11institutetext: Department of Physics, University of Helsinki, P.O. Box 64,
FI-00014, Helsinki, Finland
11email<EMAIL_ADDRESS>22institutetext: CmPA/Department of
Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
33institutetext: Instituto de Astrofísica e Ciências do Espaço, Department of
Physics, University of Coimbra, Coimbra, Portugal 44institutetext: Solar
Physics and Space Plasma Research Centre (SP2RC), School of Mathematics and
Statistics, University of Sheffield, Sheffield, S3 7RH, United Kingdom
55institutetext: NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
66institutetext: Institute of Physics, University of Maria Curie-Skłodowska,
ul. Radziszewskiego 10, 20-031 Lublin, Poland 77institutetext: Instituto de
Astrofísica e Ciências do Espaço, Department of Earth Sciences, University of
Coimbra, Coimbra, Portugal 88institutetext: CFisUC, Department of Physics,
University of Coimbra, 3004 516 Coimbra, Portugal 99institutetext: Department
of Astronomy, Eötvös Loránd University, Pázmány Péter sétány 1/A, H-1117
Budapest, Hungary 1010institutetext: Gyula Bay Zoltan Solar Observatory (GSO),
Hungarian Solar Physics Foundation (HSPF), Petőfi tér 3., H-5700 Gyula,
Hungary

# The Automatic Identification and Tracking of Coronal Flux Ropes - Part II:
New Mathematical Morphology-based Flux Rope Extraction Method and Deflection
Analysis

A. Wagner 1122 S. Bourgeois 3344 E. K. J. Kilpua 11 R. Sarkar 11 D. J. Price
11 A. Kumari 55 J. Pomoell 11 S. Poedts 2266 T. Barata 77 R. Erdélyi 44991010
O. Oliveira 88 R. Gafeira 33

(Received XXX; accepted XXX)

###### Abstract

Context. Constructing the relevant magnetic field lines from active region
modelling data is crucial to understand the underlying instability mechanisms
that trigger the corresponding eruptions.

Aims. We present a magnetic flux rope (FR) extraction tool for solar coronal
magnetic field modelling data, which builds upon the methodology from Wagner
et al. (2023). The newly developed method is then compared against its
previous iteration. Furthermore, we apply the scheme to magnetic field
simulations of active regions AR12473 (similar to our previous study) and
AR11176. We compare the method to its predecessor and study the 3D movement of
the newly extracted FRs up to heights of 200 and 300 Mm, respectively.

Methods. The extraction method is based on the twist parameter $T_{w}$ and a
variety of mathematical morphology (MM) algorithms, including the opening
transform and the morphological gradient. We highlight the differences between
the methods by investigating the circularity of the FRs in the plane we
extract from. The simulations for the active regions are carried out with a
time-dependent data-driven magnetofrictional model (TMFM; Pomoell et al.
(2019)). We investigate the FR trajectories by tracking their apex throughout
the full simulation time span.

Results. Comparing the newly developed method to the extraction scheme in
Wagner et al. (2023), we demonstrate that this upgrade provides the user with
more tools and less a-priori assumptions about the FR shape that, in turn,
leads to a more accurate set of field lines. Despite some differences, both
the newly extracted FR of AR12473 as well as the FR derived with the old
iteration of the method show a similar general appearance, confirming that
both methods indeed extract the same structure. The methods differ the most in
their emergence/formation stages, where the newly extracted FR deviates
significantly from a perfectly circular cross-section (that was the basic
assumption of the initial method). The propagation analysis yields that the
erupting FR from AR12473 showcases indeed stronger dynamics than the AR11176
FR and a significant deflection during its ascent through the domain. The
modelling results are also verified with observations, with AR12473 being
indeed dynamic and eruptive, while AR11176 only features an eruption outside
of our simulation time window.

Conclusions. We implemented a FR extraction method, incorporating mathematical
morphology algorithms for 3D solar magnetic field simulations of active region
FRs. This scheme was applied to AR12473 and AR11176. We find that the clearly
eruptive FR of AR12473 experiences significant deflection during its rise. The
AR11176 FR appears more stable, though there still is a notable deflection.
This confirms that at these low coronal heights, FRs do undergo significant
changes in the direction of their propagation even for less dynamic cases.

###### Key Words.:

sun: corona – sun: activity – methods: observational – sun: magnetic fields –
sun: coronal mass ejections (CMEs) – methods: data analysis

##  1 Introduction

Solar eruptions play a central role in many space weather studies, as their
space weather effects have direct impact on our technology and life on Earth.
In particular, coronal mass ejections (CMEs; e.g. Webb & Howard, 2012) can
cause great disruptions to both our space- and ground-based technology, and
related high energy particle radiation can pose a risk to astronauts and high-
altitude airplane crew and passengers (see e.g., Zhang et al., 2021; Bain et
al., 2023). Thus, there is broad interest in the understanding of the
triggering mechanisms of solar eruptive phenomena and their evolution from
initiation until their arrival at Earth.

Flux ropes (FRs) are the key magnetic structures related to CME eruptions,
(see e.g., Vourlidas et al., 2013; Green et al., 2018; Chen, 2017). Due to the
limited capability of observing the magnetic field in the solar corona,
modelling the corona via numerical simulations is necessary to study the
formation, destabilization and dynamics of solar FRs. There are a wide range
of numerical simulations that incorporate physics to varying degrees in many
different implementations, like potential field methods, non-linear force-free
fields or magnetohydrodynamic (MHD) models, to name a few (e.g., Kilpua et
al., 2019, and references therein). Identifying and tracking of the magnetic
FR (i.e., following the field lines that belong to the same magnetic
structure) that are either injected into these simulations or directly result
from them is not a trivial task.

Often, very approximate methods are used to determine the magnetic field lines
of FRs from the models. Manual extractions are widely performed (e.g., Kumari
et al., 2023) and only few automatic methods exist, for example Lowder &
Yeates (2017), who have constructed a FR identification and tracking procedure
for coronal magnetic field simulations. The authors base their method on
thresholding photospheric maps of field line helicity, while the tracking is
carried out by calculating the overlap of footpoints between two output
frames.

In this paper, we present a novel, semi-automatic FR extraction scheme to
identify and track the evolution of solar FRs in coronal simulations. Here, we
base the extraction on the twist parameter $T_{w}$ (e.g., Berger & Prior,
2006; Liu et al., 2016), paired with a nonlinear image processing method
called mathematical morphology (MM). A prominent advantage of the MM
methodology is that any proxy that captures the relevant FR field lines can be
employed. The incorporation of MM methods was chosen because it allows the
user to have more control over the FR extraction procedure, as well as to
remove the need for assumptions to simplify the process. This new extraction
scheme builds upon the methodology developed in Wagner et al. (2023), to which
we will be referring to as ”Paper I” hereafter. The previous method was also
focused around extracting a FR from the $T_{w}$ maps. Instead of MM algorithms
aiding the procedure, the assumption of a perfectly circular cross-section in
the plane of extraction was employed. Additionally, two parameters were
introduced to control the FR size depending on the gradient in the twist maps
($\kappa$) and the amount of overlying, non-FR twisted field lines
($\epsilon$). The tool presented in this paper removes both the restricting
circularity assumption as well as complications due to the additional
parameters. The new tools will be tested on the output of a time-dependent
data-driven magnetofrictional model (TMFM; Pomoell et al., 2019) applied to
active regions (ARs) AR12473 (passed the center of the solar disk on 28 Dec
2015) and AR11176 (passed the center of the solar disk on 28 Mar 2011). The
former AR is the same as analysed in Paper I, which allows us to directly
compare the results and highlight the improvements of the new scheme, while
AR11176 serves as an additional case to test the methodology.

The mathematical morphology method was introduced by Matheron (1967) and
Serra, J. & École nationale supérieure des mines de Paris (1969) who applied
it to probe porous media (Serra, 1982; Matheron & Serra, 2001). Based on set
theory, topology, and integral geometry, MM has turned out to be a powerful
tool to describe the geometrical shape of objects of interest in images.
According to Carvalho et al. (2020), MM is particularly suited for analyzing
structures with complex and irregular shapes and/or sizes. Quantitative
properties of these structures can be easily found with specific
implementations provided by MM software libraries. Moreover, MM is applicable
to different types of images, from low to higher resolution as it can deal
with images taken at both ground-based as well as space-borne observatories.
Therefore, it is also a suitable methodology for the field of heliophysics.
Several studies have already demonstrated that the MM analysis is able to
characterise a vast range of solar features accurately. For example, Shih &
Kowalski (2003), Qu et al. (2005) and, more recently, Koch & Rosolowsky (2015)
applied MM techniques to detect solar filaments in H-alpha full-disk and far-
infrared solar images, while Barata et al. (2018) and Carvalho et al. (2020)
uncovered sunspots and solar plages through a MM algorithm pipeline. Stenning
et al. (2013) used MM to classify sunspots.

In this paper we apply MM operations to extract and process the early-phase FR
structures from the simulation generated twist maps. In order to obtain the
external outlines of these structures, we implement an operator called the
morphological gradient (cf., Sect. 2.3).

The paper is organized as follows: In Sect. 2 we discuss the data and methods,
needed to perform the simulation and extraction. Sect. 3 showcases the
resulting flux ropes and their trajectories. Finally, Sect. 4 discusses the
method’s performance and the results of the extraction and deflection
analysis.

##  2 Data and Methods

###  2.1 Observational Data

The TMFM simulation requires a time series of photospheric vector magnetograms
(see Sect. 2.2 below). We therefore use magnetic field data from the
Helioseismic and Magnetic Imager (HMI, see Couvidat et al., 2016; Schou et
al., 2012) from the Solar Dynamics Observatory (SDO) (see Pesnell et al.,
2012) mission.

To compare with our modelling efforts, we furthermore use image material from
the Atmospheric Imaging Assembly (AIA, see Lemen et al., 2012) instrument of
SDO. In particular, we are making use of the 171, 193 and 211 Å channels.

###  2.2 The TMFM Model and Set-up

Analogously to Paper I, we apply a time-dependent data-driven
magnetofrictional model (TMFM; Pomoell et al., 2019) to active region AR12473.
In Paper I, we paired this simulation with a so-called relaxation procedure,
where the TMFM was coupled to a zero-beta magnetohydrodynamics approach (Daei
et al., 2023). In addition, in this work we also apply the tracking
methodology to a second active region, AR11176. From the HMI vector
magnetograms, electric field maps are created, which are then ultimately used
as input to the model (more details on this procedure can be found in Lumme et
al. (2017) and Pomoell et al. (2019)). The dynamics in the model is set based
on the assumption of the velocity $\mathbf{v}$ in the coronal domain being
proportional to the Lorentz force:

$\mathbf{v}=\frac{1}{\nu}\frac{\mu_{0}\mathbf{J}\times\mathbf{B}}{B^{2}},$ (1)

with $\mathbf{B}$ the magnetic field, $\mu_{0}$ the vacuum magnetic
permeability, $\mathbf{J}$ the current density and $\nu$ the magnetofrictional
coefficient.

The TMFM simulation of AR12473 is performed for the time interval from 22 Dec
2015, 23:36 UT to 2 Jan 2016, 12:36 UT, thus identical to Paper I, and of
AR11176 from 25 Mar 2011 at 4:00 UT until 1 Apr 2011, 18:00 UT. These periods
correspond to the times when the active regions were best observed on-disk and
not too close to the limbs of the Sun, which is necessary for keeping
projection effects in the input data minimal. The simulations are both
analysed at a cadence of 6 hours. The Cartesian domain for AR12473 has an
approximate size of 386x254x200 Mm while the domain for AR11176 has
approximate dimensions of 800x374x300 Mm (corresponding to $x$, $y$ and $z$,
respectively, with the plane of the magnetogram being the $z=0$ plane). In
total, the AR12473 simulation domain consists of approximately 10.6 million
cells, while the AR11176 domain contains approximately 48.8 million cells. For
AR11176 the domain height is larger (300 Mm compared to 200 Mm) due to
specifics of the FR that will be detailed later in Sect. 3.2. One fundamental
difference between the two sets of simulations is the masking of the
magnetograms: while for AR12473 no masking is applied (analogously to Price et
al. (2020) and Kumari et al. (2023)), for AR11176 we set all $B$-field values
below 250 G to zero. This step had to be done for AR11176, as the simulation
with an unmasked magnetogram yielded only a small, stable and weakly twisted
bundle of magnetic field lines close to the photosphere. Otherwise, the pre-
processing steps are analogous to Price et al. (2020).

###  2.3 Mathematical Morphology (MM) Algorithms

We use MM algorithms (introduced in Sect. 1) to aid our extraction method in
multiple ways. MM is built upon a reference object called the Structuring
Element (SE). This object has a size, shape, and orientation which are
a-priori defined by the user, and it is used as a kernel to probe images. Most
MM transforms operate by comparing the structures of interest in images to
this carefully user-selected SE. In other words, we extract, reject or modify
image structures that the SE fits or misses, respectively - hence the
importance of accurately choosing an appropriate shape and size for the SE
(Soille, 1999).

Erosion and dilation are the two fundamental MM operations that provide a
basis for all other image transforms. On the one hand, the erosion transform
is a useful tool in morphological image processing to enhance the darkest
regions of an image, called valleys, and to numb the intensity of the
brightest ones, called peaks (Barata et al., 2018). From the point of view of
set theory, the erosion of a set $X$, $\epsilon_{A}(X)=X\ominus A$, is
determined by all the centre points $x$ of the SE $A$, such that $A$ is
included in $X$. Thus, for binary images, this has the effect that pixels
around the object boundaries may be removed. On the other hand, the dilation
transform of a set $X$, $\delta_{A}(X)=X\oplus A$, is defined as the union of
all the centre points $x$ of the SE $A$, where $A$ and $X$ have non-zero
overlap. Contrary to erosion, the dilation operator adds pixels around the
original object boundaries and widens the bright areas while reducing the
valleys (Barata et al., 2018).

The morphological gradient subtracts the erosion from the dilation for a grey-
scale image $f$: $\nabla(f)=\delta_{A}(f)-\epsilon_{A}(f)$, also written as
$\nabla(f)=(f\oplus A)-(f\ominus A)$ where $f$ is the image and $A$ the SE.
The morphological gradient helps to accentuate variations in the twist number
maps, as will be shown in Sects. 2.4 and 4.

###  2.4 Extraction Method based on MM

Following the procedure employed in Paper I, the extraction starts from a 2D
slice, chosen to cut the domain close to the polarity inversion line (PIL) of
the simulated active region. In theory, any plane where the FR is expected to
pass through, is suitable. In this slice, we calculate the twist map using the
twist number $T_{w}$, according to the approach in Liu et al. (2016). $T_{w}$
counts the number of turns two infinitesimally close field lines make about
each other Berger & Prior (2006). More discussion on the choice of $T_{w}$ as
the FR proxy can be found in Sect. 4.3. Figure 1 shows as an example $T_{w}$
maps, reconstructed for a few selected frames from the TMFM simulation output
for AR12473 (the plane is placed at $x\approx 0.7$ Mm). These maps have a
resolution of approximately 0.36 Mm per pixel for both active regions.

In the next step, we apply the morphological gradient to sharpen the twist
maps. This has the effect that the subsequent application of a threshold is
more stable with regards to the resulting outlines. This allows us to use a
constant threshold throughout the whole time series to identify high-twist
regions (and ultimately reduce the twist maps to binary masks) without
removing relevant parts or creating unwanted artefacts. Given our twistmap
resolution, we used circular SEs, with sizes between 4 and 10 for this
procedure. In the resulting binary masks, we then seek to improve our
extraction on a case-by-case basis. Indeed, it may be necessary to smooth out
the extracted structures and/or to remove any spur or twisted feature that
would not be part of the actual FR. To do so, we apply a morphological
opening. The opening transform of an image $f$, $\gamma_{A}(f)$, is defined as
the dilation of the erosion of the image $f$, i.e.,
$\gamma_{A}(f)=\delta_{A}(\epsilon_{A}(f))$, also written as $\gamma=f\circ
A=(f\ominus A)\oplus A$. Generally, the initial structure is not entirely
recovered since erosion and dilation are not reversible. Thus, the opening is
effective in our case for filtering noise out of the initial FR extraction,
depending on the choice for the structuring element. One must be very careful
when applying an opening with a large SE, as it may irreversibly suppress
details that are yet relevant for an accurate extraction. To prevent this, we
only perform opening operations on the frames that require them and, in
particular, only to sub-regions of these frames where noise and unwanted
structures are visible. After trial and error, we find the optimal SE for each
time frame by incrementally increasing the SE size and checking the resulting
extraction by visual inspection. This approach ensures that we do not use
unnecessarily large SE sizes. We furthermore exclusively use circles for the
shape of our SE. The SE sizes range from 10 to 50 for early-to-mid stages of
the AR12473 simulation, while we had to use substantially larger SE sizes
($>100$) for the late-stages, as there was a significant connection with a
non-FR feature in the maps. For the AR11176 simulation, on the other hand, SE
sizes of approximately 10 were sufficient. Once the outlines appear
satisfactory, we proceed with the tracking, which is implemented similarly to
Paper I, that is, the region of interest is identified as the same structure
in subsequent frames if they overlap sufficiently.

Our method also offers some post-processing routines, for example the erosion
algorithm. Drawing the analogy to Paper I: the size of the SE of the erosion
here is analogous to the $\epsilon$-prescription, that is, a parameter
determining to which degree the extracted shape will be eroded. The difference
is that the morphological erosion works for any arbitrary shape, while our
previous $\epsilon$-method per definition was only applicable for circular
shapes. One may also use a dilation operation as a post-processing step to
fill holes in the extracted area. Finally, we compute the source points from
the resulting outlines as uniformly distributed points within the derived
binary FR mask. We note that, in practice, if no frame-specific pre- or post-
processing is needed, the extraction is practically automatic, once the
relevant parameters have been set.

To qualitatively showcase the result of the MM-based method in contrast to the
result of the method presented in Paper I, we calculate, for both AR11176 and
AR12473, the circularity of the FR cross-section. It is the ratio of the
standard deviation of the FR radius (defined as the distance between the
centroid of the extracted shape to its edge pixels) over its mean radius. In
other words, we seek to estimate the variations of the FR radius throughout
the whole structure, with a circularity of zero indicating that the FR shape
is a perfect circle. For the original implementation to work best, a
circularity close to zero is required, which we will probe with our new
extraction results. We furthermore calculate the circularity for an additional
FR extraction, where we do not apply any post-processing (except for one
problematic frame in each simulation to ensure the successful tracking of the
FR), to show the effect of these algorithms on the circularity analysis.

###  2.5 FR Propagation and Deflection

Having full access to the FR field lines enables, for example, detailed
studies of the FR trajectory through the simulation domain. We investigate
here the FR trajectory in terms of the angle of propagation (and with that the
deflection) within the simulation domain as a function of time, as well as the
evolution of all spatial coordinates. To achieve this, we calculate the FR
apex as follows: First, we compute the maximum height that any FR field line,
for a given frame, reaches. Then, we define the FR apex as the average over
all field lines that reach within 15 Mm of the maximum height. This procedure
mitigates abrupt jumps in the horizontal components the FR apex may exhibit if
many high reaching field lines are present. The results are given here as a
function of the simulation frame number with the time difference between each
frame being six hours (determined by the time cadence of the output).

We track the apex from the moment the FR forms (i.e., when there is a coherent
bundle of field lines with sufficient twist ($T_{w}\gtrapprox 0.8$ for AR12473
and $T_{w}\gtrapprox 0.4$ for AR11176) present), until it reaches the top of
the simulation domain or before it starts interacting with it. To measure how
much the FR trajectory varies during its propagation we define the propagation
angle as the angle between the $z$-axis and the vector that extends from the
first apex point to the apex point at the time step in question. The larger
the propagation angle is, the more the FR propagation direction differs from a
trajectory parallel to the $z$-axis (which, in the spherical case, would
correspond to the radial direction).

##  3 Results

###  3.1 Analysis of AR12473 simulation

A selection of snapshots from the time series of $T_{w}$-maps are shown in
Fig. 1 (the plane is located at $x\approx 0.7$ Mm). The movie showing the full
evolution of the twist map is included as Supplementary material. The twist
maps show that the FR forms within the first frames, corresponding to $\approx
1.5\;$days close to the bottom of the domain and then starts to rise upward.
The early formation phase is shown in the top panel of Fig. 1, when there is
only a weakly twisted (faintly reddish) structure at $y=0$. When the
simulation progresses, this structure evolves into a coherent, twisted FR that
moves upward.

Figure 1: Twist map evolution for AR12473. The colour intensity indicates the
magnitude of twist, where blue indicates negative twist and red indicates
positive twist. The shown snapshots correspond to simulation frame numbers: 7,
17 and 27 (from top to bottom). The shape, identified as belonging to the FR
is marked with a black contour.

The FR field lines for AR12473 are extracted following the scheme described in
Sect. 2.4, and are shown in Fig. 2 on the right, compared against the
extraction from Paper I with an $\epsilon$ value of 0.03, using the same
simulation data and frames (left panels). Comparing the results of the two
methods, one can see that throughout all stages of the evolution of the
structure, the FR appears significantly thicker when extracted with the new
method. The FRs show clear differences also in their overall coherence and
appearance for the earlier frames 7 and 17. For the mid-frame, the key
difference between the two extraction scheme outputs is (in addition to
thickness) that for the FR extracted using the MM-based method there is a set
of field lines included that connects to the adjacent positive polarity region
located at approximately $(x,y,z)=(100,0,0)$ and visible in the middle panel
on the right and indicated by a plus in the first panel (top left). In the
last frame shown, the field lines both from the old and updated method extend
to this region, indicating that this region is indeed tied to the FR.
Additionally, the FRs are more similar in this frame than at earlier times. A
closer look at the footpoint regions also reveals that the positive polarity
footpoint exhibits significant movement throughout the simulation, while the
negative polarity footpoints approximately stay rooted at the same location in
the photosphere. This confirms also our findings from Paper I, where this was
investigated in great detail.

The above findings highlight the fact that the updated method captures more
relevant field lines, while still avoiding surrounding non-FR field lines. A
top-view comparison of the FR field lines using the MM extraction to AIA
observations can be seen in Fig. 3, which shows that the overall morphology of
the observed structure is well captured.

Figure 2: FR snapshots for the TMFM AR12473 simulation, using the method from
Paper I (on the left), versus the presented extraction method (on the right).
As in Fig. 1, the frames are from top to the bottom: 7, 17, 27. The magnetic
polarity of the different footpoint regions is labelled in the top left panel
in the corresponding colour (black = negative, white = positive). Figure 3:
Comparison of FR appearance as observed in EUV (left panel) and the simulation
(right panel) for AR12473. The EUV image is the composite image constructed
from the AIA 171 Å (red), 211 Å (green) and 193 Å (blue) passbands. Blue and
red arrows indicate similar structures between the AIA images and the
simulation results.

###  3.2 Analysis of AR11176 simulation

Next, we apply the new extraction method to AR11176. The observations show
that AR11176 has a bipolar-like overall structure, featuring a predominantly
north-south running polarity inversion line (PIL) (see Figure 4 (a) & (b)).
Over the course of the simulation window, there is ongoing activity close to
the north-south part of the PIL, which may be tied to flux cancellation seen
in the photospheric magnetic field. During the simulation window, no
significant eruptions were observed, apart from localised jets forming away
from the PIL as a result of flux emergence (Solanki et al., 2020). However,
there was an eruption that took place about two days after the end of the
simulation, on 3 Apr 2011, producing clear flare ribbons along the PIL (see
Figure 4 (c)). Modelling this eruption is out of the scope of the present
study due to the AR reaching too close to the limb, thereby preventing the use
of the direct data-driven approach employed here. The observations suggest the
presence of a filament along the PIL (see Figure 4 (d) & (e)). Curiously,
significant parts of the filament appear to stay intact after the eruption on
April 3rd, and can be seen as a prominence once it rotates to the solar limb
on 5 Apr 2011 (see Figure 4 (f)). The prominence appears to reach relatively
high above the photosphere, a few tens of mega metres, suggesting a large-
scale structure.

Figure 4: The HMI line-of-sight magnetic field for AR11176 plotted in gray
scale within saturation values $\pm$ 500 Gauss (a). Panel (d) shows the
appearance of AR11176 in AIA 304 Å image within the same field-of-view as
panel (a). The region bounded by the blue box in panels (a) and (d) are shown
in panel (b) and (e) respectively. The red dashed line in panels (a) and (b)
is the approximate polarity inversion line drawn on top of the HMI
magnetogram. The green dashed lines in panels (b) and (e) indicate the
location of the filament channel as observed in AIA 304 Å. Panel (c) depicts
the flare ribbons indicated by the blue arrow, as observed in AIA 304 Å
channel during the eruption on 3 April 2011. Panel (f) shows the prominence
(indicated by the white arrow) on 5 April 2011 seen in AIA 304 Å channel.

The evolution of the twist maps (computed in the $y\approx-25.5$ Mm plane)
using the AR11176 TMFM simulation is provided in Fig. 5. In the first panel
(and associated supplementary movie) one can see the early FR formation,
featuring a significantly twisted core with weakly twisted surrounding fields.
As the simulation progresses, the distinction between the FR core and the
surrounding field lines gradually vanishes as the high twist dissipates into
the surrounding field of the same twist polarity. It is worth noting that the
FR in AR11176 features a significantly larger cross-section upon formation
than the AR12473 FR (cf., first panel of Fig. 5 to Fig. 1). This also has the
consequence that the apex of this FR is already at a greater height upon
formation, compared to the AR12473 FR. Once the whole structure has
accumulated sufficient twist such that our method finds a coherent contour
(using a low threshold of $T_{w}\approx 0.4$ due to its less twisted nature),
we identify it as FR (which happens at frame 10; cf. black contours of the top
versus the middle panel in Fig. 5). During its evolution we observe that the
structure retains its original shape rather well (middle and bottom panel),
which is very different from the evolution in the simulation of AR12473.
Furthermore, the FR is expanding, rather than rising, and consequently, the
apex of the FR rises in its early phase significantly slower than the apex of
the FR in AR12473. Note that we chose a higher domain height for the
simulation of AR11176, 300 Mm instead of 200 Mm. This was done to ensure that
the evolution of the FR is not influenced by interactions with the boundaries
in the later stages of its evolution due to its excessive expansion.

Figure 5: Same as Fig. 1, but for the AR11176 FR, showing simulation frames 8,
18 and 28 (from top to bottom).

The FR field lines of AR11176, extracted with the new method, are shown in
Fig. 6. This figure also illustrates the slow build-up of twist and its
expansion, as discussed above. Figure 6 confirms that the FR consists (in the
early to mid stages) of two parts: 1) a highly twisted core that stays
approximately at the same height, and 2) a less twisted envelope that strongly
expands. The height of the highly twisted FR core reaches only a few tens of
megameters above the photosphere during the simulation. This elevation agrees
approximately with the height of the north-south oriented prominence (Figure
4). The apex of the envelope however rises to about 200 megameters, mainly due
to its strong expansion. The results thus suggest that while the simulation
appears to be non-eruptive, the TMFM captures the magnetic field related to
the observed prominence (highly twisted core of the FR) located above the
north-south running PIL.

Figure 6: FR snapshots for the TMFM AR11176 simulation, using the presented
extraction method. The corresponding simulation frames are 8, 18 and 28 (from
top to bottom).

###  3.3 Circularity of Twisted Structures

Next, we investigate the shape of the cross-section of the FR, determined from
the twist maps using the circularity parameter (see Section 2.4). The obtained
circularity values are displayed in Fig. 7. The figure shows that while in the
early stages the FRs have clearly a non-circular shape, they gradually become
in both cases more circular as they rise/expand through the simulation domain.
Note that we do exclude the very late stage evolution from the subsequent
deflection analysis to mitigate tracking unphysical behaviour when the FR
starts to interact with the domain boundary. Similarly, we exclude the very
early stages until the FR is formed and properly detected by our method. The
relevant frames are located in-between the correspondingly coloured dashed
lines for both simulations.

While for the case of AR11176 the FR cross-section stabilises to a nearly
circular shape, for AR12473 the FR deforms as it approaches the top of the
simulation domain. We note that the sharp increase observed in Fig. 7, at the
end of the TMFM simulation for AR12473 could result from the FR deformation as
it reaches the upper boundary of the simulation box and starts to exit the
domain. Additional extractions, avoiding the use of morphological openings
(except in one frame for each simulation, which was necessary to for a
successful tracking), are shown in Fig. 7 as well. Typically, the differences
in circularity are below 0.1, with the exception when a particular sub-feature
of substantial size is purposefully removed. This becomes most apparent for
the AR12473 simulation after frame 10 as well in the very last frames. The
features that were responsible for this behaviour were long, elongated
substructures in the twistmaps, that can also be seen in the middle
(connection with the contour at around $y=75$ Mm) and bottom panel (connection
at approximately $z=75$ Mm, $y=-10$ Mm) of Fig. 1. Cutting off these sub-
features will unavoidably change the circularity of the extracted shapes, but
this is necessary for correctly identifying the FR. We thus conclude that the
circularity is only weakly affected by the applied morphological openings.

Figure 7: Circularity of the extracted cross-sectional shapes, obtained for
the TMFM FRs for AR11176 (in blue) and AR12473 (in orange). The dashed
vertical lines indicate the relevant (i.e., physical) time windows, which are
used in Fig. 8. The dotted lines show the circularity of an additional FR
extraction for each simulation, where no morphological openings were applied.

###  3.4 Deflection Analysis

The results of the derived FR apex trajectories for both AR12473 and AR11176
are displayed in Fig. 8, with the frame numbers being normalised to start from
0, once the FRs have been formed (cf., dashed lines in Fig. 7). From the twist
maps and field line extractions shown in Figs. 1 and 2, respectively, it is
already clear, that the FR in AR12473 rises throughout the simulation and by
the end of the simulation, a part of it has already left the simulation
domain. The deflection angle analysis however shows that the trajectory of the
FR features a high initial angle of propagation. The FR experiences a strong
shift of propagation direction after which it stabilises for most of its
further ascent. This can be seen in Fig. 8, both from the strong early
variation in horizontal coordinates for AR12473, as well as the initially high
angle of propagation, defined as in Sect. 2.5. This early deflection and
subsequent stabilisation of the trajectory is marked by a linear evolution of
the propagation angle and $x,y$ coordinates in this phase. In the later
stages, the structure experiences another deflection, in approximately the
inverse direction of the first one. The AR12473 (orange) graph in the bottom
right panel of Fig. 8 shows that while after the early formation stage, the
angle of propagation is high (approximately $60^{\circ}$), the subsequent
deflections counter the initial phase. In other words, the FR first propagates
clearly non-radially outward but then the propagation direction changes such
that the horizontal coordinates are close to its original ”starting position”.
This results in the FR front ultimately having a very low effective deflection
(ending up with an effective angle of propagation below $10^{\circ}$). In the
bottom left panel of Fig. 8 one can see the AR12473 FR trajectory in 3D and
how the starting and ending phases practically cancel each other.

Figure 8: Trajectory of the AR12473 (orange) and AR11176 (blue) TMFM FRs. The
top left panel shows the height evolution, while the top right panel shows the
evolution of the horizontal components during the ascend of the FRs. The
bottom left panel shows the 3D trajectory in grey, with each computed apex
point marked as orange/blue dot, except the starting and end points marked in
black for better visibility. The bottom right panel depicts the angle of
propagation for each time step. Frame numbers are normalised to start from 0
once the FR has formed.

For the FR in AR11176, on the other hand, the horizontal coordinates of the
apex stay notably constant over time (in particular the x component), although
the results give significant deflection angles (see blue graphs/points Fig.
8). Similar to AR12473, the most dramatic dynamics can be seen in the early
phase. We note that we performed a linear interpolation in the spatial
coordinates of the apex location at frame 17, as it was an outlier. The reason
for this abrupt change of location may be the removal of some overlying newly
”formed” poloidal FR field lines (i.e, either an effect of twisting untwisted
field lines and immediate reconnection thereafter or a smaller overlying
twisted structure moving in and out of the 2D plane). The evolution of the FR
also features a stable path of propagation in the latter phase of its
evolution, which strongly departs from a purely radial direction (or in our
case departs clearly from a parallel direction to the $z$-axis). This then
leads to an effective angle of propagation of about $20^{\circ}$. The full 3D
trajectory can be seen in blue in the bottom left panel of Fig. 8. When
comparing the two simulations, one can see that while AR12473 FR rises about
150 Mm in the relevant time window, the FR in the case of AR11176 rises only
100 Mm in the same time span.

##  4 Discussion

###  4.1 FR Evolution

The updated FR tracking method based on the MM algorithm and the previous
version from Paper 1 both extract the AR12473 FR eruption with largely similar
evolutionary characteristics. The FRs start as a smaller bundle of twisted
loops that continuously thicken and rise as the simulation progresses. In the
later stages, two distinct structures are identified, as seen in the bottom
panels of Fig. 2, where the blue-colored field lines wrap around the beige and
red ones in both cases. Although both tracking methods produced coherent FRs,
one of the biggest differences is that the MM based extraction method captures
more FR field lines and thus, appears notably thicker. Also in terms of field
line connectivity, there are some differences in the early and mid phases of
the simulation, particularly at the positive polarity footpoint (cf. middle
panels in Fig. 2).

The height evolution is similar for the two extraction methods (cf., orange
curve in the top left panel of Fig. 8 with Fig. 3 in Paper I), where
quantitative difference appear mostly because the new method captures more FR
field lines, and thus increases the apex height. In both cases the FR ascents
more or less steadily through the simulation domain, until it reaches a point
where the upward velocity increases. This phase is best visualised for the new
algorithm in the propagation angle plots, being the orange curves in the right
panels of Fig. 8). Due to deflection away from the intrinsic propagation
direction and then back, the FR apex ends up very close to the starting
direction.

The application of the new algorithm to another AR (AR11176) showed that it
can capture dynamical behaviour of coronal fields in two very different types
of events. In contrast to the AR12473 FR, the AR11176 FR evolves significantly
slower and shows more stable behaviour, in accordance with observations. It
forms from a twisted ”core” that steadily dissipates the twist to its
surrounding, forming a relatively large FR in the process. This twisted core
forms at a height of about 50 Mm, while the AR12473 FR forms significantly
lower in the corona, at about 10 Mm (cf. Figs. 1 for AR12473 and 5 for AR11176
as well as Fig. 8). The AR11176 FR evolves also morphologically more steadily,
as both field lines and twist maps show. Figure 7 also supports this finding,
demonstrating that after the initial phase, the FR’s cross-section is nearly
circular (i.e., circularity values are close to zero) throughout the whole mid
to late stage of the simulation. The FR consists of a set of toroidal
(directed along the axis of the FR) field lines, that are partially covered by
overlying field lines that connect to the same footpoint regions (implying
that these overlying field lines indeed belong to the same structure), but
appear as almost poloidal.

We also found that the AR11176 FR rose more steadily (most notably in the
later stages) than AR12473 FR (cf. Fig. 8), as can be seen from the lower
variation of all 3 spatial coordinates and lower variability of the
propagation angle. This also implies, that the FR rises significantly slower.
These findings imply that the FR in the simulation of AR11176 is non-eruptive,
but for a more decisive conclusion, more detailed analysis would be needed
(e.g., as in Daei et al., 2023). This is especially relevant because of the
characteristically unrealistic slow evolution of TMFM FRs (see, e.g., Pomoell
et al. (2019) or Paper I, where a temporal normalization was introduced to
deal with this problem). We however note that the slower evolution of the
AR11176 FR, compared to the AR12473 FR, matches well with the observations in
the corresponding time windows, namely the presence of a large and non-
eruptive filament/prominence that formed at the north-southward directed part
of an extended PIL in the AR11176.

###  4.2 FR Extraction Method

The new extraction scheme shows clear improvements, as discussed above and in
Sect. 2.4. Firstly, one of the key improvements is that the updated method
allows non-circular cross-sections. For the investigated events we found the
flux ropes to be most circular during the mid to late rising phase, but
otherwise clearly non-circular cross-sections were found.

Secondly, although our current MM-based implementation is semi-automated
(automated main extraction but case-by-case refinement of the parameters), it
was not only found to be more accurate (i.e., we capture more of the relevant
magnetic field lines), but also computationally more effective than the
previous method. One could further speed up the MM algorithms by implementing
entire sequences of erosion/dilation or opening/closing operations with small
SEs instead of performing a single operation with one large SE (Shih &
Kowalski, 2003). We note though that even without this possible optimisation,
all computations run within seconds, using an ordinary laptop, as long as
reasonable SE sizes are used.

###  4.3 Utility of MM algorithms for pre- and post-processing

Figure 9 provides some motivation for the choice of proxy as well as insights
into the extraction procedure as a whole. It is noteworthy that we do not
incorporate the squashing-factor (short Q-factor, see e.g., Titov et al.,
2002; Démoulin et al., 1996) into our extraction scheme here. Because the
extraction is based on thresholding, we require the whole FR in the extraction
plane to exceed some critical value. Furthermore, it is favourable, if the FR
exhibits sharp boundaries to allow for some flexibility in the choice of the
threshold. This is one particular strength of $T_{w}$ combined with the
morphological gradient: The MM gradient with different structuring elements
(panel c) and e) on the left side of Fig. 9) serves as an adjustable
sharpening routine to the twist maps (cf. original map in panel b) with the
sharpened maps in panels d) and f)). The Q-factor map in panel a) appears
structurally similar as $T_{w}$, but the outlines of highlighted features
often do not fully close (see e.g., on the top right of the FR cross-section
at $(y,z)=(0,75)$ Mm). While it may still be possible to use it in combination
with the twistmaps in a similar manner as the morphological gradient, it lacks
the tune-ability and forces the introduction of a scaling parameter since the
log(Q) maps cover different and larger ranges of values. The tune-ability with
choice of different structuring elements is distinctly different from a
scaling parameter as this also structurally affects the maps. Hence, we chose
to use $T_{w}$ and the MM gradient in this work. We avoid calculating the
winding number about a defined FR axis field line, as this formulation of the
twist number elevates the complexity significantly and hinders a feasibly fast
implementation, but note that tools for the computation of this version do
exist (e.g., in Price et al., 2022) and may very well be used in combination
with our extraction method.

The opening algorithm is used to smooth the extracted outlines and most
importantly to remove unwanted features. An example of this is the high-twist
channel seen in the panels of Fig. 9 that connects to the main FR body at a
height of about 50 Mm and $y$-location between 50 and 100 Mm. This channel
represents a set of twisted field lines that clearly do not belong to the FR.
Such connections may not only lead to an incorrect extraction at this
particular frame, but can cause the tracking to fail too (as in subsequent
frames, the algorithm may continue to track this channel-feature instead and
not the main FR in case it disconnects from it). The opening algorithm helps
to disconnect this feature entirely in the affected frame. Most importantly,
this processing can be applied only in the affected sub-region of the image,
and thus one can keep unproblematic regions unaffected. This is illustrated by
the bottom panels in Fig. 9, where one can see the original (after the
extraction) on the left and the post-processed version on the right. Here, an
opening with a small SE was applied to the whole image, to remove small
artefacts (see e.g., near $z=0\;$Mm), as well as an opening of bigger size to
a sub-region of the image to remove the twisted ”arm” of the structure. In
fact, for the shown case of the AR12473 TMFM FR, the opening algorithm had to
be applied for this exact reason in about half of the frames, thus it
significantly contributed to improving the extraction procedure. Apart from
frames, where connections like these were significant, the circularity of the
extracted contours in the twistmaps were only weakly affected by the applied
openings (cf., Fig. 7). This is in strong contrast to the AR11176 FR, where
significantly less post-processing was required.

Figure 9: Comparison of different parameters and processing for frame 16 of
the AR12473 TMFM simulation in the plane of $x\approx 0.7$ Mm: Panel a) shows
the $\log(Q)$-map, and panel b) the twist-map. The morphological gradient
using big structuring elements is shown in panel c), and its combination with
the twist-map from b) is shown in panel d). Panel e) is the same as panel c),
but with smaller structuring elements and panel f) is the combination of
$T_{w}$ from panel b) with the morphological gradient in panel e). Finally, in
panel g) there is the mask of high twist areas as extracted from the
twist+gradient map in panel d) (using a threshold of 0.8), and panel h) shows
the same mask, but after post-processing, mostly removing the twisted channel
that connects to the main structure at about $y=75\;$Mm.

##  5 Conclusion and Outlook

In this work, we have developed and tested a new flux rope extraction and
tracking method for 3D solar magnetic field simulations. The extraction is
based on the twist parameter $T_{w}$ combined with mathematical morphology
algorithms.

We applied the new algorithm to the data-driven magnetofrictional model
simulation output of the AR12473 eruption on 28 Dec 2015, i.e., the same event
that was used to test the previous method presented in Wagner et al. (2023).
To further test the new algorithm we also applied it to AR11176. We
investigated both AR AR12473 and AR11176 flux ropes in terms of their
appearance, motion through the modelling domain and deflection. These events
presented quite different dynamics, the AR AR12473 resulting in a strong
dynamics and clear flux rope eruption while the AR11176 FR was more stable in
terms of its movement and we theorise it may be a confined eruption without
further driving the model through the ARs main eruptive phase.

The results for both active regions are in accordance with the EUV
observations. Our modelling and extraction attempts thus capture both the
formation phase and early rise of the flux ropes, as well as the eruptive
phase in the case of AR12473.

While the results in Sect. 3 illustrate that the method robustly extracts the
flux ropes, fine-tuning of the parameters (probably on case-by-case basis) may
still be necessary for the best extractions. Full-automation as well as
exploring a broader space of flux rope proxies such as the Q-factor, winding
number and magnetic flux density and possible combinations of the
aforementioned quantities with each other as well as with mathematical
morphology algorithms are avenues for further improvements.

###### Acknowledgements.

This work is part of the SWATNet project funded by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 955620. The work is supported by Academy of Finland
Centre of Excellence FORESAIL (grant 336807). EK and AK acknowledges European
Research Council under the European Union’s Horizon 2020 research and
innovation programme, grant 724391 (SolMAG). RS acknowledges support from the
project EFESIS under the Academy of Finland Grant 350015. TB and RG
acknowledge the support by Fundação para a Ciência e a Tecnologia (FCT)
through the research grants UIDB/04434/2020 and UIDP/04434/2020. R.E. is
grateful to Science and Technology Facilities Council (STFC, grant No.
ST/M000826/1) UK and acknowledges NKFIH (OTKA, grant No. K142987) Hungary for
enabling this research. JP acknowledges Academy of Finland project SWATCH
(343581). We acknowledge the use of the data from the Solar Dynamics
Observatory (SDO). SDO is the first mission of NASA’s Living With a Star (LWS)
program. The SDO/AIA and SDO/HMI data are publicly available from NASA’s SDO
website https://sdo.gsfc.nasa.gov/data/). O.O. acknowledges support by the FCT
– Fundação para a Ciência e a Tecnologia, I.P., undeProjects Nos.
UIDB/04564/2020, UIDP/04564/2020 and CERN/FIS-COM/0029/2017. AK’s research was
supported by an appointment to the NASA Postdoctoral Program at the the NASA
Goddard Space Flight Center (GSFC). AK acknowledges the University of Helsinki
Three-year Grant. S.P. acknowledges support form the projects C14/19/089 (C1
project Internal Funds KU Leuven), G.0B58.23N and G.0025.23N (WEAVE) (FWO-
Vlaanderen), 4000134474 (SIDC Data Exploitation, ESA Prodex-12), and Belspo
project B2/191/P1/SWiM. We also want to thank the referee for their feedback,
which lead to improving the manuscript.

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# On some Frobenius type divisibility results in a premodular category

Sebastian Burciu  Inst. of Math. “Simion Stoilow” of the Romanian Academy P.O.
Box 1-764, RO-014700, Bucharest, Romania<EMAIL_ADDRESS>
## Abstract

In this note some new Frobenius type divisibility results are obtained for
premodular categories. In particular, we extend [Yu20, Corollary 3.4] from the
settings of super-modular categories to arbitrary pseudo-unitary premodular
categories.

##  1\. Introduction

In [Yu20] the author showed that in a slightly-degenerate braided fusion
category one has that $\frac{\dim(\mathcal{C})}{2\dim(Y)^{2}}\in\mathbb{A}$
for any simple object $Y$ of $\mathcal{C}$. The main goal of this note is to
generalize this result in the settings of a premodular fusion category with
pointed Müger center, i.e. all the objects of this center are invertible.

Our first main result is the following:

######  Theorem 1.1.

Let $\mathcal{C}$ be a premodular category and ${\mathcal{D}}$ a fusion
subcategory of $\mathcal{C}$. If
${\mathcal{D}}\cap{\mathcal{Z}_{2}(\mathcal{C})}=\mathrm{Vec}$ then

(1.2) $\frac{\dim(\mathcal{C})}{\dim(Y)^{2}}\in\mathbb{A}$

for any simple object $Y$ in ${\mathcal{D}}$.

Moreover, in the case of a pointed Müger center we prove that:

######  Theorem 1.3.

Let $\mathcal{C}$ be a pseudo-unitary premodular fusion category with
$\mathcal{Z}_{2}(\mathcal{C})\subseteq{\mathcal{C}_{\mathrm{pt}}}$.

  1. (1)

For any simple object $\mathcal{C}$ of $Y$ one has

(1.4)
$\frac{\mathrm{FPdim}(\mathcal{C})\mathrm{FPdim}(\mathcal{Z}_{2}(\mathcal{C}))}{\mathrm{FPdim}(Y)^{2}}\in\mathbb{A}.$

  2. (2)

If the action of $\mathcal{Z}_{2}(\mathcal{C})$ on the simple objects of
$\mathcal{C}$ is free then

(1.5)
$\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}(\mathcal{Z}_{2}(\mathcal{C}))\mathrm{FPdim}(Y)^{2}}\in\mathbb{A}.$

Recall that ${\mathcal{C}_{\mathrm{pt}}}$ denotes the maximal pointed
subcategory of $\mathcal{C}$. Note that the second item of the above Theorem
generalizes [Yu20, Corollary 3.4] in the case of a pseudo-unitary premodular
category.

Shortly, this note is organized as follows. In Section 2 we recall the basic
properties of fusion categories needed through this paper. In Subsection 2.1
we briefly recall the adjoint algebra and the characters of simple objects of
pivotal fusion categories as defined in [Shi17].

In Section 3 we construct a Hecke type algebras associated to the right cosets
of a fusion subcategory of a given fusion category. In the same section we
prove orthogonality relations for Hecke algebras. In Section 4 we prove the
main results stated above.

##  2\. Preliminaries

In this section we recall the main properties of premodular fusion categories
that are needed through the paper. For the basic theory of fusion categories,
we refer the reader to [EGNO15]. Recall that a fusion category is by
definition, a semisimple finite tensor category.

Throughout this note $\mathcal{C}$ denotes a fusion category over $\mathbb{C}$
with eventual additional properties. Let
$\mathrm{Irr}(\mathcal{C}):=\\{X_{0},X_{1},\dots,X_{m}\\}$ be a complete set
of representatives for the isomorphism classes of simple objects of
$\mathcal{C}$. By $K_{0}(\mathcal{C})$ we denote the Grothendieck ring of
$\mathcal{C}$ and let
$K(\mathcal{C}):=K_{0}(\mathcal{C})\otimes_{\mathbb{Z}}\mathbb{C}$.

Recall that a pivotal structure of a rigid monoidal category $\mathcal{C}$ is
an isomorphism $j:\operatorname{id}_{\mathcal{C}}\rightarrow()^{**}$ of
monoidal functors. A pivotal monoidal category is a rigid monoidal category
endowed with a pivotal structure. For any pivotal structure one can construct
categorical dimensions $\dim(X)$ of any object $X$ of $\mathcal{C}$. In fact,
the map $\dim:K(\mathcal{C})\rightarrow\mathbb{C},[X]\mapsto\dim(X)$ is an
algebra morphism.

A pivotal structure a on a fusion category $\mathcal{C}$ is called spherical,
see [EGNO15] if $\dim(V)=\dim(V^{*})$ for any object $V$ of $\mathcal{C}$. A
tensor category is spherical if it is equipped with a spherical
structure.Given a fusion category there is a unique algebra morphism
$\mathrm{FPdim}:K(\mathcal{C})\rightarrow\mathbb{C}$ such that
$\mathrm{FPdim}(X_{i})>0$ for any $X_{i}\in\mathrm{Irr}(\mathcal{C})$. Then
$\mathrm{FPdim}$ is called the Frobenius-Perron morphism.

Recall that a braided fusion category is called premodular if it has a
spherical structure. Equivalently, a premodular category is a braided fusion
category equipped with a twist (also called a balanced structure), see
[EGNO15, Section 8.10 ] for details.

A fusion category $\mathcal{C}$ is called pseudo-unitary if
$\mathrm{FPdim}(\mathcal{C})=\dim(\mathcal{C})$. If such is the case, then by
[ENO05, Proposition 8.23], $\mathcal{C}$ admits a unique spherical structure
with respect to which the categorical dimensions of each simple coincides with
its Frobenius-Perron dimension, i.e. $\mathrm{FPdim}(X)=\dim(X)$ for any
object $X$ of $\mathcal{C}$.

###  2.1. Adjoint algebra and internal characters

Let $\mathcal{C}$ be a fusion category and ${\mathcal{Z}(\mathcal{C})}$ be its
Drinfeld center. The forgetful functor
$F:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{C}$ admits a right adjoint
functor $R:\mathcal{C}\rightarrow{\mathcal{Z}(\mathcal{C})}$ and let
$Z:=FR:\mathcal{C}\rightarrow\mathcal{C}$. Then $Z$ is a Hopf comonad and by
[Shi17, Section 2.6] one has that

(2.1) $Z(V)\simeq\int_{X\in\mathcal{C}}X\otimes V\otimes X^{*}$

It is known that $A:=R(\mathbf{1})$ has the structure of central commutative
algebra in $\mathcal{Z}(\mathcal{C})$.

We denote by $m:A\otimes A\rightarrow A$ the multiplication and by
$u:\mathbf{1}\rightarrow A$ the unit of the adjoint algebra $A=Z(\mathbf{1})$.

The vector space
$\mathrm{{CE}(\mathcal{C})}:=\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},A)$ is
called the set of central elements. For $a,b\in\mathrm{{CE}(\mathcal{C})}$ ,
we set $a\textperiodcentered b:=m\circ(a\otimes b)$. Then the set
$\mathrm{{CE}(\mathcal{C})}$ is a $\mathbb{C}$-algebra with respect to this
operation. The space
$\mathrm{{CF}}(\mathcal{C}):=\mathrm{{Hom}}_{\mathcal{C}}(A,\mathbf{1})$ is
called the space of class functions of $\mathcal{C}$. The space
$\mathrm{{CF}}(\mathcal{C})$ is a $\mathbb{C}$-algebra with the multiplication
defined $f\star g:=f\circ Z(g)\circ\delta_{\mathbf{1}}.$ for any
$f,g\in\mathrm{{CF}}(\mathcal{C})$. Here $\delta:Z\rightarrow Z^{2}$ is the
comultiplication structure of the Hopf comonad $Z$, see [Shi17] for details.
To any object $X$ of $\mathcal{C}$ Shimizu assigned in [Shi17, Section 3] a
class function $\mathrm{ch}(X)$ called the internal character. It is defined
as a partial trace of the canonical left action $\rho_{X}:A\otimes
X\rightarrow\mathbf{1}$. By [Shi17, Theorem 3.10] one has that
$\mathrm{ch}(X\otimes Y)=\mathrm{ch}(X)\mathrm{ch}(Y)$ for any two objects $X$
and $Y$ of $\mathcal{C}$ and $K(\mathcal{C})\simeq\mathrm{{CF}}(\mathcal{C})$
as algebras.

There is a also non-degenerate a paring
$\langle\;,\;\rangle:\mathrm{{CF}}(\mathcal{C})\times\mathrm{{CE}(\mathcal{C})}\rightarrow\mathbf{1}$,
given by $\langle f,a\rangle\operatorname{id}_{\mathbf{1}}=f\circ a,$ for all
$f\in\mathrm{{CF}}(\mathcal{C})$ and $a\in\mathrm{{CE}(\mathcal{C})}$.

For any simple object $X_{i}$ of $\mathcal{C}$ we denote by
$\chi_{i}:=\mathrm{ch}(X_{i})$ its associated character. The central element
space $\mathrm{{CE}(\mathcal{C})}$ has a basis of primitive orthogonal
idempotents $E_{i}$ such that $<\chi_{i},E_{j}>=\delta_{i,j}d_{i}$ for all
$i,j$.

Recall $R:\mathcal{C}\rightarrow{\mathcal{Z}(\mathcal{C})}$ is a right adjoint
to the forgetful functor $F:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{C}$.
As explained in [Shi17, Theorem 3.8] this adjunction gives an isomorphism of
algebras

(2.2)
$\mathrm{{CF}}(\mathcal{C})\xrightarrow{\cong}\mathrm{End}_{{\mathcal{Z}(\mathcal{C})}}(R(\mathbf{1})),\;\;\chi\mapsto
Z(\chi)\circ\delta_{\mathbf{1}}.$

Since ${\mathcal{Z}(\mathcal{C})}$ is also fusion category we can write
$R(\mathbf{1})=\bigoplus_{j=0}^{m}\mathcal{C}^{j}$ as a direct sum of simple
objects in ${\mathcal{Z}(\mathcal{C})}$. Recall that $\mathcal{C}^{j}$ are
called conjugacy classes  for $\mathcal{C}$.

For the rest of this section let $\mathcal{C}$ be a fusion category with a
commutative Grothendieck ring $K(\mathcal{C})$. Since
$\mathrm{{CF}}(\mathcal{C})$ is also commutative semisimple
$\mathbb{C}$-algebra has

(2.3) $\mathrm{{CF}}(\mathcal{C})\simeq\bigoplus_{j=0}^{m}\mathbb{C}F_{j}$

where $F_{j}$ are the central primitive idempotents of $\mathcal{C}$. We
define $\mathcal{J}:=\\{0,\dots,m\\}$.

Since $\mathrm{{CF}}(\mathcal{C})$ is a commutative algebra it follows from
Equation (2.2) that $R(\mathbf{1})$ is multiplicity free, i.e. each conjugacy
class $\mathcal{C}^{j}$ appears with multiplicity $1$ inside $R(\mathbf{1})$.
Moreover we fix the canonical bijection $F_{j}\leftrightarrow\mathcal{C}^{j}$
for which the image of $F_{j}$ under the canonical adjunction isomorphism from
Equation (2.2) is the projection of $R(\mathbf{1})$ on $\mathcal{C}^{j}$.

With this bijection, the second orthogonality from [Shi17, Theorem 6.10] can
be written as:

(2.4)
$\sum_{i=0}^{m}\mu_{l}(\chi_{i})\mu_{k}(\chi_{i^{*}})=\delta_{l,k}\frac{\dim(\mathcal{C})}{\dim(\mathcal{C}^{k})}$

where $\mu_{j}:\mathrm{{CF}}(\mathcal{C})\rightarrow\mathbb{C}$ is the set of
algebra maps of $\mathrm{{CF}}(\mathcal{C})$. In terms of Equation (2.3) one
has $\mu_{j}(F_{k})=\delta_{j,k}$.

Following [Bur20] we use the following definition:

######  Definition 2.5.

To any fusion subcategory ${\mathcal{D}}$ we associate a set of indices
$\mathcal{J}_{\mathcal{D}}\subseteq\mathcal{J}$ such that:

(2.6) $\lambda_{{\mathcal{D}}}=\sum_{j\in\mathcal{J}_{\mathcal{D}}}F_{j}.$

In this settings, Equation (4.20) from [Bur20] gives that

(2.7)
$\sum\limits_{j\in\mathcal{J}_{{\mathcal{D}}}}\dim(\mathcal{C}^{j})=\frac{\dim(\mathcal{C})}{\dim({\mathcal{D}})}.$

##  3\. Hecke algebras of cosets

For a fusion subcategory ${\mathcal{D}}$ of $\mathcal{C}$ we denote by
$\simeq^{r}_{\mathcal{D}}$ the equivalence relation on the set of simple
objects $\mathrm{Irr}(\mathcal{C})$ of $\mathcal{C}$. It is given by
$X\simeq^{r}_{\mathcal{D}}Y$ if and only if there is a simple object $S$ of
${\mathcal{D}}$ such that $X$ is a constituent of $Y\otimes S$. An equivalence
class of $\simeq^{r}_{\mathcal{D}}$ is called a right coset of $\mathcal{C}$
with respect to ${\mathcal{D}}$. We denote by
${\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$ the set of equivalence
classes (right cosets) with respect to ${\mathcal{D}}$. It was shown in [BB15]
that

(3.1)
$X\simeq^{r}_{{\mathcal{D}}}Y\iff\frac{[X]R_{\mathcal{D}}}{\mathrm{FPdim}(X)}=\frac{[Y]R_{\mathcal{D}}}{\mathrm{FPdim}(Y)}$

where $R_{\mathcal{D}}$ is the regular element of the fusion subcategory
${\mathcal{D}}$,
$R_{\mathcal{D}}=\sum_{X\in\mathrm{Irr}({\mathcal{D}})}\mathrm{FPdim}(X)[X]\in
K(\mathcal{C})$. In this case it can be shown that

(3.2)
$\frac{[X]R_{\mathcal{D}}}{\mathrm{FPdim}(X)}=\frac{[Y]R_{\mathcal{D}}}{\mathrm{FPdim}(Y)}=\frac{R_{m}}{\mathrm{FPdim}(R_{m})}$

where $m\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$ is the right
coset corresponding to $X$ and $Y$ and and $R_{m}:=\sum_{Z\in
m}\mathrm{FPdim}(Z)[Z]\in K(\mathcal{C})$ is the regular element corresponding
to any coset $m$.

Based on Equation (3.2) it is easy to verify that if $K(\mathcal{C})$ is
commutative then for all
$m,n,p\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$

(3.3)
$\frac{R_{m}}{\mathrm{FPdim}(R_{m})}\frac{R_{n}}{\mathrm{FPdim}(R_{n})}=\sum_{p\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}}H^{p}_{mn}\frac{R_{p}}{\mathrm{FPdim}(R_{p})}$

where $H^{p}_{mn}$ are (integral) scalars given by

$H^{p}_{mn}=\big{(}\sum_{Z\in p}\mathrm{FPdim}(Z)N^{Z}_{XY}\big{)}$

where $X\in m$ and $Y\in n$ are arbitrarily chosen representatives for each
coset $m,n\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$. Thus in the
case $K(\mathcal{C})$ is commutative, the vector subspace
$\mathcal{H}({\mathcal{D}}):=\\{R_{m}\;\big{|}\;m\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}\\}$
of $K(\mathcal{C})$ generated by the regular elements $R_{m}$ is in fact a
subalgebra of $K(\mathcal{C})$. It is called the Hecke algebra of
$\mathcal{C}$ with respect to ${\mathcal{D}}$. Note that Equation (3.2)
implies $\mathcal{H}({\mathcal{D}})=K(\mathcal{C})R_{{\mathcal{D}}}$ inside
$K(\mathcal{C})$.

######  Proposition 3.4.

Let ${\mathcal{D}}$ be a fusion subcategory of a pseudo-unitary fusion
category $\mathcal{C}$ with a commutative Grothendieck ring. Then the
dimension of the Hecke algebra $\mathcal{H}({\mathcal{D}})$ equals the
cardinality of $\mathcal{J}_{\mathcal{D}}$.

###### Proof.

If $\mathcal{C}$ is pseudo-unitary then as above
$K(\mathcal{C})\simeq\mathrm{{CF}}(\mathcal{C})$ and $R_{\mathcal{D}}$
corresponds to the idempotent cointegral $\lambda_{\mathcal{D}}$. Then from
Equation (2.3) in this case one has:

$\mathcal{H}({\mathcal{D}})=\mathrm{{CF}}(\mathcal{C})\lambda_{\mathcal{D}}=\bigoplus_{j\in\mathcal{J}_{\mathcal{D}}}\mathrm{{CF}}(\mathcal{C})F_{j}=\bigoplus_{j\in\mathcal{J}_{\mathcal{D}}}\mathbb{C}F_{j}$

by Equation (2.6). ∎

### General Hecke orthogonality

For any $t\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$ let
$t^{*}:=\\{X^{*}\;|\;X\in t\\}.$ It is easy to see that in the case of a
commutative Grothendieck ring, if
$t\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$ is a right coset
then $t^{*}$ is also a right coset. Indeed, since $K(\mathcal{C})$ is
commutative one has
$([X])R_{\mathcal{D}})^{*}=R_{\mathcal{D}}^{*}[X]^{*}=R_{\mathcal{D}}[X^{*}]$
since $R_{\mathcal{D}}^{*}=R_{\mathcal{D}}$. For any
$t\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$ choose a
representative simple object $X_{t}$ of $\mathcal{C}$ belonging to this coset
such that $X_{t}^{*}={X_{t^{*}}}$. Denote also
$\chi_{t}:=\chi(X_{t})\in\mathrm{{CF}}(\mathcal{C})$.

Equation (2.6) implies that

$\mathcal{H}({\mathcal{D}})=\mathrm{{CF}}(\mathcal{C})\lambda_{\mathcal{D}}=\bigoplus_{j\in\mathcal{J}_{\mathcal{D}}}\mathbb{C}F^{j}$

Therefore $\widehat{\mathcal{H}({\mathcal{D}})}=\\{\mu_{j},\;|\;j\in
J_{{\mathcal{D}}}\\}$ are the all linear characters of
$\mathcal{H}({\mathcal{D}})$.

######  Theorem 3.5.

Let ${\mathcal{D}}\subseteq\mathcal{C}$ be a fusion subcategory of a pseudo-
unitary fusion category $\mathcal{C}$ with commutative Grothendieck ring. Then
we have the following:

  1. (1)

The first orthogonality relation for $\mathcal{H}({\mathcal{D}})$:

(3.6)
$\sum_{t\in{{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}}}\frac{\mathrm{FPdim}(R_{t})}{\mathrm{FPdim}(X_{t})^{2}}\mu_{k}(\chi_{t})\mu_{l}(\chi_{t^{*}})=\delta_{l,k}\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}(\mathcal{C}^{k})},\;\text{for
any}\;k,l\in{\mathcal{J}_{{\mathcal{D}}}},$

where $R_{t}=\sum_{\chi_{i}\in\mathcal{D}_{t}}d_{i}\chi_{i}$.

  2. (2)

The second orthogonality relation for $\mathcal{H}({\mathcal{D}})$:

(3.7)
$\sum_{k=0}^{r}\mathrm{FPdim}(\mathcal{C}^{k})\mu_{k}({\chi_{t}})\mu_{k}(\chi_{s^{*}})=\delta_{s,t}\frac{\mathrm{FPdim}(X_{t})\mathrm{FPdim}(X_{s})\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}(R_{t})},\;\text{for
all}\;t,s\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}.$

###### Proof.

Since $\mathcal{C}$ is a pseudo-unitary fusion category one has
$\lambda_{\mathcal{D}}=\frac{R_{\mathcal{D}}}{\mathrm{FPdim}(R_{\mathcal{D}})}$
for any fusion subcategory and
$\dim(\mathcal{C})=\mathrm{FPdim}(\mathcal{C})$. Moreover
${\mathcal{Z}(\mathcal{C})}$ is also pseudo-unitary and therefore
$\dim(\mathcal{C}^{k})=\mathrm{FPdim}(\mathcal{C}^{k})$ for any conjugacy
class of $\mathcal{C}$.

Let $j\in\mathcal{J}_{\mathcal{D}}$. If $X_{i}\in t$ for some
$t\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$ by applying
$\mu_{j}$ to Equation (3.1) one has

(3.8)
$\frac{\mu_{j}(\chi_{i})}{\mathrm{FPdim}(X_{i})}=\frac{\mu_{j}(\chi_{t})}{\mathrm{FPdim}(X_{t})}=\mu_{j}(\frac{R_{t}}{\mathrm{FPdim}(R_{t})}).$

Note that since $j\in{\mathcal{J}_{{\mathcal{D}}}}$ Equation (2.6) implies
that $\mu_{j}(\lambda_{\mathcal{D}})=1$.

The second orthogonality relation for $\mathcal{C}$ from Equation (2.4) can be
written as

$\sum_{t\in{{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}}}\big{(}\sum_{X_{i}\in
t}\mu_{l}(\chi_{i})\mu_{k}(\chi_{i^{*}})\big{)}=\delta_{l,k}\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}(\mathcal{C}^{k})}.$

If $l,k\in\mathcal{J}_{{\mathcal{D}}}$, using Equation (3.8) one obtains
Equation (3.6) from above. Indeed,

$\displaystyle\delta_{l,k}\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}(\mathcal{C}^{k})}$
$\displaystyle=$
$\displaystyle\sum_{t\in{{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}}}\sum_{X_{i}\in
t}\mu_{l}(\chi_{i})\mu_{k}(\chi_{i^{*}})$ $\displaystyle=$
$\displaystyle\sum_{t\in{{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}}}\sum_{X_{i}\in
t}\frac{\mathrm{FPdim}(X_{i})\mu_{l}(\chi_{t})}{\mathrm{FPdim}(X_{t})}\frac{\mathrm{FPdim}(X_{i})\mu_{k}(\chi_{t^{*}})}{\mathrm{FPdim}({X_{t^{*}}})}$
$\displaystyle=$
$\displaystyle\sum_{t\in{{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}}}\frac{\mu_{k}(\chi_{t^{*}})\mu_{l}(\chi_{t})}{\mathrm{FPdim}(X_{t})^{2}}\big{(}\sum_{X_{i}\in
t}\mathrm{FPdim}(X_{i})^{2}\big{)}=$ $\displaystyle=$
$\displaystyle\sum_{t\in{{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}}}\frac{\mu_{k}(\chi_{t^{*}})\mu_{l}(\chi_{t})}{\mathrm{FPdim}(X_{t})^{2}}\mathrm{FPdim}(R_{t})$

The second orthogonality relation follows as usually from the fact that for
complex matrices $(BA=I_{n}\implies AB=I_{n})$. Indeed, denote

$a_{tk}:=\sqrt{\frac{\mathrm{FPdim}(\mathcal{C}^{k})\mathrm{FPdim}(R_{t})}{\mathrm{FPdim}(\mathcal{C})}}\frac{\mu_{k}(\chi_{t})}{\mathrm{FPdim}(X_{t})},\;k\in\mathcal{J}_{\mathcal{D}},t\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$

and

$b_{lt}:=\sqrt{\frac{\mathrm{FPdim}(\mathcal{C}^{l})\mathrm{FPdim}(R_{t})}{\mathrm{FPdim}(\mathcal{C})}}\frac{\mu_{l}(\chi_{t^{*}})}{\mathrm{FPdim}({X_{t^{*}}})},\;l\in\mathcal{J}_{\mathcal{D}},t\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}.$

With these notations it is easy to see that the equality $BA=I_{n}$ (where
$n:=|{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}|=|\mathcal{J}_{\mathcal{D}}|$)
is equivalent to the first orthogonality relations. Then the second
orthogonality relations are equivalent to $AB=I_{n}$.

Indeed,

$\displaystyle\delta_{s,t}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{m}a_{tk}b_{ks}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{m}\sqrt{\frac{\mathrm{FPdim}(\mathcal{C}^{k})\mathrm{FPdim}(R_{t})}{\mathrm{FPdim}(\mathcal{C})}}\frac{\mu_{k}(\chi_{t})}{\mathrm{FPdim}(X_{t})}\sqrt{\frac{\mathrm{FPdim}(\mathcal{C}^{k})\mathrm{FPdim}(R_{s})}{\mathrm{FPdim}(\mathcal{C})}}\frac{\mu_{k}(\chi_{s^{*}})}{\mathrm{FPdim}({X_{s^{*}}})}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{\mathrm{FPdim}(R_{t})\mathrm{FPdim}(R_{s})}}{\mathrm{FPdim}(X_{t})\mathrm{FPdim}(X_{s})\mathrm{FPdim}(\mathcal{C})}\sum_{k=0}^{m}\mathrm{FPdim}(\mathcal{C}^{k})\mu_{k}(\chi_{t})\mu_{k}(\chi_{s^{*}})$

∎

######  Corollary 3.9.

Let ${\mathcal{D}}$ be a fusion subcategory of a pseudo-unitary fusion
category $\mathcal{C}$. With the above notations one has:

  1. (1)

For any right coset $t\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$
and any representative $X_{t}\in t$ one has

(3.10)
$\frac{\mathrm{FPdim}(X_{t})^{2}\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}(R_{t})}\in\mathbb{A}.$

  2. (2)

Suppose that ${\mathcal{D}}\subseteq{\mathcal{C}_{\mathrm{pt}}}$ acts freely
on the simple objects of $\mathcal{C}$. Then

(3.11)
$\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}({\mathcal{D}})\mathrm{FPdim}(\mathcal{C}^{j})}\in\mathbb{A}$

for all $j\in\mathcal{J}_{{\mathcal{D}}}$.

###### Proof.

  1. (1)

It follows from Equation (3.7) for $s=t$ since
$\mathrm{FPdim}(\mathcal{C}^{k})\in\mathbb{A}$ and
$\mu_{k}(\chi_{t})\in\mathbb{A}$.

  2. (2)

Note that in this case, since ${\mathcal{D}}$ acts freely, one has
$\mathrm{FPdim}(R_{m})=\mathrm{FPdim}({\mathcal{D}})\mathrm{FPdim}(X_{m})^{2}.$
Then equation (3.6) can be written as

$\sum_{t\in{{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}}}\mu_{k}(\chi_{t})\mu_{l}(\chi_{t^{*}})=\delta_{l,k}\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}({\mathcal{D}})\mathrm{FPdim}(\mathcal{C}^{k})},\;\text{for
any}\;k,l\in{\mathcal{J}_{{\mathcal{D}}}}.$

Put $k=l$ above, and since $\mu_{k}(\chi_{t})\in\mathbb{A}$ the divisibility
result follows.

∎

######  Lemma 3.12.

Suppose that ${\mathcal{D}},\mathcal{A}$ are fusion subcategories of a fusion
category $\mathcal{C}$. If
$m\in{\big{(}\mathcal{C}\big{/}{\mathcal{D}}\big{)}_{r}}$ is a right coset of
$\mathcal{C}$ with respect to ${\mathcal{D}}$ then the set $\mathcal{A}\cap
m$, if not empty, is a right coset $\mathcal{A}$ with respect to
$\mathcal{A}\cap{\mathcal{D}}$.

###### Proof.

Let $X,Y\in\mathcal{A}\cap m$ be any two simple objects. We have to show that
$X\simeq^{r}_{\mathcal{A}\cap{\mathcal{D}}}Y$ as objects of $\mathcal{A}$.
Since $X,Y$ are in the same right coset with respect to ${\mathcal{D}}$, by
definition, there is $Z\in{\mathcal{D}}$ such that $m(X,Y\otimes Z)>0$. This
implies $m(Z,Y^{*}\otimes X)>0$ which in turn gives that $Z\in\mathcal{A}$
since both $X,Y\in\mathcal{A}$. Thus $Z\in{\mathcal{D}}\cap\mathcal{A}$. It
follows that $X,Y$ are in the same coset of $\mathcal{A}$ with respect to
$\mathcal{A}\cap{\mathcal{D}}$. ∎

##  4\. Divisibility results for premodular categories

###  4.1. Definition of the braided partition function $M$

Let $\mathcal{C}$ be a premodular (i.e. braided and spherical) fusion
category. We keep all the notations from the previous section, in particular
$\mathrm{Irr}(\mathcal{C})=\\{X_{0},X_{1},\dots,X_{m}\\}$ and
$d_{i}:=\dim(X_{i})$ for all $i$. By [Shi17, Example 6.14] there is
$\mathbb{C}$-algebra map
$f_{Q}:{\mathrm{{CF}}(\mathcal{C})}\rightarrow\mathrm{{CE}(\mathcal{C})}$
given by the following formula:

(4.1)
$f_{Q}(\chi_{i})=\sum_{i^{\prime}=0}^{m}\frac{s_{ii^{\prime}}}{d_{i^{\prime}}}E_{i^{\prime}}.$

where $S=(s_{ij})$ is the $S$-matrix of $\mathcal{C}$.

Given a fusion subcategory ${\mathcal{D}}$ of a braided fusion category
$\mathcal{C}$, the notion of Müger centralizer of ${\mathcal{D}}$ was
introduced in [DGNO10]. Two objects $X$ and $Y$ of $\mathcal{C}$ centralize
each other if $c_{X,Y}c_{Y,X}=\mathrm{id}_{X\otimes Y}$.

The centralizer ${\mathcal{D}}^{\prime}$ is defined as the fusion subcategory
${\mathcal{D}}^{\prime}$ of $\mathcal{C}$ generated by all simple objects $Y$
of $\mathcal{C}$ centralizing any object objects $X$ of ${\mathcal{D}}$. In
the premodular case $X_{i}$ centralizes $X_{i^{\prime}}$ if and only if
$s_{ii^{\prime}}=d_{i}d_{i^{\prime}}$, (see also [Müg03]). In particular, the
centralizer $\mathcal{C}^{\prime}$ of $\mathcal{C}$ is also denoted by
$\mathcal{Z}_{2}(\mathcal{C})$ and it is called the Müger center of
$\mathcal{C}$.

Since $f_{Q}(F_{j})$ is an idempotent element of $\mathrm{{CE}(\mathcal{C})}$
one may write:

$f_{Q}(F_{j})=\sum_{i\in\mathcal{A}_{j}}E_{i}$

for some subset $\mathcal{A}_{j}\subseteq\\{0,\dots,m\\}$. Note that the set
$\mathcal{A}_{j}$ might be empty precisely when $f_{Q}(F_{j})=0$. Denote by
$\mathcal{J}_{2}\subseteq\mathcal{J}:=\\{0,1,\dots,m\\}$ the set of all
indices $j$ with $\mathcal{A}_{j}$ not a empty set. Since $f_{Q}(1)=1$ we
obtain in this way a partition for the set of all irreducible representations
$\mathrm{Irr}(\mathcal{C})=\bigsqcup_{j\in\mathcal{J}_{2}}\mathcal{\tilde{A}}_{j}$
where, to be precise,
$\mathcal{\tilde{A}}_{j}=\\{[X_{i}]\;|\;i\in\mathcal{A}_{j}\\}$. For any index
$0\leq i\leq m$ we denoted by $M(i)$ the unique index $j\in\mathcal{J}_{2}$
such that $i\in\mathcal{A}_{j}$. One obtains a (unique) function
$M:\\{0,1\dots,m\\}\rightarrow\mathcal{J}_{2}$ with the property that
$E_{i}f_{Q}(F_{M(i)})\neq 0$ for all $i\in\\{0,1\dots,m\\}$.

Since $\mathrm{{CF}}(\mathcal{C})$ is a semisimple commutative algebra it
follows that $F_{j}$ form a $\mathbb{C}$-linear basis for
$\mathrm{{CF}}(\mathcal{C})$. Then for any irreducible character
$\chi_{i}=\sum_{j=0}^{m}\alpha_{ij}F_{j}$ for some $\alpha_{ij}\in\mathbb{C}$.

The following lemma generalizes [CW10, Equation (22)].

######  Lemma 4.2.

With the above notations, for all $0\leq i,i^{\prime}\leq m$ one has

(4.3)
$\frac{\alpha_{{}_{iM(i^{\prime})}}}{d_{i}}=\frac{s_{ii^{\prime}}}{d_{i}d_{i^{\prime}}}=\frac{\alpha_{{}_{i^{\prime}M(i)}}}{d_{i^{\prime}}}.$

###### Proof.

Since $f_{Q}(F_{j})=0$ if $j\notin\mathcal{J}$ it follows that
$f_{Q}(\chi_{i})=\sum_{j=0}^{m}\alpha_{ij}f_{Q}(F_{j})=\sum_{j\in\mathcal{J}_{2}}\sum_{i^{\prime}\in\mathcal{A}_{j}}\alpha_{{}_{iM(i^{\prime})}}E_{i^{\prime}}=\sum_{i^{\prime}=0}^{r}\alpha_{{}_{iM(i^{\prime})}}E_{i^{\prime}}.$
Thus

(4.4)
$f_{Q}(\chi_{i})=\sum_{i^{\prime}=0}^{m}\alpha_{{}_{iM(i^{\prime})}}E_{i^{\prime}}.$

for all $i$. Comparing with Equation (4.1), for all indices  $i$ and
$i^{\prime}$ one has
$\alpha_{{}_{iM(i^{\prime})}}=\frac{s_{ii^{\prime}}}{d_{i^{\prime}}},$ Now
Equation (4.3) follows since $s_{ii^{\prime}}=s_{i^{\prime}i}$. ∎

Recall from [EGNO15, Proposition 8.13.11], that for any simple object
$X_{i}\in\mathrm{Irr}(\mathcal{C})$ there is a character
$\psi_{{}_{[X_{i}]}}:\mathrm{{CF}}(\mathcal{C})\rightarrow\mathbb{C}$ given by
$\psi_{{}_{[X_{i}]}}(\chi_{i^{\prime}})=\frac{s_{ii^{\prime}}}{d_{i}}$.

Using Equation (4.3) it can be easily seen that

(4.5) $\psi_{[X_{i}]}=\mu_{{}_{M(i)}},\text{for all}\;i\in\mathcal{I}.$

Indeed, for all indices $i^{\prime}$ one has:

$\psi_{{}_{[X_{i}]}}(\chi_{i^{\prime}})=\frac{s_{ii^{\prime}}}{d_{i}}=\alpha_{{}_{i^{\prime}M(i)}}=\mu_{{}_{M(i)}}(\chi_{i^{\prime}}).$

Next Theorem is a generalization of the corresponding result for semisimple
Hopf algebras obtained in [CW10, Theorem 4.3].

######  Theorem 4.6.

With the above notations one has

(4.7)
$f_{Q}(\chi_{i})=\frac{d_{i}}{\dim(\mathcal{C}^{M(i)})}\mathrm{C}_{{}_{M(i)}}$

###### Proof.

Let $\chi_{i}=\sum_{j=0}^{m}\alpha_{ij}F_{j}$ as above. Using Equation (4.4)
and Equation (4.3) it follows that

(4.8)
$f_{Q}(\chi_{i})=\sum_{i^{\prime}=0}^{m}\alpha_{{}_{iM(i^{\prime})}}E_{i^{\prime}}=\sum_{i^{\prime}=0}^{m}\frac{\alpha_{{}_{i^{\prime}M(i)}}d_{i}}{d_{i^{\prime}}}E_{i^{\prime}}=d_{i}(\sum_{i^{\prime}=0}^{m}\frac{\alpha_{{}_{i^{\prime}M(i)}}}{d_{i^{\prime}}}E_{i^{\prime}}).$

Note that by [Bur20, Equations (4.3) and (4.8)] one has

(4.9)
$\mathrm{C}_{j}={\dim(\mathcal{C}^{j})}(\sum_{i^{\prime}=0}^{m}\frac{1}{d_{i^{\prime}}}\alpha_{{}_{i^{\prime}j}}E_{i^{\prime}})$

Then for $j=M(i)$ Equation (4.8) becomes
$f_{Q}(\chi_{i})=\frac{d_{i}}{\dim(\mathcal{C}^{M(i)})}\mathrm{C}_{{}_{M(i)}}$.
∎

###  4.2. Cosets with respect to $\mathcal{Z}_{2}(\mathcal{C})$

Recall the notion of cosets with respect to a fusion subcategory from Section
3.

######  Theorem 4.10.

Two simple objects $X_{i},C_{i^{\prime}}$ of a pseudo premodular fusion
category $\mathcal{C}$ are in the same coset with respect to
$\mathcal{Z}_{2}(\mathcal{C})$ if and only if $M(i)=M(i^{\prime})$.

###### Proof.

Let $\chi_{i}:=\chi(X_{i})$ and $\chi_{i^{\prime}}:=\chi(X_{i^{\prime}})$.
Suppose that the two characters $\chi_{i}$ and $\chi_{i^{\prime}}$ are in the
same coset with respect to $\mathcal{Z}_{2}(\mathcal{C})$. Then as above from
Equation (3.2) one has:

$\frac{\chi_{i}\lambda_{\mathcal{C}^{\prime}}}{d_{i}}=\frac{\chi_{i^{\prime}}\lambda_{\mathcal{C}^{\prime}}}{d_{i^{\prime}}}.$

On the other hand $f_{Q}(\lambda_{\mathcal{C}^{\prime}})=1$ by [Bur20,
Corollary 5.8], and applying $f_{Q}$ to the above Equation one has:

$f_{Q}(\frac{\chi_{i}\lambda_{\mathcal{C}^{\prime}}}{d_{i}})=\frac{1}{\dim(\mathcal{C}^{M(i)})}\mathrm{C}_{{}_{M(i)}}=f_{Q}(\frac{\chi_{i^{\prime}}\lambda_{\mathcal{C}^{\prime}}}{d_{i^{\prime}}})=\frac{1}{\dim(\mathcal{C}^{M(i^{\prime})})}\mathrm{C}_{{}_{M(i^{\prime})}}$

which proves that $M(i)=M(i^{\prime})$.

Therefore every set
$\mathcal{\widetilde{A}}_{j}:=\\{X_{i}\;\big{|}\;M(i)=j\\}$ is a union of
right cosets with respect to $\mathcal{C}^{\prime}$. Clearly, by its
definition, the number of the non-empty sets $\mathcal{A}_{j}$ equals the
cardinality of $\mathcal{J}_{2}$.

Note that the relation $f_{Q}(\lambda_{\mathcal{Z}_{2}(\mathcal{C}}))=1$
implies also that
$\mathcal{J}_{2}=\mathcal{J}_{\mathcal{Z}_{2}(\mathcal{C})}$. Then from
Proposition (3.4) it follows that

(4.11)
$\dim\mathcal{H}(\mathcal{Z}_{2}(\mathcal{C}))=\big{|}\mathcal{J}_{2}\big{|}.$

On the other hand, Equation (4.11) implies that the number of right cosets
also equals the cardinal of $\mathcal{J}_{2}$. Thus each $\mathcal{A}_{j}$
consists of a single right coset with respect to $\mathcal{C}^{\prime}$ and
the proof is finished. ∎

For any $j\in\mathcal{J}_{2}$ we denote by
$\mathcal{R}_{j}:=\\{X_{i}\;|\;M(i)=j\\}$. The above theorem implies that
$\mathcal{R}_{j}$ with $j\in\mathcal{J}_{2}$ are exactly the right cosets of
$\mathcal{C}$ with respect to $\mathcal{Z}_{2}(\mathcal{C})$. Denote also
${\mathrm{R}}_{j}:=\sum\limits_{X_{i}\in\mathcal{R}_{j}}d_{i}\chi_{i}\in\mathrm{{CF}}(\mathcal{C})$
the regular part of their characters.

###  4.3. On the dimension of the cosets

For any premodular fusion category ${\mathcal{D}}$ of $\mathcal{C}$ we denote
by $\mathcal{R}({\mathcal{D}})_{j}:={\mathcal{D}}\cap\mathcal{R}_{j}$, the
intersection of ${\mathcal{D}}$ with each coset $\mathcal{R}_{j}$ of
$\mathcal{C}$ with respect to $\mathcal{C}^{\prime}$. We denote also their
regular parts by

${\mathrm{R}}({\mathcal{D}})_{j}:=\sum\limits_{X_{i}\in\mathcal{\mathrm{R}}({\mathcal{D}})_{j}}d_{i}\chi_{i}\in\mathrm{{CF}}(\mathcal{C}).$

######  Proposition 4.12.

Let $\mathcal{C}$ be a premodular fusion category and ${\mathcal{D}}$ a fusion
subcategory of $\mathcal{C}$. With the above notations it follows that

(4.13)
$\mathcal{J}_{{{{\mathcal{D}}^{\prime}}}}=\\{M(i)\;|\;\chi_{i}\in\mathrm{Irr}({\mathcal{D}})\\}$

and

(4.14)
$\dim({\mathrm{R}}({\mathcal{D}})_{j})=\dim({\mathcal{D}}\cap{{\mathcal{C}^{\prime}}})\dim(\mathcal{C}^{j}),\;\text{for
all}\;j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}.$

###### Proof.

Since $\mathcal{C}$ is spherical one has $d_{i}=d_{i^{*}}$ for all $i$. Recall
by [Shi17, Equation (6.8)] that
$\lambda_{{\mathcal{D}}}=\frac{1}{\dim({\mathcal{D}})}\big{(}{\sum}_{\chi_{i}\in\mathrm{Irr}({\mathcal{D}})}d_{i^{*}}\chi_{i}\big{)}$
and Theorem 4.7 gives

$f_{Q}(\lambda_{{\mathcal{D}}})=\frac{1}{\dim({\mathcal{D}})}\bigg{(}\sum_{\chi_{i}\in\mathrm{Irr}({\mathcal{D}})}\frac{d_{i}d_{i}^{*}}{\dim(\mathcal{C}^{M(i)})}\mathrm{C}_{{}_{M(i)}}\bigg{)}.$

Applying now the Fourier transform to the last equality, since
$\mathrm{C}_{j}:=\mathcal{F}^{-1}(F_{j})$ it follows that

$\displaystyle\mathcal{F}(f_{Q}(\lambda_{{\mathcal{D}}}))$ $\displaystyle=$
$\displaystyle\frac{1}{\dim({\mathcal{D}})}\bigg{(}\sum_{\chi_{i}\in\mathrm{Irr}({\mathcal{D}})}\frac{d_{i}d_{i^{*}}}{\dim(\mathcal{C}^{M(i)})}F_{{}_{M(i)}}\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\dim({\mathcal{D}})}\bigg{(}{\sum}\limits_{j\in\mathcal{J}_{2}}\big{(}{\sum}\limits_{\\{\chi_{i}\in\mathrm{Irr}({\mathcal{D}})\;|\;M(i)=j\\}}d_{i}d_{i^{*}}\big{)}\frac{F_{j}}{\dim(\mathcal{C}^{j})}\bigg{)}.$
$\displaystyle=$
$\displaystyle\frac{1}{\dim({\mathcal{D}})}{\sum}\limits_{\\{j\in\mathcal{J}_{2}\;|\;j=M(i),\;\chi_{i}\in\mathrm{Irr}({\mathcal{D}})\\}}\frac{\dim({\mathrm{R}}({\mathcal{D}})_{j})}{\dim(\mathcal{C}^{j})}F_{j}.$

since, from its definition, one has
$\dim({\mathrm{R}}({\mathcal{D}})_{j}):=\bigg{(}{\sum}\limits_{\\{\chi_{i}\in\mathrm{Irr}({\mathcal{D}})\;|\;M(i)=j\\}}d_{i}^{2}\bigg{)}$.
On the other hand by [Bur20, Theorem 1] one can write

$\mathcal{F}(f_{Q}(\lambda_{{\mathcal{D}}}))=\frac{\dim({{{\mathcal{D}}^{\prime}}})}{\dim(\mathcal{C})}\lambda_{{{{\mathcal{D}}^{\prime}}}}=\frac{\dim({{{\mathcal{D}}^{\prime}}})}{\dim(\mathcal{C})}\bigg{(}{\sum}\limits_{j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}}F_{j}\bigg{)}$

Note that the sphericality of $\mathcal{C}$ implies by [EGNO15, Prop 4.8.4]
that the coefficients of $F_{j}$ in both above formulae for
$\mathcal{F}(f_{Q}(\lambda_{\mathcal{D}}))$ are non-zero scalars. Comparing
the coefficients of $F_{j}$ in the above two formulae for
$\mathcal{F}(f_{Q}(\lambda_{{\mathcal{D}}}))$ it follows that
$\mathcal{J}_{{{{\mathcal{D}}^{\prime}}}}=\\{M(i)\;|\;\chi_{i}\in\mathrm{Irr}({\mathcal{D}})\\}$
and

$\dim({\mathrm{R}}({\mathcal{D}})_{j})={\dim(\mathcal{C}^{j})}\frac{\dim({{{\mathcal{D}}^{\prime}}}){\dim({\mathcal{D}})}}{\dim(\mathcal{C})}$

which implies the result since by [DGNO10, Theorem 3.10] one has

(4.15)
$\dim({\mathcal{D}})\dim({{{\mathcal{D}}^{\prime}}})=\dim(\mathcal{C})\dim({\mathcal{D}}\cap{{\mathcal{C}^{\prime}}}).$

∎

######  Corollary 4.16.

With the above notations it follows that

(4.17)
$\frac{\dim(\mathcal{C})\dim(\mathcal{C}^{\prime}\cap{\mathcal{D}})}{{\dim({\mathrm{R}}({\mathcal{D}})_{j})}}\in\mathbb{A}$

for all $j\in J_{{\mathcal{D}}^{\prime}}$.

###### Proof.

Note that equation (4.14) gives that
$\frac{\dim(\mathcal{C})}{\dim(\mathcal{C}^{j})}=\frac{\dim(\mathcal{C})\dim(\mathcal{C}^{\prime}\cap{\mathcal{D}})}{{\dim({\mathrm{R}}({\mathcal{D}})_{j})}}$.
Since $\mathcal{C}_{j}$ is a simple object of ${\mathcal{Z}(\mathcal{C})}$ the
result follows. ∎

######  Corollary 4.18.

If $\mathcal{C}$ is an integral braided fusion category of free square
dimension and ${\mathcal{D}}\cap\mathcal{Z}_{2}(\mathcal{C})=\mathrm{Vec}$
then ${\mathcal{D}}$ is pointed.

######  Corollary 4.19.

With the above notations it follows that

(4.20)
$\dim({\mathrm{R}}_{j})=\dim({\mathcal{C}^{\prime}})\dim(\mathcal{C}^{j}),\;\text{for
all}\;j\in\mathcal{J}_{2}.$

###### Proof.

It is Equation (4.14) for ${\mathcal{D}}=\mathcal{C}$. ∎

###  4.4. Proof of Theorem 1.1

######  Proposition 4.21.

Let ${\mathcal{D}}$ be a fusion subcategory of a pseudo-unitary premodular
category $\mathcal{C}$. With the above notations one has:

${\mathcal{D}}=\bigoplus_{j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}}\mathcal{\mathrm{R}}({\mathcal{D}})_{j}$

is the decomposition of ${\mathcal{D}}$ in cosets with respect to
${\mathcal{D}}\cap\mathcal{C}^{\prime}$.

###### Proof.

By Lemma 3.12 each non-empty set $\mathcal{\mathrm{R}}({\mathcal{D}})_{j}$ is
a coset of ${\mathcal{D}}$ with respect to
${\mathcal{D}}\cap\mathcal{Z}_{2}(\mathcal{C})$. On the other hand we will
show that

$\sum\limits_{j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}}\mathrm{FPdim}(\mathcal{\mathrm{R}}({\mathcal{D}})_{j})=\mathrm{FPdim}({\mathcal{D}}).$

This proves that $\mathcal{\mathrm{R}}({\mathcal{D}})_{j}$ with
$j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}$ are all the right cosets. Indeed,
adding all the Equations (4.14) for $j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}$
one obtains that

$\sum_{j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}}\mathrm{FPdim}(\mathcal{\mathrm{R}}({\mathcal{D}})_{j})=\mathrm{FPdim}({\mathcal{D}}\cap\mathcal{C}^{\prime})\big{(}\sum\limits_{j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}}\mathrm{FPdim}(\mathcal{C}^{j})\big{)}.$

On the other hand, since $\mathcal{C}$ is pseudo-unitary note that by Equation
2.7 one has

(4.22)
$\sum\limits_{j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}}\mathrm{FPdim}(\mathcal{C}^{j})=\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}({\mathcal{D}}^{\prime})}=\frac{\mathrm{FPdim}({\mathcal{D}})}{\mathrm{FPdim}({\mathcal{D}}\cap\mathcal{Z}_{2}(\mathcal{C}))}$

and Equation (4.15) gives that
$\sum\limits_{j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}}\mathrm{FPdim}(\mathcal{\mathrm{R}}({\mathcal{D}})_{j})=\mathrm{FPdim}({\mathcal{D}}).$
∎

Next one can obtain a proof of Theorem 1.1 as follows:

###### Proof.

By Proposition 4.21 we may suppose that
$Y\in\mathcal{\mathcal{missing}}{\mathrm{R}}({\mathcal{D}})_{j}$ for some
$j\in\mathcal{J}_{{\mathcal{D}}^{\prime}}$. If
${\mathcal{D}}\cap\mathcal{Z}_{2}(\mathcal{C})=\mathrm{Vec}$ then
$\mathcal{\mathrm{R}}({\mathcal{D}})_{j}$ is a coset of $\mathrm{Vec}$ inside
${\mathcal{D}}$ and therefore it consists of a single element $Y$ of
${\mathcal{D}}$. Thus, in this case
$\mathrm{FPdim}({\mathrm{R}}({\mathcal{D}})_{j})=\dim(Y)^{2}$ and the result
follows from Equation (4.17). ∎

###  4.5. Proof of Theorem 1.3

In this subsection we prove our main second result:

###### Proof.

For the first item suppose that $Y=X_{i}$ and $M(i)=j$. Since
$\mathcal{C}^{\prime}$ is pointed all the simple objects in $\mathcal{R}_{j}$
have the same dimension, namely $\mathrm{FPdim}(Y)$. If
$r_{j}:=|\mathcal{R}_{j}|$ and $G_{Y}$ is the stabilizer of $Y$ under the
action of $\mathcal{Z}_{2}(\mathcal{C})$ it follows that

$\mathrm{FPdim}({\mathrm{R}}_{j})=r_{j}\mathrm{FPdim}(Y)^{2}=\frac{\mathrm{FPdim}(\mathcal{C}^{\prime})}{|G_{Y}|}\mathrm{FPdim}(Y)^{2}$

since $r_{j}|G_{Y}|=\mathrm{FPdim}(\mathcal{C}^{\prime})$. Then by Proposition
4.12 one has:

(4.23)
$\mathrm{FPdim}(\mathcal{C}^{j})=\frac{\mathrm{FPdim}(\mathrm{R}_{j})}{\mathrm{FPdim}(\mathcal{C}^{\prime})}=\frac{\mathrm{FPdim}(Y)^{2}}{|G_{Y}|}$

Note that
$\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}(\mathcal{C}^{j})}\in\mathbb{A}$
since $\mathcal{C}^{j}$ is a simple object of ${\mathcal{Z}(\mathcal{C})}$.
Thus

(4.24)
$\frac{\mathrm{FPdim}(\mathcal{C})|G_{Y}|}{\mathrm{FPdim}(Y)^{2}}\in\mathbb{A}$

and the first divisibility result follows.

If the action of $\mathcal{C}^{\prime}$ is free then by the second item of
Corollary 3.9

$\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}(\mathcal{Z}_{2}(\mathcal{C}))\mathrm{FPdim}(\mathcal{C}^{j})}\in\mathbb{A}$

for all $j\in\mathcal{J}_{\mathcal{Z}_{2}(\mathcal{C})}$. Since in this case
$|G_{Y}|=1$ it follows by Equation (4.23) that
$\mathrm{FPdim}(\mathcal{C}^{j})=\mathrm{FPdim}(Y)^{2}$ and therefore

$\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}(\mathcal{Z}_{2}(\mathcal{C}))\mathrm{FPdim}(Y)^{2}}\in\mathbb{A}.$

∎

######  Remark 4.25.

Note that for any simple object $X_{i}$ of $\mathcal{C}$ Equation (3.10) gives
that

(4.26)
$\frac{\mathrm{FPdim}(X_{i})^{2}\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}(\mathcal{C}^{\prime})\mathrm{FPdim}(\mathcal{C}^{M(i)})}\in\mathbb{A}.$

since
$\mathrm{FPdim}(R_{M(i)})=\mathrm{FPdim}(\mathcal{Z}_{2}(\mathcal{C}))\mathrm{FPdim}(\mathcal{C}^{M(i)})$

######  Remark 4.27.

As mentioned in the introduction, the divisibility of Equation (1.5)
generalizes [Yu20, Corollary 3.4.] in the pseudo-unitary settings.

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