}-e^{-{e^{-\frac{(t_{1}-\alpha)}{\beta}}}}\right]}$
(10)

With the above $C$ value we can normalize the Gumbel distribution function for
any values of $t_{1}$ and $t_{2}$. Note that when $t_{1}\rightarrow-\infty$
and $t_{2}\rightarrow\infty$ the constant $C\rightarrow 1$.

Figure S5: a Kolmogorov-Smironov test KS values for Gumbel distribution fits.
Lower values indicate better fits. b Corresponding p-values for the KS-test.
Values of $p>0.05$ indicate a plausible fit. c Scatter plot of the fitted vs
empirical values of the temporal distributions for the 72 selected fields. The
Spearman correlation is $\rho=0.81$, with $p<1e-16$. d Same as Fig 2e, in a
log scale. We show both the Gumbel and Gaussian fits. The Gumbel fit provides
a better description of the tails. Figure S6: Same as Fig 3a, for references
(a) and citations (b) within the same field than the article. Citations are
limited to the 5 years following the article.

Figure S7: Article-centric and author-centric properties calculated with the
Gaussian distribution fits. Figure S8:  Example of a shortest path linking the
distant fields of Quantitative finance and High Energy Physics in the field
co-occurrence network.


††footnotetext: Email addresses<EMAIL_ADDRESS>(John MacQuarrie),
<EMAIL_ADDRESS>(Marlon Stefano)

# Butler’s method applied to ${}_{p}[C_{p}\times C_{p}]$-permutation modules

John W. MacQuarrie  Universidade Federal de Minas Gerais, Belo Horizonte, MG,
Brazil  Marlon Stefano  Universidade Federal de Minas Gerais, Belo Horizonte,
MG, Brazil

###### Abstract

Let $G$ be a finite $p$-group with normal subgroup $N$ of order $p$. The first
author and Zalesskii have previously given a characterization of permutation
modules for ${}_{p}G$ in terms of modules for $G/N$, but the necessity of
their conditions was not known. We apply a correspondence due to Butler to
demonstrate the necessity of the conditions, by exhibiting highly non-trivial
counterexamples to the claim that if both the $N$-invariants and the
$N$-coinvariants of a given lattice $U$ are permutation modules, then so is
$U$.

##  1 Introduction

Let $R$ be a complete discrete valuation ring whose residue field has
characteristic $p$ ($R$ will almost always be p or $\mathbb{F}_{p}$) and let
$G$ be a finite $p$-group. An $RG$-module is a _lattice_ if it is free of
finite $R$-rank. An $RG$-lattice is a _permutation module_ if it possesses an
$R$-basis that is set-wise preserved by multiplication from $G$. Given an
$RG$-module $U$ and a normal subgroup $N$ of $G$, the _$N$ -invariants_ of
$U$, $U^{N}$, is the largest submodule of $U$ on which $N$ acts trivially, and
the _$N$ -coinvariants_ of $U$, $U_{N}$, is the largest quotient module of $U$
on which $N$ acts trivially.

In [MZ20], extending the famous sufficient condition of Weiss [Wei88, Theorem
2] to a characterization, the first author and Pavel Zalesskii proved the
following:

######  Theorem 1.1.

Let $G$ be a finite $p$-group, $U$ a ${}_{p}G$-lattice and $N$ a normal
subgroup of $G$ of order $p$. Then $U$ is a permutation module if, and only if

  1. 1.

$U^{N}$ and $U_{N}$ are permutation ${}_{p}[G/N]$-modules and

  2. 2.

$(U/U^{N})_{N}$ is a permutation $\mathbb{F}_{p}[G/N]$-module.

Simple examples are given in [MZ20] showing that we cannot remove the
conditions of Part 1. It was however left open whether Condition 2 is
redundant. That is, whether $U$ is a permutation module if, and only if, both
$U^{N}$ and $U_{N}$ are permutation modules. We will show here that Condition
2 cannot be removed, by exhibiting examples of lattices $U$ for
${}_{p}[C_{p}\times C_{p}]$ ($p$ odd) and for ${}_{2}[C_{2}\times C_{2}\times
C_{2}]$ for which both $U^{N}$ and $U_{N}$ are permutation modules, but $U$ is
not. The examples seem to be highly non-trivial.

We believe that the methods used to construct these examples are interesting
in their own right, utilizing in a new way a wonderful correspondence due to
Butler [But74a, But74b], wherein he associates to a lattice what he calls a
“diagram”. For Butler, the idea was to analyze the _category_ of lattices for
a given abelian $p$-group, using the correspondence to compare it with a
category that is better understood. The novelty of our approach is to use
Butler’s correspondence to analyze the structure of a _given_ lattice, by
analyzing how prescribed properties of the lattice affect the corresponding
diagram. In particular, we will characterize the diagrams of those
${}_{p}[C_{p}\times C_{p}]$-lattices $U$ for which $U$, $U^{N}$ and $U_{N}$
are permutation modules. Using these characterizations, we find a diagram
whose corresponding lattice $U$ has the properties that both $U^{N}$ and
$U_{N}$ are permutation modules, but $U$ is not. In the future, we believe
that analyses of this type will yield further insights in the study of
permutation modules.

##  2 Preliminaries

Notations introduced here will be used throughout our discussion. In what
follows, $G$ is always a finite abelian $p$-group and $N$

is a subgroup of $G$ of order $p$.

Given a ${}_{p}G$-lattice $U$, the ${}_{p}[G/N]$-modules of $N$-invariants
$U^{N}$ and $N$-coinvariants $U_{N}$ are, respectively, the largest submodule
and the largest quotient module of $U$ on which $N$ acts trivially. Explicitly

$\displaystyle U^{N}$ $\displaystyle=\\{u\in U\,|\,nu=u\,\,\forall n\in N\\}$
$\displaystyle U_{N}$ $\displaystyle=U/I_{N}U$

where $I_{N}$, the augmentation ideal of ${}_{p}N$, is the ideal generated by
the kernel of the map ${}_{p}N\to{}_{p}$ sending $\sum_{n\in N}\lambda_{n}n$
to $\sum\lambda_{n}$.

By [HR62, Theorem 2.6] there are three isomorphism classes of indecomposable
${}_{p}N$-lattice, being the trivial module p, the free module ${}_{p}N$ and a
non-permutation module $S$ of p-rank $p-1$, which can be described in either
of the following equivalent ways:

$S=I_{N}={}_{p}N/({}_{p}N)^{N}.$

######  Lemma 2.1 ([MZ20, Lemma 8]).

Let $H$ be a cyclic group of order $p$ and $U$ a ${}_{p}H$-lattice. Then
$U_{H}$ is a lattice if, and only if, $U$ is a permutation module.

If ever $H$ is a subgroup of $G$, we use the symbol $\widehat{H}$ to denote
the element $\sum_{h\in H}h$ in ${}_{p}G$.

######  Lemma 2.2.

Let $G=C_{p}\times C_{p}$ and $H,K$ subgroups of $G$. In ${}_{p}G$ we have
$\widehat{H}\cdot\widehat{K}=|H\cap K|\cdot\widehat{HK}$.

###### Proof.

This follows by direct calculations.

∎

##  3 Butler’s Correspondence for ${}_{p}[C_{p}\times C_{p}]$

We describe the parts we require from [But74b] for the special case of
${}_{p}[C_{p}\times C_{p}]$, which is what we require to construct examples
for $p$ odd. The case of $p=2$ must be treated separately, but the methods are
similar and we only present the example, in Section 6.

In what follows, $G=C_{p}\times C_{p}=N\times C$ . We denote as usual by
${}_{p},\mathbb{F}_{p}$ the field of fractions and the residue field of the
$p$-adic integers p, respectively. The group algebra
${}_{p}G={}_{p}\otimes_{{}_{p}}{}_{p}G$ has a decomposition

${}_{p}G=e_{0}{}_{p}G\times\prod_{\begin{subarray}{c}H\leqslant G\\\
|H|=p\end{subarray}}{e_{H}{}_{p}G},$

where the $e_{i}$ are the $p+2$ primitive orthogonal idempotents of ${}_{p}G$,
given as

$e_{0}=\frac{1}{p^{2}}\widehat{G}\,,\quad
e_{H}=\frac{1}{p^{2}}(p\widehat{H}-\widehat{G}).$

Denote by $I$ the indexing set $\\{0,H_{1},\ldots,H_{p+1}\\}$ of the primitive
idempotents, where the $H_{i}$ are the subgroups of $G$ of order $p$. For each
$i\in I$ we denote by $\Lambda_{(i)}$ both the ${}_{p}G$-algebra
$e_{i}{}_{p}G$ and the free $e_{i}{}_{p}G$-module of rank 1iiiwe write
$\Lambda_{(i)}$ rather than $\Lambda_{i}$ to avoid confusion with
coinvariants.. Recall that ${}_{p}G$ is a local algebra, with maximal ideal
$I_{G}+p{{}_{p}G}$.

######  Lemma 3.1.

Let $H,K$ be distinct subgroups of order $p$.

  1. 1.

$\widehat{H}\cdot e_{K}=0$,

  2. 2.

$\Lambda_{(K)}$ has no non-zero $H$-fixed points,

  3. 3.

$J(\Lambda_{(K)})=I_{H}\Lambda_{(K)}$.

###### Proof.

  1. 1.

By Lemma 2.2,

$\widehat{H}\cdot
e_{K}=\frac{1}{p^{2}}(p\widehat{H}\widehat{K}-\widehat{H}\widehat{G})=\frac{1}{p^{2}}(p\widehat{G}-p\widehat{G})=0.$

  2. 2.

$\widehat{H}$ acts as multiplication by $p$ on a fixed point of
$\Lambda_{(K)}$, and so ${\Lambda_{(K)}}^{H}=0$ by Part 1.

  3. 3.

$p\Lambda_{(K)}\subseteq I_{H}\Lambda_{(K)}$ because by Part 1

$pe_{K}=pe_{K}-\widehat{H}e_{K}=\sum_{h\in H}(1-h)e_{K}\in
I_{H}\Lambda_{(K)}.$

∎

######  Definition 3.2.

The ${}_{p}G$-lattice $U$ is _reduced_ if it has neither free direct summands
nor summands isomorphic to any $\Lambda_{i}$.

######  Definition 3.3 ([But74b, §2.2]).

A _diagram_ for $C_{p}\times C_{p}$ is a $(p+3)$-tuple
$V_{*}=(V;V_{(i)}\,,\,i\in I)$, where

  * •

$V$ is a finitely generated $\mathbb{F}_{p}[C_{p}\times C_{p}]$-module,

  * •

the $V_{(i)}$ are $\mathbb{F}_{p}[C_{p}\times C_{p}]$-submodules of $V$,

  * •

for each $j\in I$ we have $\sum_{i\neq j}V_{(i)}=V$,

  * •

for each $i\in I$, the induced action of ${}_{p}[C_{p}\times C_{p}]$ on
$V_{(i)}$ factors through the canonical surjection ${}_{p}[C_{p}\times
C_{p}]\to\Lambda_{(i)}$ sending $1$ to $e_{i}$.

A _map of diagrams_ $V_{*}\to V^{\prime}_{*}$ is an
$\mathbb{F}_{p}[C_{p}\times C_{p}]$-module homomorphism $\rho:V\to V^{\prime}$
such that $\rho(V_{(i)})\subseteq V^{\prime}_{(i)}$ for each $i\in I$.

We make explicit how one obtains a diagram from a reduced lattice, and vice-
versa, fixing as we do some notation that will remain in force throughout the
rest of the article. As we follow Butler’s arguments in [But74b], simply
specialising to the case of ${}_{p}[C_{p}\times C_{p}]$ (which simplifies
things slightly), we don’t give any justifications.

  * •

Getting a diagram from a reduced lattice: Let $U$ be a reduced lattice. We
regard $U$ as the sublattice $1\otimes U$ of the ${}_{p}G$-module
${}_{p}U={}_{p}\otimes_{{}_{p}}U$. Defining for each $i\in I$ the submodule
$U_{(i)}:=e_{i}U$ of ${}_{p}U$, the sum

$U_{*}:=\sum_{i\in I}U_{(i)}$

is direct, and since $U$ is reduced we have

$pU_{*}\subsetneq U\subsetneq U_{*}.$

Defining

$V=U_{*}/U\,,\quad V_{(i)}=(U_{(i)}+U)/U,$

the tuple $V_{*}=(V;V_{(i)}\,,\,i\in I)$ is the diagram corresponding to $U$.

  * •

Getting a reduced lattice from a diagram: Let $V_{*}=(V;V_{(i)}\,,\,i\in I)$
be a diagram. For each $i\in I$, let $F_{(i)}$ be the direct sum of
$\textnormal{dim}_{\mathbb{F}_{p}}(V_{(i)}/\textnormal{Rad}(V_{(i)}))$ copies
of $\Lambda_{(i)}$ and denote by $F$ the direct sum $\bigoplus_{i\in
I}F_{(i)}$. For each $i$ let $f_{i}:F_{(i)}\to V_{(i)}$ be a surjective
homomorphism of $\Lambda_{(i)}$-modules inducing an isomorphism
$F_{(i)}/\textnormal{Rad}(F_{(i)})\to V_{(i)}/\textnormal{Rad}(V_{(i)})$. We
thus obtain for each $i$ a map
$F_{(i)}\xrightarrow{f_{i}}V_{(i)}\hookrightarrow V$ and hence by summing
these maps, a map $f:F\to V$. The reduced lattice corresponding to $V_{*}$ is
$\textnormal{Ker}(f)$.

These processes are mutually inverse in the sense that if $U^{\prime}$ is the
reduced lattice obtained from the diagram of $U$, then $U^{\prime}\cong U$ as
${}_{p}G$-lattices, and if $V^{\prime}_{*}$ is the diagram coming from the
reduced lattice associated to $V_{*}$, then $V^{\prime}_{*}\cong V_{*}$.

It follows that given a reduced lattice $U$, we may associate a unique (up to
isomorphism) diagram $V_{*}$ and from this, the modules $F_{(i)},F$, with $U$
being considered as a submodule of $F$. We will use these notations freely in
what follows.

##  4 Characterizations using diagrams

Throughout this section, $U$ is a reduced ${}_{p}G$-lattice with diagram
$V_{*}=(V;V_{(i)}\,,\,i\in I)$ and $F_{(i)},F$ are the corresponding modules
defined in the previous section. Given an element $x\in F$, we denote by
$\overline{x}$ its image in $V$. Throughout, the letters $H,K$ will always
refer to subgroups of order $p$. If an element of $\bigoplus F_{(i)}$ is
written like $x=\sum_{H\neq N}x_{H}$, this should be taken to mean that
$x\in\bigoplus_{\begin{subarray}{c}H\leqslant G\\\ |H|=p\\\ H\neq
N\end{subarray}}F_{(H)}$ with each $x_{H}\in F_{(H)}$, and similarly in $V$.

######  Theorem 4.1.

The reduced lattice $U$ is a permutation module if, and only if, $G$ acts
trivially on each $V_{(i)}$ and
$V=V_{(0)}=\bigoplus_{\begin{subarray}{c}H\leqslant G\\\
|H|=p\end{subarray}}V_{(H)}$.

###### Proof.

A direct calculation shows that the reduced permutation lattice
${}_{p}[G/H]=\langle u\rangle$ has diagram

$(\mathbb{F}_{p};\mathbb{F}_{p},0,\ldots,0,\mathbb{F}_{p},0,\ldots,0)$

where the last non-zero entry occurs in the position indexed by $H$: taking
$K$ to be any subgroup of order $p$ different from $H$ we have by Lemma 2.2:

$\displaystyle e_{0}\cdot u$
$\displaystyle=\frac{1}{p^{2}}\widehat{G}u=\frac{1}{p^{2}}\widehat{K}\widehat{H}u=\frac{1}{p}\widehat{K}u$
$\displaystyle e_{H}\cdot u$
$\displaystyle=\frac{1}{p^{2}}(p\widehat{H}-\widehat{K}\widehat{H})u=\frac{1}{p}(p\widehat{1}-\widehat{K})u$
$\displaystyle e_{K}\cdot u$
$\displaystyle=\frac{1}{p^{2}}(p\widehat{K}-\widehat{H}\widehat{K})u=\frac{1}{p^{2}}(p\widehat{1}-\widehat{H})\widehat{K}u=\frac{1}{p^{2}}(p\widehat{1}-p\widehat{1})\widehat{K}u=0.$

Since $e_{0}\cdot u+e_{H}\cdot u=u\in U$, $U_{*}/U$ has dimension 1 and the
claim follows. A reduced permutation module is a direct sum of modules
${}_{p}[G/H]$ for different $H$ of order $p$, and the diagram of a direct sum
is the direct sum of the diagrams. The result follows. ∎

######  Lemma 4.2.

  1. 1.

$U^{N}=U\cap(F_{(0)}+F_{(N)})$

  2. 2.

For any subgroup $H\neq N$ of order $p$ we have $I_{H}U^{N}\subseteq U\cap
F_{(N)}$, with equality if, and only if, $U^{N}$ is a permutation module.

###### Proof.

  1. 1.

$N$ acts trivially on $F_{(0)},F_{(N)}$, while $F_{(H)}$ has no $N$-fixed
points for $H\neq N$ by Lemma 3.1. Part 1 now follows because $F$ is the
direct sum of the $F_{(i)}$.

  2. 2.

The inclusion follows from Part 1 because $I_{H}F_{(0)}=0$. For any $u\in
U\cap F_{(N)}$ we have $\widehat{H}u=\widehat{H}e_{N}u=0$ by Lemma 3.1 and so

$pu=pu-\widehat{H}u=\sum_{h\in H}(1-h)u\in I_{H}U^{N}.$

If $U^{N}$ is a permutation module, then $U^{N}/I_{H}U^{N}$ is a lattice by
Lemma 2.1 and so $u\in I_{H}U^{N}$. On the other hand, if $I_{H}U^{N}=U\cap
F_{(N)}$ then

$U^{N}/I_{H}U^{N}=U^{N}/(U^{N}\cap F_{(N)}),$

which is a lattice since $F_{(N)}$ is a summand of $F$, and so $U^{N}$ is a
permutation module by Lemma 2.1 again.

∎

######  Theorem 4.3.

The lattice $U^{N}$ is a permutation module if, and only if,
${V_{(N)}}^{G}\subseteq V_{(0)}$.

###### Proof.

Suppose first that $U^{N}$ is a permutation module and fix
$\overline{x}\in{V_{(N)}}^{G}$, with lift $x\in F_{N}$. Since
$(1-c)\overline{x}=0$, we have $(1-c)x\in U\cap F_{(N)}$. But by Lemma 4.2,
$U\cap F_{(N)}=I_{C}U^{N}$ so by Part 1 of Lemma 4.2 there are $x_{0}\in
F_{(0)},x_{N}\in F_{(N)}$ with $x_{0}+x_{N}\in U$ and
$(1-c)x=(1-c)(x_{0}+x_{N})=(1-c)x_{N}$. As $(1-c)(x-x_{N})=0$ and $F_{(N)}$
has no non-zero $C$-fixed points, $x=x_{N}$, so that $x_{0}+x\in U$. Hence
$\overline{x}=-\overline{x_{0}}\in V_{(0)}$, as required.

Suppose now that ${V_{(N)}}^{G}\subseteq V_{(0)}$. To conclude that $U^{N}$ is
a permutation module, it is enough to check that $U\cap F_{(N)}\subseteq
I_{C}U^{N}$, by Lemma 4.2. So fix $y\in U\cap F_{(N)}$. Since $U$ is reduced,
$y\in\textnormal{Rad}(F_{(N)})=I_{C}F_{(N)}$, by Lemma 3.1, so we may write
$y=(1-c)x$ with $x\in F_{(N)}$. Since $(1-c)\overline{x}=0$,
$\overline{x}\in{V_{(N)}}^{G}$, so in $V_{(0)}$ by hypothesis. Thus there is
$x_{0}\in F_{(0)}$ with $\overline{x_{0}}=\overline{x}$, hence $x-x_{0}\in
U^{N}$ and so $y=(1-c)(x-x_{0})\in I_{C}U^{N}$ as required. ∎

######  Lemma 4.4.

We have

$I_{N}U\subseteq U\cap\Big{(}\bigoplus_{H\neq N}F_{(H)}\Big{)}$

with equality if, and only if, $U_{N}$ is a lattice.

###### Proof.

The inclusion is clear because $(1-n)(F_{(0)}+F_{(N)})=0$. Suppose that
$U_{N}$ is a lattice and take $x$ in the right hand set. Since
$\widehat{N}x=0$ by Lemma 3.1, $px=\sum_{n\in N}(1-n)x\in I_{N}U$ and hence
$x\in I_{N}U$ since $U_{N}$ is a lattice. On the other hand, the right hand
set is a pure submodule of $U$ because $\bigoplus_{H\neq N}F_{(H)}$ is a
summand of $F$, and hence if this set is equal to $I_{N}U$ then $U_{N}$ is a
lattice. ∎

We need a technical lemma:

######  Lemma 4.5.

Fix $a\in I$ and $B\subseteq I\backslash\\{a\\}$. The homomorphism

$\displaystyle\varphi:U\cap\bigoplus_{i\in B\cup\\{a\\}}F_{(i)}$
$\displaystyle\to V_{(a)}\cap\sum_{i\in B}V_{(i)}$ $\displaystyle\sum
x_{i}\quad$ $\displaystyle\mapsto\quad\overline{x_{a}}$

is well-defined and surjective.

###### Proof.

The element $\overline{x_{a}}$ is in $V_{(a)}\cap\sum_{i\in B}V_{(i)}$ because
$\sum x_{i}\in U$, and hence $\overline{x_{a}}=-\sum_{i\in
B}\overline{x_{i}}$. Given $v\in V_{(a)}\cap\sum_{i\in B}V_{(i)}$, write
$v=-\sum_{i\in B}v_{i}$ and lift $v$ to $x_{a}$ in $F_{(a)}$ and each $v_{i}$
to $x_{i}$ in $F_{(i)}$. The element $\sum_{i\in B\cup\\{a\\}}x_{i}\in
U\cap\bigoplus_{i\in B\cup\\{a\\}}F_{(i)}$ and maps to $v$ by $\varphi$. ∎

Recall that by Lemma 2.1, $U_{N}$ is a lattice if, and only if, the
restriction $U\downarrow_{{N}}$ of $U$ to $N$ is a permutation module. Thus
the following theorem is also a characterization of those reduced $U$ for
which $U\downarrow_{{N}}$ is a permutation module.

######  Theorem 4.6.

The module $U_{N}$ is a lattice if, and only if

  1. 1.

For each $H\neq N$ of order $p$, $V_{(H)}\cap(\sum_{K\neq
N,H}V_{(K)})\subseteq I_{N}V_{(H)}$, and

  2. 2.

$V^{N}\subseteq V_{(0)}+V_{(N)}$.

###### Proof.

Suppose first that $U_{N}$ is a lattice. We check the first condition, so fix
an element $v\in V_{(H)}\cap(\sum_{K\neq N,H}V_{(K)})$. By Lemma 4.5 there is
$u=\sum_{K\neq N}u_{K}\in U\cap\bigoplus_{K\neq N}F_{(K)}$ such that
$\overline{u_{H}}=v$. By Lemma 4.4, there is $u^{\prime}=\sum_{i\in
I}u^{\prime}_{i}\in U$ such that $u=(1-n)u^{\prime}=\sum(1-n)u^{\prime}_{i}$.
As the sum of the $F_{(i)}$ is direct, we have $u_{H}=(1-n)u_{H}^{\prime}$ and
hence

$v=\overline{u_{H}}=(1-n)\overline{u_{H}^{\prime}}\in I_{N}V_{(H)}.$

To show the second condition, it is enough to confirm that $(\sum_{H\neq
N}V_{(H)})^{N}\subseteq V_{(0)}+V_{(N)}$ so fix $v=\sum_{H\neq
N}\overline{x_{H}}\in V^{N}$ and a lift $\sum_{H\neq N}x_{H}$ of $v$. Then
$(1-n)\sum_{H\neq N}x_{H}\in U\cap\bigoplus_{H\neq N}F_{(H)}$, which is
$I_{N}U$ by Lemma 4.4 and so

$(1-n)\sum_{H\neq N}x_{H}=(1-n)\sum_{i\in I}y_{i}$

with $\sum y_{i}\in U$. As $F_{(H)}$ has no non-zero $N$-fixed points,
$x_{H}=y_{H}$ for each $H\neq N$. Hence

$V_{(0)}+V_{(N)}\ni-(\overline{y_{0}}+\overline{y_{N}})=\sum_{H\neq
N}\overline{y_{H}}=\sum_{H\neq N}\overline{x_{H}}=v.$

Suppose conversely that Conditions $1$ and $2$ hold for $U$. We will check the
conditions of Lemma 4.4. Given $u=\sum x_{H}\in U\cap\bigoplus_{H\neq
N}F_{(H)}$, Lemma 4.5 and Condition 1 imply that $\overline{x_{H}}\in
I_{N}V_{(H)}=\textnormal{Rad}(V_{(H)})$ for each $H\neq N$ and hence, since
$F_{(H)}/\textnormal{Rad}(F_{(H)})\to V_{(H)}/\textnormal{Rad}(V_{(H)})$ is an
isomorphism, $x_{H}\in\textnormal{Rad}(F_{(H)})=I_{N}F_{(H)}$. For each $H\neq
N$, write $x_{H}=(1-n)y_{H}$ for some $y_{H}\in F_{(H)}$. Then
$(1-n)\sum_{H\neq N}\overline{y_{H}}=0$ so that $\sum_{H\neq
N}\overline{y_{H}}\in V^{N}$ and hence

$\sum_{H\neq N}\overline{y_{H}}=-\overline{y_{0}}-\overline{y_{N}}\in
V_{(0)}+V_{(N)}$

for some $y_{0}\in F_{(0)}$ and $y_{N}\in F_{(N)}$, by Condition 2. Thus
$y=\sum_{i\in I}y_{i}\in U$ and

$I_{N}U\ni(1-n)y=\sum_{H\neq N}x_{H}=u.$

∎

######  Lemma 4.7.

Let $U$ be a reduced lattice such that $U_{N}$ is a lattice. Then

$I_{G}U\subseteq U\cap\Big{(}\bigoplus_{H}F_{(H)}\Big{)}$

with equality if, and only if, $U_{N}$ is a permutation module.

###### Proof.

The inclusion is clear because $(1-g)F_{(0)}=0$ for every $g\in G$. Since
$U_{N}$ is a lattice, Lemma 2.1 says that $(U_{N})_{C}=U/I_{G}U$ is a lattice
if, and only if, $U_{N}$ is a permutation module. Suppose that $U_{N}$ is a
permutation module and fix $u=\sum_{H}x_{H}\in U\cap\bigoplus_{H}F_{(H)}$. By
Lemma 2.2, $(p\cdot 1-\widehat{K})u=\sum_{H\neq K}px_{H}\in I_{G}U$, and
hence, summing over the subgroups of order $p$ we get that
$\sum_{H}p^{p}x_{H}=p^{p}u\in I_{G}U$, and hence $u\in I_{G}U$ since
$U/I_{G}U$ is a lattice. On the other hand, the right hand submodule is pure
in $U$ because $\bigoplus_{H}F_{(H)}$ is a summand of $F$, so if the sets are
equal then $I_{G}U$ is pure in $U$ and $U_{N}$ is a permutation module. ∎

######  Remark 4.8.

Lemma 4.7 is false without the condition that $U_{N}$ be a lattice: the given
equality applies for the ${}_{2}[C_{2}\times C_{2}]$-lattice $U$ whose diagram
has
$V=\mathbb{F}_{2}\oplus\mathbb{F}_{2},V_{(0)}=\langle(1,1)\rangle,V_{(N)}=0,V_{(C)}=\langle(1,0)\rangle$
and $V_{(\langle nc\rangle)}=\langle(0,1)\rangle$. Here $U_{N}$ is not a
lattice.

######  Theorem 4.9.

Let $U$ be a reduced lattice such that $U_{N}$ is a lattice. Then $U_{N}$ is a
permutation module if, and only if $V_{(H)}\cap\sum_{K\neq H}V_{(K)}\subseteq
I_{G}V_{(H)}$ for every subgroup $H$ of order $p$.

###### Proof.

Suppose that $U_{N}$ is a permutation module. Given an element
$\overline{x_{H}}=\sum_{K\neq H}\overline{x_{K}}$ in $V_{(H)}\cap\sum_{K\neq
H}V_{(K)}$, then $x_{H}-\sum_{K\neq H}x_{K}\in
U\cap\Big{(}\bigoplus_{H}F_{(H)}\Big{)}$, hence in $I_{G}U\subseteq I_{G}F$ by
Lemma 4.7. Thus $x_{H}\in I_{G}F_{(H)}$ and hence $\overline{x_{H}}\in
I_{G}V_{(H)}$.

Suppose now that $V_{(H)}\cap\sum_{K\neq H}V_{(K)}\subseteq I_{G}V_{(H)}$ for
every $H$. By Lemma 4.7, to prove that $U_{N}$ is a permutation module, it is
enough to consider an element $u\in U\cap\bigoplus_{K}F_{(K)}$ and check that
it is in $I_{G}U$. The given inclusion implies that
$U\cap(\bigoplus_{K}F_{(K)})\subseteq I_{G}(\bigoplus_{K}F_{(K)})$, so we can
write

$u=(1-c)y_{N}-(1-n)\sum_{K\neq N}y_{K},$

so that $(1-c)\overline{y_{N}}=(1-n)\sum_{K\neq N}\overline{y_{K}}$. But by
the definition of a diagram, we have $V=V_{(0)}+\sum_{K\neq N}V_{(K)}$ and so
we can write $\overline{y_{N}}=\overline{x_{0}}+\overline{x_{C}}+\sum_{K\neq
N,C}\overline{x_{K}}$, and hence

$(1-c)\overline{y_{N}}=(1-c)\sum_{K\neq N,C}\overline{x_{K}}.$

Thus the element $w:=(1-c)y_{N}-(1-c)\sum_{K\neq N,C}x_{K}$ is in $U$, and
indeed in $I_{C}U$ because $y_{N}-\sum_{K\neq N,C}x_{K}-x_{0}-x_{C}\in U$ and

$w=(1-c)(y_{N}-x_{0}-x_{C}-\sum_{K\neq N,C}x_{K}).$

On the other hand, since

$(1-n)\sum_{K\neq N}\overline{y_{K}}=(1-c)\overline{y_{N}}=(1-c)\sum_{K\neq
N,C}\overline{x_{K}},$

the element $w^{\prime}:=(1-n)\sum_{K\neq N}y_{K}-(1-c)\sum_{K\neq
N,C}x_{K}\in U\cap\bigoplus_{K\neq N}F_{(K)}$, so $w^{\prime}\in I_{N}U$ by
Lemma 4.4 since $U_{N}$ is a lattice. It follows that $u=w-w^{\prime}\in
I_{C}U+I_{N}U=I_{G}U$, as required. ∎

##  5 Examples for $C_{p}\times C_{p}$

If $p=2$, it follows from Theorem 1.1 that a ${}_{2}[C_{2}\times
C_{2}]$-lattice $U$ is a permutation module if, and only if, both $U^{N}$ and
$U_{N}$ are permutation modules, because every $\mathbb{F}_{2}[G/N]$-module is
a permutation module so that Condition 2 is automatically satisfied. But this
is not the case for odd primes. Here we present a counterexample for each odd
prime $p$, using the results from the previous section.

######  Example 5.1.

Fix an odd prime $p$ and let $G=C_{p}\times C_{p}=\langle
n\rangle\times\langle c\rangle$. We construct a diagram for ${}_{p}G$. Let $V$
be the $\mathbb{F}_{p}[C_{p}\times C_{p}]$-module with $\mathbb{F}_{p}$-basis
$v_{1},\ldots,v_{p+2}$ and action from $G$ given as follows:

$n\cdot v_{1}=v_{1},\quad n\cdot v_{2}=v_{2},\quad n\cdot
v_{j}=v_{1}+v_{j}\quad\forall j\geqslant 3,$ $c\cdot v_{1}=v_{1},\quad c\cdot
v_{2}=v_{1}+v_{2},\quad c\cdot v_{3}=v_{3},\quad c\cdot
v_{j}=-k_{j}v_{1}+v_{j}\quad\forall j\geqslant 4,$

where $k_{j}\in\\{1,\ldots,p-1\\}$ is such that $(j-3)k_{j}=1$ modulo $p$. The
submodules $V_{(i)}$ are defined as follows:

$\displaystyle V_{(0)}$ $\displaystyle=\langle
v_{1},\,v_{2}-v_{3}+v_{4},\,(k_{j+1}-k_{j})v_{2}-v_{j}+v_{j+1}\hbox{ for
}4\leqslant j\leqslant p+1\rangle,$ $\displaystyle V_{(N)}$
$\displaystyle=\langle v_{1},v_{2}\rangle,$ $\displaystyle V_{(C)}$
$\displaystyle=\langle v_{1},v_{3}\rangle,$ $\displaystyle V_{(\langle
nc^{a}\rangle)}$ $\displaystyle=\langle v_{1},v_{a+3}\rangle,\qquad
a\in\\{1,\ldots,p-1\\}$

The tuple $(V;V_{(i)}\,,\,i\in I)$ is a diagram, with corresponding reduced
lattice $U$. Then

  * •

$U^{N}$ is a permutation module by Theorem 4.3, because ${V_{(N)}}^{G}=\langle
v_{1}\rangle\subseteq V_{(0)}$.

  * •

$U_{N}$ is a lattice by Theorem 4.6: the first condition is satisfied because
$V_{(H)}\cap\sum_{K\neq N,H}V_{(K)}=\langle v_{1}\rangle$ and
$v_{1}=(n-1)v_{j}$ for any $j\geqslant 3$. The second condition is satisfied
because the element $\sum_{j=1}^{p+2}\lambda_{j}v_{j}\in V^{N}$ if, and only
if, $p\mid\sum_{j=3}^{p+2}\lambda_{j}$, and hence $V^{N}$ is generated by
$v_{1},v_{2}$ and the elements $v_{j}-v_{j+1}$ for $j\in\\{3,\ldots,p+1\\}$,
which are in $V_{(0)}+V_{(N)}$.

  * •

$U_{N}$ is a permutation module by Theorem 4.9: $V_{(H)}\cap\sum_{K\neq
H}V_{(K)}=\langle v_{1}\rangle$ which is in $I_{N}V_{(H)}$ if $H\neq N$ and,
since $v_{1}=(c-1)v_{2}$, is in $I_{C}V_{(H)}$ if $H=N$.

  * •

But $U$ is not a permutation module by Theorem 4.1, since $G$ does not act
trivially on $V$.

The lattice $U$ has p-rank $p^{2}+p-1$. In the smallest case, when $p=3$, it
has 3-rank $11$ and with respect to some 3-basis of $U$, the matrices of the
actions of $n$ and $c$ respectively, are

$\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&0&-1&-1&-1&-1&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&-2&0&0&-1&1&0&1&0&0\\\
0&0&0&-2&0&-1&-1&0&0&1&0\\\ 0&0&0&0&-2&-1&0&0&0&0&1\\\
0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0&0\\\
0&0&-3&0&0&-1&1&0&1&0&0\\\ 0&0&0&-3&0&-1&-1&0&0&1&0\\\
0&0&0&0&-3&-1&0&0&0&0&1\end{pmatrix}\hbox{ and
}\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&-1&0&1&-1&0&0&0&0&-1&0\\\
0&-2&0&0&0&0&-1&1&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&-1&0\\\
0&0&0&0&-2&-1&0&0&0&0&1\\\ 0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0&0\\\
0&-3&0&0&0&0&-1&1&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&3&0&1&1&0&0&-2&0\\\
0&0&0&0&-3&-1&0&0&0&0&1\end{pmatrix}.$

##  6 Example for $C_{2}\times C_{2}\times C_{2}$

A similar application of Butler’s Correspondence to the group $G=C_{2}\times
C_{2}\times C_{2}=\langle n\rangle\times\langle b\rangle\times\langle
c\rangle$ yields the following lattice $U$ for which $U^{N}$ and $U_{N}$ are
permutation modules, but $U$ is not. The matrices of $n,b,c$ respectively,
are:

$\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0&0\\\
0&1&0&0&1&0&0&1&1&1&1\\\ 0&1&0&0&0&1&0&1&1&1&1\\\
0&-1&0&0&0&0&1&-1&-1&-1&-1\\\ 0&-1&0&0&0&0&0&-1&0&0&0\\\
0&0&-1&1&0&0&0&0&-1&0&0\\\ 0&-1&1&0&0&0&0&0&0&-1&0\\\
0&0&0&-1&0&0&0&0&0&0&-1\end{pmatrix}\,,\,\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0&0\\\
0&0&0&1&0&1&0&0&0&1&1\\\ 0&1&-1&0&1&0&0&0&0&1&1\\\
1&0&1&-1&-1&-1&-1&0&0&-1&-1\\\ 0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&1&0&0\\\ 0&-1&1&0&0&0&0&0&0&-1&0\\\
0&0&0&-1&0&0&0&0&0&0&-1\end{pmatrix}\,,\,\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0&0\\\
0&0&0&1&0&0&1&-1&0&-1&0\\\ 1&1&0&0&-1&-1&-1&-1&0&-1&0\\\
0&0&0&-1&1&0&0&1&0&1&0\\\ 0&0&0&0&0&0&0&1&0&0&0\\\ 0&0&-1&1&0&0&0&0&-1&0&0\\\
0&0&0&0&0&0&0&0&0&1&0\\\ 0&0&0&-1&0&0&0&0&0&0&-1\end{pmatrix}.$

## References

  * [But74a] M. C. R. Butler.  The $2$-adic representations of Klein’s four group.  In Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973), pages 197–203. Lecture Notes in Math., Vol. 372, 1974.
  * [But74b] M. C. R. Butler.  On the classification of local integral representations of finite abelian $p$-groups.  In Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974), Paper No. 6, pages 18 pp. Carleton Math. Lecture Notes, No. 9, 1974.
  * [HR62] A. Heller and I. Reiner.  Representations of cyclic groups in rings of integers. I.  Ann. of Math. (2), 76:73–92, 1962.
  * [MZ20] John W. MacQuarrie and Pavel A. Zalesskii.  A characterization of permutation modules extending a theorem of Weiss.  Doc. Math., 25:1159–1169, 2020.
  * [Wei88] Alfred Weiss.  Rigidity of $p$-adic $p$-torsion.  Ann. of Math. (2), 127(2):317–332, 1988.