# Personalized QoE Enhancement for Adaptive Video Streaming: A Digital Twin-
Assisted Scheme

Xinyu Huang1, Conghao Zhou1, Wen Wu2, Mushu Li1, Huaqing Wu1, and Xuemin
(Sherman) Shen1  1Department of Electrical & Computer Engineering, University
of Waterloo, Canada
2Frontier Research Center, Peng Cheng Laboratory, China
Email: {x357huan, c89zhou, m475li, h272wu<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we present a digital twin (DT)-assisted adaptive video
streaming scheme to enhance personalized quality-of-experience (PQoE). Since
PQoE models are user-specific and time-varying, existing schemes based on
universal and time-invariant PQoE models may suffer from performance
degradation. To address this issue, we first propose a DT-assisted PQoE model
construction method to obtain accurate user-specific PQoE models.
Specifically, user DTs (UDTs) are respectively constructed for individual
users, which can acquire and utilize users’ data to accurately tune PQoE model
parameters in real time. Next, given the obtained PQoE models, we formulate a
resource management problem to maximize the overall long-term PQoE by taking
the dynamics of users’ locations, video content requests, and buffer statuses
into account. To solve this problem, a deep reinforcement learning algorithm
is developed to jointly determine segment version selection, and communication
and computing resource allocation. Simulation results on the real-world
dataset demonstrate that the proposed scheme can effectively enhance PQoE
compared with benchmark schemes.

##  I Introduction

With the rapid popularization of emerging video applications, video streaming
data accounts for the majority of global mobile data [1], which has placed a
growing strain on wireless networks. To guarantee the user’s continuous
playback under dynamic channel conditions, adaptive bitrate (ABR) technology
[2] that splits a complete video sequence into multiple segments with
different bitrates enables adaptive video streaming. However, the traditional
ABR schemes usually adopt users’ predicted throughput as the principle of
bitrate selection, which may not improve users’ personalized watching
experience that depends on multiple factors [3]. To handle this issue, the
personalized quality-of-experience (PQoE) model is proposed, which can
characterize the user-specific perception of different QoE factors, including
rebuffer time, video quality, and quality variation [4]. Based on the PQoE
model, appropriate segments and network resources are allocated to the user to
enhance its personalized watching experience.

In the literature, significant research efforts have been put to construct the
PQoE model. Wang et al. proposed a PQoE construction method that combined
user’s service preferences and network layer configurations, which can
effectively characterize user’s resource demands [4]. Gao et al. enhanced the
PQoE model accuracy by leveraging the sensing information from the network
[5]. To perform resource management based on the constructed PQoE model, a
deep reinforcement learning (DRL)-based scheme was proposed to solve the PQoE-
oriented resource allocation problem [6]. The above works focus on time-
invariant PQoE models. However, PQoE models may vary across different video
contents and playback statuses, such as buffer occupancy, video quality, and
quality variation. Such dynamics make the previous constructed accurate PQoE
models outdated over time, thereby rendering sub-optimal resource management
decisions and degrading PQoE performance. Hence, constructing a real-time and
accurate PQoE model is paramount.

To tackle this challenge, digital twin (DT) technology is a potential
solution. DT is a digital representation of a physical entity (PE) that can
accurately reflect its status and feature via real-time synchronization
between the DT and PE [7]. DT technology has been widely applied to fault
diagnosis and predictive maintenance [8]. In adaptive video streaming, DT
technology can be leveraged to store and analyze users’ data, such that the
accurate user-specific PQoE model can be established in real time and then
utilized to make resource management decisions for enhancing PQoE.

In this paper, we present a DT-assisted adaptive video streaming scheme to
enhance PQoE. Specifically, we propose a DT-assisted PQoE model construction
method to tune PQoE model parameters in real time, and design a tailored DRL-
based algorithm for resource management. Firstly, user DTs (UDTs) are
established to construct real-time PQoE models by storing and analyzing users’
data related to adaptive video streaming. The constructed PQoE model not only
adopts the linear weighting combination of QoE factors, but also incorporates
the Ebbinghaus memory effect to reflect the fading effect of user’s watching
experience. The personalization is reflected in PQoE model parameters, i.e.,
relative memory length and sensitivity degree of QoE factors. The PQoE model
parameters are tuned in real time by adopting a data fitting method. Secondly,
we develop a DRL-based resource management algorithm to efficiently facilitate
the PQoE-oriented adaptive video streaming. The objective is to maximize the
overall long-term PQoE considering the dynamics of users’ locations, video
content requests, and buffer statuses, by jointly optimizing the segment
version selection, and communication and computing resource allocation. To
reduce the algorithm training complexity, we narrow down the action dimension
by splitting the range of transmission and transcoding variables into multiple
parts to represent the decisions of different segment versions. Extensive
simulation results on the real-world dataset demonstrate that the proposed DT-
assisted scheme, including the PQoE model construction and the corresponding
resource management algorithm, can effectively enhance PQoE compared with
benchmarks. The main contributions of this paper are summarized as follows:

  * $\bullet$

We propose a DT-assisted PQoE model construction method, which can obtain
user’s real-time and accurate PQoE model parameters.

  * $\bullet$

We develop a DRL-based resource management algorithm to jointly determine the
segment version selection, and communication and computing resource
allocation.

The remainder of this paper is organized as follows. The DT-assisted adaptive
video streaming scheme is presented in Section II. The DRL-based resource
management algorithm is proposed in Section III. Simulation results are
provided in Section IV, followed by the conclusion in Section V.

##  II DT-Assisted Adaptive Video Streaming Scheme

Figure 1: DT-assisted adaptive video streaming.

###  II-A System Model

As shown in Fig. 1, we consider the DT-assisted adaptive video streaming
framework consisting of three parts: PEs, DTs, and interaction links.

  * $\bullet$

PEs: PEs consist of a base station (BS), an edge server, a cloud server, and
multiple users. The set of users and the set of video sequences are denoted by
$\mathcal{U}$ and $\mathcal{F}$, respectively, where $u$ and $f$ represent the
corresponding indexes. Video sequences are encoded into multiple segments with
different versions. The set of all segment versions is denoted by
$\mathcal{L}=\\{1,...,l,...,L\\}$. The edge server caches popular videos for
low-latency content delivery due to limited storage capacity, while the cloud
server caches all videos.

  * $\bullet$

DTs: There are multiple UDTs deployed at the edge server. Each UDT corresponds
to a user, which stores the corresponding user’s data and extracted PQoE
model. The set of UDTs is denoted by $\hat{\mathcal{U}}$.

  * $\bullet$

Interaction links: Interaction between PEs and DTs is bidirectional
coordinated by the network controller, including data synchronization and PQoE
models transmission.

The proposed framework operates as follows. When users request adaptive video
streaming services, the network controller deployed at the BS will coordinate
the BS, the edge server, the cloud server, and UDTs to transmit and transcode
segments. Firstly, the network controller is used to collect users’ data,
including requested videos, rebuffer time, video quality, quality variation,
and engagement time. Specifically, the rebuffer time and engagement time data
are collected from users, while the rest data are collected from the BS.
Secondly, user’s data are uploaded to respective UDTs through the network
controller to update the stored data. Based on which, the real-time PQoE model
parameters is obtained by data analysis method, which are used to update
respective PQoE models. Thirdly, the real-time and accurate PQoE models in
UDTs are provided to the network controller to make resource management
decisions, including segment version selection, communication and computing
resource allocation. Specifically, the cloud server transmits the required
video segments to the BS based on the segment version selection decision. The
edge server transcodes cached segments of high versions to those of targeted
versions based on the computing resource allocation decision. The BS transmits
the processed segments to users based on the communication resource allocation
decision.

###  II-B Service Delay Model

The adaptive video streaming system operates in each scheduling slot, indexed
by $t$. The segment can be delivered to the user in the following three cases,
and the corresponding service delay is analyzed.

Case 1: If the segment is downloaded from the edge server without transcoding,
the service delay, $D_{t,u}^{(1)}$, refers to the transmission delay from the
BS to user $u$ at scheduling slot $t$, given by

$D_{t,u}^{(1)}=\sum\limits_{l=1}^{L}\frac{{g_{k_{t,u}}^{l}}\chi_{k_{t,u}}^{l}\varsigma_{k_{t,u}}^{l}}{r_{t,u}^{BS}}.$
(1)

Here, $k_{t,u}$ represents the next segment index of the maximum segment index
at user $u$’s buffer at scheduling slot $t$. Here, decision variable
$g_{k_{t,u}}^{l}\in\\{0,1\\}$ indicates whether to transmit segment $k_{t,u}$
of version $l$ to user $u$ at scheduling slot $t$. If the segment of version
$l$ is transmitted, $g_{k_{t,u}}^{l}=1$; Otherwise, $g_{k_{t,u}}^{l}=0$. The
parameter $\chi_{k_{t,u}}^{l}\in\\{0,1\\}$ indicates whether segment $k_{t,u}$
of version $l$ for user $u$ has been stored in the edge server. If segment
$k_{t,u}$ of version $l$ has been stored, $\chi_{k_{t,u}}^{l}=1$; Otherwise,
$\chi_{k_{t,u}}^{l}=0$. Here, $\varsigma_{k_{t,u}}^{l}$ denotes the file size
of segment $k_{t,u}$ of version $l$, and ${{r}_{t,u}^{BS}}$ denotes the
average transmission capacity of user $u$ at scheduling slot $t$. Here,
${{r}_{t,u}^{BS}}={{\xi}_{t,u}}W\log(1+{{P}_{t,u}}/{{N}_{o}})$, where
${{P}_{t,u}}$ is user $u$’s received signal power, ${{N}_{o}}$ is the noise
power, and decision variable ${{\xi}_{t,u}\in\left[0,1\right]}$ represents the
ratio of total bandwidth $W$ allocated to user $u$ at scheduling slot $t$.

Case 2: If the segment is downloaded from the edge server with transcoding,
the service delay, $D_{t,u}^{(2)}$, includes transcoding delay in the edge
server and the transmission delay from the BS to user $u$ at scheduling slot
$t$, given by

$\displaystyle D_{t,u}^{(2)}$
$\displaystyle=\sum\limits_{l=1}^{L}{g_{k_{t,u}}^{l}(1\\!-\\!\chi_{k_{t,u}}^{l})o_{k_{t,u}}^{l}}\left(\frac{\mu\varsigma_{k_{t,u}}^{l}}{{{\omega}_{t,u}}{{c}_{t}}}\\!+\\!\frac{\varsigma_{k_{t,u}}^{l}}{r_{t,u}^{BS}}\right).$
(2)

Here, decision variable $o_{k_{t,u}}^{l}\in\\{0,1\\}$ indicates whether
segment $k_{t,u}$ of version $l$ can be obtained by transcoding. If segment
$k_{t,u}$ of version $l$ can be obtained by transcoding, $o_{k_{t,u}}^{l}=1$;
Otherwise, $o_{k_{t,u}}^{l}=0$. The computing intensity for transcoding per
unit file size is denoted by $\mu$, and $c_{t}$ is the computing capacity of
the edge server at scheduling slot $t$. Here, decision variable
$\omega_{t,u}\in\left[0,1\right]$ represents the ratio of computing resources
allocated to user $u$ at scheduling slot $t$.

Case 3: If the segment is downloaded from the cloud server, the service delay,
$D_{t,u}^{(3)}$, refers to the transmission delay from the cloud server to the
BS, and the BS to user $u$ at scheduling slot $t$, given by

$D_{t,u}^{(3)}=\sum\limits_{l=1}^{L}{g_{k_{t,u}}^{l}(1-\chi_{k_{t,u}}^{l})}(1-o_{k_{t,u}}^{l})\left(\frac{\varsigma_{k_{t,u}}^{l}}{r^{C}}+\frac{\varsigma_{k_{t,u}}^{l}}{r_{t,u}^{BS}}\right),$
(3)

where ${{r}^{C}}$ is the transmission capacity between the cloud server and
the BS. Due to the wired link connection between the cloud server and the BS,
${{r}^{C}}$ is assumed to be a constant.

Based on these, service delay $D_{t,u}$ can be represented by

$D_{t,u}=D_{t,u}^{(1)}+D_{t,u}^{(2)}+D_{t,u}^{(3)},$ (4)

which impacts the rebuffer time in the PQoE model.

###  II-C PQoE Model

The PQoE model consists of the following three factors.

Rebuffer time: The rebuffer time is related to service delay (as analyzed in
Eq. (4)) and playback buffer occupancy. The playback buffer occupancy111Note
that if the user stops watching the current video and switches to another
video, the video packets in the current playback buffer will not be used and
the playback buffer occupancy will be empty. at the current scheduling slot
depends on the playback buffer occupancy at the previous scheduling slot,
current received segment time length, and scheduling slot length. Let
$B_{t,u}$ denote user $u$’s playback buffer occupancy at scheduling slot $t$,
which is updated via

${{B}_{t+1,u}}={{\left({{B}_{t,u}}+\sum\nolimits_{l=1}^{L}{g_{k_{t,u}}^{l}}e-d\right)}^{+}}.$
(5)

Here, $e$ is the time length of segment222The video sequence is encoded into
multiple equal-length segments., and $d$ is the scheduling slot length. The
function ${{(x)}^{+}}=\max\\{x,0\\}$.

When the user’s service delay exceeds the current buffer occupancy, the
rebuffering event occurs. Therefore, the rebuffer time of user $u$ at
scheduling slot $t$ is defined by

$R_{t,u}=\left(D_{t,u}-B_{t,u}\right)^{+}.$ (6)

Video quality: Since one video sequence consists of multiple video segments
with different video qualities, the higher video quality can usually bring the
higher PQoE to user. At scheduling slot $t$, the video quality of the new
segment added to user $u$’s playback buffer is defined by

$\displaystyle V{{}_{t,u}}$
$\displaystyle=\sum\nolimits_{l=1}^{L}{g_{k_{t,u}}^{l}{\wp}_{k_{t,u}}^{l}}.$
(7)

Here, ${\wp}_{k_{t,u}}^{l}$ is the peak signal-to-noise ratio (PSNR) of
segment $k_{t,u}$.

Quality variation: Since the content-coding complexity of adjacent segments is
different, segments with the same version also exhibits different PSNRs [9].
However, in this case, the user cannot feel the quality variation. Therefore,
we use switching magnitude of segment versions to represent quality variation,
which is defined by

${H_{t,u}}=\sum\nolimits_{l=1}^{L}{g_{k_{t,u}}^{l}}|{{l}_{k_{t,u}}}-{{l}_{k_{t,u}-1}}|.$
(8)

Based on these factors, we can formulate the PQoE model. Specifically, since
video playback is usually a long-term process for users, the impact of
previous bad or good watching experiences, such as rebuffer time, video
quality, and quality variation, on users’ current watching experience
decreases gradually. Therefore, we incorporate the Ebbinghaus memory effect
[10] into the linear weighting combination of PQoE factors [3] to obtain an
accurate PQoE model, i.e.,

$\displaystyle{Z_{t,u}}({\mathbf{g}_{t,u}},{\mathbf{o}_{t,u}},{{\omega}_{t,u}},{{\xi}_{t,u}})$
$\displaystyle=\exp\left(-\frac{({p}_{u}-t)}{{{\lambda}_{t,u}}}\right)[{{\alpha}_{t,u}}V{{}_{t,u}}$
(9) $\displaystyle-{{\beta}_{t,u}}{{H}_{t,u}}-{{\gamma}_{t,u}}{R_{t,u}}],$

where ${\mathbf{g}_{t,u}}=\left\\{g_{k_{t,u}}^{l}\right\\}_{l\in\mathcal{L}}$
and ${\mathbf{o}_{t,u}}=\left\\{o_{k_{t,u}}^{l}\right\\}_{l\in\mathcal{L}}$.
Here, ${{\lambda}_{t,u}}$ is the relative memory length of user $u$ at
scheduling slot $t$, and ${p}_{u}$ is the requested video sequence length.
Here, $\alpha_{t,u}$, $\beta_{t,u}$, and $\gamma_{t,u}$ are user $u$’s
sensitivity degrees of QoE factors at scheduling slot $t$. A relatively small
$\alpha_{t,u}$ indicates that user $u$ is not particularly concerned about
video quality, while a large $\alpha_{t,u}$ means that more communication
resources need to be allocated to transmit the segment of high video quality.
A large $\beta_{t,u}$, relatively to the other parameters, indicates that user
$u$ is deeply concerned about quality variation. The segment with suitable
version needs to be transmitted to achieve smoother changes of video quality.
In cases where user $u$ prefers low rebuffer time, a large $\gamma_{t,u}$
should be used.

###  II-D DT-Assisted PQoE Model

To obtain user $u$’s PQoE model parameters $\lambda_{t,u}$, $\alpha_{t,u}$,
$\beta_{t,u}$, $\gamma_{t,u}$ in real time, we utilize the UDT to analyze the
user’s data related to the adaptive video streaming. The user’s engagement
time is employed by UDT to calculate the objective PQoE reference value [11].
Specifically, assume that a user only stops watching the video due to the long
rebuffer time, low video quality, and frequent quality variation. If the
proportion of the user’s engagement time to the total video playback time is
high, the user is insensitive to PQoE factors. Correspondingly, the PQoE is
high; Otherwise, the PQoE is low. Therefore, the objective PQoE reference
value is defined by

$Z_{\hat{u},f}^{ref}=\frac{5\cdot q_{\hat{u},f}}{{{K}_{f}}\cdot
e+{R_{\hat{u},f}}},$ (10)

where ${q_{\hat{u},f}}$ is the engagement time of user $\hat{u}$ watching
video $f$. Here, $K_{f}$ is the total number of segments of video $f$, and
$R_{\hat{u},f}$ is the total rebuffer time for user $\hat{u}$ watching video
$f$, which can be calculated based on the historical statistics on
$R_{t,\hat{u}}$. The PQoE range is from 0 to 5, aligning with the common
principle [12]. To build the mapping relationship between ${Z}_{t,\hat{u}}$
and $Z_{\hat{u},f}^{ref}$, we need to accumulate the ${{Z}_{t,\hat{u}}}$ of
video $f$ in the time domain. Since the established PQoE model is a non-linear
function, we employ the nonlinear regression method to obtain the PQoE model
parameters. The input is the user’s historical rebuffer time, video quality,
quality variation, and objective PQoE reference value, and the output is PQoE
model parameters in UDT, i.e., $\widetilde{\lambda}_{t,\hat{u}}$,
$\widetilde{\alpha}_{t,\hat{u}}$, $\widetilde{\beta}_{t,\hat{u}}$,
$\widetilde{\gamma}_{t,\hat{u}}$. Here, $\hat{u}$ corresponds to $u$. By
substituting these model parameters into Eq. (9), we can obtain the DT-
assisted PQoE model,
$\widetilde{Z}_{t,{u}}({\mathbf{g}_{t,{u}}},{\mathbf{o}_{t,{u}}},{{\omega}_{t,{u}}},{{\xi}_{t,{u}}})$.

##  III DRL-Based Resource Management Algorithm

###  III-A Problem Formulation

Our objective is to maximize the overall long-term PQoE over $T$ scheduling
slots. Correspondingly, the optimization problem is formulated as

$\displaystyle{\text{P}_{0}}:$
$\displaystyle\underset{\begin{smallmatrix}\\{{\mathbf{g}_{t,{u}}},{\mathbf{o}_{t,{u}}},\\\
{\mathbf{\omega}_{t,{u}}},{\mathbf{\xi}_{t,{u}}}\\}\end{smallmatrix}}{\mathop{\max}}\,\underset{T\to\infty}{\mathop{\lim}}\,\frac{1}{T}\sum\limits_{t=1}^{T}{\sum\nolimits_{{u}\in{{\mathcal{U}}}}{{\widetilde{Z}}_{t,{u}}}}({\mathbf{g}_{t,{u}}},{\mathbf{o}_{t,{u}}},{{\omega}_{t,{u}}},{{\xi}_{t,{u}}})$
(11) s.t. $\displaystyle\sum\nolimits_{l=1}^{L}{g_{k_{t,{u}}}^{l}}\leq
1,g_{k_{t,{u}}}^{l}\in\\{0,1\\},$ (11a)
$\displaystyle\sum\nolimits_{l=1}^{L}{o_{k_{t,{u}}}^{l}}\leq
1,o_{k_{t,{u}}}^{l}\in\\{0,1\\},$ (11b)
$\displaystyle\sum_{l=1}^{L}{o_{k_{t,{u}}}^{l}\varsigma_{k_{t,{u}}}^{l}}\leq\underset{l}{\mathop{\max}}\,\\{\chi_{k_{t,{u}}}^{l}\varsigma_{k_{t,{u}}}^{l},\forall
l\in\mathcal{L}\\},$ (11c)
$\displaystyle\sum\nolimits_{{u}\in{\mathcal{U}}}{{\omega_{t,{u}}}\leq
1},\omega_{t,{u}}\in[0,1],$ (11d)
$\displaystyle\sum\nolimits_{{u}\in{\mathcal{U}}}{{\xi_{t,{u}}}\leq
1},\xi_{t,{u}}\in[0,1].$ (11e)

Constraint (11a) indicates that a user can only receive the segment with one
version at each scheduling slot. Constraint (11b) guarantees that a segment
can only be transcoded for one target version for a user at each scheduling
slot. Constraint (11c) guarantees that a segment can only be transcoded from
the high version to the low version in the edge server at each scheduling
slot. Constraint (11d) guarantees that the total allocated computing resources
cannot exceed the computing capacity of the edge server. Constraint (11e)
guarantees that total allocated bandwidth cannot exceed overall bandwidth.

###  III-B Proposed Algorithm

The formulated problem is mixed-integer nonlinear programming, which is hard
to be directly solved. Considering that the user’s playback status satisfies
Markov chain and the optimization objective is to maximize the long-term
performance, the optimization problem can be modeled as a Markov decision
process (MDP). To solve this MDP, we adopt the deep deterministic policy
gradient (DDPG) algorithm.

####  III-B1 MDP

MDP is a discrete-time stochastic control process, which consists of four
elements, i.e., state, action, state transition probability, and reward. At
each step (scheduling slot) $t$, state $s_{t}$ can be transformed to $s_{t+1}$
by taking action $a_{t}$. Correspondingly, the reward is $r_{t}$.

State: The state includes users’ rebuffer time, buffer occupancy, segment
version, video quality, and segment bitrate:

$\small{{s}_{t}}=\left\\{{\left\\{R_{t,{u}}\right\\}_{{u}\in{\mathcal{U}}}},{\left\\{B_{t,{u}}\right\\}_{{u}\in{\mathcal{U}}}},{\left\\{l_{t,{u}}\right\\}_{{u}\in{\mathcal{U}}}},{\left\\{V_{t,{u}}\right\\}_{{u}\in{\mathcal{U}}}},\left\\{\varsigma_{t,{u}}\right\\}_{{u}\in{\mathcal{U}}}\right\\}.$
(12)

Action: The action includes all optimization variables in Eq. (11), which is
defined by

$\small{{a}_{t}}=\left\\{\left\\{\mathbf{g}_{t,{u}}\right\\}_{{u}\in{\mathcal{U}}},\left\\{\mathbf{o}_{t,{u}}\right\\}_{{u}\in{\mathcal{U}}},\left\\{\omega_{t,{u}}\right\\}_{{u}\in{\mathcal{U}}},\left\\{\xi_{t,{u}}\right\\}_{{u}\in{\mathcal{U}}}\right\\}.$
(13)

State Transition Probability: At each step, the current segment version
selection, and communication and computing resource allocation affect user’s
next state. Therefore, the state transition probability between $s_{t}$ and
$s_{t+1}$ is given by

$\small\begin{split}&\Pr({{s}_{t+1}}|{{s}_{t}},{{a}_{t}})\\!=\\!\\!\prod\limits_{u\in\mathcal{U}}{\Pr(R_{t+1,{u}}|R_{t,{u}},{\mathbf{g}_{t,{u}}},{\mathbf{o}_{t,{u}}},{{\omega}_{t,{u}}},{{\xi}_{t,{u}}})}\cdot\\\
&\prod\limits_{{u}\in{\mathcal{U}}}{\Pr(V_{t+1,{u}}|V_{t,{u}},{\mathbf{g}_{t,{u}}})}\cdot\prod\limits_{{u}\in{\mathcal{U}}}{\Pr(\varsigma_{t+1,{u}}|\varsigma_{t,{u}},{\mathbf{g}_{t,{u}}},{\mathbf{o}_{t,{u}}})}\cdot\\\
&\prod\limits_{{u}\in{\mathcal{U}}}{\Pr(B_{t+1,{u}}|B_{t,{u}},{\mathbf{g}_{t,{u}}})}\cdot\prod\limits_{{u}\in{\mathcal{U}}}{\Pr(l_{t+1,{u}}|l_{t,{u}},{\mathbf{g}_{t,{u}}})}.\end{split}$
(14)

Reward: The reward at step $t$ is designed to maximize overall PQoE, which is
defined by

$r_{t}(s_{t},a_{t})=\sum\nolimits_{{u}\in{{\mathcal{U}}}}\widetilde{Z}_{t,{u}}.$
(15)

####  III-B2 DRL-Based Communication and Transcoding Resource Allocation
(DCTRA) Algorithm

Based on DT-assisted PQoE models and formulated MDP, we adopt the DDPG
algorithm to obtain segment version selection, and communication and computing
resource allocation decisions [13][14]. The training process is as shown in
Algorithm 1. The DDPG contains three modules, i.e., primary network, target
network, and replay memory. Primary network consists of one actor network,
i.e., $\pi(s_{t}|\theta_{\pi})$, and one critic network, i.e.,
$Q(s_{t},a_{t}|\theta_{Q})$, where $\theta_{\pi}$ and $\theta_{Q}$ are primary
actor and critic network parameters, respectively. Primary network is mainly
responsible for finding the best action $a_{t}$ for the current state $s_{t}$.
The target network is the same as the primary network in terms of network
structure but with different parameters $\theta_{\pi^{\prime}}$ and
$\theta_{Q^{\prime}}$, which is responsible for generating target values for
training the primary actor and critic networks. At each step $t$,
$(s_{t},a_{t},s_{t+1},r_{t})$ is stored at replay memory $M$ as an experience
tuple for random sampling. The size of random sampling is the mini-batch. In
each step, the ornstein-uhlenbeck noise added to action is denoted by $X_{t}$.
The network learning rate is denoted by $\tau$.

The primary actor network parameter, i.e., $\theta_{\pi}$, is updated based on
the sampled policy gradient, as follows

$\bigtriangledown_{\theta_{\pi}}J\approx\frac{1}{N}\sum_{t}\bigtriangledown_{a}Q(s,a|\theta_{Q})|_{s=s_{t},a=\pi(s_{t})}\bigtriangledown_{\theta_{\pi}\pi(s|\theta_{\pi})|_{s=s_{t}}},$
(16)

where $Q(s,a|\theta_{Q})$ is the action-value function. The parameter
$\theta_{Q}$ of primary critic network is updated by minimizing the loss
function ${G}(\theta_{Q})$ as follows

${G}(\theta_{Q})=\frac{1}{N}\sum_{t}(y_{t}-Q(s_{t},a_{t}|\theta_{Q}))^{2}.$
(17)

Here, $y_{t}$ is the target value that combines the current reward and the
estimated Q value, i.e., $Q^{\prime}$, as follows

$y_{t}=r_{t}+\epsilon
Q^{\prime}(s_{t+1},\pi^{\prime}(s_{t+1}|\theta_{\pi^{\prime}})|\theta_{Q^{\prime}}),$
(18)

where $\epsilon$ is the discounting factor.

Since the output values of the DDPG algorithm are continuous, we need to
discretize a part of DDPG output values to obtain $\mathbf{g}_{t,{u}}$ and
$\mathbf{o}_{t,{u}}$. To reduce the dimension of $\mathbf{g}_{t,{u}}$ and
$\mathbf{o}_{t,{u}}$ caused by multiple segment versions, we utilize $g_{t,u}$
and $o_{t,u}$ to replace $\mathbf{g}_{t,{u}}$ and $\mathbf{o}_{t,{u}}$.
Specifically, we first map the output value of the Tanh function from $[-1,1]$
to $[0,1]$. Then, we split the range of ${g}_{t,{u}}$ into five parts, i.e.,
$[0,0.2),[0.2,0.4),[0.4,0.6),[0.6,0.8),[0.8,1]$, which indicate no segment
transmission, and segment transmission for version 1, 2, 3, 4 for user ${u}$
at step $t$, respectively. Similarly, the range of ${o}_{t,{u}}$ is split into
two parts, i.e., $[0,0.5),[0.5,1]$, which indicate no segment transcoding, and
segment transcoding for user ${u}$ at step $t$, respectively.

1 Initialize: primary and target network with parameters
$\theta_{\pi},\theta_{Q},\theta_{\pi^{\prime}},\theta_{Q^{\prime}}$, and
replay memory $M$.

2for  _each episode_ do

3 Initialize video list in edge server and cloud server, users’ request
videos, buffer queues and locations.

4 for  _each step $t\in\\{1,...,t_{max}\\}$_ do

5 Update each PQoE model based on parameters $\widetilde{\lambda}_{t,{u}}$,
$\widetilde{\alpha}_{t,{u}}$, $\widetilde{\beta}_{t,{u}}$,
$\widetilde{\gamma}_{t,{u}}$ provided by UDTs;

6 Discretize actions $g_{t,u}$ and $o_{t,u}$ based on the designed principle
in Section III-B2;

7 BS and edge server execute action $a_{t}$ based on
$a_{t}=\pi(s_{t}|\theta_{\pi})+{X}_{t}$;

8 Obtain $r_{t}$ based on Eq. (15), update $s_{t+1}$;

9 Store $(s_{t},a_{t},r_{t},s_{t+1})$ into $M$, and sampling;

10 Compute tagret value $y_{t}$ based on Eq. (18);

11 Update primary critic network parameter $\theta_{Q}$ by minimizing Eq.
(17);

12 Update primary actor network parameter $\theta_{\pi}$ by minimizing Eq.
(16);

13 Update target network parameters via
$\theta_{\pi^{\prime}}\leftarrow\tau\theta_{\pi}+(1-\tau)\theta_{\pi^{\prime}},\theta_{Q^{\prime}}\leftarrow\tau\theta_{Q}+(1-\tau)\theta_{Q^{\prime}}$;

14 end for

15

16 end for

Algorithm 1 DCTRA

##  IV Performance Evaluation

In this section, simulations are conducted to evaluate the performance of the
proposed DCTRA algorithm. We adopt the real-world dataset “Waterloo Streaming
Quality-of-Experience Database-III”333https://ieee-dataport.org/open-
access/waterloo-streaming-quality-experience-database-iii., which includes
users’ rebuffer time, video quality and quality variation. The dataset
contains 450 videos, and the length of each video is 10 seconds. The user’s
leave probability in [15] is used to obtain the engagement time for
calculating the PQoE reference value. To obtain PQoE model parameters, we
employ the lsqcurvefit
function444https://www.mathworks.com/help/optim/ug/lsqcurvefit.html. in Matlab
to fit the function curve. The main simulation parameters are presented in
Table I.

TABLE I: Simulation Parameters Parameter | Value | Parameter | Value
---|---|---|---
BS radius | 600 m | $U$ | 12
Hidden layer shape | (512, 256, 128) | $c$ |  $10^{9}$ cycles/s
Critic learning rate | $10^{-4}$ | $W$ | 200 MHz
Actor learning rate | $10^{-6}$ | $r^{C}$ | 200 Mbps
Mini-batch size | 128 | $P$ | 25 dBm
Replay memory | $6\times 10^{3}$ | $\mu$ | 10 cycles/bit
Scheduling slot length | 100 ms | $L$ | 4
Number of episodes | 1500 | $t_{\text{max}}$ | 100

We compare the proposed DTCRA algorithm with the following benchmark schemes.

  * $\bullet$

Round-Robin (RR): In each scheduling slot, communication and computing
resources are equally allocated to 3 users randomly selected from 12 users.
The segment version is randomly selected from all versions.

  * $\bullet$

Proportional Fair (PF): Resources are allocated based on the scheduling
priority considering user’s current channel state and buffer occupancy. The
segment version is selected based on the previous average version.

  * $\bullet$

Joint Resource AllocaTion (JRAT)[9]: The communication and transcoding
resources are sequentially allocated to the user who can achieve the highest
PQoE enhancement. The segment version selection is determined by the branch
and bound method.

  * $\bullet$

Communication and Transcoding Resource Allocation (CTRA): The proposed DCTRA
algorithm does not use DT-assisted PQoE models, but uses static PQoE models.

Figure 2: PQoE model parameters extracted by UDT.

Figure 2 shows the dynamic PQoE model parameters of Users 1, 4, 7, and 11,
whose average relative memory lengths are $12.80$ seconds, $11.93$ seconds,
$2.02$ seconds, $11.68$ seconds, respectively. It can be observed that User
$1$ has the highest sensitivity degree of all QoE factors, while User $11$
owns the lowest one. In addition, the user’s sensitivity degree of QoE factors
changes with varying video contents and playback statuses.

Figure 3: Performance comparison among different algorithms: (a) convergence
performance of the DCTRA algorithm, (b) PQoE (normalized) of different users,
and (c) average PQoE (normalized) among different numbers of users.

Figure 3 shows the convergence property of the proposed DCTRA algorithm, which
conducts three trials of training to draw the corresponding envelope curve and
the mean curve. It can be observed that our algorithm can converge after near
1,100 episodes. In Fig. 3, we select 6 users from 12 users to show the
individual user’s normalized PQoE among different algorithms. It can be
observed that the proposed DCTRA algorithm can always achieve the highest PQoE
for each user compared with other algorithms. This is because accurate user-
specific PQoE models can be provided to the network controller in real time to
perform appropriate resource management. Fig. 3 shows the proposed algorithm
can well adapt to different user scales and always achieve the highest PQoE.
The gap between the JRAT algorithm and the CTRA algorithm decreases gradually,
because these two algorithms cannot find better resource management decisions
based on static PQoE models.

##  V Conclusion

In this paper, we have investigated a PQoE enhancement problem for adaptive
video streaming. We have proposed a DT-assisted PQoE model construction method
to obtain accurate user-specific PQoE models, and then developed a DRL-based
resource management algorithm to enhance the overall long-term PQoE. The
proposed scheme can extract users’ features through DT, and leverage them to
enhance users’ video watching experience. In the future, we will investigate
an efficient data collection and model update scheme for UDT to reduce
communication and computing overhead.

## References

  * [1] A. A. Barakabitze, N. Barman, A. Ahmad, S. Zadtootaghaj, L. Sun, M. G. Martini, and L. Atzori, “QoE management of multimedia streaming services in future networks: A tutorial and survey,” _IEEE Commun. Surveys Tuts._ , vol. 22, no. 1, pp. 526–565, 2019.
  * [2] C. Liu, I. Bouazizi, and M. Gabbouj, “Rate adaptation for adaptive HTTP streaming,” in _Proc. ACM MMSys_ , 2011, San Jose CA, USA, pp. 169–174.
  * [3] X. Yin, A. Jindal, V. Sekar, and B. Sinopoli, “A control-theoretic approach for dynamic adaptive video streaming over HTTP,” in _Proc. ACM SIGCOMM_ , 2015, New York, NY, USA, pp. 325–338.
  * [4] Y. Wang, P. Li, L. Jiao, Z. Su, N. Cheng, X. Shen, and P. Zhang, “A data-driven architecture for personalized QoE management in 5G wireless networks,” _IEEE Wireless Commun._ , vol. 24, no. 1, pp. 102–110, 2017\.
  * [5] Y. Gao, X. Wei, and L. Zhou, “Personalized QoE improvement for networking video service,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 10, pp. 2311–2323, 2020.
  * [6] F. Wang, C. Zhang, F. Wang, J. Liu, Y. Zhu, H. Pang, and L. Sun, “DeepCast: Towards personalized QoE for edge-assisted crowdcast with deep reinforcement learning,” _IEEE Trans. Netw._ , vol. 28, no. 3, pp. 1255–1268, 2020.
  * [7] X. Shen, J. Gao, W. Wu, M. Li, C. Zhou, and W. Zhang, “Holistic network virtualization and pervasive network intelligence for 6G,” _IEEE Commun. Surveys Tuts._ , vol. 24, no. 1, pp. 1–30, 2022.
  * [8] T. H. Luan, R. Liu, L. Gao, R. Li, and H. Zhou, “The paradigm of digital twin communications,” _arxiv preprint arXiv: 2105.07182_ , 2021.
  * [9] X. Huang, L. He, L. Wang, and F. Li, “Towards 5G: Joint optimization of video segment caching, transcoding and resource allocation for adaptive video streaming in a multi-access edge computing network,” _IEEE Trans. Veh. Technol._ , vol. 70, no. 10, pp. 10 909–10 924, 2021.
  * [10] Z. Duanmu, K. Zheng, K. Ma, A. Rehman, and Z. Wang, “A quality-of-experience index for streaming video,” _IEEE J. Sel. Topics Signal Process._ , vol. 11, no. 1, pp. 154–166, 2017.
  * [11] F. Dobrian, V. Sekar, A. K. Awan, I. Stoica, D. Joseph, A. Ganjam, J. Zhan, and H. Zhang, “Understanding the impact of video quality on user engagement,” in _Proc. ACM SIGCOMM_ , 2011, Toronto, ON, CA, pp. 362–373.
  * [12] Y. Chen, K. Wu, and Q. Zhang, “From QoS to QoE: A tutorial on video quality assessment,” _IEEE Commun. Surveys Tuts._ , vol. 17, no. 2, pp. 1126–1165, 2015.
  * [13] T. P. Lillicrap, J. J. Hunt, A. Pritzel, N. Heess, T. Erez, Y. Tassa, D. Silver, and D. Wierstra, “Continuous control with deep reinforcement learning,” in _Proc. ICLR_ , 2016, San Juan, Puerto Rico.
  * [14] C. Zhou, W. Wu, H. He, P. Yang, F. Lyu, N. Cheng, and X. Shen, “Deep reinforcement learning for delay-oriented IoT task scheduling in SAGIN,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 2, pp. 911–925, 2020.
  * [15] H. Nam, K. H. Kim, and H. Schulzrinne, “QoE matters more than QoS: Why people stop watching cat videos,” in _Proc. IEEE INFOCOM_ , 2016, San Francisco, CA, USA.


# Mirror symmetry on levels of non-abelian Landau–Ginzburg orbifolds

Wolfgang Ebeling and Sabir M. Gusein-Zade  Keywords: invertible polynomial,
group action, dual pairs, orbifold Euler characteristic, orbifold monodromy
zeta function, E-function. Math. Subject Classification: 14R20, 14J33, 57R18,
58K10.

###### Abstract

We consider the Berglund–Hübsch–Henningson–Takahashi duality of
Landau–Ginzburg orbifolds with a symmetry group generated by some diagonal
symmetries and some permutations of variables. We study the orbifold Euler
characteristics, the orbifold monodromy zeta functions and the orbifold
E-functions of such dual pairs. We conjecture that we get a mirror symmetry
between these invariants even on each level, where we call level the conjugacy
class of a permutation. We support this conjecture by giving partial results
for each of these invariants.

##  0 Introduction

Mirror symmetry is the observation originating from physics that there are
pairs of objects (originally Calabi–Yau manifolds) with symmetries of certain
invariants (originally Hodge numbers). A well-known instance of mirror
symmetry is the Berglund–Hübsch–Henningson duality of Landau–Ginzburg
orbifolds with abelian groups of diagonal symmetries. A Landau–Ginzburg
orbifold is a pair $(f,G)$ consisting of a quasihomogeneous polynomial $f$ of
a special type (a so-called invertible polynomial) and a finite subgroup $G$
of the full group of symmetries of $f$ [17]. In the case that $G$ consists
only of diagonal symmetries, P. Berglund, T. Hübsch, and M. Henningson
constructed a dual pair $(\widetilde{f},\widetilde{G})$ [3, 4]. In a series of
papers [5, 7, 12] the authors and A. Takahashi showed that there are
symmetries between certain invariants of such dual pairs such as the orbifold
Euler characteristic of the Milnor fibre, the orbifold zeta function of the
monodromy, and the orbifold E-function.

The object of this paper is to study these symmetries for non-abelian symmetry
groups $G$. As the group $G$, we consider the semi-direct product of a finite
group $G$ of diagonal symmetries and a subgroup $S$ of the group of
permutations of the variables. Takahashi suggested a method to associate a
dual pair in this situation. This is called the BHHT-dual pair. We started the
study in [9] where we found that we obtain symmetries only under a special
condition on the permutation group called parity condition (PC). One can
conjecture that there are natural generalizations of the results for the
abelian case to this non-abelian case. Here we give some evidence for the
conjecture that the above invariants are not only symmetric for BHHT-dual
pairs with PC, but also their distributions among the levels of $S$, namely
the conjugacy classes of the elements of $S$.

The plan of the paper is the following. In Section 1, we introduce the
orbifold invariants which we want to consider, namely, the orbifold Euler
characteristic, the orbifold zeta function and the orbifold E-function and we
discuss relations between these invariants. Section 2 is devoted to results
about these invariants for BHH-dual pairs and the conjectural generalizations
to BHHT-dual pairs. In Section 3, we formulate our main conjectures about the
distribution of the invariants among levels. In the remaining sections we
indicate some results which support these conjectures. In Section 4, we show
that the orbifold Euler characteristics of BHHT-dual pairs coincide up to sign
on any level for which the centralizer is cyclic and satisfies PC. In Section
5, we show that, depending on the dimension, the orbifold zeta functions of
BHHT-dual pairs with PC either coincide or are inverse to each other on the
level of an element which generates its centralizer. For the level 1, we show
that the mirror map yields a symmetry of the orbifold E-functions of BHHT-dual
pairs under certain conditions, see Section 6. Finally, in Section 7, we
compute the orbifold E-function for a rather simple example. Our conjecture is
true for this example, but it turns out to be difficult (if possible) to
define an analogue of the mirror map for other levels than 1.

##  1 Orbifold Euler characteristic, zeta function, and E-function

Let $X$ be a topological space (good enough, e.g., homeomorphic to a locally
closed union of cells in a finite CW-complex) and let $G$ be a finite group
acting on $X$ (also with some goodness properties, e.g., with the action of
elements of $G$ being cellular maps). Then the orbifold Euler characteristic
of the pair $(X,G)$ is defined by

$\chi^{{\,{\rm orb}}}(X,G)=\frac{1}{|G|}\sum_{{(g_{1},g_{2})\in
G^{2}:}\atop{g_{1}g_{2}=g_{2}g_{1}}}\chi(X^{\langle g_{1},g_{2}\rangle})\,,$

where, for a subgroup $H\subset G$, $X^{H}$ is the fixed point set $\\{x\in
X:hx=x\mbox{ for all }h\in H\\}$ of the subgroup $H$, $\langle
g_{1},g_{2},\dots\rangle$ is the subgroup of $G$ generated by the elements
$g_{1},g_{2},\ldots$ (see, e.g., [16], [1]). By $\chi(\cdot)$ we always mean
the additive Euler characteristic defined as the alternating sum of dimensions
of the cohomology groups $H^{q}_{c}(X;{\mathbb{R}})$ with compact support.
(For complex quasiprojective varieties it coincides with the “usual” Euler
characteristic: the alternating sum of dimensions of the cohomology groups
$H^{q}(X;{\mathbb{R}})$.) One has an equivalent equation for the orbifold
Euler characteristic:

$\chi^{{\,{\rm orb}}}(X,G)=\sum_{[g]\in{{\rm Conj\,}}G}\chi(X^{\langle
g\rangle}/C_{G}(g))\,,$

where ${{\rm Conj\,}}G$ is the set of conjugacy classes of the elements of
$G$, $g$ is a representative of the conjugacy class $[g]$, $C_{G}(g)$ is the
centralizer of $g$ (see, e.g., [16]). The reduced orbifold Euler
characteristic is defined by

$\overline{\chi}^{{\,{\rm orb}}}(X,G)=\chi^{{\,{\rm orb}}}(X,G)-\chi^{{\,{\rm
orb}}}({\rm pt},G)\,,$

where ${\rm pt}$ is the one-point set with the only (trivial) action of the
group $G$.

For a topological space $X$ and a proper self-map $h:X\to X$, the zeta
function of $h$ is the rational function in $t$ defined by

$\zeta_{h}(t)=\prod_{k\geq 0}\left(\det({\rm
id}-th^{*}_{|H^{k}_{c}(X;{\mathbb{R}})})\right)^{(-1)^{k}}\,,$

where $h^{*}$ is the action of $h$ on the cohomology groups
$H^{k}_{c}(X;{\mathbb{R}})$ with compact support.

Let $X$ be a complex $n$-dimensional $G$-manifold with a proper
$G$-equivariant (i.e., commuting with all the elements of $G$) self-map
$h:X\to X$. For an element $g\in G$ and a fixed point $x\in X^{\langle
g\rangle}$ the age (or the shift number) of the element $g$ at $x$ is ${{\rm
age}_{x}}(g)=\sum_{i=1}^{n}\theta_{i}$, where $\theta_{i}\in[0,1)$ and
${\mathbf{e}}[\theta_{i}]:=\exp{2\pi i\theta_{i}}$, $i=1,\ldots,n$, are the
eigenvalues of the action of $g$ on the tangent space $T_{x}X$. The age of an
element is constant on a connected component of the fixed point set
$X^{\langle g\rangle}$. Let $X^{\langle g\rangle}_{\alpha}$ be the set of
points of $X^{\langle g\rangle}$ with the age of $g$ equal to $\alpha$. Then
the orbifold zeta function of $(h,X,G)$ is defined by

$\zeta^{{\,{\rm orb}}}_{h,X,G}(t)=\prod_{[g]\in{{\rm
Conj\,}}G}\prod_{\alpha\in{\mathbb{Q}}}\zeta_{h_{|X^{\langle
g\rangle}_{\alpha}/C_{G}(g)}}({\mathbf{e}}[-\alpha]\cdot t)\,.$

Let $f(x_{1},\ldots,x_{n})$ be a quasihomogeneous polynomial:

$f(\lambda^{w_{1}}x_{1},\ldots,\lambda^{w_{n}}x_{n})=\lambda^{d}f(x_{1},\ldots,x_{n})$
(1)

with the weights $w_{1}$, …, $w_{n}$ and the quasidegree $d$ being natural
numbers. Let $f$ be endowed with a finite group $G\subset GL(n,{\mathbb{C}})$
of linear automorphisms preserving $f$, let $X$ be the Milnor fiber
$V_{f}=\\{f=1\\}$ of $f$ and let $h:V_{f}\to V_{f}$ be the monodromy
transformation

$h(x_{1},\ldots,x_{n})=\left({\mathbf{e}}\left[\frac{w_{1}}{d}\right]x_{1},\ldots,{\mathbf{e}}\left[\frac{w_{n}}{d}\right]x_{n}\right)\,.$

(In the context of Landau–Ginzburg models the monodromy transformation $h$ is
often called “the exponential grading operator”.) In this case the age of an
element $g\in G$ does not depend on the point $x\in X^{\langle g\rangle}$ and
one has the age function ${{\rm age}}:G\to{\mathbb{Q}}_{\geq 0}$, where ${{\rm
age}}(g)=\sum_{i=1}^{n}\theta_{i}$, $0\leq\theta_{i}<1$, with
${\mathbf{e}}[\theta_{i}]$ being the eigenvalues of $g$ on ${\mathbb{C}}^{n}$.
(This coincides with the age of the element $g$ considered as a map $V_{f}\to
V_{f}$.) In this case, the definition of the zeta function $\zeta^{{\,{\rm
orb}}}_{f,G}(t)=\zeta^{{\,{\rm orb}}}_{h,V_{f},G}(t)$ of $(f,G)$ was given in
[7]. The reduced orbifold zeta function of the monodromy is defined by

$\overline{\zeta}^{{\,{\rm orb}}}_{f,G}(t)=\zeta^{{\,{\rm
orb}}}_{f,G}(t)\left/\prod_{[g]\in{\rm Cong\,}G}(1-{\mathbf{e}}[-{{\rm
age}}(g)]t)\right.{.}$

One can see that the (reduced) orbifold Euler characteristic of $(V_{f},G)$ is
the degree of the (reduced) orbifold zeta function as a rational function in
$t$ (that is the degree of the numerator minus the degree of the denominator).

One can say that the orbifold zeta function of the monodromy is the zeta
function of the action of the monodromy transformation on the Fan–Jarvis–Ruan
quantum cohomology group associated to $(f,G)$ ([14]). A generating polynomial
of the bigrading on this space is the orbifold E-function defined for an
invertible polynomial with an abelian group of (diagonal) symmetries in [12].
Here we give a general definition for an arbitrary quasihomogeneous polynomial
$f$ and for an arbitrary $G$ as it was made in [13] for abelian subgroups
$G\subset{\rm SL}(n,{\mathbb{C}})$ acting diagonally on the coordinates.

Let $f(x_{1},\ldots,x_{n})$ be a quasihomogeneous polynomial with an isolated
critical point at the origin and let $G\subset GL(n,{\mathbb{C}})$ be a finite
group of transformations preserving $f$. The cohomology group
$H^{n-1}(V_{f};{\mathbb{C}})$ is endowed with the mixed Hodge structure
introduced by J. H. M. Steenbrink in [21]. Let us consider the
${\mathbb{Q}}\times{\mathbb{Q}}$-graded vector space
${\mathcal{H}}_{f}=\displaystyle\bigoplus_{p,q\in{\mathbb{Q}}}{\mathcal{H}}_{f}^{p,q}$
defined by

  1. 1)

if $p+q\neq n$, then ${\mathcal{H}}_{f}^{p,q}:=0$;

  2. 2)

if $p+q=n$, then ${\mathcal{H}}_{f}^{p,q}:={\rm
Gr}_{F^{\bullet}}^{[p]}H^{n-1}(V_{f};{\mathbb{C}})_{{\mathbf{e}}[p]}$, where
$H^{n-1}(V_{f};{\mathbb{C}})_{{\mathbf{e}}[p]}$ is the eigenspace of the
monodromy transformation $h$ corresponding to the eigenvalue
${\mathbf{e}}[p]$, $[p]$ is the integer part of $p$.

If $n=1$, one has to consider the reduced cohomology group
$\overline{H}^{n-1}(V_{f};{\mathbb{C}})$. If $n=0$ (this case is important for
the definition below), we assume $H^{-1}(V_{f};{\mathbb{C}})$ to be one
dimensional with the trivial action of $G$. This means that one considers the
“critical point” of the function of zero variables to be non-degenerate and
thus to have the dimension of the cohomology group in dimension $(-1)$ to be
equal to 1.

For any $g\in G$, the restriction $f^{g}$ of the polynomial $f$ to the fixed
point set $({\mathbb{C}}^{n})^{\langle g\rangle}$ has an isolated critical
point as well: see, e.g., [22]. Let $n_{g}$ be the dimension of the fixed
locus $({\mathbb{C}}^{n})^{\langle g\rangle}$. The centralizer $C_{G}(g)$ of
the element $g$ acts on $({\mathbb{C}}^{n})^{\langle g\rangle}$ and on the
Milnor fiber of $f^{g}$. For $i=0,1$, let

${\mathcal{H}}_{f,G,\overline{i}}:=\bigoplus_{{[g]\in{\rm
Conj\,}G}\atop{n_{g}\equiv i\bmod 2}}({\mathcal{H}}_{f^{g}})^{C_{G}(g)}(-{\rm
age}(g),-{\rm age}(g))\,,$

where $({\mathcal{H}}_{f^{g}})^{C_{G}(g)}$ is the $C_{G}(g)$-invariant part of
${\mathcal{H}}_{f^{g}}$, $(-{{\rm age}}(g),-{{\rm age}}(g))$ means the shift
of the bigrading ($\overline{0}$ means even, $\overline{1}$ means odd). Let

$E_{\overline{i}}(f,G)(t,\overline{t}):=\sum_{p,q\in{\mathbb{Q}}}\dim_{{\mathbb{C}}}({\mathcal{H}}_{f,G,\overline{i}})^{p,q}t^{p-\frac{n}{2}}\overline{t}^{q-\frac{n}{2}}\,,$

where $t$ and $\overline{t}$ are two (independent) variables.

###### Definition 1

The (orbifold) E-function of $(f,G)$ is

$E(f,G)(t,\overline{t})=E_{\overline{0}}(f,G)(t,\overline{t})-E_{\overline{1}}(f,G)(t,\overline{t}).$

For an exponent $s:=q-\frac{n}{2}$ of the variable $\overline{t}$ in
$E(f,G)(1,\overline{t})$ with non-zero coefficient,
${\mathbf{e}}[q]=(-1)^{n}{\mathbf{e}}[s]$ is an eigenvalue of the
transformation induced by the monodromy transformation $h$ on
$(H^{n_{g}-1}(V_{f^{g}};{\mathbb{C}}))^{C_{G}(g)}$ multiplied by
${\mathbf{e}}[-{\rm age}(g)]$. It gives the summand
$(-1)^{n_{g}}\overline{t}^{s}$ in $E(f,G)(1,\overline{t})$ and the factor
$(1-(-1)^{n}{\mathbf{e}}[s]t)^{(-1)^{n_{g}-1}}$ in
$\overline{\zeta}_{f,G}^{{\,{\rm orb}}}(t)$. This implies that, if

$E(f,G)(1,\overline{t})=\sum_{s\in{\mathbb{Q}}}a_{s}\overline{t}^{s}\quad(a_{s}\in{\mathbb{Z}}),$

then

$\overline{\zeta}_{f,G}^{{\,{\rm
orb}}}(t)=\prod_{s\in{\mathbb{Q}}}(1-(-1)^{n}{\mathbf{e}}[s]t)^{-a_{s}}.$

In particular, the orbifold $E$-function determines the orbifold monodromy
zeta-function.

###### Remark 1

As it was mentioned above, the orbifold $E$-function is a generating
polynomial of the bigrading on the quantum cohomology group of
Fan–Jarvis–Ruan. One can consider other polynomials, e.g.,

$E^{\prime}(f,G)(t,\overline{t})=E_{\overline{0}}(f,G)(t,\overline{t})+E_{\overline{1}}(f,G)(t,\overline{t})$

defined in [8] or the Poincaré polynomial defined in [19].

##  2 Duality

An attempt to construct mirror symmetric pairs led Berglund, Hübsch, and
Henningson to the following concept. A quasihomogeneous polynomial
$f(x_{1},\ldots,x_{n})$ is called invertible if the number of monomials in it
(with non-zero coefficients) is equal to the number $n$ of variables, i.e.,

$f(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}c_{i}\prod_{j=1}^{n}x_{j}^{E_{ij}}$ (2)

with $c_{i}\neq 0$, $E_{ij}\in{\mathbb{Z}}_{\geq 0}$, and the matrix
$E=(E_{ij})$ is invertible (over ${\mathbb{Q}}$, i.e., $\det E\neq 0$).
Without loss of generality one may assume that $c_{i}=1$ for all $i$ (this can
be achieved by rescaling the variables).

###### Remark 2

Usually, in the definition of an invertible polynomial, it is assumed in
advance that an invertible polynomial is non-degenerate, i.e., has an isolated
critical point at the origin. We prefer not to include this assumption into
the initial definition since some statements about invertible polynomials and
symmetries of their invariants hold without it (e.g., Equations (3) and (4)
below). However, in this paper we shall only consider non-degenerate
invertible polynomials dropping the adjective non-degenerate.

One can show that a polynomial is invertible if and only if it is the
Sebastiani–Thom sum (i.e., sum of polynomials in non-intersecting sets of
variables) of polynomials of the form:

  1. 1)

Fermat type: $x_{1}^{a_{1}}$, $a_{1}\geq 2$;

  2. 2)

chain type: $x_{1}^{a_{1}}x_{2}+x_{2}^{a_{2}}x_{3}+\cdots+x_{m}^{a_{m}}$,
$a_{m}\geq 2$;

  3. 3)

loop type: $x_{1}^{a_{1}}x_{2}+x_{2}^{a_{2}}x_{3}+\cdots+x_{m}^{a_{m}}x_{1}$
(if $m$ is even, then neither $a_{1}=a_{3}=\ldots=a_{m-1}=1$ nor
$a_{2}=a_{4}=\ldots=a_{m}=1$).

###### Remark 3

a) Depending on the desire, the Fermat type polynomial can be considered as a
particular case either of chain type or of loop type.
b) If $a_{1}=1$ in 1) or $a_{m}=1$ in 2), the corresponding polynomial is
regular (does not have a critical point) at the origin.
c) If, in 3), the restriction in parentheses does not hold, the polynomial
either is not quasihomogeneous (if there exists $a_{i}$ different from $1$, in
(1) some weights $w_{i}$ are forced to be equal to zero) or is not invertible
(if all $a_{i}$ are equal to $1$, $\det E=0$).

The group of diagonal symmetries of an invertible polynomial
$f(x_{1},\ldots,x_{n})$ is

$G_{f}=\\{(\lambda_{1},\ldots,\lambda_{n})\in({\mathbb{C}}^{*})^{n}:f(\lambda_{1}x_{1},\ldots,\lambda_{n}x_{n})=f(x_{1},\ldots,x_{n})\\}\,.$

One can show that $|G_{f}|=|\det E|$.

The dual (or the transpose) of the invertible polynomial (2) is

$\widetilde{f}(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}\prod_{j=1}^{n}x_{j}^{E_{ji}}.$

One can show that there is a canonical isomorphism between the group
$G_{\widetilde{f}}$ of diagonal symmetries of $\widetilde{f}$ and the group
${\rm Hom}(G_{f},{\mathbb{C}}^{*})$ of characters of $G_{f}$ (see details,
e.g., in [6]). For a subgroup $G\subset G_{f}$, its dual $\widetilde{G}$ is
the subgroup of $G_{\widetilde{f}}$ consisting of the characters of $G_{f}$
vanishing (i.e., having the value $1$) on $G$. The pair
$(\widetilde{f},\widetilde{G})$ is called Berglund–Hübsch–Henningson\- (BHH\-
for short) dual to the pair $(f,G)$.

In [5, 7, 12] it was shown that, for BHH-dual pairs $(f,G)$ and
$(\widetilde{f},\widetilde{G})$, one has

$\displaystyle\overline{\chi}^{{\,{\rm orb}}}(V_{f},G)$ $\displaystyle=$
$\displaystyle(-1)^{n}\,\overline{\chi}^{{\,{\rm
orb}}}(V_{\widetilde{f}},\widetilde{G})\,;$ (3)
$\displaystyle\overline{\zeta}^{{\,{\rm orb}}}_{f,G}(t)$ $\displaystyle=$
$\displaystyle\left(\overline{\zeta}^{{\,{\rm
orb}}}_{\widetilde{f},\widetilde{G}}(t)\right)^{(-1)^{n}}\,;$ (4)
$\displaystyle E(f,G)(t,\overline{t})$ $\displaystyle=$
$\displaystyle(-1)^{n}E(\widetilde{f},\widetilde{G})(t^{-1},\overline{t})\,.$
(5)

Usually similar invariants of pairs are considered under a (physically-
motivated) assumption on the group $G$ to be “admissible”. Note that the
relations (3) –(5) hold for an arbitrary subgroup $G\subset G_{f}$ (without
the assumption to be admissible).

The group $S_{n}$ of permutations on $n$ elements acts on the space
${\mathbb{C}}^{n}$ by permuting the variables. Let $f$ be an invertible
polynomial in $n$ variables and $G$ a group of diagonal symmetries of $f$
(i.e., $G\subset G_{f}$) and let $S$ be a subgroup of $S_{n}$ preserving $f$
and $G$ ($\sigma^{-1}G\sigma=G$ for $\sigma\in S$). The action of $S$ on $G$
(by conjugation) defines the semidirect product $G\rtimes S$ with its action
on ${\mathbb{C}}^{n}$. (The precise equations for the group structure on
$G\rtimes S$ and for its action on ${\mathbb{C}}^{n}$ can be found, e.g., in
[11, Remark 2.1].) In this case the dual polynomial $\widetilde{f}$ and the
dual subgroup $\widetilde{G}\subset G_{\widetilde{f}}$ are also preserved by
$S$ and, for the pair $(f,G\rtimes S)$, one gets the so-called
Berglund–Hübsch–Henningson–Takahashi- (BHHT\- for short) dual pair
$(\widetilde{f},\widetilde{G}\rtimes S)$: see [9]. In [9], it was shown that,
for BHHT-dual pairs, the obvious analogues of the symmetries (3) – (5) do not
hold in general, but may hold only under a special condition on the subgroup
$S\subset S_{n}$ (called “PC”: parity condition), namely, for any subgroup
$T\subset S$, $\dim({\mathbb{C}}^{n})^{T}\equiv n\bmod 2$.

One can conjecture that, under PC, one has symmetries like (3), (4), and (5),
i.e.,

$\displaystyle\overline{\chi}^{{\,{\rm orb}}}(V_{f},G\rtimes S)$
$\displaystyle=$ $\displaystyle(-1)^{n}\,\overline{\chi}^{{\,{\rm
orb}}}(V_{\widetilde{f}},\widetilde{G}\rtimes S)\,;$ (6)
$\displaystyle\overline{\zeta}^{{\,{\rm orb}}}_{f,G\rtimes S}(t)$
$\displaystyle=$ $\displaystyle\left(\overline{\zeta}^{{\,{\rm
orb}}}_{\widetilde{f},\widetilde{G}\rtimes S}(t)\right)^{(-1)^{n}}\,;$ (7)
$\displaystyle E(f,G\rtimes S)(t,\overline{t})$ $\displaystyle=$
$\displaystyle(-1)^{n}E(\widetilde{f},\widetilde{G}\rtimes
S)(t^{-1},\overline{t})\,.$ (8)

Equation (6) was proved in [10] for polynomials of loop type with $S$
satisfying PC and in [11] for a cyclic subgroup $S$ (also satisfying PC).

From the computations in [20] one can derive the symmetries (6) and (7) for
the following pairs $(f,G\rtimes S)$ (with Fermat type polynomials $f$):

  1. 1)

$f=x_{1}^{4}+\cdots+x_{4}^{4}$, $G=\langle J\rangle$, $S=\langle(123)\rangle$,
where $J=({\mathbf{e}}[1/4],\ldots,{\mathbf{e}}[1/4])$ is the exponential
grading operator (the monodromy transformation);

  2. 2)

$f=x_{1}^{5}+\cdots+x_{5}^{5}$, $G=\langle J\rangle$,
$S=\langle(12)(34)\rangle$, where
$J=({\mathbf{e}}[1/5],\ldots,{\mathbf{e}}[1/5])$.

A symmetry similar to (8) was proved in [19] for the pairs $(f,G\rtimes S)$
with $f=x_{1}^{p-1}x_{2}+\cdots+x_{p}^{p-1}x_{1}$ (a loop), an arbitrary
subgroup $G$ of $G_{f}\cong{\mathbb{Z}}_{(p-1)^{p}+1}$ and $S=\langle(12\ldots
p)\rangle$, where $p$ is an odd prime number. Moreover, one can derive from
[2] a symmetry similar to the symmetry of [19] for pairs $(f,G\rtimes S)$ with
$f=x_{1}^{p}+\cdots+x_{p}^{p}$ (a Fermat polynomial), $G$ is either $G_{f}$ or
$G_{f}\cap{\rm SL}(p,{\mathbb{C}})$, $S$ satisfies PC, and $p$ is a prime
number.

##  3 Distribution of invariants among levels

Let $f$ be an invertible polynomial in $n$ variables and let $G\rtimes S$ be a
group of its symmetries ($G\subset G_{f}$, $S\subset S_{n}$). The conjugacy
class of an element $(\underline{\lambda},\sigma)\in G\rtimes S$ is contained
in $G\times[\sigma]$, where $[\sigma]\subset S$ is the conjugacy class of
$\sigma$. Therefore each orbifold invariant considered in Section 1 is the sum
or the product of the parts corresponding to different conjugacy classes of
the elements of $S$. Namely,

$\chi^{{\,{\rm orb}}}(V_{f},G\rtimes S)=\sum_{[\sigma]\in{{\rm
Conj\,}S}}\chi^{{\,{\rm orb}},[\sigma]}(V_{f},G\rtimes S)\,.$

where

$\chi^{{\,{\rm orb}},[\sigma]}(V_{f},G\rtimes S)=\sum_{{[g]\in{{\rm
Conj\,}}G\rtimes S:}\atop{[g]\subset G\times[\sigma]}}\chi(V_{f}^{\langle
g\rangle}/C_{G}(g))\,;$

and

$\overline{\zeta}^{{\,{\rm orb}}}_{f,G\rtimes S}(t)=\prod_{[\sigma]\in{{\rm
Conj\,}S}}\overline{\zeta}^{{\,{\rm orb}},[\sigma]}_{f,G\rtimes S}(t)\,.$

where

$\overline{\zeta}^{{\,{\rm orb}},[\sigma]}_{f,G\rtimes
S}(t)=\prod_{[g]\in{{\rm Conj\,}}G\rtimes S:\atop[g]\subset
G\times[\sigma]}\overline{\zeta}_{h_{|V_{f^{g}}/C_{G}(g)}}({\mathbf{e}}[-{\rm
age}(g)]t)\,.$

Finally

$E(f,G\rtimes S)(t,\overline{t})=\sum_{[\sigma]\in{\rm
Conj\,}S}E^{[\sigma]}(f,G\rtimes S)(t,\overline{t})\,,$

where

$E^{[\sigma]}(f,G\rtimes
S)(t,\overline{t})=\sum_{p,q\in{\mathbb{Q}}}\left(\dim_{{\mathbb{C}}}({\mathcal{H}}^{[\sigma]}_{f,G\rtimes
S,\overline{0}})^{p,q}-\dim_{{\mathbb{C}}}({\mathcal{H}}^{[\sigma]}_{f,G\rtimes
S,\overline{1}})^{p,q}\right)t^{p-\frac{n}{2}}\overline{t}^{q-\frac{n}{2}}\,,$
${\mathcal{H}}^{[\sigma]}_{f,G\rtimes S,\overline{i}}:=\bigoplus_{{[g]\in{\rm
Conj\,}G\rtimes S,[g]\subset G\times[\sigma]}\atop{n_{g}\equiv i\bmod
2}}({\mathcal{H}}_{f^{g}})^{C_{G\rtimes S}(g)}(-{\rm age}(g),-{\rm
age}(g))\,.$

There are some indications for the following conjectures ($S$ satisfying PC).

###### Conjecture 1

$\chi^{{\,{\rm orb}},[\sigma]}(V_{f},G\rtimes S)=(-1)^{n}\chi^{{\,{\rm
orb}},[\sigma]}(V_{\widetilde{f}},\widetilde{G}\rtimes S)\,.$ (9)

###### Conjecture 2

$\overline{\zeta}^{{\,{\rm orb}},[\sigma]}_{f,G\rtimes
S}(t)=\left(\overline{\zeta}^{{\,{\rm
orb}},[\sigma]}_{\widetilde{f},\widetilde{G}\rtimes S}(t)\right)^{(-1)^{n}}.$
(10)

###### Conjecture 3

$E^{[\sigma]}(f,G\rtimes
S)(t,\overline{t})=(-1)^{n}E^{[\sigma]}(\widetilde{f},\widetilde{G}\rtimes
S)(t^{-1},\overline{t})\,.$ (11)

We shall call levels the conjugacy classes of the elements of $S$. The
conjectures state that not only the discussed invariants are symmetric for
BHHT-dual pairs, but also their distributions among the levels. In what
follows, we shall show some examples supporting Conjectures 1–3.

##  4 Orbifold Euler characteristic

The proof of Theorem 4.1 in [11] gives equation (9) for a cyclic group $S$
with PC. Note that in this case the conjugacy class $[\sigma]$ in $S$ consists
only of $\sigma$ itself. Here we prove a somewhat stronger statement. Let $f$,
$G$, and $S$ be as above.

Suppose that $K\subset H$ are finite groups. One has an operation ${\rm
Ind}_{K}^{H}$ which converts $K$-spaces to $H$-spaces. Namely, for a $K$-space
$X$, the $H$-space ${\rm Ind}_{K}^{H}X$ is $(H\times X)/\sim$, where the
equivalence relation $\sim$ is defined by $(h_{1},x_{1})\sim(h_{2},x_{2})$ if
there exists $g\in K$ such that $h_{1}=h_{2}g^{-1}$, $x_{1}=gx_{2}$; the
action of $H$ on ${\rm Ind}_{K}^{H}X$ is defined in a natural way:
$h^{\prime}(h,x)=(h^{\prime}h,x)$. (A useful example: for $G\subset K\subset
H$, one has ${\rm Ind}_{K}^{H}K/G=H/G$. In particular, ${\rm
Ind}_{K}^{H}K/K=H/K$.)

###### Theorem 1

Assume that $C_{S}(\sigma)$ is cyclic and satisfies PC (that is, it is
contained in $A_{n}$). Then Equation (9) holds.

###### Proof.

One has the decompositions

${\mathbb{C}}^{n}\setminus\\{0\\}=\coprod_{I\subset
I_{0},\,I\neq\emptyset}({\mathbb{C}}^{*})^{I},\quad V_{f}=\coprod_{I\subset
I_{0},\,I\neq\emptyset}V_{f}^{I}\,,$

where $I_{0}=\\{1,\ldots,n\\}$, $V_{f}^{I}=V_{f}\cap({\mathbb{C}}^{*})^{I}$.
Using the action of $S$ on $2^{I_{0}}$, one can write

$V_{f}=\coprod_{{\cal J}\in(2^{I_{0}}\setminus{\\{\emptyset\\}})/S}{\
}\coprod_{J\in{\cal J}}V_{f}^{J}\,.$

Let $S^{I}\subset S$ be the isotropy subgroup of $I$ for the $S$-action on
$2^{I_{0}}$. For any $I$ from ${\cal
J}\in(2^{I_{0}}\setminus{\\{\emptyset\\}})/S$, one has

$\left(\coprod_{J\in{\cal J}}V_{f}^{J},G\rtimes S\right)={\rm Ind}_{G\rtimes
S^{I}}^{G\rtimes S}(V_{f}^{I},G\rtimes S^{I})\,.$

Elements of the form $(\underline{\lambda},\sigma)$ may have fixed points in
$\coprod\limits_{J\in{\cal J}}({\mathbb{C}}^{*})^{J}$ only if $\sigma\in
S^{I}$ for some $I\in{\cal J}$ and, in this case,

$\chi^{{\,{\rm orb}},[\sigma]}({\rm Ind}_{G\rtimes S^{I}}^{G\rtimes
S}(V_{f}^{I},G\rtimes S^{I})=\chi^{{\,{\rm orb}},[\sigma]}(V_{f}^{I},G\rtimes
S^{I})\,.$

(This is a simple version of Theorem 1 from [15]; cf. Theorem 3 below.)

For $I\subset I_{0}$, let $f^{I}$ be the restriction of $f$ to
${\mathbb{C}}^{I}$. If $f^{I}$ has less than $|I|$ monomials, then
$\chi^{{\,{\rm orb}},[\sigma]}(V_{f}^{I},G\rtimes S^{I})=0$. This follows from
the fact that, in this case, there exists a free ${\mathbb{C}}^{*}$-action on
$({\mathbb{C}}^{*})^{I}$ preserving $f^{I}$ and commuting with the action of
$G_{f^{I}}\rtimes S^{I}$ (cf. [9, Lemma 1]). If $f^{I}$ has less than $|I|$
monomials, then $\widetilde{f}^{\overline{I}}$ has less than $|\overline{I}|$
monomials ($\overline{I}:=I_{0}\setminus I$). Therefore in this case

$\chi^{{\,{\rm orb}},[\sigma]}(V_{f}^{I},G\rtimes S^{I})=\chi^{{\,{\rm
orb}},[\sigma]}(V_{\widetilde{f}}^{\overline{I}},\widetilde{G}\rtimes
S^{\overline{I}})=0\,.$

(Pay attention that $S^{\overline{I}}=S^{I}$.)

Now let $f^{I}$ have $|I|$ monomials (and thus be invertible). The arguments
in the proof of [11, Theorem 4.1] give

$\chi^{{\,{\rm orb}},[\sigma]}(V_{f}^{I},G\rtimes
S^{I})=\frac{1}{|S^{I}|}\sum_{{\sigma^{\prime}\in
S^{I}}\atop{\sigma\sigma^{\prime}=\sigma^{\prime}\sigma}}\chi^{I}_{f,G}(\sigma,\sigma^{\prime})\,,$

where

$\chi_{f,G}^{I}(\sigma,\sigma^{\prime})=\frac{1}{|G|}\sum_{{(\underline{\lambda},\underline{\lambda}^{\prime})\in
G^{2}:}\atop{(\underline{\lambda},\sigma)(\underline{\lambda}^{\prime},\sigma^{\prime})=(\underline{\lambda}^{\prime},\sigma^{\prime})(\underline{\lambda},\sigma)}}\chi\left((V_{f}^{I})^{\langle(\underline{\lambda},\sigma),(\underline{\lambda}^{\prime},\sigma^{\prime})\rangle}\right).$

If one defines

$\chi_{f,G}^{\emptyset}(\sigma,\sigma^{\prime}):=-\frac{1}{|G|}|\\{(\underline{\lambda},\underline{\lambda}^{\prime})\in
G^{2}:(\underline{\lambda},\sigma)(\underline{\lambda}^{\prime},\sigma^{\prime})=(\underline{\lambda}^{\prime},\sigma^{\prime})(\underline{\lambda},\sigma)\\}|,$

one has

$\overline{\chi}^{{\,{\rm orb}},[\sigma]}(V_{f},G\rtimes S)=\sum_{{\cal
J}=[I]\in 2^{I_{0}}/S:\atop\sigma\in
S^{I}}\frac{1}{|S^{I}|}\sum_{\sigma^{\prime}\in
S^{I}:\atop\sigma\sigma^{\prime}=\sigma^{\prime}\sigma}\chi_{f,G}^{I}(\sigma,\sigma^{\prime})\,.$

Let $s$ be the generator of the centralizer $C_{S^{I}}(\sigma)\subset
C_{S}(\sigma)$ and let $\sigma=s^{m}$, $\sigma^{\prime}=s^{m^{\prime}}$. Then
[11, Proposition 4.2] gives

$\chi_{f,G}^{I}(\sigma,\sigma^{\prime})=\chi_{f,G}^{I}(s^{m^{*}},1)$

with $m^{*}=\gcd(m,m^{\prime},|C_{S^{I}}(\sigma)|)$.

The fact that, for all $I$ (including $I=\emptyset$),

$\chi_{f,G}^{I}(\sigma,1)=(-1)^{n}\chi_{\widetilde{f},\widetilde{G}}^{\overline{I}}(\sigma,1)$

is shown in the proof of [11, Theorem 4.1] since [11, Proposition 4.3] and
Equation [11, (4.5)] do not demand $S$ to be cyclic. The fact that $S$ is
cyclic was used in [11] only for the reduction of
$\chi^{I}_{f,G}(\sigma,\sigma^{\prime})$ to $\chi_{f,G}^{I}(s^{m^{*}},1)$
([11, Proposition 4.2]). ∎

###### Corollary 1

If $S\subset S_{n}$ (satisfying PC) is such that the centralizer of each non-
unit element is cyclic, then

$\overline{\chi}^{{\,{\rm orb}}}(V_{f},G\rtimes
S)=(-1)^{n}\cdot\overline{\chi}^{{\,{\rm
orb}}}(V_{\widetilde{f}},\widetilde{G}\rtimes S)\,.$ (12)

###### Proof.

Equation (12) follows from its analogue for all the levels $[\sigma]\in{\rm
Conj}\,S$. For $\sigma\neq 1$, this is just Theorem 1. For $\sigma=1$ this is
a direct consequence of Equation [11, (4.5)] (since
$\chi^{I}_{f,G}(1,\sigma^{\prime})=\chi^{I}_{f,G}(\sigma^{\prime},1)$). ∎

###### Remark 4

One can indicate the following examples of groups $S$ possessing the mentioned
property, i.e., the centralizer of each non-unit element is cyclic:

  1. 1)

$S$ is cyclic.

  2. 2)

$S=D_{k}$ (the dihedral group) with $k$ odd. (Pay attention that
$D_{3}=S_{3}$, but in order to satisfy PC, the number of variables $n$ should
not be 3.)

  3. 3)

Semidirect products ${\mathbb{Z}}_{p}\rtimes{\mathbb{Z}}_{q}$ with prime $p$
and $q$ such that $p\equiv 1\,{\rm mod}\,q$. (There is only one non-abelian
group of order $pq$ in this case.)

##  5 Orbifold zeta function

Let $f$, $G$, and $S$ be as above, $S$ satisfies PC.

###### Theorem 2

If $\sigma\in S$ is such that the centralizer $C_{S}(\sigma)$ is generated by
$\sigma$, then

$\overline{\zeta}^{{\,{\rm orb}},[\sigma]}_{f,G\rtimes
S}(t)=\left(\overline{\zeta}^{{\,{\rm
orb}},[\sigma]}_{\widetilde{f},\widetilde{G}\rtimes
S}(t)\right)^{(-1)^{n}}\,.$ (13)

###### Remark 5

Among the groups listed in Remark 4, those satisfying the condition of Theorem
2 are

  1. 1)

cyclic groups of prime order;

  2. 2)

the dihedral group $D_{k}$ with $k$ prime;

  3. 3)

the non-abelian semidirect products ${\mathbb{Z}}_{p}\rtimes{\mathbb{Z}}_{q}$.

We shall use the following generalization of [15, Theorem 1]. Let $G$ be a
subgroup of a finite group $K$, let $X$ be a complex $G$-manifold with a
proper self-homeomorphism $h:X\to X$ of finite order commuting with actions of
the elements of $G$. The action of $h$ on $X$ can in a natural way be extended
to $\widehat{h}:{\rm Ind}_{G}^{K}X\to{\rm Ind}_{G}^{K}X$
($\widehat{h}(k,x)=(k,h(x))$ for $k\in K$, $x\in X$).

###### Theorem 3

One has

$\overline{\zeta}^{{\,{\rm orb}}}_{h,X,G}(t)=\overline{\zeta}^{{\,{\rm
orb}}}_{\widehat{h},{\rm Ind}_{G}^{K}X,K}(t)\,.$ (14)

###### Proof.

To use the arguments similar to those in [15, Lemma 1 & Theorem 1], let us
consider triples $(X,\\{G_{i}\\},h)$ consisting of a (topological) space $X$
with actions of different finite groups $G_{i}$ on different parts and with a
self-homeomorphism $h$ of finite order commuting with all the actions. This
means that $X$ is a finite disjoint union $X=\coprod_{i=1}^{s}X_{i}$ with the
finite group $G_{i}$ acting on $X_{i}$ and $h:X_{i}\to X_{i}$ is such that
$g\circ h=h\circ g$ for any $g\in G$ (cf. [15, Definition 2]). We shall write
$(X,\\{G_{i}\\},h)=\coprod_{i=1}^{s}(X_{i},G_{i},h)$. Two triples
$(X,\\{G_{i}\\},h)$ and $(Y,\\{G_{i}^{{}^{\prime}}\\},h^{\prime})$ are called
equivalent if there exist partitions
$(X,\\{G_{i}\\},h)=\coprod_{j=1}^{N}(X_{(j)},G_{(j)},h)$ and
$(Y,\\{G_{i}^{{}^{\prime}}\\},h^{\prime})=\coprod_{j=1}^{N}(Y_{(j)},G_{(j)}^{{}^{\prime}},h^{\prime})$
such that there exist homeomorphisms $\psi_{j}:X_{(j)}\to Y_{(j)}$ and
isomorphisms $\phi_{j}:G_{(j)}\to G^{\prime}_{(j)}$ such that
$\psi_{j}(gx)=\phi_{j}(g)\psi_{j}(x)$ and
$(h^{\prime}\circ\psi_{j})(x)=(\psi_{j}\circ h)(x)$ for $x\in X_{(j)}$ and
$g\in G_{(j)}$.

Let $X,G,K,h$ be as above. An obvious version of [15, Lemma 1] says that, for
$g\in G$,

$(({\rm Ind}_{G}^{K}X)^{\langle g\rangle},C_{K}(g),h)\mbox{ and
}\coprod_{[g^{\prime}]\in{\rm Conj}\,G:\atop[g^{\prime}]_{K}=[g]_{K}}({\rm
Ind}_{C_{G}(g^{\prime})}^{C_{K}(g^{\prime})}X^{\langle
g^{\prime}\rangle},C_{K}(g^{\prime}),\widehat{h})$ (15)

are equivalent ($[g]_{K}$ is the conjugacy class of $g$ in $K$).

It is obvious that

$\zeta_{\widehat{h}_{|({\rm Ind}_{G}^{K}X/K)}}(t)=\zeta_{h_{|X/G}}(t)\,.$

One has

$\zeta^{{\,{\rm orb}}}_{h,{\rm
Ind}_{G}^{K}X,K}(t)=\prod_{\alpha\in{\mathbb{Q}}}\prod_{[k]\in{\rm
Conj}\,K}\zeta_{h_{|({\rm Ind}_{G}^{K}X)^{\langle
k\rangle}_{\alpha}/C_{K}(k)}}({\mathbf{e}}[-\alpha]t)\,.$

It is easy to see that, for $k\in K$, $({\rm Ind}_{G}^{K}X)^{\langle
k\rangle}$ is not empty if (and only if) there exists $g\in G$ such that
$[g]_{K}=[k]_{K}$ (and $X^{\langle g\rangle}$ is not empty). Equation (15)
implies that

$\displaystyle\zeta^{{\,{\rm orb}}}_{h,{\rm Ind}_{G}^{K}X,K}(t)$
$\displaystyle=$
$\displaystyle\prod_{\alpha\in{\mathbb{Q}}}\prod_{[g^{\prime}]\in{\rm
Conj}\,G}\zeta_{h_{|({\rm
Ind}^{C_{K}(g^{\prime})}_{C_{G}(g^{\prime})}X)^{\langle
g^{\prime}\rangle}_{\alpha}/C_{K}(g^{\prime})}}({\mathbf{e}}[-\alpha]t)$
$\displaystyle=$
$\displaystyle\prod_{\alpha\in{\mathbb{Q}}}\prod_{[g^{\prime}]\in{\rm
Conj}\,G}\zeta_{h_{|X^{\langle
g^{\prime}\rangle}_{\alpha}/C_{K}(g^{\prime})}}({\mathbf{e}}[-\alpha]t)$
$\displaystyle=$ $\displaystyle\zeta^{{\,{\rm orb}}}_{h,X,G}(t)\,.$

∎

Proof of Theorem 2. We shall use the decomposition

$V_{f}=\coprod_{I\subset I_{0},\,I\neq\emptyset}V_{f}^{I}$

described in the proof of Theorem 1. (The monodromy operator $h$ respects this
decomposition.) Therefore

$\zeta^{{\,{\rm orb}},[\sigma]}_{h,V_{f},G\rtimes S}(t)=\prod_{{\cal
J}\in(2^{I_{0}}\setminus{\\{\emptyset\\}})/S}\zeta^{{\,{\rm
orb}},[\sigma]}_{h,\coprod_{J\in{\cal J}}V_{f}^{J},G\rtimes S}(t)\,.$

Theorem 3 together with the remark that the parts of the zeta functions in
Equation (14) corresponding to different levels coincide gives

$\zeta^{{\,{\rm orb}},[\sigma]}_{h,V_{f},G\rtimes S}(t)=\prod_{{\cal
J}\in(2^{I_{0}}\setminus{\\{\emptyset\\}})/S}\zeta^{{\,{\rm
orb}},[\sigma]}_{h,V_{f}^{I},G\rtimes S^{I}}(t)$

for a representative $I$ of ${\cal J}$.

To prove Equation (13), it is sufficient to show that

$\zeta^{{\,{\rm orb}},[\sigma]}_{h,V_{f}^{I},G\rtimes
S^{I}}(t)=\left(\zeta^{{\,{\rm
orb}},[\sigma]}_{\widetilde{h},V_{\widetilde{f}}^{\overline{I}},\widetilde{G}\rtimes
S^{\overline{I}}}(t)\right)^{(-1)^{n}}$

for $\sigma\in S^{I}=S^{\overline{I}}$, $\sigma\neq 1$. We shall show that

$\zeta^{{\,{\rm orb}},[\sigma]}_{h,V_{f}^{I},G\rtimes
S^{I}}(t)=(1-t^{\ell_{I}})^{R_{I}}$

for some exponent $R_{I}$, where the exponent $\ell_{I}$ is the one given by
Equation [7, (4.6)], i.e., the exponent in the equation of the orbifold zeta
function on $V_{f}^{I}$ for the group $G$ itself (i.e., for trivial $S^{I}$).

Let $(\underline{\lambda},\sigma)\in G\rtimes S^{I}$, $\sigma\neq 1$. First
let us show that

$\zeta_{h_{|(V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle}/C_{G\rtimes
S}(\underline{\lambda},\sigma)}}(t)=(1-t^{m_{I}})^{R^{\prime}_{I}}$

for some exponent $R^{\prime}_{I}$, where $m_{I}$ is the same exponent as in
Equation [7, (4.3)]. The centralizer of the element
$(\underline{\lambda},\sigma)$ is generated by the elements
$(\underline{\nu},1)$ commuting with $(\underline{\lambda},\sigma)$ and the
element $(\underline{\lambda},\sigma)$ itself. (This follows from the
condition that $C_{S}(\sigma)=\langle\sigma\rangle$.) The element
$(\underline{\nu},1)$ commutes with $(\underline{\lambda},\sigma)$ if and only
if $\underline{\nu}\in{\rm Ker}\,A_{\sigma}$, where
$A_{\sigma}(\underline{\mu}):=\underline{\mu}(\sigma(\underline{\mu}))^{-1}$
(see [11, Definition 3.2]). Therefore

$C_{G\rtimes S}(\underline{\lambda},\sigma)=\langle{\rm Ker}\,A_{\sigma}\cap
G,(\underline{\lambda},\sigma)\rangle\,.$

The monodromy operator $h$ acts on
$(V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle}$. The exponent in
the binomial $(1-t^{\bullet})$ in
$\zeta_{h_{|(V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle}/C_{G\rtimes
S}(\underline{\lambda},\sigma)}}$ corresponding to a point
$x\in(V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle}$ is the minimal
positive $d$ such that $h^{d}x\in C_{G\rtimes
S}(\underline{\lambda},\sigma)x$. Since $(\underline{\lambda},\sigma)$ acts
trivially on $(V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle}$, one
has $C_{G\rtimes S}(\underline{\lambda},\sigma)x=({\rm Ker}\,A_{\sigma}\cap
G)x$. For $\underline{\nu}\in G$, one has

$\displaystyle\underline{\nu}(V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle}$
$\displaystyle=$
$\displaystyle(V_{f}^{I})^{\langle(\underline{\nu},1)(\underline{\lambda},\sigma)(\underline{\nu}^{-1},1)\rangle}$
$\displaystyle=$
$\displaystyle(V_{f}^{I})^{\langle(A_{\sigma}(\underline{\nu})\underline{\lambda},\sigma)\rangle}\,.$

If $\underline{\nu}\not\in{\rm Ker}\,A_{\sigma}\cap G$, then
$A_{\sigma}(\underline{\nu})\underline{\lambda}\neq\underline{\lambda}$ and
therefore
$(V_{f}^{I})^{\langle(A_{\sigma}(\underline{\nu})\underline{\lambda},\sigma)\rangle}\cap(V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle}=\emptyset$.
Therefore, for $x\in(V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle}$,
$h^{d}x\in C_{G\rtimes S}(\underline{\lambda},\sigma)$ if and only if
$h^{d}x\in Gx$. This minimal $d$ is just $m_{I}$ computed in [7]. This implies
that

$\zeta_{h_{|(V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle}/C_{G\rtimes
S}(\underline{\lambda},\sigma)}}(t)=(1-t^{m_{I}})^{r_{I}(\underline{\lambda},\sigma)}\,,$

where

$r_{I}(\underline{\lambda},\sigma)=\frac{\chi((V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle})}{m_{I}|{\rm
Ker}\,A_{\sigma}\cap G|}\,.$

According to [11, Proposition 3.6],
$\chi((V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle})$ does not
depend on $\underline{\lambda}$. Therefore

$\zeta_{h_{|(V_{f}^{I})^{\langle(\underline{\lambda},\sigma)\rangle}/C_{G\rtimes
S}(\underline{\lambda},\sigma)}}(t)=\prod_{[\underline{\lambda}]\in G/({\rm
Ker}\,A_{\sigma}\cap G)}(1-{\mathbf{e}}[-{\rm
age}(\underline{\lambda})]t^{m_{I}})^{r_{I}}\,,$

where $r_{I}=r_{I}(\underline{\lambda},\sigma)$. The conjugacy classes of
$(\underline{\lambda},\sigma)$ are of the same size equal to
$|A_{\sigma}(G)|$. The exponents ${\mathbf{e}}[-{\rm
age}(\underline{\lambda})]$ are roots of degree $k_{I}$ with equal
multiplicities (see [7]). Therefore

$\zeta^{{\,{\rm orb}},[\sigma]}_{h,V_{f}^{I},G\rtimes
S^{I}}(t)=(1-t^{\ell_{I}})^{R_{I}}\,,$

where $\ell_{I}={\rm lcm}(m_{I},k_{I})$ (see [7, Lemma 3.5]).

In the same way,

$\zeta^{{\,{\rm
orb}},[\sigma]}_{\widetilde{h},V_{\widetilde{f}}^{\overline{I}},\widetilde{G}\rtimes
S^{\overline{I}}}(t)=(1-t^{\widetilde{\ell}_{\overline{I}}})^{\widetilde{R}_{\overline{I}}}\,.$

In [7], it was shown that $\widetilde{\ell}_{\overline{I}}=\ell_{I}$.
Moreover,

$\ell_{I}R_{I}=\chi^{{\,{\rm orb}},[\sigma]}(V_{f}^{I},G\rtimes
S^{I}),\quad\widetilde{\ell}_{\overline{I}}\widetilde{R}_{\overline{I}}=\chi^{{\,{\rm
orb}},[\sigma]}(V_{\widetilde{f}}^{\overline{I}},\widetilde{G}\rtimes
S^{\overline{I}})\,.$

By Theorem 1,
$\ell_{I}R_{I}=(-1)^{n}\widetilde{\ell}_{\overline{I}}\widetilde{R}_{\overline{I}}$.
This proves the statement for proper $I$.

The proof of

$\zeta^{{\,{\rm orb}},[\sigma]}_{h,V_{f}^{I_{0}},G\rtimes
S}(t)=\left(\zeta^{{\,{\rm orb}},[\sigma]}_{{\rm id},{\rm
pt},\widetilde{G}\rtimes S}(t)\right)^{(-1)^{n}}$ (16)

follows the same scheme. The corresponding zeta functions are equal to
$(1-t^{\ell_{I_{0}}})^{R_{I_{0}}}$ and
$(1-t^{\ell_{\emptyset}})^{R_{\emptyset}}$ respectively. The exponent
$\ell_{I_{0}}$ computed above (in this case $k_{I_{0}}=1$ and
$\ell_{I_{0}}=m_{I_{0}}$) is equal to the exponent $\ell_{\emptyset}$ computed
in [7] (denoted by $\widetilde{k}$ there; one can say that
$\widetilde{k}=k_{\emptyset}$). The fact that
$\ell_{I_{0}}R_{I_{0}}=(-1)^{n}\ell_{\emptyset}R_{\emptyset}$ is again a
consequence of Theorem 1. ∎

##  6 Orbifold E-function

We shall now show that, under certain conditions, the conjecture for the
orbifold E-function, namely Conjecture 3, holds for level 1. For this we need
the mirror map [12, 18]. The arguments in [12] are a little bit sketchy, here
we give the precise definition.

Let $f(x_{1},\ldots,x_{n})$ be an invertible polynomial and $G\subset G_{f}$.
For $I\subset I_{0}:=\\{1,\ldots,n\\}$, let
${\mathbb{C}}^{I}:=\\{(x_{1},\ldots,x_{n})\in{\mathbb{C}}^{n}:x_{i}=0\mbox{
for }i\not\in I\\}$ and let $\Omega^{p}({\mathbb{C}}^{I})$ be the
${\mathbb{C}}$-vector space of regular $p$-forms on ${\mathbb{C}}^{I}$. For
$\underline{\lambda}\in G$, the restriction $f^{\underline{\lambda}}$ of the
polynomial $f$ to the fixed point set
${\mathbb{C}}^{I_{\underline{\lambda}}}$, where $I_{\underline{\lambda}}$ is
the set of the indices of coordinates fixed by $\underline{\lambda}$, has an
isolated critical point at the origin as well. Therefore the Milnor algebra
$A_{f^{\underline{\lambda}}}:={\mathbb{C}}[x_{i}:i\in
I_{\underline{\lambda}}]/(\frac{\partial f}{\partial x_{i}}:i\in
I_{\underline{\lambda}})$ is finite dimensional. Recall that
$n_{\underline{\lambda}}=|I_{\underline{\lambda}}|$. and let
$\Omega_{f^{\underline{\lambda}}}:=\Omega^{n_{\underline{\lambda}}}({\mathbb{C}}^{I_{\underline{\lambda}}})/(df^{\underline{\lambda}}\wedge\Omega^{n_{\underline{\lambda}}-1}({\mathbb{C}}^{I_{\underline{\lambda}}}))$.
This is a free $A_{f^{\underline{\lambda}}}$-module of rank one. We define a
map $\psi:\bigoplus_{\underline{\lambda}\in
G}\Omega_{f^{\underline{\lambda}}}\to{\mathbb{C}}[G_{\widetilde{f}}]$ on the
natural basis of $\bigoplus_{\underline{\lambda}\in
G}\Omega_{f^{\underline{\lambda}}}$ as follows: Let $\wedge_{i\in
I_{\underline{\lambda}}}x_{i}^{k_{i}}dx_{i}\in\Omega_{f^{\underline{\lambda}}}$.
Define the element $(b_{1},\ldots,b_{n})\in{\mathbb{Z}}^{n}$ as follows. Let

$b_{i}:=\left\\{\begin{array}[]{cl}k_{i}+1&\mbox{for }i\in
I_{\underline{\lambda}},\\\ 0&\mbox{for }i\not\in
I_{\underline{\lambda}}.\end{array}\right.$

and let

$(\beta_{1},\ldots,\beta_{n}):=(b_{1},\ldots,b_{n})E^{-1},$

where $E$ is the matrix defining $f$. We set

$\psi(\wedge_{i\in
I_{\underline{\lambda}}}x_{i}^{k_{i}}dx_{i}):=({\mathbf{e}}[\beta_{1}],\ldots,{\mathbf{e}}[\beta_{n}]).$

From the definition of $G_{\widetilde{f}}$, it follows that
$({\mathbf{e}}[\beta_{1}],\ldots,{\mathbf{e}}[\beta_{n}])\in
G_{\widetilde{f}}$.

Let $\underline{\lambda}\in G$. The space
${\mathbb{C}}^{I_{\underline{\lambda}}}$ admits a natural $G$-action by
restricting the $G$-action on ${\mathbb{C}}^{n}$ to
${\mathbb{C}}^{I_{\underline{\lambda}}}$ (well-defined since $G$ acts
diagonally on ${\mathbb{C}}^{n}$). The form $\wedge_{i\in
I_{\underline{\lambda}}}x_{i}^{k_{i}}dx_{i}$ is $G$-invariant if and only if
$\psi(\wedge_{i\in
I_{\underline{\lambda}}}x_{i}^{k_{i}}dx_{i})\in\widetilde{G}$. Therefore the
map $\psi$ restricts to a map $\bigoplus_{\underline{\lambda}\in
G}\Omega^{G}_{f^{\underline{\lambda}}}\to{\mathbb{C}}[\widetilde{G}]$. This is
the mirror map.

Let $f=f_{1}\oplus\cdots\oplus f_{s}$ be the decomposition of $f$ into a
Sebastiani-Thom sum of atomic polynomials (blocks) such that each $f_{\nu}$,
$\nu=1,\ldots,s$ is either of Fermat, chain, or loop type. An element of the
symmetry group $S$ can either permute blocks, act as the identity on a block
or as a rotation on a block of loop type, see [9, p. 12312].

Let $f_{\nu}$ be a loop type polynomial. It can be characterized by its length
$L$ and the sequence $a_{1}$, …, $a_{L}$ of the exponents. That is

$f_{\nu}(x_{1},\ldots,x_{L})=x_{1}^{a_{1}}x_{2}+x_{2}^{a_{2}}x_{3}+\ldots,x_{L}^{a_{L}}x_{1}\,.$

The sequence $a_{1}$, …, $a_{L}$ of the exponents considered up to cyclic
order is called the type of the loop.

###### Theorem 4

Let $f(x_{1},\ldots,x_{n})$ be an invertible polynomial with a symmetry group
$G\rtimes S$ ($G\subset G_{f}$) such that

  1. 1)

in the Sebastiani-Thom decomposition $f=f_{1}\oplus\cdots\oplus f_{s}$ into
atomic polynomials, there is not more than one loop of any type with the
length $L$ even;

  2. 2)

For any subset $I\subset I_{0}:=\\{1,\ldots,n\\}$, the group $S^{I}$ consists
of permutations $\sigma$ with $\sigma|_{I}$ being an even permutation of $I$.

Then one has

$E^{[1]}(f,G\rtimes
S)(t,\overline{t})=(-1)^{n}E^{[1]}(\widetilde{f},\widetilde{G}\rtimes
S)(t^{-1},\overline{t})\,.$

###### Proof.

Let $\underline{\lambda}\in G$. By Assumption 2), the group
$S^{I_{\underline{\lambda}}}$ acts on $\Omega^{G}_{f^{\underline{\lambda}}}$
with determinant 1. Let $C((\underline{\lambda},1)):=C_{G\rtimes
S^{I_{\underline{\lambda}}}}((\underline{\lambda},1))$ be the centralizer of
the element $(\underline{\lambda},1)$ in $G\rtimes
S^{I_{\underline{\lambda}}}$. Hence the orbit of a monomial $\wedge_{i\in
I_{\underline{\lambda}}}x_{i}^{k_{i}}dx_{i}$ in
$\Omega^{C((\underline{\lambda},1))}_{f^{\underline{\lambda}}}$ with respect
to the group $S^{I_{\underline{\lambda}}}$ corresponds to the element
$\sum_{s\in S^{I_{\underline{\lambda}}}}\wedge_{i\in
I_{\underline{\lambda}}}x_{s^{-1}(i)}^{k_{s^{-1}(i)}}dx_{s^{-1}(i)}$ in
$\Omega^{C((\underline{\lambda},1))}_{f^{\underline{\lambda}}}$. Since
$(1,s)^{-1}(\underline{\mu},1)(1,s)=(s^{-1}(\underline{\mu}),1)$ for
$\underline{\mu}\in\widetilde{G}$ and $s\in S$, this orbit is mapped by the
map $\psi$ to the conjugacy class of the element $(\psi(\wedge_{i\in
I_{\underline{\lambda}}}x_{i}^{k_{i}}dx_{i}),1)$ in $\widetilde{G}\rtimes S$.

The group $G_{f}$ is the direct sum of the groups $G_{f_{\nu}}$,
$\nu=1,\ldots,s$. Therefore an element $\underline{\lambda}\in G_{f}$ can be
written as
$\underline{\lambda}=(\underline{\lambda}_{1},\ldots,\underline{\lambda}_{s})$
with $\underline{\lambda}_{\nu}\in G_{f_{\nu}}$. One can see that the map
$\psi$ can be applied separately to each part of a monomial corresponding to
the atomic summands of $f$. Two orbits of monomials can have the same image
under the map $\psi$ only if their parts corresponding to a loop of even
length are mapped to $1$ (under the restriction of $\psi$ to this loop).
According to Assumption 1), the group $S$ preserves a loop $f_{\nu}$ of even
length. Therefore $S$ preserves the monomials in the variables of $f_{\nu}$
whose image under the map $\psi$ is equal to $1$. The two elements of a loop
of type $a_{1},\dots,a_{L}$ which are mapped to 1 by $\psi$ are the elements
$\underline{k}=(k_{1},\ldots,k_{L})$ with $k_{2i}=0$, $k_{2i-1}=a_{2i-1}-1$
and $k_{2i-1}=0$, $k_{2i}=a_{2i}-1$ for $i=1,\ldots,\frac{L}{2}$ (see the
proof of [12, Proposition 12]). They are exchanged by a shift of an odd period
$\ell$: $a_{j+\ell}=a_{j}$, $j=1,\ldots,L$, where $j+\ell$ is considered
modulo $L$. The order of this shift is even. By Assumption 2), such a shift is
excluded, see also Remark 7. Therefore $\psi$ is injective on an orbit of
$S^{I_{\underline{\lambda}}}$.

As in [12], define, for $\underline{\lambda}\in G$ and
$\widetilde{\underline{\lambda}}\in\widetilde{G}$, the number
$m_{\underline{\lambda},\widetilde{\underline{\lambda}}}$ as
$m_{\underline{\lambda},\widetilde{\underline{\lambda}}}:=2^{r}$ where $r$ is
the number of indices $\nu$ such that $f_{\nu}$ is a loop with an even number
of variables and both components $\underline{\lambda}_{\nu}$ and
$\widetilde{\underline{\lambda}}_{\nu}$ are equal to 1. Define

$\widehat{m}_{\underline{\lambda},\widetilde{\underline{\lambda}}}:=\left\\{\begin{array}[]{cl}0&\mbox{if
}\widetilde{\underline{\lambda}}\not\in\widetilde{G}_{(I_{\underline{\lambda}})}\,,\\\
m_{\underline{\lambda},\widetilde{\underline{\lambda}}}&\mbox{otherwise},\end{array}\right.$

where $\widetilde{G}_{(I_{\underline{\lambda}})}$ is the set of elements of
$\widetilde{G}$ which are images under the map $\psi$ of
$\Omega_{f^{\underline{\lambda}}}^{G}$. Note that the number
$\widehat{m}_{\underline{\lambda},\widetilde{\underline{\lambda}}}$ only
depends on the conjugacy classes of the elements $(\underline{\lambda},1)$ and
$(\widetilde{\underline{\lambda}},1)$.

Let $[(\underline{\lambda},1)]$ denote the conjugacy class of
$(\underline{\lambda},1)\in G\rtimes S$. As in [12], one can derive an
analogue of [12, Proposition 14]:

$E^{[1]}(f,G\rtimes
S)(t,\overline{t})=\sum\widehat{m}_{\underline{\lambda},\widetilde{\underline{\lambda}}}(-1)^{n_{\underline{\lambda}}}(t\overline{t})^{{\rm
age}(\underline{\lambda})-\frac{n-n_{\underline{\lambda}}}{2}}\left(\frac{\overline{t}}{t}\right)^{{\rm
age}(\widetilde{\underline{\lambda}})-\frac{n-n_{\widetilde{\underline{\lambda}}}}{2}},$
(17)

where the sum is over pairs of conjugacy classes
$([(\underline{\lambda},1)],[(\widetilde{\underline{\lambda}},1)])$ in ${\rm
Conj}(G\rtimes S)\times{\rm Conj}(\widetilde{G}\rtimes S)$. As in [12], one
can see that this gives the formula of the theorem. ∎

###### Remark 6

The (somewhat strange) assumptions in Theorem 4 are explained by the fact that
they permit to make the proof close to the one in [12] (or rather using the
arguments from there), in particular to get Equation (17) (a version of
Equation (4.3) from [12]). Assumption 2) permits to identify collections of
monomials with invariant elements in the cohomology group. Assumption 1) is
needed to control the symmetry of preimages of unit elements. Just for loops
of even length the map $\psi$ is surjective, but the element 1 has two
preimages. If there is more than one loop of a type of this sort, there are
several pairs of the preimages of 1 and it seems to be very difficult to
control the action of $S$ on them: they may have different isotropy subgroups.

###### Remark 7

Assumption 2) (under the condition that Assumption 1) holds) means the
following: in the Sebastiani-Thom decomposition $f=f_{1}\oplus\cdots\oplus
f_{s}$ into atomic polynomials,

  1. a)

there is no loop of even length $L$ such that an element of $S$ acts on it by
a permutation of order $k$ with $L/k$ odd;

  2. b)

for any loop of odd length, its orbit under the group $S$ consists of an odd
number of its copies (including the loop itself);

  3. c)

for any chain, its orbit under the group $S$ consists of an odd number of its
copies (including the chain itself).

By the remarks at the end of Section 1, Theorem 4 implies the following
corollary.

###### Corollary 2

Under the assumptions of Theorem 4, one has

$\overline{\zeta}^{{\,{\rm orb}},[1]}_{f,G\rtimes
S}(t)=\left(\overline{\zeta}^{{\,{\rm
orb}},[1]}_{\widetilde{f},\widetilde{G}\rtimes S}(t)\right)^{(-1)^{n}}\,.$

Combining Theorem 2 and Corollary 2, we obtain the following result.

###### Theorem 5

Let $f(x_{1},\ldots,x_{n})$ be an invertible polynomial satisfying the
conditions of Theorem 4. Let $G\subset G_{f}$ and let $S$ be a subgroup of
$S_{n}$ satisfying PC and such that the centralizer $C_{S}(\sigma)$ of each
element $\sigma\in S,\sigma\neq 1$, is generated by $\sigma$. Then one has

$\overline{\zeta}^{{\,{\rm orb}}}_{f,G\rtimes
S}(t)=\left(\overline{\zeta}^{{\,{\rm
orb}}}_{\widetilde{f},\widetilde{G}\rtimes S}(t)\right)^{(-1)^{n}}\,.$

##  7 Example

Here we compute the E-function for a rather simple example with
$S={\mathbb{Z}}_{2}=\langle s\rangle$. We showed that under some conditions at
the level 1 the symmetry is essentially given by the mirror map defined by
Krawitz. There exists the hope that on other levels the symmetry may be given
by certain analogues of the Krawitz mirror map. It appears to be difficult (if
possible) to find this analogue even for this simple example.

Let

$f(x_{1},x_{2},x_{3},x_{4})=x_{1}^{3}x_{2}+x_{2}^{5}x_{3}+x_{3}^{3}x_{4}+x_{4}^{5}x_{1}$

(of loop type), $S={\mathbb{Z}}_{2}=\langle(13)(24)\rangle$,
$G_{f}={\mathbb{Z}}_{224}=\langle\sigma\rangle$, where $\sigma$ can be taken
of the form $\sigma=\frac{1}{224}(1,-3,15,-45)$. Here we use the notation
$(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$ ($\alpha_{i}\in{\mathbb{Q}}$)
for the element
$({\mathbf{e}}[\alpha_{1}],{\mathbf{e}}[\alpha_{2}],{\mathbf{e}}[\alpha_{3}],{\mathbf{e}}[\alpha_{4}])\in({\mathbb{C}}^{\ast})^{4}$.
Note that $\widetilde{f}=f$. Any subgroup of $G_{f}$ is preserved by $S$. Let
us take $G={\mathbb{Z}}_{8}=\langle\delta\rangle$, where
$\delta=\sigma^{28}=\frac{1}{8}(1,-3,-1,3)$ (and therefore
$\widetilde{G}={\mathbb{Z}}_{28}$).

At the level 1, one has

$\displaystyle E^{[1]}(f,G\rtimes
S)(t,\overline{t})=E^{[1]}(\widetilde{f},\widetilde{G}\rtimes
S)(t^{-1},\overline{t})$ $\displaystyle=$
$\displaystyle\left(\frac{\overline{t}}{t}\right)^{-\frac{8}{7}}+\left(\frac{\overline{t}}{t}\right)^{-\frac{6}{7}}+2\left(\frac{\overline{t}}{t}\right)^{-\frac{4}{7}}+\left(\frac{\overline{t}}{t}\right)^{-\frac{3}{7}}+3\left(\frac{\overline{t}}{t}\right)^{-\frac{2}{7}}+\left(\frac{\overline{t}}{t}\right)^{-\frac{1}{7}}+8$
$\displaystyle{}+\left(\frac{\overline{t}}{t}\right)^{\frac{1}{7}}+3\left(\frac{\overline{t}}{t}\right)^{\frac{2}{7}}+\left(\frac{\overline{t}}{t}\right)^{\frac{3}{7}}+2\left(\frac{\overline{t}}{t}\right)^{\frac{4}{7}}+\left(\frac{\overline{t}}{t}\right)^{\frac{6}{7}}+\left(\frac{\overline{t}}{t}\right)^{\frac{8}{7}}.$

At the level $[s]$, we have two conjugacy classes of elements $(\lambda,s)\in
G\rtimes S$, namely, $[(1,s)]$ and $[(\delta,s)]$. For the representatives
$(1,s)$ and $(\delta,s)$ of them, the fixed point sets are 2-dimensional and
are given by the equations $x_{3}=ax_{1}$, $x_{4}=bx_{2}$ for certain complex
numbers $a,b$, where $a=b=1$ for $(1,s)$. Therefore, in the coordinates
$x_{1}$ and $x_{2}$ the Milnor fibres inside them are given by the equation
$a^{\prime}x_{1}^{3}x_{2}+b^{\prime}x_{2}^{5}x_{1}=1$
($a^{\prime}=b^{\prime}=2$ for $(1,s)$). A basis of
$\Omega_{f^{\langle(\bullet,s)\rangle}}$ is given by the monomials
$x_{1}^{k_{1}}x_{2}^{k_{2}}dx_{1}dx_{2}$ with $0\leq k_{1}\leq 2$, $0\leq
k_{2}\leq 4$. The centralizers of $(1,s)$ and of $(\delta,s)$ are
$\langle(\delta^{4},1),(1,s)\rangle$ and
$\langle(\delta^{4},1),(\delta,s)\rangle$ respectively, where
$\delta^{4}=\frac{1}{2}(1,1,1,1)$. Therefore the monomials which give a basis
of $\Omega_{f^{\langle(\bullet,s)\rangle}}^{C_{G\rtimes S}(\bullet,s)}$ are
those of even order. The ages of the elements $(1,s)$ and $(\delta,s)$ are
equal to 1. Therefore, in the E-function on level $[s]$, we only have to count
the weights of the monomials in Table 1.

Monomials | Weights | Elements of $\widetilde{G}\rtimes S$
---|---|---
$dx_{1}dx_{2}$ | $-\frac{4}{7}$ | $\left\\{\begin{array}[]{c}(\frac{1}{28}(18,2,18,2),s)\sim(\frac{1}{28}(4,16,4,16),s),\\\ (\frac{1}{28}(25,23,11,9),s)\sim(\frac{1}{28}(11,9,25,23),s)\end{array}\right\\}$
$x_{1}^{2}dx_{1}dx_{2}$ | $0$ |
$x_{1}x_{2}dx_{1}dx_{2}$ | $-\frac{1}{7}$ | $\left\\{\begin{array}[]{c}(\frac{1}{28}(8,4,8,4),s)\sim(\frac{1}{28}(22,18,22,18),s),\\\ (\frac{1}{28}(15,25,1,11),s)\sim(\frac{1}{28}(1,11,15,25),s)\end{array}\right\\}$
$x_{2}^{2}dx_{1}dx_{2}$ | $-\frac{2}{7}$ | $\left\\{\begin{array}[]{c}(\frac{1}{28}(16,8,16,8),s)\sim(\frac{1}{28}(2,22,2,22),s),\\\ (\frac{1}{28}(9,15,23,1),s)\sim(\frac{1}{28}(23,1,9,15),s)\end{array}\right\\}$
$x_{1}^{2}x_{2}^{2}dx_{1}dx_{2}$ | $\frac{2}{7}$ | $\left\\{\begin{array}[]{c}(\frac{1}{28}(26,6,26,6),s)\sim(\frac{1}{28}(12,20,12,20),s),\\\ (\frac{1}{28}(5,27,19,13),s)\sim(\frac{1}{28}(19,13,5,27),s)\end{array}\right\\}$
$x_{1}x_{2}^{3}dx_{1}dx_{2}$ | $\frac{1}{7}$ | $\left\\{\begin{array}[]{c}(\frac{1}{28}(6,10,6,10),s)\sim(\frac{1}{28}(20,24,20,24),s),\\\ (\frac{1}{28}(27,17,13,3),s)\sim(\frac{1}{28}(13,3,27,17),s)\end{array}\right\\}$
$x_{2}^{4}dx_{1}dx_{2}$ | $0$ |
$x_{1}^{2}x_{2}^{4}dx_{1}dx_{2}$ | $\frac{4}{7}$ | $\left\\{\begin{array}[]{c}(\frac{1}{28}(24,12,24,12),s)\sim(\frac{1}{28}(10,26,10,26),s),\\\ (\frac{1}{28}(17,19,3,5),s)\sim(\frac{1}{28}(3,5,17,19),s)\end{array}\right\\}$
Table 1: Correspondence between monomials and elements of
$\widetilde{G}\rtimes S$

The weight of an element $(\widetilde{g},s)\in\widetilde{G}\rtimes S$ is equal
to ${\rm age}(\widetilde{g})-\frac{n-n_{\widetilde{g}}}{2}$. For the weight 0
there are two conjugacy classes
$\\{(\widetilde{\delta}^{14},s),(\widetilde{\delta}^{0},s)\\}$ and
$\\{(\widetilde{\delta}^{7},s),(\widetilde{\delta}^{21},s)\\}$, where

$\widetilde{\delta}^{14}=\frac{1}{2}(1,1,1,1),\quad\widetilde{\delta}^{7}=\frac{1}{4}(1,-1,-1,1),\quad\widetilde{\delta}^{21}=\frac{1}{4}(-1,1,1,-1).$

Representatives of them have 2-dimensional fixed point sets and they give 4
generators with weight $-\frac{n_{g}}{2}=0$ (two of them each).

From Table 1 one gets

$\displaystyle E^{[s]}(f,G\rtimes
S)(t,\overline{t})=E^{[s]}(\widetilde{f},\widetilde{G}\rtimes
S)(t^{-1},\overline{t})$ $\displaystyle=$ $\displaystyle
2\left(\left(\frac{\overline{t}}{t}\right)^{-\frac{4}{7}}+\left(\frac{\overline{t}}{t}\right)^{-\frac{2}{7}}+\left(\frac{\overline{t}}{t}\right)^{-\frac{1}{7}}+2+\left(\frac{\overline{t}}{t}\right)^{\frac{1}{7}}+\left(\frac{\overline{t}}{t}\right)^{\frac{2}{7}}+\left(\frac{\overline{t}}{t}\right)^{\frac{4}{7}}\right).$

Looking at the table, one sees that the mirror map (if it exists) should send,
e.g., the monomial $dx_{1}dx_{2}$ originated from one of the two conjugacy
classes to the conjugacy class either of $(\frac{1}{28}(18,2,18,2),s)$ or of
$(\frac{1}{28}(25,23,11,9),s)$. We did not succeed to find an analogue of the
mirror map for this case.

## Acknowledgements

This work has been partially supported by DFG. The work of the second author
(Sections 1, 3, 5, 7) was supported by the grant 21-11-00080 of the Russian
Science Foundation. We would like to thank the referee for carefully reading
our paper and for their useful comments which helped to improve the paper.

## References

  * [1] M. Atiyah, G. Segal: On equivariant Euler characteristics. J. Geom. Phys. 6 (1989), no.4, 671–677.
  * [2] A. A. Basalaev, A. A. Ionov: Mirror map for Fermat polynomials with a nonabelian group of symmetries. (Russian) Teoret. Mat. Fiz. 209 (2021), no. 2, 205–223; translation in Theoret. and Math. Phys. 209 (2021), no. 2, 1491–1506.
  * [3] P. Berglund, T. Hübsch: A generalized construction of mirror manifolds. Nuclear Physics B 393 (1993), 377–391.
  * [4] P. Berglund, M. Henningson: Landau-Ginzburg orbifolds, mirror symmetry and the elliptic genus. Nuclear Physics B 433 (1995), 311–332.
  * [5] W. Ebeling, S. M. Gusein-Zade: Orbifold Euler characteristics for dual invertible polynomials. Mosc. Math. J. 12 (2012), no.1, 49–54.
  * [6] W. Ebeling, S. M. Gusein-Zade: Saito duality between Burnside rings for invertible polynomials. Bull. Lond. Math. Soc. 44 (2012), no.4, 814–822.
  * [7] W. Ebeling, S. M. Gusein-Zade: Orbifold zeta functions for dual invertible polynomials. Proc. Edinb. Math. Soc. 60 (2017), no.1, 99–106.
  * [8] W. Ebeling, S. M. Gusein-Zade: Orbifold Milnor lattice and orbifold intersection form. Manuscripta Math. 155 (2018), no.3-4, 335–353.
  * [9] W. Ebeling, S. M. Gusein-Zade: A version of the Berglund–Hübsch–Henningson duality with non-abelian groups. Int. Math. Res. Not. IMRN 2021, no. 16, 12305–12329.
  * [10] W. Ebeling, S. M. Gusein-Zade: On the orbifold Euler characteristics of dual invertible polynomials with non-abelian symmetry groups. Pure Appl. Math. Q. 16 (2020), no.4, 1099–1113.
  * [11] W. Ebeling, S. M. Gusein-Zade: Dual invertible polynomials with permutation symmetries and the orbifold Euler characteristic. SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 051, 15 pp.
  * [12] W. Ebeling, S. M. Gusein-Zade, A. Takahashi: Orbifold E-functions of dual invertible polynomials. J. Geom. Phys. 106 (2016), 184–191.
  * [13] W. Ebeling, A. Takahashi: Mirror symmetry between orbifold curves and cusp singularities with group action. Int. Math. Res. Not. 2013, 2240–2270.
  * [14] H. Fan, T. Jarvis, Y. Ruan: The Witten equation, mirror symmetry, and quantum singularity theory. Ann. of Math. (2) 178 (2013), no.1, 1–106.
  * [15] S.M. Gusein-Zade, I. Luengo, A. Melle-Hernández: Grothendieck ring of varieties with actions of finite groups. Proc. Edinb. Math. Soc. (2) 62 (2019), no.4, 925–948.
  * [16] F. Hirzebruch, Th. Höfer: On the Euler number of an orbifold. Math. Ann. 286 (1990), no.1–3, 255–260.
  * [17] K. Intriligator, C. Vafa: Landau-Ginzburg orbifolds. Nuclear Phys. B 339 (1990), no. 1, 95–120.
  * [18] M. Krawitz: FJRW-rings and Landau–Ginzburg mirror symmetry. Preprint arXiv: 0906.0796.
  * [19] D. Mukai: Mirror symmetry of nonabelian Landau–Ginzburg orbifolds with loop type potentials. J. Geom. Phys. 159 (2021), Paper No.103877, 12 pp.
  * [20] N. Priddis, J. Ward, M.M. Williams: Mirror symmetry for nonabelian Landau-Ginzburg models. SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No.059, 31 pp.
  * [21] J.H.M. Steenbrink: Mixed hodge structure on the vanishing cohomology. In: Real and Complex Singularities, Proc. Ninth Nordic Summer School, Oslo, 1976, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525-563.
  * [22] C.T.C. Wall: A note on symmetry of singularities. Bull. London Math. Soc. 12 (1980), no. 3, 169–175.

Leibniz Universität Hannover, Institut für Algebraische Geometrie,
Postfach 6009, D-30060 Hannover, Germany
E-mail<EMAIL_ADDRESS>

Moscow State University, Faculty of Mechanics and Mathematics,
Moscow Center for Fundamental and Applied Mathematics,
Moscow, GSP-1, 119991, Russia
& National Research University “Higher School of Economics”,
Usacheva street 6, Moscow, 119048, Russia.
E-mail<EMAIL_ADDRESS>