11institutetext: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
D-53121 Bonn, Germany
11email<EMAIL_ADDRESS>22institutetext: Nicolaus Copernicus
Astronomical Centre, Polish Academy of Sciences, Bartycka 18, 00-716 Warsaw,
Poland 33institutetext: Black Hole Initiative at Harvard University, 20 Garden
Street, Cambridge, MA 02138, USA 44institutetext: Department of
Astrophysics/IMAPP, Radboud University,P.O. Box 9010, 6500 GL Nijmegen, The
Netherlands 55institutetext: Center for Astrophysics — Harvard & Smithsonian,
60 Garden Street, Cambridge, MA 02138, USA 66institutetext: Departament
d’Astronomia i Astrofísica, Universitat de València, C. Dr. Moliner 50,
E-46100 Burjassot, València, Spain 77institutetext: Observatori Astronòmic,
Universitat de València, C. Catedrático José Beltrán 2, E-46980 Paterna,
València, Spain 88institutetext: Las Cumbres Observatory, 6740 Cortona Drive,
Suite 102, Goleta, CA 93117-5575, USA 99institutetext: Department of Physics,
University of California, Santa Barbara, CA 93106-9530, USA 1010institutetext:
Italian ALMA Regional Centre, INAF-Istituto di Radioastronomia, Via P. Gobetti
101, I-40129 Bologna, Italy 1111institutetext: Dipartimento di Fisica,
Università degli Studi di Cagliari, SP Monserrato-Sestu km 0.7, I-09042
Monserrato, Italy 1212institutetext: Universidade de São Paulo, Instituto de
Astronomia, Geofísica e Ciências Atmosféricas, Departamento de Astronomia, São
Paulo, SP 05508-090, Brazil 1313institutetext: Joint ALMA Observatory, Alonso
de Cordova 3107, Vitacura 763-0355, Santiago de Chile, Chile

# Orbital motion near Sagittarius A∗

Constraints from polarimetric ALMA observations

M. Wielgus 112233 M. Moscibrodzka 44 J. Vos 44 Z. Gelles 5533 I. Martí-Vidal
6677 J. Farah 8899 N. Marchili 101011
C. Goddi 11111212 H. Messias 1313

We report on the polarized light curves of the Galactic Center supermassive
black hole Sagittarius A∗, obtained at millimeter wavelength with the Atacama
Large Millimeter/submillimeter Array (ALMA). The observations took place as a
part of the Event Horizon Telescope campaign. We compare the observations
taken during the low variability source state on 2017 Apr 6 and 7 with those
taken immediately after the X-ray flare on 2017 Apr 11. For the latter case,
we observe rotation of the electric vector position angle with a timescale of
$\sim\,70$ min. We interpret this rotation as a signature of the equatorial
clockwise orbital motion of a hot spot embedded in a magnetic field dominated
by a dynamically important vertical component, observed at a low inclination
$\sim\,20^{\circ}$. The hot spot radiates strongly polarized synchrotron
emission, briefly dominating the linear polarization measured by ALMA in the
unresolved source. Our simple emission model captures the overall features of
the polarized light curves remarkably well. Assuming a Keplerian orbit, we
find the hot spot orbital radius to be $\sim$ 5 Schwarzschild radii. We
observe hints of a positive black hole spin, that is, a prograde hot spot
motion. Accounting for the rapidly varying rotation measure, we estimate the
projected on-sky axis of the angular momentum of the hot spot to be
$\sim\,60^{\circ}$ east of north, with a 180∘ ambiguity. These results suggest
that the accretion structure in Sgr~A$^∗$ is a magnetically arrested disk
rotating clockwise.

###### Key Words.:

black holes – galaxies: individual: Sgr A* – Galaxy: center – techniques:
interferometric

##  1 Introduction

Sagittarius A∗ (Sgr~A$^∗$) is associated with a supermassive black hole
located in the center of our galaxy, with a mass
$M_{\mathbin{\vbox{\hbox{\scalebox{1.35}{$\bullet$}}}}}\approx
4\times$10${}^{6}\,M_{\odot}$ (Do et al., 2019a; Gravity Collaboration et al.,
2022; EHTC et al., 2022a). The source exhibits rapid variability across the
electromagnetic spectrum, with particularly strong flaring events in the
infrared (IR; e.g., Genzel et al., 2003; Eckart et al., 2006; Do et al.,
2019b) and X-ray (e.g., Baganoff et al., 2001; Porquet et al., 2003; Haggard
et al., 2019), during which the observed flux density rises by 1-2 orders of
magnitude. While the detailed physics of these energetic episodes is not fully
understood, magnetic reconnection in a radiatively inefficient magnetized
accretion flow (Yuan & Narayan, 2014) constitutes a plausible theoretical
framework to interpret the flaring activity of Sgr~A$^∗$ (Yuan et al., 2003;
Dodds-Eden et al., 2010; Dexter et al., 2020). In a reconnection event, the
magnetic energy can be rapidly dissipated, heating plasma locally. The exhaust
of the reconnection can form a transient feature in the orbiting accretion
flow – a hot spot of low-density plasma confined in a flux tube of a vertical
magnetic field (e.g., Porth et al., 2021; Ripperda et al., 2022). It has been
proposed that investigating the dynamics of such transient flow features would
constitute a powerful probe into the physics of gravity and accretion
(Broderick & Loeb, 2005, 2006; Hamaus et al., 2009; Zamaninasab et al., 2010).

Figure 1: Full Stokes 229 GHz ALMA light curves of Sgr~A$^∗$ obtained on 2017
Apr 6, 7, and 11. Top row: linear polarization represented on a
$\mathcal{Q}$-$\mathcal{U}$ plane. The color of the curve indicates the time
from the beginning of the observation. Crosses ”+” show the locations of the
daily mean linear polarization $\overline{P}$ with the EVPA $\overline{\chi}$.
The pattern observed on 2017 Apr 11 suggests the presence of coherent
clockwise $\mathcal{Q}$-$\mathcal{U}$ loops, emphasized with arrows. Bottom
rows: full Stokes Sgr~A$^∗$ light curves. On Apr 11, ALMA started observing
immediately after the X-ray flare (gray band region). Chandra X-ray data are
plotted in counts per second and were arbitrarily scaled. The red-shaded
region corresponds to the period in which polarimetric loops are apparent.

Studying hot spots associated with Sgr~A$^∗$ flares became possible with the
GRAVITY near-IR interferometer, achieving an astrometric precision of tens of
microarcseconds (Gravity Collaboration et al., 2017). Several observations of
hot spots’ signatures during the Sgr~A$^∗$ flaring state were reported
(Gravity Collaboration et al., 2018). In all cases, a clockwise, nearly
circular motion of the brightness centroid on a timescale of 0.5-1.0 h was
detected, which is consistent with the interpretation of an orbiting hot spot
in an innermost region of the accretion flow, viewed at low inclination.
Simultaneously with the brightness centroid rotation, polarimetric signatures
of a hot spot threaded by the magnetic field were found in the form of an
electric vector position angle (EVPA) rotation (Gravity Collaboration et al.,
2018, 2020a, 2020b).

The activity of Sgr~A$^∗$ appears less dramatic at millimeter wavelengths,
with a flux density increase of several tens of percent lagging behind the IR
and X-ray flares (Marrone et al., 2008; Wielgus et al., 2022). Resolving
relative motions with angular resolution comparable to that of GRAVITY is only
possible at millimeter wavelengths using very-long-baseline interferometry
(VLBI) with the Event Horizon Telescope (EHT) array (EHTC et al., 2019).
However, the EVPA rotation in the unresolved compact source can be traced with
connected-element interferometric arrays such as ALMA or the SubMillimeter
Array (SMA). Hints of the coherent EVPA evolution in the SMA observations were
already reported by Marrone et al. (2006).

In this paper we present full Stokes light curves of millimeter emission from
Sgr~A$^∗$ observed by ALMA on 2017 Apr 6, 7, and 11. We focus on the Apr 11
observations, in the period immediately following an X-ray flare observed by
Chandra (EHTC et al., 2022b; Wielgus et al., 2022). We identify polarimetric
signatures of an orbiting hot spot and, using a simple emission model, we
infer system parameters broadly consistent with the findings of GRAVITY
(Gravity Collaboration et al., 2018).

##  2 Observations and data properties

###  2.1 ALMA observations

Sgr~A$^∗$ was observed with ALMA as a part of the EHT campaign in 2017. ALMA
participated in the EHT VLBI observations as a phased array (Goddi et al.,
2019; EHTC et al., 2022a). In parallel to the VLBI data reduction pipeline,
the phased array observations were processed to recover the connected
interferometer ALMA-only measurements (Goddi et al., 2021). The initial
calibration was performed following the ALMA QA2 procedures described in Goddi
et al. (2019), benefiting from the long duration of observing tracks and the
utilization of multiple calibrator sources that are necessary to enable full
Stokes VLBI imaging with the EHT (Martí-Vidal et al. 2016; EHTC et al. 2021a;
Issaoun, Wielgus et al. 2022, ). Subsequently, an additional correction of
time-dependent amplitude gains was performed, after modeling the time
variability of the compact Sgr~A$^∗$ source against the static parsec-scale
Galactic Center minispiral (Lo & Claussen, 1983; Mus et al., 2022). This self-
calibration procedure corresponds to the A1 data reduction pipeline described
in Wielgus et al. (2022), where a detailed discussion of the related
algorithms, extraction of the compact unresolved source signal, and quality
control can be found. The resulting full Stokes light curves of Sgr~A$^∗$ have
a cadence of 4 s and an exquisite signal-to-noise ratio (S/N) formally
reaching $\sim\,1000$ (for the discussion of systematic uncertainties, see
Appendix E in Goddi et al. 2021, and Sections 4-5 in Wielgus et al. 2022). The
light curves correspond to four frequency bands of 2 GHz width each, with
central frequencies of 213.1, 215.1, 227.1, and 229.1 GHz.

###  2.2 Mean light curves properties

The light curves corresponding to all three observing epochs (2017 Apr 6, 7,
and 11) are shown in Fig. 1. Time-averaged parameters of the Stokes components
are also provided. The observations took place during a period of a
particularly low total intensity of Sgr~A$^∗$. The properties of the total
intensity (Stokes $\mathcal{I}$) light curves are discussed in detail in
Wielgus et al. (2022). Here, we focus on the linear polarization (LP),
$\mathcal{P}=\mathcal{Q}+i\,\mathcal{U}$, and circular polarization (CP),
$\mathcal{V}$. There is a good consistency between the three observing days on
the average LP, corresponding to 7-8% of the total intensity, as well as the
average CP, corresponding to about -1% of the total intensity. The EVPA
(denoted as $\chi$), defined as $\chi=0.5\arg\,\mathcal{P}$, corresponds to
$-57\pm 11$ deg, with Apr 6-7 corresponding to the average of
$\chi\approx-60^{\circ}$, and Apr 11 corresponding to the average
$\chi\approx-47^{\circ}$ in the 229.1 GHz band. The EVPA differs between
frequency bands because of the significant rotation measure (RM), see Appendix
A. The measured polarimetric parameters are broadly consistent with the
historical measurements, for example from Bower et al. (2003); Marrone et al.
(2008); Muñoz et al. (2012); Bower et al. (2018).

###  2.3 Polarimetric loops

A notable difference between the LP signatures on 2017 Apr 6-7 and 2017 Apr 11
is the presence of coherent loops on the $\mathcal{Q}$-$\mathcal{U}$ plane in
the latter data set – see the top row of Fig. 1. These signatures appear in
the time period following the X-ray flare detected by Chandra, which peaked at
8:48 UT. After about 2 h this ”loopy period” ends, and the variability of the
LP component returns to a more stochastic character, similar to that observed
on 2017 Apr 6-7. This suggests a causal connection between the X-ray flare and
the $\mathcal{Q}$-$\mathcal{U}$ plane loops observed at a millimeter
wavelength. In Fig. 2 we inspect data corresponding to the loopy period. We
observe two large clockwise loops, first $A_{\rm big}$ with a period slightly
above 1 h, executed between $T_{0}=$ 9:20 UT and $T_{5}=T_{0}+68$ min, and a
second one with a reduced period, observed between $T_{5}$ and
$T_{7}=T_{5}+35$ min. The first loop is very close to a round shape, with
asymmetry, defined as the largest ratio of perpendicularly projected
diameters, $A\sim\\!1.2$. The second loop is smaller and significantly more
distorted. We also observe a small inner loop $A_{\rm tiny}$, traversed in
just $\sim$10 min between $T_{1}$ and $T_{2}$. The ratio of the first large
loop area to the small loop area is $R_{a}\sim\\!550$, and the ratio of the
full loop period to the time spent inside a small loop is $R_{t}\sim\\!7$. We
interpret these loops as signatures of an orbiting hot spot and use the three
dimensionless observables ($A$, $R_{t}$, and $R_{a}$) to crudely constrain the
system geometry – see Appendix B.

Figure 2: Polarimetric loops observed by ALMA at 229 GHz on 2017 Apr 11. The
observations began at $T_{0}=$ 9:20 UT (MJD 57854.39), 30 min after the peak
of the X-ray flare. We highlight the small inner loop $A_{\rm tiny}$.
Following the full loop $A_{\rm big}$ between $T_{0}$ and $T_{5}$, a similar
pattern continues between $T_{5}$ and $T_{7}$, with a decreased period and
reduced LP.

##  3 Interpretation and modeling

###  3.1 General model specification

ALMA light curves from the 2017 EHT campaign correspond to the observed Stokes
components averaged over the region unresolved by the instrument $\sim$1
arcsec, or $\sim\\!10^{5}r_{\rm g}$ for $r_{\rm
g}=GM_{\mathbin{\vbox{\hbox{\scalebox{1.35}{$\bullet$}}}}}/c^{2}$. The
millimeter flux density is strongly dominated by the most compact region of
the accretion flow, confined to the field of view of $\sim$100 $\mu$as, or
$\sim$20 $r_{\rm g}$ (Doeleman et al., 2008; EHTC et al., 2022b). The total
intensity image of Sgr~A$^∗$ in a nonflaring state corresponds to a ring of
$\sim\\!50\,\mu$as diameter (EHTC et al., 2022c, d), which is consistent with
the expected appearance of a Kerr black hole surrounded by a radiatively
inefficient accretion flow (EHTC et al., 2022e). We refer to this feature as
the observable shadow of a black hole (EHTC et al., 2022f). Its total
millimeter flux density fluctuates as a red noise stochastic process, which is
qualitatively (but not necessarily quantitatively, see EHTC et al. 2022e for
details) consistent with the variability of the numerical models of turbulent
accretion flow (Georgiev et al., 2022; Wielgus et al., 2022). We assume that
this single stochastic component, fluctuating around $\mathcal{I}_{\rm
shadow}\\!\equiv\\!\mathcal{I}_{\rm sha}\\!\approx\\!2.4$ Jy,
$|\mathcal{P}_{\rm shadow}|\\!\equiv\\!|\mathcal{P}_{\rm
sha}|\\!\approx\\!0.2$ Jy and $\mathcal{V}_{\rm
shadow}\\!\equiv\\!\mathcal{V}_{\rm sha}\\!\approx\\!-0.03$ Jy suffices to
understand light curves observed outside of the loopy period.

To interpret observations in the loopy phase, we consider a presence of an
additional component, an equatorial orbiting hot spot, very similar to that
envisioned by Broderick & Loeb (2005). From the size of the
$\mathcal{Q}$-$\mathcal{U}$ loop shown in Fig. 2, we conclude a polarized flux
density of $|\mathcal{P}_{\rm hot\,spot}|\equiv|\mathcal{P}_{\rm hsp}|\lesssim
0.15$ Jy, and $\mathcal{I}_{\rm hot\,spot}\equiv\mathcal{I}_{\rm
hsp}\lesssim\\!0.3$ Jy, assuming a strong fractional polarization of 50%. This
implies that while $\mathcal{I}_{\rm sha}\gg\mathcal{I}_{\rm hsp}$, we may
expect $|\mathcal{P}_{\rm hsp}|\approx|\mathcal{P}_{\rm sha}|$. Hence, our
best chance of characterizing the hot spot component is through LP, rather
than through the total intensity. Furthermore, we assume that during the 68
min between $T_{0}$ and $T_{5}$ in Fig. 2 we can approximate $\mathcal{P}_{\rm
sha}$ with a constant value with $|\mathcal{P}_{\rm sha}|\approx 0.2$ Jy, and
since the measured LP is a coherent sum of the polarization of the two
components, we can assume that a variation in the observed LP in the loopy
period is strongly dominated by the rapidly changing hot spot component,
$\mathcal{P}_{\rm hsp}$. We note that the loop component tends to average out
when coherently averaged in time, explaining the roughly consistent daily mean
polarization observed by ALMA on 2017 Apr 6, 7, and 11.

From the physical point of view, a larger relative importance of the hot spot
component in LP rather than in total intensity seems very reasonable. We
expect the energized hot spot to indicate an increased temperature, lower
density, and lower optical depth than the surrounding flow, which should
result in an increase in the fractional polarization of the produced radiation
(Rybicki & Lightman, 1979). Furthermore, we generally expect the hot spot to
be smaller than the observable shadow (e.g., Gravity Collaboration et al.,
2020a), and hence to experience less depolarization when coherently averaged
over a spatially varying magnetic field structure.

As a hot spot moves along its orbit, the trajectories of photons reaching a
distant observer are lensed because of the spacetime curvature, while the
transported flux density is affected by the Doppler effect and gravitational
redshift. The polarization vector, perpendicular to the direction of
propagation and the magnetic field in the emission frame, is parallel-
transported along the null geodesics and influenced by Faraday effects (e.g.,
see Appendix A). Hence, the imprint of the system geometry (spacetime
curvature, observer’s inclination, and magnetic field configuration) on the
polarimetric pattern on a $\mathcal{Q}$-$\mathcal{U}$ plane can be used to
constrain parameters of the source model. Under the assumption of axisymmetry,
and particularly for low inclinations, these signatures resemble circular loop
patterns. A pedagogical discussion of the observable LP signatures and impact
of different effects (Doppler and lensing) and parameters (magnetic field
topology and emitter velocity) is given in Section 3 of Narayan et al. (2021),
and more detailed analyses can be found in Gelles et al. (2021) and in Vos et
al. (2022). We notice that the reported ALMA $\mathcal{Q}$-$\mathcal{U}$ loop
is traversed clockwise, consistent with the observations of Gravity
Collaboration et al. (2018). In all models considered in this paper, this
implies a clockwise physical motion of the orbiting hot spot in the sky.

Figure 3: Comparison between ALMA data in the loopy period on 2017 Apr 11
(blue dots) and hot spot model prediction for the inclination $i=158^{\circ}$,
the orbital radius $r_{\rm orb}=11M$, the vertical magnetic field, and the
spin $a_{*}=0$. We compare predictions of a simplified semi-analytic model
(dashed red lines, Appendix B and Gelles et al. 2021) with a slow light, full
radiative transfer model (continuous red line, Vos et al., 2022). A static,
linearly polarized component of $|\mathcal{P}_{\rm sha}|=|\mathcal{Q}_{\rm
sha}+i\,\mathcal{U}_{\rm sha}|=0.16$ Jy and $\chi_{\rm sha}=-37^{\circ}$ was
coherently added to the hot spot models, and the loop position angle was
adjusted to match the observations.

###  3.2 Constraints on the system geometry

Unlike GRAVITY, phased ALMA lacks the resolving power to constrain the
brightness centroid motion coinciding with the polarimetric loops. Hence, to
break the degeneracy between the orbital period and radius, we assume that the
motion of the putative hot spot is Keplerian. Some discussions of the impact
of Keplerianity on models and the interpretation of the observations are given
in Appendix C. While we generally expect the innermost flow to be sub-
Keplerian (e.g., Porth et al., 2021), super-Keplerian pattern motion was also
proposed as an interpretation of the GRAVITY observations (Matsumoto et al.,
2020). The period of Keplerian orbits in Kerr spacetime is

$T(x)=2\pi t_{\rm g}\left(x^{3/2}+a_{*}\right)\ ,$ (1)

where $t_{\rm
g}=GM_{\mathbin{\vbox{\hbox{\scalebox{1.35}{$\bullet$}}}}}/c^{3}=r_{\rm g}/c$,
$x=r_{\rm orb}/r_{\rm g}$, and $a_{*}$ is the dimensionless black hole spin.
Assuming $M_{\mathbin{\vbox{\hbox{\scalebox{1.35}{$\bullet$}}}}}=4.155\times
10^{6}M_{\odot}$, which is between the two competing results of Do et al.
(2019a) and Gravity Collaboration et al. (2022), we find $t_{\rm g}=20.46$ s.
We estimate the period of the large loop shown in Fig. 2 to be $74\pm 6$ min.
This range implies a Keplerian orbit radius $r_{\rm orb}$ in the
$10.0-11.2\,r_{\rm g}$ range for a Schwarzschild black hole ($a_{*}=0$) or
$9.8-11.0\,r_{\rm g}$ for a maximally spinning Kerr black hole ($a_{*}=1$).

We used a simplified semi-analytic model of an equatorial Keplerian hot spot
developed by Gelles et al. (2021) to survey the parameter space of the orbital
radius $r_{\rm orb}$, observer’s inclination $i$, and magnetic field geometry
$\boldsymbol{B}=[B_{r},B_{\phi},B_{z}]$, expressed in the cylindrical
coordinate system tied to the hot spot orbital plane. Using the observables
described in Section 2.3 we were able to constrain these parameters – see the
details provided in Appendix B. We concluded that only models with low
inclination angle $i\sim\\!20^{\circ}$ (or, equivalently,
$i\sim\\!160^{\circ}$) and dominance of the vertical magnetic field $B_{z}$
can reproduce the observed data features. These findings are remarkably
consistent with the conclusions from the IR observations of hot spots by
GRAVITY (Gravity Collaboration et al., 2018, 2020a, 2020b).

###  3.3 Direct comparison to data

Having constrained the parameters of the geometrical hot spot model, we
attempted a direct comparison with data using a more physical numerical
modeling framework, described in detail in Vos et al. (2022). Here we consider
a Gaussian hot spot on an equatorial orbit around a Kerr black hole, simulated
with the relativistic radiative transfer code ipole (Mościbrodzka & Gammie,
2018). We accounted for the finite velocity of light, hence the impact of the
secondary images was correctly taken into account (see also Appendix D). We
performed a complete radiative transfer calculation, accounting for emission,
self-absorption, internal Faraday rotation, and Faraday conversion, in order
to compute the full $\mathcal{I,Q,U,V}$ Stokes vector of the observed
emission. While we only discuss the LP here, comments on the total intensity
and CP are given in Appendix E.

The characteristic (but not unique) parameters that allowed us to match the
model shown in Fig. 3 to the observed polarized flux density are the
following: the number density of $n_{\rm e}\\!=\\!5\times 10^{5}\,{\rm
cm}^{-3}$, the magnetic field $B\\!=\\!10$ G, the dimensionless electron
temperature of $\Theta_{\rm e}=k_{\rm B}T_{\rm e}/m_{\rm e}c^{2}\\!=\\!50$,
and the Gaussian hot spot diameter of $\sim$ 6 $r_{\rm g}$. For simplicity, we
assumed a relativistic thermal distribution of the energy of electrons. We
find that for such parameters, the system is both optically and Faraday thin.
Given the turbulent, time-dependent character of the emission, we attempted to
primarily reproduce the observational appearance of the small inner loop
$A_{\rm tiny}$, as it is a complex feature that corresponds to a short time
duration of 10 min and hence is likely strongly dominated by the dynamical hot
spot signature rather than by the stochastic, time-correlated variation of the
observable shadow feature. In Fig. 3 we impose the fiducial model with a spin
$a_{*}\\!=\\!0.0$, Keplerian orbit at $r_{\rm orb}\\!=\\!11r_{\rm g}$, the
purely vertical magnetic field $B_{z}$, and the observer’s inclination of 158∘
on the ALMA data. We adjusted the position angle (PA) of the
$\mathcal{Q}$-$\mathcal{U}$ loop and a constant component $\mathcal{P}_{\rm
sha}$ to approximately match the data features. We fixed the black hole spin
to $a_{*}=0$, but some hints of a larger spin are discussed in Appendix C. The
observed PA is corrupted by the Faraday rotation, and we discuss the intrinsic
source orientation in Appendix F, where we estimate the PA of the hot spot
spin axis projected on the sky to be $\sim 60^{\circ}$ east of north. We find
a rather remarkable correspondence between the model and observations during
the primary polarimetric loop ($T_{0}$-$T_{5}$). While more fine-tuned models
could likely provide a better fit to data, the residuals can easily be
explained with the corrupting effects and model limitations discussed in
Section 3.4.

###  3.4 Limitations of the modeling

There are several important limitations to our modeling framework, which are
listed in the order of relevance for the presented results:

  1. 1.

Emission from the turbulent observable shadow component $\mathcal{P}_{\rm
sha}$ is stochastically variable and correlated in time. With a standard
deviation of $\sim\\!0.08$ Jy (see Fig. 1), this component may contribute to
the majority of the present discrepancies.

  2. 2.

We assume that hot spot emission is constant in the comoving frame. The
synchrotron cooling timescale for electrons is111The synchrotron cooling
timescale for electrons $t_{\mathrm{cool}}\equiv u_{\rm e}/\Lambda$ where
$u_{\rm e}=3n_{\rm e}\Theta_{e}m_{\rm e}c^{2}$ is the electron internal energy
and $\Lambda\simeq 5.4B^{2}e^{4}n_{\rm e}\Theta_{\rm e}^{2}/(c^{3}m_{\rm
e}^{2})$ is the synchrotron cooling rate for a thermal population of electrons
with $\Theta_{\rm e}\gtrsim 1$ (see Appendix A in Mościbrodzka et al. 2011).
$t_{\rm cool}\approx 11\,{\rm h}\times(B/10\,{\rm G})^{-2}\times(\Theta_{\rm
e}/50)^{-1}$. Hence, if the magnetic field is strong and the temperature is
high, the hot spot may radiatively cool down appreciably on a timescale of
$\sim\\!1$ h. This would increase the energetic output in the millimeter
wavelengths toward the end of the observations (see also Appendix G).

  3. 3.

Hot spots may be subject to shearing in a differentially rotating accretion
flow (e.g., Zamaninasab et al., 2010; Tiede et al., 2020), which in turn
causes depolarization. We expect that this affects the second polarimetric
loop, between $T_{5}$ and $T_{7}$ in Fig. 2, which is smaller and more
distorted than the first one. For the shearing to be important on a relevant
timescale, the hot spot must be relatively large, with a radius $r_{\rm
hsp}>r_{\rm g}$.

  4. 4.

Fluctuations of the Faraday screen result in the time dependence of RM, as
discussed in Appendix A. During the loopy period, a RM variation may
contribute to the $(\mathcal{Q}_{\rm hsp},\mathcal{U}_{\rm hsp})$ vector PA by
as much as 20∘, but the exact angle depends on the unknown details of the
Faraday screen geometry.

  5. 5.

We assume a circular motion, but there could be a non-negligible radial
component of the hot spot velocity, either following the infall of matter onto
a black hole, or perhaps even as an ejection of a magnetic flux tube from the
system (Dexter et al., 2020; Porth et al., 2021).

  6. 6.

We assume a spherically symmetric Gaussian structure of the hot spot, while in
reality it is more likely an irregular region of the accretion flow (see,
e.g., Ripperda et al. 2022). This impacts how the polarized emission
integrated over the hot spot volume decorrelates with the spatially variable
magnetic field.

  7. 7.

We only consider a hot spot moving in the equatorial plane, but the orbit
could be displaced vertically, and the observed feature could, in principle,
be located in the jet stream (Ball et al., 2021).

  8. 8.

The magnetic field structure may be nonaxisymmetric or time-dependent, and
itself subject to the turbulence of the accretion flow.

Because of these important limitations of the model specification and
applicability translating into poorly characterized systematics, we did not
attempt a formal fit to the data. However, with a reasonably general model,
accounting for the time-correlated corruption effects, this could be a
fruitful path forward.

##  4 Summary and discussion

A hot spot orbiting black hole in a clockwise direction in the sky, observed
at low inclination, is a simple scenario which explains the linear
polarization of ALMA millimeter light curves of flaring Sgr~A$^∗$ remarkably
well. Models with vertical magnetic fields threading the hot spot represent
data characteristics far better than those with purely toroidal or radial
magnetic fields. While in this paper the model is postulated for a single
observed event, the hot spot interpretation of millimeter light curves’
variability following X-ray flares will be tested through future observations
of Sgr~A$^∗$.

In this paper, we adopted simple geometric models to interpret observations. A
more physically complete model can be developed using the time-dependent
general relativistic magnetohydrodynamic simulations (GRMHD) of radiatively
inefficient accretion flows representing magnetically arrested disks (MADs;
Narayan et al., 2003; Dexter et al., 2020). In such systems magnetic
reconnection episodes occurring near the black hole event horizon can expel
magnetic flux in large eruptions. For example, Ripperda et al. (2022) reported
orbiting hot spots in the inner 10-30 $r_{\rm g}$, corresponding to flux tubes
of a vertical magnetic field, produced by the reconnection. The flux tubes
remain coherent for $\sim$1 orbit and confine hot, low-density plasma, in
comparison to the surrounding colder, higher-density accretion disk with a
mostly toroidal magnetic field. Individual plasmoids are less suitable as a
model for the observed $\mathcal{Q}$-$\mathcal{U}$ loops since they remain
smaller than 1 $r_{\rm g}$, have a higher density, and originate closer to the
black hole event horizon. Additionally, individual plasmoids in no-guide-field
reconnection, as occurs near the horizon, mainly contain a helical magnetic
field (e.g, Ripperda et al., 2020; Nathanail et al., 2022). We note, however,
that the temporal behavior and hot spot formation in the GRMHD models is
stochastic and may depend on resolution or initial plasma conditions and
magnetic field configuration, so finding the solution matching Sgr~A$^∗$ data
quantitatively is challenging. Hence, in this paper we do not attempt to model
the source with GRMHD simulations, but we hope that the presented simplistic
model will guide more complex work in the future.

The EHT Collaboration published VLBI images of Sgr~A$^∗$ total intensity,
taken on 2017 Apr 6 and 7 (EHTC et al., 2022a, c). Comparisons with the GRMHD
models revealed a preference for viewing angles of $i\leq 30^{\circ}$ (EHTC et
al., 2022e), in agreement with inclination inferred from our polarimetric ALMA
data. The direction of rotation in the EHT images is currently unconstrained,
but the analysis presented in this paper suggests that the ring in Sgr~A$^∗$
should be rotating in the clockwise direction in the sky, assuming that its
direction should be consistent with the inferred rotation of the hot spot.
Additionally, GRMHD MAD models with significant vertical magnetic fields are
preferred in the EHT analysis. Nonimaging, proto-EHT polarimetry of Sgr~A$^∗$
also points toward organized magnetic fields on the relevant scales (Johnson
et al., 2015). We conclude that it is very probable that the X-ray flare and
the subsequent millimeter bright spot produced in Sgr~A$^∗$ on 2017 Apr 11
were generated by magnetic reconnection within the magnetically arrested
accretion flow. In the future, polarimetric EHT images and movies of Sgr~A$^∗$
should give us clearer information on the geometry and evolution of magnetic
fields near the event horizon of the black hole; for example, readers can see
how polarimetric images of event horizon scale emission in M87∗ in EHTC et al.
2021a are strongly constraining for GRMHD simulations in EHTC et al. 2021b.

The models that we employed are similar to the ones used to interpret IR
observations of Sgr~A$^∗$ by the GRAVITY instrument, and so are the inferred
parameters of the observed system (Gravity Collaboration et al., 2018, 2020a,
2020b). The two instruments may be observing the same phenomenon, perhaps with
millimeter polarimetric loops appearing with a delay, after a hot spot cools
down. There are, however, interesting discrepancies. IR observations suggest
shorter periods, and a brightness centroid loop shifted from the gravity
center, which may be an indication of a nonequatorial motion in the base of
the jet (e.g., Ball et al., 2021). Future simultaneous multiwavelength
observations will play an essential role in revealing the relationship between
orbiting hot spots observed at different frequencies.

## Acknowledgements

We thank M. Bauböck, A. Broderick, K. Chatterjee, J. Dexter, F. Eisenhauer, S.
von Fellenberg, D. Haggard, E. Himwich, S. Issaoun, A. Jiménez-Rosales, M.
Johnson, D. Marrone, D. Palumbo, D. Pesce, B. Ripperda, P. Tiede, M. Zajaček,
and the EHT Polarimetry Working Group for helpful comments, discussions, and
encouragements. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2017.1.00797.V. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. This work was supported by the Black Hole Initiative at Harvard
University, which is funded by grants the John Templeton Foundation and the
Gordon and Betty Moore Foundation to Harvard University. MM acknowledges
support by the Dutch Research Council (NWO) grant No. OCENW.KLEIN.113. This
work has been partially supported by the Generalitat Valenciana GenT Project
CIDEGENT/2018/021 and by the MICINN Research Project PID2019-108995GB-C22. We
also thank A. Elbakyan for her contributions to the open science initiative.

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##  Appendix A Variable rotation measure

Polarized radiation emitted by the compact Sgr~A$^∗$ source is affected by the
Faraday effect. Under simplifying assumptions of an external character of the
Faraday screen, it causes rotation of the EVPA by the angle proportional to
the square of the wavelength with respect to the intrinsic (zero wavelength)
EVPA $\chi_{0}$,

$\chi=\chi_{0}+\text{RM}\,\lambda^{2}\ .$ (2)

The proportionality constant RM is the rotation measure, which is dependent on
the gas density, magnetic field along the line of sight, and the plasma
temperature (if the plasma is relativistically hot; Quataert & Gruzinov 2000).
We confirm the previously reported value of RM at $\sim\\!220$ GHz, to be
$\approx-5\times 10^{5}\,{\rm rad/m^{2}}$ (e.g., Bower et al. 2018), with our
measurement of $(-4.3\pm 1.3)\times 10^{5}\,{\rm rad/m^{2}}$ corresponding to
the mean of the entire observing campaign. Given the uniquely high cadence and
S/N of our observations, we can study the RM variability on short timescales,
elaborating on the results of Goddi et al. (2021), where single values per day
were provided for the same data set. As shown in Fig. 4, it is not unusual for
the RM to change by $\sim\\!1\times 10^{5}\,{\rm rad/m^{2}}$ within $\sim$ 30
min, which is a dynamical timescale at the innermost stable circular orbit
(ISCO) of Sgr~A$^∗$ modeled as a nonspinning black hole. Moreover, the intra-
day variability of RM is generally larger than the variability between the
three observing days. This implies a large intrinsic component of the Faraday
screen, most likely corresponding to the rapidly fluctuating gas density in a
turbulent accretion flow very close to the black hole event horizon. We note
that future observations of hot spots, performed by ALMA at higher available
frequency, could greatly reduce systematic uncertainties related to the RM
variation.

Figure 4: Rapid variation of the RM at millimeter wavelengths in Sgr~A$^∗$.
The plot corresponds to the RM fit with linear regression across all four
bands on 4 s cadence, smoothed with a Gaussian filter with 6 min standard
deviation, with color bands around the mean RM indicating the local signal
standard deviation. Dashed lines represent the mean daily values reported by
Goddi et al. (2021). We observe rapid swings of the RM on timescales $\sim$0.5
h, indicating a presence of the intrinsic Faraday screen component
corresponding to the innermost region of the accretion flow. The vertical gray
and red bands indicate the time range of the X-ray flare and polarimetric
loops observed on 2017 Apr 11.

Persistence of the RM sign, along with the persistence of the sign of the
circular polarization and the fractional LP magnitude (see Section 2.2),
suggest that there is a structured magnetic field of well-defined geometry
present in the Sgr~A$^∗$ system, which does not change dramatically with the
accretion flow turbulence. The presence of such dynamically important magnetic
fields near the black hole event horizon is a characteristic theoretical
expectation from MAD systems (Narayan et al. 2003; EHTC et al. 2022e).

##  Appendix B Semi-analytic model constraints

Figure 5: Three observables constrained with ALMA (dashed horizontal lines),
evaluated for the semi-analytic model of Gelles et al. (2021). The thick,
semi-transparent lines correspond to Keplerian models with a zero spin and
orbital radius ranging from $10\,r_{\rm g}$ to $11\,r_{\rm g}$ and a fixed
magnetic field orientation, following the legend. Additionally, we show the
Keplerian models with a spin of $a_{*}=0.99$ and $r_{\rm orb}=11\,r_{\rm g}$
with dashed lines, and two cases of mixed magnetic fields with a spin of
$a_{*}=0.0$ and $r_{\rm orb}=11\,r_{\rm g}$: $B_{z}=B_{r},B_{\phi}=0$ (dashed
blue and dark red) and $B_{z}=B_{\phi},B_{r}=0$ (dashed blue and magenta). We
also show predictions of the numerical model of Vos et al. (2022), the
fiducial model shown in Fig. 3 (red circles), and several models deviating
from the fiducial one by changing a single parameter: a sub-Keplerian orbit at
$r_{\rm orb}=8\,r_{\rm g}$ (black circles), super-Keplerian orbit at $r_{\rm
orb}=14\,r_{\rm g}$ (green circles), a combined vertical and radial magnetic
field (blue circle), a spin of $a_{*}=0.99$ (black ”$\times$”), and a spin of
$a_{*}=0.5$ (black ”+”).

The most elementary model of the hot spot emission corresponds to a
synchrotron point source in an equatorial Keplerian (circular geodesic) orbit
of radius $r_{\rm orb}$, moving through a static axisymmetric magnetic field
$\boldsymbol{B}=[B_{r},B_{\phi},B_{z}]$ in vacuum, viewed by a distant
observer at an inclination angle $i$. In such a case, null geodesics in Kerr
spacetime can be readily computed semi-analytically (e.g., Gralla & Lupsasca
2020), with LP being evaluated following the conservation of the Penrose-
Walker constant (Walker & Penrose 1970). Such a simplified model, implemented
by Gelles et al. (2021), ignores the effects related to the radiative
transfer, the optical and Faraday depth, the finite size of the hot spot, as
well as those related to the finite time of light propagation – it employs a
fast light approximation, implying that the secondary images, emerging from
photons looping around the black hole in a strongly curved spacetime (Darwin
1959) are not treated correctly222A procedure to account for the time delay in
the zero spin case with a single numerical time integral is discussed by
Gelles et al. (2021), but was not used here.. Since these effects are expected
to be subdominant, we nevertheless used the semi-analytic model to compare its
predictions with the observations, and to obtain general constraints on the
system geometry, before verifying our findings with a more physical model of
Vos et al. (2022). A direct comparison between the models of Gelles et al.
(2021) and Vos et al. (2022) is given in Fig. 3, showing a high degree of
consistency for the considered low Faraday depth example. A clear benefit of
the semi-analytic model is that it is very computationally cheap, allowing us
to survey a broad parameter space efficiently.

We employed the model of Gelles et al. (2021) to compare its predictions to
observables described in Section 2.3. None of the considered quantities
$(A,R_{t},R_{a})$ distinguish between the inclination of $i$ and
$180^{\circ}-i$, or between the clockwise and counterclockwise rotation of the
hot spot. Our findings are summarized in Fig. 5. In the first panel, we
quantify how the small observed degree of asymmetry of the
$\mathcal{Q}$-$\mathcal{U}$ loop generally disfavors large inclination,
particularly for poloidal magnetic field configurations with $B_{\phi}=0$. As
noticed by Gravity Collaboration et al. (2018), a purely toroidal (azimuthal)
magnetic field implies a time ratio between the total period (the hot spot
orbital period) and the inner loop of $R_{t}=2$. This is a consequence of the
origin of the inner loop in such a configuration being related to the Doppler
deboost. Hence, our observation of $R_{t}\sim\\!7$ rules out the dominance of
the azimuthal magnetic field and implies that a poloidal magnetic field
component is needed – see the middle panel of Fig. 5. With a vertical magnetic
field and low inclination, the inner loop originates from geometry between the
magnetic field and the line of sight (subject to lensing), rather than from a
Doppler effect, see also Appendix D and Vos et al. (2022). Similar conclusions
follow from the loop area ratio $R_{a}$ measurement. The impact of the black
hole spin is already strongly subdominant at $r_{\rm orb}\sim\\!10\,r_{\rm
g}$, particularly at low inclinations, for which trajectories of photons do
not approach the event horizon, see also Appendix C. If we put these
constraints together, we conclude that only models with a low inclination
angle $i\sim\\!20^{\circ}$ (or, equivalently, $i\sim\\!160^{\circ}$) and
dominance of the vertical magnetic field $B_{z}$ can reproduce the observed
properties of the polarimetric loop. We notice, however, that a configuration
with a slightly higher inclination and added nonvertical magnetic field
component could also reproduce the observed properties – a broader exploration
of the parameter space would be most helpful here.

##  Appendix C Impact of other model parameters

Figure 6: Impact of modifying parameters of the fiducial model: Keplerianity
and orbital radius with a fixed orbital period (left), black hole spin
(center), and inclination (right). The presence and size of the small inner
loop puts strong constraints on the inclination and orbital radius admitted by
the vertical magnetic field model. Additionally, in the vertical magnetic
field model framework, the observed loop size may suggest a large positive
spin of Sgr~A$^∗$.

For the low inclination models dominated by the vertical magnetic field, which
we identified as ones that are capable of matching the features observed by
ALMA, we found that the inner loop becomes smaller ($R_{a}$ increases) with
increasing spin, smaller with growing inclination (i.e., away from $0^{\circ}$
or $180^{\circ}$), smaller with a growing radius of a Keplerian orbit, larger
($R_{a}$ decreases) with a growing nonvertical component of the magnetic
field, and larger for more super-Keplerian orbits of the same period. At the
same time, the loop asymmetry $A$ increases with spin and inclination, but
decreases with a nonvertical magnetic field and for more super-Keplerian
motion of the same period. Some of these dependencies are illustrated in Fig.
6 for the pretzel-like $\mathcal{Q}$-$\mathcal{U}$ loop corresponding to the
fiducial hot spot model.

The observations suggest a $\mathcal{Q}$-$\mathcal{U}$ loop asymmetry slightly
lower, and the inner loop relatively smaller than the fiducial model
parameters. In Fig. 6 we observe that for a vertical magnetic field
configuration ($B=B_{z}$) and low optical depth, loop parameters are very
sensitive to Keplerianity and inclination. Increasing the orbital radius to
$r_{\rm orb}=14\,r_{\rm g}$ (super-Keplerian model) or changing inclination
from $i=158^{\circ}$ to $i=161^{\circ}$ renders the inner loop far too large.
Decreasing $r_{\rm orb}$ to $8\,r_{\rm g}$ (sub-Keplerian model) or changing
inclination from $i=158^{\circ}$ to $i=155^{\circ}$ is a sufficient change to
make the inner loop extremely small or to destroy it altogether. These
considerations can be used to obtain more definite limits on Keplerianity and
inclination in the framework of our model, thus we can claim $r_{\rm
orb}=11\pm 3r_{\rm g}$ and $i=158\pm 3$ deg with a degree of confidence. At
the same time, a relatively smaller observed inner loop (larger $R_{a}$ and
$R_{t}$) than for a fiducial Schwarzschild black hole model may be interpreted
as a hint of a larger positive black hole spin, that is, a prograde motion of
the hot spot with respect to the black hole rotation. This suggestion is
further supported by our observation that deviating the magnetic field from
purely vertical configuration generally also increases the inner loop size.
Nevertheless, obtaining strong conclusions require more systematic inspection
of the parameter space and, in any case, would be rather speculative at this
time given the large systematic model uncertainties. For example, modifying
the magnetic field to include a radial or azimuthal component while
simultaneously increasing the model inclination may yield a similar effect as
the spin increase, see also Fig. 5. Regardless of the current difficulties, it
should be stressed that aggregating constraints from multiple observed hot
spots, or obtaining additional constraints on the inclination and the magnetic
field geometry, could lead to a robust measurement of the Sgr~A$^∗$ black hole
spin in the future through the analysis of the $\mathcal{Q}$-$\mathcal{U}$
loops’ shape.

##  Appendix D Appearance of the resolved model

Figure 7: Spatially resolved appearance of the fiducial Keplerian model at
times $T_{1}-T_{5}$ matching observed times defined in Fig. 2. The first panel
shows the prediction for the observed $\mathcal{Q}$-$\mathcal{U}$ loop shape
with locations of individual snapshots being indicated (dark magenta line), as
well as the intrinsic loop shape, after correcting for the Faraday rotation
(dashed green line; see Appendix F). The remaining panels show the resolved
system geometry at times $T_{1}-T_{5}$ as a model prediction for the EHT
observations on 2017 Apr 11. The projected angular momentum of the hot spot is
indicated with $\mathcal{L}$ and it points into the screen ($i=158^{\circ}$).
Dashed black circles correspond to radii of 12 $r_{\rm g}/D\approx 60\,\mu$as
and $3\sqrt{3}r_{\rm g}/D\approx 26\,\mu$as (critical curve in the
Schwarzschild spacetime), where $D$ is the distance toward Sgr~A$^∗$. Dashed
red lines correspond to the primary and secondary image trajectories on the
observer’s screen in the case of a point source emission. We report the total
intensities and LP of the primary (direct image) and secondary (lensed image)
components, as well as those of the unresolved image. Dark magenta ticks
indicate the EVPA as it would be observed (corrupted by the Faraday rotation),
and green ticks correspond to the model prediction with the Faraday rotation
removed. Polarization ticks are shown for the primary and secondary image
components, and for a total unresolved source (in the image center). Red and
blue curved arrows indicate the receding and approaching side of the hot spot
orbit, respectively.

The calculations performed for the model of Vos et al. (2022) not only
generate full Stokes light curves, but also produce resolved images of the hot
spot model. This is illustrated in Fig. 7, for which images of a pure hot spot
model, with no accretion disk (no observable shadow component) present, were
computed. We used the fiducial model, with a spin of $a_{*}=0$, a vertical
magnetic field, a Keplerian orbit of $r_{\rm orb}=11r_{\rm g}$, and an
inclination angle of $i=158^{\circ}$. Following Appendix F, we fixed the PA of
the projected hot spot angular momentum $\mathcal{L}$ to 57∘ east of north.
The model of the observed $\mathcal{Q}$-$\mathcal{U}$ loop, shown in the top
left panel of Fig. 7 (dark magenta continuous line), is rotated with respect
to the model of the intrinsic source $\mathcal{Q}$-$\mathcal{U}$ (dashed green
line) clockwise because of the Faraday rotation effect. The primary image
appears as a polarized blob rotating at $\sim 60\,\mu$as radius with a total
intensity $\mathcal{I}_{\rm hsp,0}$ fluctuating between $0.02$ and $0.2$ Jy.
The secondary image appears as a weaker component with a PA shifted by $\sim
140^{\circ}$, rotating at a radius of $\sim 30\,\mu$as. It is interesting to
notice that the Doppler effect on the system appearance is subdominant with
respect to the magnetic field and line-of-sight geometry relevant for the
synchrotron emission, since the orbital velocity projected onto the line of
sight is at most $\sim\\!0.1c$ for this low inclination model. The resolved
model is also helpful to highlight the role of the secondary image for a
detailed geometry of the small inner $\mathcal{Q}$-$\mathcal{U}$ loop, where
the secondary image flux density becomes comparable to that of the suppressed
primary image. This effect was discussed by Gelles et al. (2021) in more
detail, and it can also be seen in the model comparison shown in the right
panel of Fig. 3. Given the relative importance of this effect, we expect that
studying detailed geometry of $\mathcal{Q}$-$\mathcal{U}$ loops may, in the
future, deliver a proof of the existence of secondary images, thus providing
an interesting new test of gravity in the strong-field regime.

If the general appearance of Sgr~A$^∗$ on 2017 Apr 11 is similar as reported
by EHTC et al. (2022a) on 2017 Apr 6 and 7, that is if it corresponds to a
roughly uniform ring of diameter $\sim 52\,\mu$as, then the secondary image
location would overlap with the ring of the observable black hole shadow,
possibly being swamped by the ring emission. However, the primary image could
possibly be detected in the simultaneous VLBI data as a distinct component, a
rotating polarized blob appearing between $T_{0}$ and $T_{5}$, that is between
9:20 UT and 10:38 UT. Thus, our findings may constitute a prediction or a
prior for the EHT VLBI data analysis and interpretation. On the other hand,
VLBI data analysis could allow one to measure the size of the hot spot orbit
$r_{\rm orb}$, thus constraining the Keplerianity of the flow and breaking the
degeneracies from which our analysis suffers.

##  Appendix E Total intensity and CP

The presence of Faraday effects breaks the symmetries of the perfectly
optically thin system. In particular, only in the Faraday thin limit (when
both Faraday rotation and conversion are weak) does a change of the magnetic
field sign $B\rightarrow-B$ result in $\mathcal{V}\rightarrow-\mathcal{V}$
with no effect on $\mathcal{P}$. Since the models matching the expected hot
spot polarized flux density $|\mathcal{P}_{\rm hsp}|\lesssim 0.2$ Jy
correspond to a low optical and Faraday depth, the departure from the
optically thin model is rather small. For that reason, while we can fix the
observed hot spot rotation to the clockwise direction based on the direction
of motion on the $\mathcal{Q}$-$\mathcal{U}$ plane, we are not able to
robustly differentiate between the orbital plane inclination of $i$ and
$180^{\circ}-i$ based on the $\mathcal{Q}$-$\mathcal{U}$ loop pattern alone.
Modeling total intensity and circular polarization could be most helpful in
breaking more system degeneracies.

Figure 8: Total intensity $\mathcal{I}$ and circular polarization
$\mathcal{V}$ comparison between the fiducial hot spot model presented in Fig.
3 and the observations. A static source with a total intensity of
$\mathcal{I}_{\rm sha}=2.4$ Jy and a circular polarization of
$\mathcal{V}_{\rm sha}=-0.025$ Jy, representing the contribution from the
observable black hole shadow, was added to the model. For circular
polarization, we show the results corresponding to two opposite magnetic field
orientations (red continuous and dashed lines).

For the fiducial model shown in Fig. 3, we show the corresponding total
intensity and circular polarization light curves compared to ALMA observations
in Fig. 8. A general lack of agreement is not surprising, following the
discussion in Section 3.1. The total intensity rise during the loopy period
may be related to the system cooling down and recovering a normal millimeter
emission level, after the flaring event that pushed emission to higher
energies (Wielgus et al. 2022). This flux density increase may affect the hot
spot component as well, as we noted in Section 3.4. Perhaps the flux density
rise is less apparent in LP, as it is countered by the depolarizing effect of
the differential shearing.

In the case of CP, there is a hint of correlation between the data and the
model prediction for the magnetic field polarity denoted with a dashed line in
Fig. 8. We can obtain decent agreement between the data and model by
multiplying the model light curves with a factor linearly increasing in time,
which would be consistent with a notion of a hot spot cooling down on
dynamical timescales, increasing radiative output in millimeter wavelengths.
See Appendix G for additional hints of plasma cooling being important for the
full interpretation of these observations.

##  Appendix F Intrinsic source orientation

Figure 9: Intrinsic LP in the loopy period, after removing the mean effect of
the Faraday rotation. The 229 GHz band data are shown with a blue color, while
the three remaining ALMA frequency bands are shown with a gray color. The
implied PA of the line of nodes (l.o.n.) of the hot spot orbit as seen in the
sky was estimated at about 147∘ east of north. The intrinsic PA of the hot
spot angular momentum $\mathcal{L}$ was found to be $\sim 57^{\circ}$ east of
north (or 227∘, given the 180∘ ambiguity).

The observed EVPA of Sgr~A$^∗$ is corrupted by the time-dependent Faraday
rotation, as discussed in Appendix A. While we argued for a significant
intrinsic component of the Faraday screen, potentially varying rapidly both in
time and space, the mean value of RM can be used to approximate the
contribution of the external Faraday screen. Hence, by correcting the effect
of the mean RM, we can obtain the orientation of the
$\mathcal{Q}$-$\mathcal{U}$ loop better corresponding to the intrinsic
properties of the compact Sgr~A$^∗$ system. For the mean RM of $-3.28\times
10^{5}$ rad/m2, which we measured on 2017 Apr 11 with ALMA, this corresponds
to rotating the observed $\mathcal{Q}$-$\mathcal{U}$ loop pattern
counterclockwise by 64.4∘ (or rotating the EVPA by 32.2∘) in the 229 GHz band.
The result of this procedure, applied to the data and to the fiducial model
shown in Fig. 3, is presented in Fig. 9. In the figure we also rotated the
remaining three ALMA frequency bands, and aligned the small loops between
bands by removing a frequency-dependent, yet constant in time, component
attributed to the LP of the observable black hole shadow $\mathcal{P}_{\rm
sha}$. The inter-band consistency between loop patterns, with the small
residual differences potentially resulting from the intrinsic Faraday screen
effects and dependence of emission on frequency, further confirms the
robustness of our results. We infer the PA of the hot spot angular momentum
axis projected onto the observer’s screen to be $\sim 57^{\circ}$ east of
north, with a 180∘ ambiguity. These two possible cases correspond to an
inclination of $158^{\circ}$ (as shown in Fig. 7) or $202^{\circ}$ between the
line of sight and the hot spot angular momentum vector. Our results are
reasonably consistent with the hot spot spin axis PA inferred from GRAVITY IR
hot spots data at about 50∘ (corresponding to their reported line of nodes
angle $\Omega$ reduced by 90∘; Gravity Collaboration et al. 2018).

##  Appendix G Delay between 229 and 213 GHz bands

We investigated ALMA light curves of Sgr~A$^∗$ for a presence of delayed
correlation between the frequency bands at 229 and 213 GHz. In the tests
presented in the Section 4.5 of Wielgus et al. (2022) no total intensity
delayed correlations for this data set were detected, which was interpreted as
an indication of low optical depth of the system. For this study, we repeated
the same exercise for the LP component $|\mathcal{P}|$. We used the locally
normalized discrete correlation function (LNDCF), as defined by Lehar et al.
(1992). While on 2017 Apr 6 and 7 we found no indication of a delay in
$|\mathcal{P}|$, on 2017 Apr 11 we identified a delay of $45\pm 15$ s, with a
213 GHz signal lagging behind the 229 GHz. The delay was present during the
loopy period on 2017 Apr 11, and it disappeared later that day, see Fig. 10.
While we do not have a reliable quantitative interpretation of this delay, we
can speculate that it is a feature related to the hot spot, and not to the
accretion flow, given that it only appears in the aftermath of the X-ray
flare, and only in LP, dominated by the hot spot component. Since we have
excluded the optically thick interpretation corresponding to a difference in
the photosphere location at the two frequencies, an alternative source of a
delay could correspond to the rapid cooling of the hot spot (which could be
radiative and/or adiabatic). The emission shifts toward lower frequencies as
the plasma cools down, which can manifest as an apparent correlated delay with
a higher frequency signal leading the lower frequency counterpart. More
quantitative analysis of the observed delays is beyond the scope of this
paper; nevertheless, we notice that interpreting results in a more general
modeling framework that incorporates radiative cooling could potentially shed
light on the collisionless plasma cooling mechanisms and timescales.

Figure 10: Polarized flux density $|\mathcal{P}|$ correlations between 229 GHz
(HI band) and 213 GHz (B1 band) frequencies (blue curves). Locations of the
correlation maxima with associated uncertainties are indicated by blue shaded
regions. For 2017 Apr 6 and 7, no significant delay has been detected. On 2017
Apr 11, the B1 band lags behind the HI band by $45\pm 15$ s. The result is
consistent if only the first 2 h of the Apr 11 observations are used (i.e.,
before 11.0 UT, corresponding to the loopy period, red curve). The lag
disappears if only the latter part of the observations is analyzed (after 11.0
UT, orange curve).


Structure of the tensor product of two simple modules of quantum $GL_{2}$
M. S. Datt
School of Mathematics & Statistics
University of Hyderabad, P.O Central University, Hyderabad, India-500046.
e-mail<EMAIL_ADDRESS>
Abstract: In this article, we consider the tensor product of two simple
modules of quanum $GL_{2}$ over a field of characteristic $p\neq 0$. We show
that it can be expressed as a direct sum of indecomposable twisted tilting
modules. This problem has been studied by Henke and Doty [1] for $SL_{2}$ and
also later on for $SL_{3}$ by S R Doty, Chris Bowman and Stuart Martin, ([7,
8]).

Key words: Simple Modules, Tilting modules, Twisted tilting modules.

##  1 Preliminaries

In this section, we describe terminology and notation. We fix a field $k$ of
characteristic $p\neq 0$. Let $q$ be a primitive $\ell^{th}$ root of unity in
$k$. By a quantum $k$-group $G$, we mean a Hopf algebra $k[G]$ over $k$. We
are concern herewith the quantum $GL_{n}$ as introduced by R. Dipper and S.
Donkin in [3]. Let $A_{q}(n)$ denote the $k$\- algebra generated by $x_{ij}$,
$1\leq i,j\leq n$ subject to the relations (i) $x_{ir}x_{is}=x_{is}x_{ir}$ for
all $1\leq i,r,s,\leq n$ (ii) $x_{is}x_{jr}=q^{-1}x_{jr}x_{is}$ for all $1\leq
i<j\leq n$ and $1\leq r\leq s\leq n$ and finally
$x_{js}x_{ir}=x_{ir}x_{js}+(q-1)x_{is}x_{jr}$ for all $1\leq i<j\leq n$. The
algebra $A_{q}(n)$ is a biialgebra with the comodule structure maps
$x_{ij}\mapsto\sum_{r=1}^{n}x_{ir}\otimes x_{rj}$ (comultiplication) and
counit $\epsilon(x_{ij})=\delta_{ij}$ (the kronecker delta). Let $S_{n}$ be
the group of permutations on $\\{1,2,3\cdots,n\\}$ and $d_{q}=\sum_{\sigma\in
S_{n}}sgn(\sigma)x_{i,i\sigma}x_{2,2\sigma}\cdots x_{n,n\sigma}$, the quantum
determinant. The localisation $A_{q}(n)_{d_{q}}$ of $A_{q}(n)$ at $d_{q}$ (as
$\\{1,d_{q},d_{q}^{2}\cdots\\}$ is a Ore set) is a Hopf algebra. By the
quantum $GL_{n}(k)$ we mean the Hopf algebra $A_{q}(n)_{d_{q}}$. From now on
we denote the quantum $GL_{n}$ over $k$ by $G$ and its corresponding Hopf
algebra by $k[G]$. We also call the Hopf algebra $k[G]$ as the coordinate ring
of $G$. We denote the coordinate functions of $k[G]$ by $c_{ij}$, $1\leq
i,j\leq n$. By a quantum subgroup $H$ of $G$ we mean the Hopf algebra
$k[G]/I_{H}$ for some Hopf ideal $I_{H}$ of $k[G]$. We are concern herewith
two important quantum subgroups of $G$, namely a Borel subgroup of $G$ and
torus contained in $B$. Let $I_{B}$ be the Hopf ideal of $k[G]$ generated by
$\\{c_{ij}~{}|~{}1\leq i<j\leq n\\}$ and the corresponding quantum subgroup of
$G$ is called Borel subgroup of $G$. The quantum subgroup $T$ with the
corresponding Hopf ideal $I_{T}$ of $k[G]$ generated by $\\{c_{ij}~{}|~{}i\neq
j\\}$ is called torus. By definition, we have $T\subset B\subset G$. Let ${\rm
mod}(H)$ denote the category of finite dimesnional $G$-modules, for $H=G,B,T$.
We need to discuss some representation theory of quantum $GL_{n}(k)$ as we
need in the sequel.

We let $X(T)={\mathbb{Z}}^{n}$ and ${\mathbb{Z}}X(T)$ be the group algebra
with basis $\\{e(\lambda)~{}|~{}\lambda\in X(T)\\}$ and multiplication is
given by the rule $e(\lambda)e(\mu)=e(\lambda+\mu)$. By [7], for each
$\lambda=(\lambda_{1},\lambda_{2}\cdots,\lambda_{n})\in X(T)$ we have the
$T$-module $k_{\lambda}$, with the comodule structure map $a\mapsto a\otimes
c_{11}^{\lambda_{1}}c_{22}^{\lambda_{2}}\cdots c_{nn}^{\lambda_{n}}$. Also the
set $\\{k_{\lambda}~{}|~{}\lambda\in X(T)\\}$ is a complete set of mutually
non-isomorphic simple $T$-modules. Any finite dimensional $T$\- module is
completely reducible. For $V\in{\rm mod}(T)$, let $V(\lambda)$ be the sum of
all submodules of $V$ isomorphic to $k_{\lambda}$. We denote the formal
character $\chi(V)$ of $V$ by $\sum_{\lambda\in X(T)}V(\lambda)e(\lambda)$. We
say that an element $\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in
X(T)$ is a dominant weight if
$\lambda_{1}\geq\lambda_{2}\geq\cdots\lambda_{n}$. We denote the set of all
dominant weights by $X^{+}(T)$.

We have a homomorphism $\phi:B\longrightarrow T$, whose comorphism
$\widetilde{\phi}:k[T]\longrightarrow k[B]$ takes $c_{ii}$ to $c_{ii}+$ the
defining ideal of $B$. Hence any $T$\- module can be treated as a $B$-module
via $\phi$. For each $\lambda\in X(T)$, we have the induced module
$\nabla(\lambda)={\rm Ind}_{B}^{G}k_{\lambda}$. The module
$\nabla(\lambda)\neq 0$ if and only if $\lambda\in X^{+}(T)$. For any
$\lambda\in X^{+}(T)$, $\nabla(\lambda)$ is called costandard modules. The
socle $L(\lambda)$ of $\nabla(\lambda)$ is simple, for any $\lambda\in
X^{+}(T)$. Moreover, the set $\\{L(\lambda)~{}|~{}\lambda\in X^{+}(T)\\}$ is a
complete set mutually non-isomorphic simple $G$-modules. For $\lambda\in
X^{+}(T)$, let $\Delta(\lambda)=\nabla(-w_{0}(\lambda))^{*}$, where $w_{0}$ is
the longest element of $S_{n}$ and $\nabla(-w_{0}(\lambda))^{*}$ is the dual
of $\nabla(-w_{0}(\lambda))$. The modules $\Delta(\lambda)$ are called
standard/Weyl modules. For $V\in{\rm mod}(G)$, we call a filtration $0\subset
V_{0}\subset V_{1}\subset\cdots\subset V_{r}\subset\cdots$ of submodules of
$V$ is a good filtration, if $V=\cup_{i=1}^{\infty}V_{i}$, and for each $i\geq
1$, we have $V_{i}/V_{i-1}$ is either 0 or isomorphic to $\nabla(\lambda)$,
for some $\lambda\in X^{+}(T)$. Similarly, if a module $V\in{\rm mod}(G)$ has
a filtration $(0)=V_{0}\subset V_{1}\subset\cdots\subset V_{r}\subset\cdots$
of submodules of $V$ such that the quotient $V_{i}/V_{i+1}$ is either $(0)$ or
isomorphic to $\Delta(\lambda)$ for some $\lambda\in X^{+}(T)$, then we say
that $V$ has a standard filtration. A module $V\in{\rm mod}(G)$ is said to be
a tilting module if $V$ has a good filtration as well as a Weyl filtration.

By a theorem of C. M. Ringel [10], for each $\lambda\in X^{+}(T)$ we have a
unique(upto isomorphism) indecomposable tilting module $T(\lambda)$ of highest
weight $\lambda$. The set $\\{T(\lambda)~{}|~{}\lambda\in X^{+}(T)\\}$ is a
full set of mutually non-isomorphic indecomposable tilting modules. These
tilting modules are called partial tilting modules. And also any tilting
module is a direct sum of partial tilting modules $T(\lambda)$, $\lambda\in
X^{+}(T)$. To discuss representation theory of quantum $GL_{n}(k)$, we also
need to recall some representation theory of general linear group $GL_{n}$
over $k$.

We write $\overline{G}$ for the algebraic group $GL_{n}(k)$. Let $y_{ij}$ be
the coordinate functions for $\overline{G}$. Let $\overline{B}$ be a Borel
subgroup of $\overline{G}$ containing a torus $\overline{T}$. Let
$X^{+}(\overline{T})$ be the set of all dominants weights of $\overline{G}$.
Let $\overline{F}:\overline{G}\longrightarrow\overline{G}$ be the Frobenius
morphism(ordinary). By [3, 1.3.2 Corollary] we have the Frobenius morphism
$F:G\longrightarrow\overline{G}$ whose comorphism takes $y_{ij}$ to
$c_{ij}^{\ell}$. Hence any $\overline{G}$-module can be treated as $G$ module
via $F$. In a paper, by Ann Henke and S R Doty [1], it is shown that the
tensor product of two simple $SL_{2}$\- modules can be decomposed as a direct
sum of twisted tensor product of certain tilting modules. In this paper, we
show that the tensor product of two simple quantum $GL_{2}$-modules is a
direct sum of certain indecomposable modules. And each indecomposable summand
is the tensor product of a certain tilting module with the Frobenius
twist($F$) of twisted tilting module of $\overline{G}$. Our methods are
tilting modules, Steinberg’s tensor product theorem for quantum groups,
Clebsch-Gordan formula.

##  2 The Quantum $GL_{2}.$

In this section, we shall specialize to quantum $GL_{2}(k)$. Let $G$ be the
quantum $GL_{2}(k)$. Let $X_{1}=\\{(a,b)\in X^{+}(T)~{}|~{}0\leq
a-b\leq\ell-1,a,b\geq 0\\}$ and $\pi=\\{(a,b)\in X^{+}(T)~{}|~{}a,b\geq
0,~{}0\leq a-b\leq 2(\ell-1)\\}$. The set $\pi$ is a saturated subset of
$X^{+}(T)$ in the sense that for $\lambda\in\pi$ and $\mu\in X^{+}(T)$, if
$\mu\leq\lambda$ then $\mu\in\pi$. Therefore the category of finite
dimensional $G$-modules $V$ whose composition factors belonging to
$\\{L(\lambda)~{}|~{}\lambda\in\pi\\}$ is equivalent to the category of finite
dimensional modules for the generalized $q$-schur algebra(see [9]). We have
the following Lemma.

Lemma 2.1  (a) If $\lambda\in X_{1}$, then
$L(\lambda)=\Delta(\lambda)=\nabla(\lambda)=T(\lambda)$.
(b) Let $\lambda\in\pi\setminus X_{1}$ and $\lambda=\ell+r$, $0\leq
r\leq\ell-2$. Then the composition factors of $T(\lambda)$ are isomorphic to
$L(\lambda)$ or $L(\lambda-(r+1)(\epsilon_{1}+\epsilon_{2})$ and the character
of $T(\lambda)$ is given by
$\chi(\lambda)+\chi(\lambda-(r+1)(\epsilon_{1}-\epsilon_{2})$.

Proof: (a) This follows by [7] or [11].

(b) We follow the arguments given in [7]. Let $\lambda\in\pi\setminus X_{1}$.
By [8, 3.2], we know that $\nabla(\lambda)$ is uniserial. Let
$\lambda=(a,b)\in\pi,~{}a-b=\ell+r$, $0\leq r\leq\ell-2$ and
$\mu=\lambda-(r+1)(\epsilon_{1}-\epsilon_{2})$. Then by [7], the injective
hull $I(\lambda)$ of $L(\lambda)$ in mod$(\pi)$ is $\nabla(\lambda)$. Now by
[3], we have $((I(\lambda):\nabla(\tau))=[\nabla(\tau),L(\lambda)]$. As
$\lambda\in\pi\setminus X_{1}$, $(I(\mu):L(\lambda^{\prime}))=1$ if
$\lambda^{\prime}=\lambda-(r+1)(\epsilon_{1}-\epsilon_{2})$ or
$\lambda^{\prime}=\lambda$. By [2], the block in mod$(\pi)$ containing
$\lambda$ is $\\{\lambda,\lambda-(r+1)(\epsilon_{1}-\epsilon_{2})\\}$.
Therefore $T(\lambda)$ has $\nabla$-filtration with sections
$\nabla(\lambda),\nabla(\lambda-(r+1)(\epsilon_{1}+\epsilon_{2}))$.

Let St$=(\ell-1+b,b)$ be the Steiberg weight and $\mu=(a-b-(\ell-1),0)$. Let
$\lambda={\rm St}+\mu$. By (a), St and $L(\mu)$ are tilting. Hence by [7] the
module $\nabla({\rm St})\otimes\nabla(\mu)$ is tilting and has highest weight
$\lambda$. Therefore the module $T(\lambda)$ will be the block component of
$\nabla({\rm St})\otimes\nabla(\mu)$ of highest weight $\lambda$ [4
Prop(1.2)]. By [7], we note that two tilting modules are isomorphic if and
only if they have same character. By Clebsch-Gordan formula, we have
$\chi({\rm
St})\chi(\mu)=\chi(\lambda)+\chi(\lambda-(\epsilon_{1}-\epsilon_{2})\cdots+\chi(\lambda-(r+1)(\epsilon_{1}-\epsilon_{2}))$.
As any tilting module is a direct sum partial tilting modules, by putting
altogether, we see that
$\chi(\lambda)+\chi(\lambda-(r+1)(\epsilon_{1}-\epsilon_{2}))$ is the
character of $T(\lambda)$. $\Box$

Let $\mathbb{T}=\\{T(\lambda)~{}|~{}\lambda\in\pi\\}$ and we call them as
special tilting modules of quantum $GL_{2}$. By [5, 2.2] any tilting module of
$SL_{2}$ can be expressed as a direct sum of twisted tensor product of special
tilting modules. This remark still holds for the general linear group $GL_{2}$
as well as quantum $GL_{2}$ also(see [7]). In the case of quantum $GL_{2}(k)$,
we call these modules as $(\overline{F})^{F}$-twisted tilting modules of $G$.
In Theorem 3.2, we show that the tensor product $L\otimes L^{\prime}$ of two
simple module $L,L^{\prime}$ of quantum $GL_{2}$ can expressed as a direct sum
of $(\overline{F})^{F}$ twisted tilting of $G$. All of these summands are
indecomposable. In general, a direct summand of this expression need not
tilting. But $L,L^{\prime}$ are tilting if and only if every indecomposable
summand is tilting. To prove this, it is enough consider the dominant weights
of the form $(a,0)$. We can see this as follows: let $\lambda=(a,b)\in X_{1}$
and $d_{q}=L(\epsilon_{1}+\epsilon_{2})$. We can write
$\lambda=(a-b,0)+b(\epsilon_{1}+\epsilon_{2})$. Then we have
$L(\lambda)=d_{q}\otimes L(a-b,0)=d_{q}L(a-b,0)$.

Let $\lambda=(a,0),\mu=(c,0)\in X_{1}$. Then $a+b<\ell$ or $a+b=\ell+r,$
$r\leq\ell-2$. By Clebsch-Gordan formula, we have
$\chi(\lambda)\chi(\mu)=\chi(a+b,0)+\chi(a+b-1,1)+\cdots+\chi(a,b)$ or
$\chi(\lambda)\chi(\mu)=\sum_{i=0}^{r+1}\chi(a+b)-i(\epsilon_{1}+\epsilon_{2})$,
if $a+b=\ell+r$. We let
$W(\lambda,\mu)=\\{(a+b,0),(a+b-1,1),\cdots(a,b)-(r+1)(\epsilon_{1}+\epsilon_{2})\\}$
and $W_{S}(\lambda,\mu)=\\{(c,d)\in W(\lambda,\mu)~{}|~{}c-d\geq\ell\\}$. We
need the following lemma.

Lemma 2.2(2): Let $(a,0),(b,0)\in X_{1}$ and $L=L(a,0)$ and
$L^{\prime}=L(b,0)$ be two simple modules of highest weights $(a,0)$ and
$(b,0)$ respectively. Then $L\otimes L^{\prime}$ is a tilting module and it is
isomorphic to the direct sum of tilting modules $T(\lambda)$, $\lambda$ varies
over $W(L,L^{\prime})\setminus W_{S}(L,L^{\prime})$.

Proof: We know that the tensor product of two tilting modules is a tilting
module, therefore $L\otimes L^{\prime}$ is a tilting module. Now by the Lemma
2.1, if $W(L,L^{\prime})\subset X_{1}$, then $L\otimes L^{\prime}$ is
isomorphic to the direct sum $\oplus_{\gamma\in W(L,L^{\prime})}T(\gamma)$.
Suppose $W(L,L^{\prime})\not\subset X_{1}$. If $(c,d)\in W_{S}(L,L^{\prime})$,
then as $L\otimes L^{\prime}$ is tilting,
$\chi(c,d)+\chi((c,d)-(c-d+1)(\epsilon_{1}+\epsilon_{2}))$ is the character of
$T(c,d)$, by the Lemma 2.1. Hence $L\otimes L^{\prime}$ is isomorphic to the
direct sum $\oplus_{\gamma\in W(L,L^{\prime})\setminus
W_{S}(L,L^{\prime})}T(\gamma)$. $\Box$

Lemma 2.3(3): Let $(a,0),(b,0)\in X_{1}$ and let $L=L(a,0)$ and
$L^{\prime}=(b,0)$, where $(a,0),(b,0)\in X_{1}$. Then the tilting moddule
$L\otimes L^{\prime}$ is indecomposable if and only if $a=0$ or
$((a,b)=(\ell-1,1)$.

Proof: If $a=0$ then $L\otimes L$ is isomorphic to $L^{\prime}$ and it is
indecomposable by the Lemma 2.1. In case of $(a,b)=(\ell-1,1)$, the
comoposition factors of $L(\ell-1,0)\otimes L(1,0)$ are $L(\ell,0)$ and
$L(\ell-(\epsilon_{1}+\epsilon_{2}))$. The modules $L(\ell-1,0)$ and $L(1,0)$
are tilting and hence $L(\ell-1,0)\otimes L(1,0)$ is also tilting. Now by
Lemma 1.2, the character of $L(\ell-1,0)\otimes L(1,0)$ is
$\chi(\ell,0)+\chi(\ell-(\epsilon_{1}+\epsilon_{2}))$ and this is same as the
character of $T(\ell,0)$ and hence it is indecmposable tilting, as required.

Conversely, suppose $L\otimes L^{\prime}$ is tilting. Suppose
$(a,b)\neq(\ell-1,1)$. First we consider the case $a=\ell-1$ and $b>1$. In
this case we have $a+b=\ell+r$, $1\leq r\leq\ell-2$. By Clebsch-Gordan
formula, $\sum_{i=1}^{r+1}\chi(a+b-i(\epsilon_{1}+\epsilon_{2})$ is the
character of $L\otimes L^{\prime}$ . From this we can conclude that the
indecomposbale tilting modules $T(\ell+1,0)$ and $T(\ell-1,1)$ appear as
direct summands in the decompostion of $L\otimes L^{\prime}$ as direct sum of
indeomposable tilting modules, which is contradiction. Similarly, we can
conclude that $L(a,0)\otimes L(b,0)$ cannot be indecomposbale if $b=1$ and
$a<\ell-1$ $\Box$

Example (i) For $\ell=5$, we have $\ell-1=4$. Let $a=4$ and $b=2$. Then we
have $L(4,0)\otimes L(2,0)=T(6,0)\oplus T(5,1)$.
(ii) Suppose $\ell=5$. Let $a=3$ and $b=1$. Then $L(3,0)\otimes
L(1,0)=T(4,0)\oplus T(3,1)$.

For any $\lambda\in X_{1}$ and $\mu\in X^{+}$, by the Steinberg’s tensor
product theorem [See [7]], we hve $L(\lambda)\otimes L(\mu)^{F})\simeq
L(\lambda+\ell\mu)$, where $F$ is the Frobenius (quantum) morphism. Given any
$a\in{\mathbb{N}}^{+}$, we can write $a=\tau+\ell(\sum_{i=0}^{r}a_{i}p^{i})$,
$0\leq\tau\leq\ell-1$ and $0\leq a_{i}\leq p-1$, uniquely. We call this as the
$(\ell,p)$ expnasion of $a$. Now for $\lambda=(a,0)$ and
$\mu=(\ell\sum_{i=0}^{r}a_{i}p^{r},0)$, we have $L(a,0)\simeq L(\tau,0)\otimes
L(\sum_{i=0}^{r}a_{i}p^{i},0)$. Since $0\leq a_{i}\leq p-1$, we have
$\overline{L}(a_{i},0)=\overline{T}(a_{i},0)$ for all $i$ and similarly
$L(\tau,0)=T(\tau,0)$ as $0\leq\tau\leq\ell-1$. Hence we have the following
Lemma.

Corollary 2.4 Suppose $a\in{\mathbb{Z}}^{+}$ and
$a=\tau+\ell(\sum_{i=0}^{r}a_{i}p^{i})$ be the $(\ell,p)$ expansion of $a$.
Then
$L(a,0)=L(\tau)\otimes(\otimes_{i=0}^{r}\overline{\displaystyle}{{\overline{L(a_{i})}}^{\overline{F}^{i}})}^{F}$.
$\Box$

##  3  Structure of the tensor product of Simple Modules

In this section, we show that the tensor product of two simple $G$-modules can
be expressed as a finite direct sum of $F$-twist of $\overline{F}$-twisted
tensor product of special tilting modules. As we discussed above, we prove
this for simple $G$-modules of the form $L(a,0)$, $a\in{\mathbb{Z}}^{+}$. Let
$\overline{X_{1}}=\\{(a,b)~{}|~{}0\leq a-b\leq p-1\\}$ and
$\overline{\pi}=\\{(a,b)\in X^{+}({\overline{T}})~{}|~{}0\leq a-b\leq
2(p-1)\\}$. First we prove the following Lemma:

Lemma 3.1 For $M\in{\rm mod}(G)$ and $N\in{\rm mod}(\overline{G})$, we have
$M\otimes N^{F}\simeq N^{F}\otimes M$.

Proof: Let $\\{x_{1},x_{2},\cdots,x_{m}\\}$ and
$\\{y_{1},y_{2},\cdots,y_{n}\\}$ be bases for $M$ and $N$ respectively. We
shall show that $M\otimes N^{F}\simeq N^{F}\otimes M$. Let
$\psi_{M}:M\longrightarrow M\otimes k[G]$, $\psi(x_{i})=\sum_{r}x_{r}\otimes
f_{ri}$ (resp. $\psi_{N}(y_{j})=\sum_{s}y_{s}\otimes g_{sj}$) be the structure
map for M (resp. for $N$). Let $\varphi:M\otimes N^{F}\longrightarrow
N^{F}\otimes M$ be the map defined as $\varphi(x\otimes y)=y\otimes x$.

Let $\psi$ be the structure map for $M\otimes N$, i.e., $\psi(x\otimes
y)=\sum_{r,s}x_{r}\otimes y_{s}\otimes f_{ri}g_{sj}$. Now consider
$(\varphi\otimes 1)(\psi(x_{i}\otimes y_{j}))=(\varphi\otimes 1)(\sum
x_{r}\otimes y_{s}\otimes f_{ri}g_{sj})=\sum_{r,s}y_{s}\otimes x_{i}\otimes
f_{ri}g_{sj}$. The module $N^{F}$ is a $G$-module with the structure map given
by $\psi_{N^{F}}(y_{j})=\sum_{s}y_{s}\otimes F(g_{sj})$. By [3], we have
$F(f)$ is in the center, for any $f\in k[G]$.

Let $z_{j}$ be $y_{j}$ when we viewed as an element of $N^{F}$. Hence
$\psi(\varphi\otimes 1)(x_{i}\otimes z_{j})=\sum_{r,s}z_{s}\otimes
x_{r}\otimes F(g_{sj})f_{ri}=\sum_{r,s}z_{s}\otimes x_{r}\otimes
F(g_{sj})f_{ri}$. On the otherhand, we have, $(\varphi\otimes
1)\psi(x_{i}\otimes z_{j})=(\varphi\otimes 1)(\sum_{r,s}x_{r}\otimes
z_{s}\otimes f_{ri}F(g_{sj})=\sum_{r,s}z_{s}\otimes x_{r}\otimes
F(g_{sj})f_{ri}$. Hence $M\otimes N^{F}\simeq N^{F}\otimes M$. $\Box$

Theorem 3.2  Let $a,b\in{\mathbb{Z}}^{+}$. Then the tensor product
$L(a,0)\otimes L(b,0)$ can be expressed is a finite direct sum of
indecomposable modules of the form

$T(\lambda_{-1})\otimes(\overline{T}(\lambda_{0})\otimes\overline{T}(\lambda_{1})^{\overline{F}}\otimes\cdots\otimes\overline{T}(\lambda_{r})^{\overline{F}^{r}})^{F}$

where $\lambda_{-1}\in X_{1}$ and $\lambda_{i}\in\overline{\pi}$, for $0\leq
i\leq r$

Proof: Let $a=\tau_{a}+\ell(\sum_{i=0}^{m}a_{i}p^{i})$ and
$b=\tau_{b}+\ell(\sum_{j=0}^{n}b_{j}p^{j})$ be $(\ell,p)$ expansion of $a$ and
$b$ respectively. Without loss of generality we can assume that $m=n$. By the
the Steinberg’s tensor product theorem, we have
$L(a,0)=L(\tau_{a},0)\otimes(\otimes_{i=0}^{m}\overline{L}(a_{i},0)^{\overline{F}^{i}})^{F}$
and
$L(b,0)=L(\tau_{b},0)\otimes(\otimes_{i=0}^{n}\overline{L}(b_{i},0)^{\overline{F^{i}}})^{F}$.
By the Lemma 3.1, we have $L(a,0)\otimes L(b,0)=(L(\tau_{a},0)\otimes
L(\tau_{b},0))\otimes(\otimes_{i=0}^{n}((L(a_{i},0)\otimes
L(b_{i},0))^{\overline{F}^{i}})^{F}.$

Asuume that $\tau_{a}\geq\tau_{b}$. By Clebsch-Gordan formula, the character
of $L(\tau_{a},0)\otimes L(\tau_{b},0)$ is given by
$\chi(\tau_{a}+\tau_{b},0)+\chi(\tau_{a}+\tau_{b}-1,1)+\cdots+\chi(\tau_{a},\tau_{b})$.
Now as $(\tau_{a},0),(\tau_{b},0)\in X_{1}$, by Lemma 2.2, we have
$L(\tau_{a},0)\otimes L(\tau_{b},0)$ is a direct sum of tilting modules
$T(\lambda)$, where $\lambda\in W(L(\tau_{a},0),L(\tau_{b},0))\setminus
W_{S}(L(\tau_{a},0),L(\tau_{b},0))$. Similarly, as the weights
$(a_{i},0),(b_{i},0)\in\overline{X}_{1}$, we can express
$\overline{L}(a_{i},0)\otimes\overline{L}(b_{i},0)$ as a direct sum of tilting
modules $\overline{T}(\mu)$, where $\mu$ varies over
$\overline{W}(L(a_{i},0),L(b_{i},0))$, for all $0\leq i\leq n$.

Let $L=L(\tau_{a},0)$, $L^{\prime}=L(\tau_{b},0)$ and
$\overline{L}_{i}=\overline{L}(a_{i},0)$,
$\overline{L}_{i}^{\prime}=\overline{L}(b_{i},0)$,and
$I=W(L,L^{\prime})\setminus W_{S}(L,L^{\prime})$ and
$I_{i}=W(L_{i},L_{i}^{\prime})\setminus W_{S}(L.L^{\prime})$, for $0\leq i\leq
n$. Then we have $L\otimes L^{\prime}=\oplus_{\lambda\in I}T(\lambda)$ and
$\overline{L}_{i}\otimes\overline{L}^{\prime}_{i}=\oplus_{\mu\in
I_{i}}\overline{T}(\mu)$, for $0\leq i\leq n$. Thus we have,

$L(a,0)\otimes L(b,0)=(\oplus_{\lambda\in
I}T(\lambda))\otimes(\otimes_{i=0}^{n}(\oplus_{\mu\in
I_{i}}\overline{T}(\mu))^{\overline{F}^{i}})^{F}$

By interchanging tensor prodcut with direct sums, we can express
$L(a,0)\otimes L(b,0)$ can be expressed as a direct sum of modules of the form

$T(\lambda_{-1})\otimes(\overline{T}(\lambda_{0})\otimes\overline{T}(\lambda_{1})^{\overline{F}}\otimes\cdots\otimes\overline{T}(\lambda_{r})^{\overline{F}^{r}})^{F}$

By [$q$-Schur], we see that each term in the sum is indecomposable.

Let
$\underline{\lambda}=(\lambda_{-1},\lambda_{0},\lambda_{1},\cdots\lambda_{n})$,
where $\lambda_{-1}\in\pi$ and $\lambda_{i}\in\overline{\pi}$. Let
$\lambda_{i}=(a_{i1},a_{i2})$, for $-1\leq i\leq r$. Let
$M(\underline{\lambda})=T(\lambda_{-1})\otimes(\otimes_{i=0}^{n}\overline{T}(\lambda_{i})^{\overline{F}^{r}})^{F}$.
$\Box$

Acknowledgements: It gives me great pleasure to thank my teacher S. Donkin for
suggesting the problem and valuable guidance throughout the work.

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