#  Do Machine-Learning Atomic Descriptors and Order Parameters Tell the Same
Story? The Case of Liquid Water.

Edward Danquah Donkor  The Abdus Salam International Center for Theoretical
Physics, Strada Costiera 11, 34151 Trieste, Italy.  Alessandro Laio  SISSA –
via Bonomea 265, 34136 Trieste, Italy<EMAIL_ADDRESS>Ali Hassanali  The Abdus
Salam International Center for Theoretical Physics, Strada Costiera 11, 34151
Trieste, Italy<EMAIL_ADDRESS>
###### Abstract

Machine-learning (ML) has become a key workhorse in molecular simulations.
Building an ML model in this context, involves encoding the information of
chemical environments using local atomic descriptors. In this work, we focus
on the Smooth Overlap of Atomic Positions (SOAP) and their application in
studying the properties of liquid water both in the bulk and at the
hydrophobic air-water interface. By using a statistical test aimed at
assessing the relative information content of different distance measures
defined on the same data space, we investigate if these descriptors provide
the same information as some of the common order parameters that are used to
characterize local water structure such as hydrogen bonding, density or
tetrahedrality to name a few. Our analysis suggests that the ML description
and the standard order parameters of local water structure are not equivalent.
In particular, a combination of these order parameters probing local water
environments can predict SOAP similarity only approximately, and viceversa,
the environments that are similar according to SOAP are not necessarily
similar according to the standard order parameters. We also elucidate the role
of some of the metaparameters entering in the SOAP definition in encoding
chemical information.

###### keywords:

American Chemical Society, LaTeX

SISSA – via Bonomea 265, 34136 Trieste, Italy. IR,NMR,UV

##  1 Introduction

The last decade has seen a tremendous spurt in both the development and
application of machine-learning (ML) approaches to study molecular systems1,
2, 3. ML has become a mainstream component of atomistic modeling. In
particular, for the construction of interaction potentials 1, 4, 5, and for
the analysis of molecular dynamics simulations6, 7, 8, 9. Establishing an
understanding of the underlying physical and chemical principles that make ML
useful, accurate and in short, avoiding its _black box_ usage, remains an open
challenge.

A critical first step in designing a ML-based model for molecular systems, is
identifying atomic descriptors that encode information of the local
environments of a chemical moiety10, 11. Over the recent years, various
flavours of atomic descriptors have been developed including atomic-density
ones which focus on characterizing _local_ environments12, 13, topology based
descriptors relying on graph-theoretical approaches14 to extract the
connectivity patterns in data and many-body tensor network representations
which serve to characterize the global structure of materials15, to name a
few. Most of these descriptors require human intervention in the selection of
various parameters. For example, the size of the local environment or the
number of basis functions one needs to describe the local configurations.
Since ML-based descriptors are designed to capture the physical and chemical
nature of molecular systems, interpreting and understanding their meaning is
important, and can lead to a more rational and also physical choice of the
hyper-parameters entering their definition.

In this work, we address the issue of interpretability, focusing on a specific
class of local-atomic descriptors namely; the Smooth Overlap of Atomic
Positions (SOAP)13, 16. SOAP has become widely popular for identifying
structural finger prints in systems such as liquid water17, 18, 19 and
inorganic crystals20, 21 and more recently, to develop coarse-grained
intermolecular potentials22. Here we examine how SOAP fingerprints are related
to standard order parameters (Figure 1, right panel) that are typically used
to describe the local structure of liquid water23. Moreover, we study how this
relationship changes from bulk liquid water to the air-water interface; a
prototypical system used to study the environmental effects of hydrophobic
interactions24, 25, 26.

To measure the relationship between SOAP-based descriptors and standard order
parameters, we deploy a recently developed technique, dubbed as the
Information Imbalance27 (IB). IB is a statistical test that determines the
relative information content between different distance measures defined on
the same data space. If applied to two different distances, it allows
determining if these distances are equivalent, unrelated, or if one of the two
is more predictive than the other. Specifically, we explore the IB between
SOAP descriptors and a wide variety of order parameters28, 29, 30, 31, 32, 33
that have been built on physical and chemical intuition to characterize
aqueous environments.

Using the IB, we first investigate if a suitable combination of the order
parameters is able to predict the similarity of local environments measured
using SOAP features. We find that even the best combination of order
parameters is able to predict the SOAP similarity only approximately. The
quality of the prediction is better for configurations close to the surface of
water, and is significantly improved by choosing a SOAP length scale parameter
$\sigma=0.25$ Å, which is much smaller than the value typically used
($\sigma=1$ Å). This result may not be too surprising as the structural
information embedded in SOAP descriptors likely contains a richer
characterisation of the local environments in water than the standard order
parameters which are often fine-tuned to capture specific chemical
interactions.

We also investigate the reverse problem, namely if SOAP descriptors are able
to predict the similarity as measured using chemical-intuition based order
parameters. Rather surprisingly, we find that most of these standard order
parameters can be predicted rather approximately. For example, the IB between
the SOAP similarity and the similarity measured by the number of hydrogen
bonds or the Local Structural Index (LSI), appears to be close to the value
observed for distances which are unrelated to each other. Interestingly, SOAP
predicts the standard coordination number the most accurately out of all the
order parameters we examine. The IB can be improved rather marginally by
reducing the value of $\sigma$. We provide some perspectives on the possible
origins of these discrepancies.

The paper is organized as follows. We begin in Section 1 with the Methods
employed in this work, including a summary of both SOAP and order parameters
we study, as well as the Information Imbalance technique. We then move on in
Section 2 to the Results where we illustrate the relationships we unravel
between the SOAP and order parameters that are obtained using the IB method
for both bulk and interfacial water. We then end in Section 3 with some
conclusions of our work.

##  2 Methods

###  2.1 Molecular Dynamics Simulations

Molecular Dynamics (MD) simulations are carried out using the LAMMPS package.
34 We use the TIP4P/2005 35 rigid water model for most of our work. We also
repeat some of our analysis using the TIP3P36 water model which is commonly
used in bio-molecular simulations.

Our initial simulation setup consists of a bulk water system with 729 water
molecules equilibrated at ambient temperature and pressure in a box with sides
27.9$\times$ 27.9$\times$ 27.9 Å. To construct an interface, we add a vaccuum
region of 139.5 Å in the z-direction and then equilibrate within the NVT
ensemble for 10 ns at 300 K with a timestep of 2 fs. This is followed by a
production run of 20 ns. The velocity-rescaling 37 thermostat is used with a
time constant of 100 fs. In our simulations, the real space cut-off for the
Coloumb and Lennard Jones (LJ) interactions is 15 Å. Long range corrections
are treated using the Particle-Particle Particle-Mesh (PPPM)38 solver for both
the Coulomb and LJ interactions. In order to validate the use of our model for
the air-water interface, we computed the surface tension for our simulation,
obtaining a value of (67.98 $\pm$ 0.74)mN/m, consistent with previous reports.
39. For the TIP3P model, we obtain a surface tension of (47.29 $\pm$ 0.40)
mN/m, which is also consistent with previous studies. 39

###  2.2 Identifying Bulk vs Surface Water Environments

Using our MD simulations, a total of approximately 5000 local environments are
sampled from the water-surface system, on which the SOAP power spectrum is
computed using the DScribe software package40. These environments are chosen
by randomly selecting a water molecule every 4 ps. Since we were interested in
understanding how the the relationship between SOAP fingerprints and order
parameters evolves from bulk water to the hydrophobic surface of water, we
characterised the interface using the Willard-Chandler Interface (WCI)41. In
brief, a coarse-grained density field is defined as a sum of Gaussian
functions, with a specified smoothening parameter ($\xi$) centered on all the
atoms. The interface is then chosen as the set of points for which the coarse-
grained density is half of the bulk density. The $\xi$ value used for the WCI
construction was 2.4 Å consistent with previous studies. The water density as
a function of distance from the WCI is then built, yielding the distribution
shown in the Supporting Information (SI Figure 1).

For our analysis, we define various layers from the density profile as
performed in several prior works42, kessler2015structure, which allows for
identifying surface and bulk water molecules. In most of our analysis, we
focus on comparing the bulk and surface as defined by those waters in layer 4
(Bulk) and layer 1 (L1) respectively (see SI Figure 1 for a visual depiction
of these layers relative to the WCI).

###  2.3 Descriptors for the Local Structure of Water

Figure 1: SOAP descriptors and order parameters (left and right respectively)
used in this work. The left panel (top) shows a cartoon of the orientational
and radial extent of the SOAP descriptors and the bottom panel summarizes the
definition of the SOAP power spectrum - see main text for more details. The
right panel shows snapshots of local water environments and some of the
species, distance and angular based criteria that goes into defining the
various order parameters. The symbols next to each variable is what is used
throughout the manuscript to refer to each variable.

In this work, we examine the relationship between chemically inspired order
parameters and SOAP-based descriptors using the Information Imbalance method.
In the following, we begin by summarizing the theory underlying the
construction of SOAP descriptors, (see Section 1.3.1) and subsequently, the
collection of different order parameters that we examine (see Section 1.3.2).
Finally, we also discuss the principles behind the Information Imbalance
method (see Section 1.4). Figure 1 illustrates the SOAP and order parameters
along with the respective symbols that are used throughout the manuscript.

####  2.3.1 Smooth Overlap of Atomic Positions (SOAP)

SOAP has emerged in the last few years as a powerful method to describe the
local environments of atoms and molecules, allowing for a wide range of
applications from the study of structural properties of organic molecules44,
45, 13 and very recently, the properties of liquid water19, 18, 17.

In the context of SOAP, the local density of an atomic environment $\chi$ is
written as a sum of Gaussian functions with variance $\sigma^{2}$, centered on
all species that are neighbours of the central atom:

$\rho_{\chi}(\vec{r})=\sum_{i\in\chi}\exp{\left(\frac{-|\vec{r_{i}}-\vec{r}|^{2}}{2\sigma^{2}}\right)}$
(1)

In this work, we will show that the choice of the value of $\sigma$ parameter
plays a very important role in the ability of SOAP to predict chemical
properties. Specifically, this parameter determines the resolution of chemical
details of the water hydrogen bond network.

The default value of $\sigma$ in the DScribe package is 1.0 Å which to the
best of our knowledge, is the value used in previous studies using SOAP to
study the structure of liquid-water 18, 17.

The atomic neighbour density in equation 1 can be expanded on a basis of
radial basis functions and real spherical harmonics such that:

$\rho_{\chi}(\vec{r})=\sum_{n=0}^{nmax}\sum_{l=0}^{lmax}\sum_{m=-l}^{l}c_{nlm}g_{n}(r)Y_{lm}(\theta,\phi)$
(2)

For practical purposes, one defines the environment $\chi$ by a cut-off radius
($r_{cut}$) and also limits the number of radial and angular basis functions
($n_{max}$, $l_{max}$) used. In the following work, we have examined the
sensitivity of our results to changing $r_{cut}$ from 3.7 to 5.5 Å as well as
varying $n_{max}$ and $l_{max}$ between 10-12 and 6-8 respectively (see SI
Figures 2-5). We find a marginal drop of chemical information contained in the
SOAP descriptors (as reflected in the increase in the IB between SOAP and the
order parameters) as we increase $r_{cut}$, owing to the fact that the order
parameters we have examined, consider fluctuations only within the first
solvation shell of a given water environment. We also find that there is no
significant dependence of our results to the choice of $n_{max}$ and
$l_{max}$. Unless otherwise stated, the $r_{cut}$, $n_{max}$ and $l_{max}$ are
3.7 Å, 10 and 6 respectively.

By accumulating the expansion coefficients, a rotationally invariant power
spectrum can be defined such that,

$p_{nn^{\prime}l}(\chi)=\pi\sqrt{\frac{8}{2l+1}}\sum_{m}(c_{nlm})^{\dagger}c_{n^{{}^{\prime}}lm}$
(3)

Equation 3 defines the components of the SOAP features we will use in our
analysis.

####  2.3.2 Order parameters

Orientational Tetrahedral Order (qtet):

The $q_{tet}$ order parameter is one of the most used local structural
quantities to describe liquid water29, 46, 28, 47, 48, 49. It measures how
much a reference water environment deviates from an ordered tetrahedron whose
vertices are defined by the bond vectors between the four nearest neighbouring
oxygen atoms of a central one. It takes as input the angles between the O-O
bond vectors and gives a value of 1 for a perfectly tetrahedral environment
and closer to zero for environments which are not tetrahedral. Specifically,
the tetrahedrality of a reference water molecule is defined as:

$q_{tet}=1-\frac{3}{8}\sum_{i=1}^{3}\sum_{j=i+1}^{4}\left(\cos\phi_{ij}+\frac{1}{3}\right)^{2}$
(4)

Where $\phi_{ij}$ is the angle between the oxygen molecule of the reference
water and its nearest four neighbours (indexed with $i$ and $j$)

Translational Tetrahedral Order ($S_{k}$):

The translational tetrahedral order parameter is another measure of how much a
reference water environment deviates from a regular tetrahedron29. Whereas
$q_{tet}$ focuses on the angles between the O-O bond vectors, $S_{k}$ is
computed as the variance between the O-O distances of the four nearest water
molecules to a central water:

$S_{k}=1-\frac{1}{3}\sum_{k=1}^{4}\frac{\left(r_{k}-\bar{r}\right)^{2}}{4\bar{r}^{2}}$
(5)

$r_{k}$ is the distance between a reference water molecule and its $kth$
neighbour and $\bar{r}$ is the mean of these distances. It has been shown in
previous works that $S_{k}$ is more sensitive to the local density variations
compared to $q_{tet}$50.

Local Structural Index (LSI):

The LSI is another important variable which has been used to study the
structure of water in the bulk under different thermodynamic conditions51, 30,
52. It is obtained by ranking the O-O distances from an $ith$ central water
molecule such that $r_{1}<r_{2}<...<r_{i}<r_{i+1}<...r_{n}<3.7$ Å$<r_{n+1}$,
and then subsequently, the LSI is computed as:

$\text{LSI}=\frac{1}{n}\sum_{i=1}^{n}\left(\Delta(i)-\bar{\Delta}\right)^{2}$
(6)

where $\Delta(i)=r_{i+1}-r_{i}$ and $\bar{\Delta}$ is the arithmetic mean of
$\Delta(i)$. A high value of LSI implies a larger separation between the first
and second solvation shell and points to a more ordered water environment,
while a lower value is interpreted as a more disordered environment.

Distance to the fifth Oxygen ($d_{5}$):

The $d_{5}$ is defined as the distance between a central oxygen atom and its
fifth closest oxygen atom31, 53, 54. A large value of $d_{5}$ points to a high
separation between the first and second solvation shell of a water environment
and is interpreted as a locally ordered structure. A smaller value indicates a
smaller separation between the first and second solvation shells and with a
similar logic, a disordered local environment.

Coordination Number (C.N.):

The coordination number quantifies the average number of atoms that surround a
chosen central site within some radial cutoff. It can be computed from the
radial integral of the radial distribution function (RDF). In order to have a
smooth and continuous definition of the coordination number, we use a
switching function, commonly done in the construction of different types of
collective variables55, 56:

$C.N.=\sum_{j=1}^{N}\frac{1-\left(\frac{r_{j}}{r_{cut}}\right)^{12}}{1-\left(\frac{r_{j}}{r_{cut}}\right)^{28}}$
(7)

where $r_{j}$ is the distance between a central oxygen atom and atom $j$.

Number of Hydrogen Bonds ($N_{H.B.}$):

We adopt the definition of hydrogen bonding by Luzar and Chandler32, 33. This
definition is based on a geometrical criterion and considers two water
molecules to be hydrogen bonded when the distance between the donating and
accepting oxygen atoms (OD and OA) is within 3.5 Å and the angle formed by the
bond vector between the donating hydrogen and oxygen (HD and OD) and the bond
vector between OD and OA is less than 30∘.

Voronoi Density ($\rho_{voro}$):

A quantitative measure of the local density ($\rho_{voro}$) in water can be
extracted using a Voronoi tessellation of the water network 57, 58. The
$\rho_{voro}$ is then defined as the inverse of the Voronoi volume associated
with a single water molecule which is a sum of the atomic contributions coming
from the oxygen and two hydrogen atoms. The Voronoi tesselation is carried out
using the Voro$++$ code 59.

SOAP distance from ice (d${}_{\text{ice}}$):

From the SOAP descriptors (power spectrum) show in equation 3, it is then
possible to define distances between structures. Several groups including ours
have examined how SOAP environments in liquid water compare to different
phases of ice18, 19, 17. In this work, one of these variables we use is
comparing SOAP environments in liquid water to those in hexagonal ice (ice 1h)
which is referred to as $d_{ice}$.

$d_{ice}=\sqrt{1-\frac{\vec{p}(\chi_{\text{water}})\cdot\vec{p}(\chi_{\text{ice
1h}})}{|\vec{p}(\chi_{\text{water}})||\vec{p}(\chi_{\text{ice 1h}})|}}$ (8)

Where $\vec{p}(\chi_{\text{water}})$ and $\vec{p}(\chi_{\text{ice 1h}})$ are
the SOAP feature vectors for the liquid water and hexagonal ice environments
respectively.

###  2.4 Information Imbalance (IB)

The Information Imbalance (IB) is a recently developed method which can be
used to quantify the relative amount of information between different types of
variables which may or may not have the same measures of distance. The reader
is referred to the original work for details27. Here, we summarize the key
ideas behind the method and why it provides a powerful tool for application in
the context of the problems we address here.

Given a dataset with $N$ data points and characterised by $F$ features, we can
define distance measures A and B between the data points, such that A and B
are computed using any subset of the feature space $F$ of choice. With these
distances in hand, the IB is then defined as:

$\Delta\left(A\rightarrow B\right)=\frac{2}{N}\langle
R^{B}|R^{A}=1\rangle=\frac{2}{N^{2}}\sum_{i,j:R_{ij}^{A}=1}R_{ij}^{B}$ (9)

Where $R_{ij}^{A}$ and $R_{ij}^{B}$ are the rank matrices obtained from
distances A and B respectively, such that $R_{ij}^{A}=1$ if $j$ is the first
nearest neighbour of $i$ and $R_{ij}^{A}=2$ if $j$ is the second nearest
neighbour of $i$ and so on. With this definition, if $\Delta\left(A\rightarrow
B\right)\sim 0$ this means A can fully predict B, while
$\Delta\left(A\rightarrow B\right)\sim 1$ implies that A cannot predict B, as
the ranks estimated with B are uncorrelated to those estimated with A. It is
important to note that the IB is by definition asymmetric and thus one can
examine predictability between different distances in both directions; if for
example, $\Delta(A\rightarrow B)\sim 0.1$ and $\Delta(B\rightarrow A)\sim
0.4$, A will be able to predict B with better reliability than the reverse.

##  3 Results

###  3.1 Predicting SOAP from Order Parameters

In the ensuing analysis, we explore the relative information content between
SOAP and order parameters using the IB. We begin by determining the ability of
the order parameters to predict the full SOAP feature space and specifically,
how sensitive this predictability is to the choice of the parameter $\sigma$
used in SOAP. Furthermore, we also examine how the IB changes as one moves
from bulk water to the air-water interface.

Figure 2: Information Imbalance between combinations of order parameters and
the SOAP space. Comparison for Bulk (Red) and Surface (Black) and for
$\sigma=1.0$ Å (Dotted line ) and $\sigma=0.25$ Å (Full line). For
$\sigma=1.0$ Å one needs a minimum of about 3 order parameters to describe the
local fluctuations of the water environments contained in the full SOAP space,
both in the bulk and at the Surface. For $\sigma=0.25$, the number of order
parameters needed increases by approximately 3.

In Figure 2 we show the IB between order parameters and SOAP, which encodes
how many different order parameters are needed to predict the distances
computed with the full high-dimensional SOAP vector. In order to obtain the IB
for different combinations, we employ a greedy optimization whereby one
iteratively selects the best combination of order parameters to minimize the
IB between that combination and the SOAP space. This protocol can also be
employed to select the $d$-dimensional compact SOAP descriptor that minimizes
the Information Imbalance between SOAP and a target variable. For more
details, we refer the reader to the original manuscript on the IB method27.

Figure 2 shows that using $\sigma=1.0$ Å, for both bulk and surface, a
combination of three order parameters minimizes the IB to approximately 0.45
and 0.35 respectively. Recall, that an IB of 0.45 implies that if the data
space included, say, 1000 local environments, the first neighbour environments
according to the order parameters will be, on average the 225th neighbour
according to SOAP. This is rather far from optimal since it suggests that
local water environments predicted to be similar in the space of the order
parameters, are quite distant using SOAP.

Using $\sigma=0.25$ Å, we notice a consistent increase in the number of order
parameters needed to obtain the minimum IB; five order parameters are needed
to obtain roughly the same IB as with $\sigma=1.0$ Å. This shows that by
reducing the coarse-graining length of the Gaussian density from which the
SOAP power spectrum was built, we essentially add more chemical information
and consequently, this requires more order parameters to correctly describe.
The sensitivity of the interplay between the size of the basis set and the
choice of $\sigma$ in describing atomic environments has recently been
discussed by Pozdnyakov and co-workers60. They show that in order to obtain
the same sensitivity of the density expansion coefficients to vibrational
distortions in a single methane molecule, the $\sigma$ should be approximately
half that of the minimal interatomic distance.

Figure 3: Information Imbalance between combinations of order parameters and
the full SOAP space for $\sigma=0.25$, showing the selected variables for both
bulk and surface. It is noteworthy that the difference between the Bulk and
Surface for $\sigma=0.25$ Å is reflected in the difference between the
combinations of variables needed to minimize the IB between the order
parameters and the SOAP space. We can also see that $\rho_{voro}$ and
$d_{ice}$ are consistently important to describe the SOAP space for both the
bulk and surface whereas $q_{tet}$ for example becomes more important for the
surface due to the enhanced orientational correlations at the Surface compared
to the Bulk.

While Figure 2 shows that with $\sigma=0.25\ \text{\AA}$, the gap in the IB
between the bulk and surface becomes much less pronounced compared to
$\sigma=1.0\ \text{\AA}$, the combination of order parameters that are needed
to describe the fluctuations in the bulk and the surface are not the same. In
Figure 3, we show the order parameters that lead to the most predictive
distance in the SOAP space. $\rho_{voro}$ and the one-dimensional SOAP based
descriptor ($d_{ice}$) consistently appear to be important for both the bulk
and surface. Remarkably, the LSI parameter which is often used to study local-
water structure in the bulk61, 30, does not appear in any of the chemical
combinations that lowers the IB. Interestingly, the tetrahedral order
parameter, $q_{tet}$ plays a more important role at the surface due to the
fact that there is an enhanced orientational order. These trends are
consistently reproduced for simulations of the surface of water using the
TIP3P water model (see SI Figure 8).

###  3.2 Predicting Order Parameters from SOAP

The preceding analysis examined how well the distances in the order parameter
space can predict the full SOAP distances. Next we move on to discover the
ability of SOAP to capture the information contained in single order
parameters and at the same time, we use the IB to identify the optimal number
of SOAP components required to perform a prediction.

Figure 4 shows the IB between SOAP space and the individual order parameters
for both bulk and surface (left and right panels respectively), as a function
of the number of components. Note that for the SOAP parameters deployed here,
the full power spectrum is a vector with dimension $D=4410$. The easiest
variable to predict is the coordination number ($C.N.$) and the most difficult
to predict is the total number of hydrogen bonds ($N_{H.B.}$). Overall, the
effect of changing from $\sigma=1.0\ \text{\AA}$ to $\sigma=0.25\ \text{\AA}$
is consistent with the previous observations seen in Figure 2 and Figure 3.
The IB is reduced for smaller $\sigma$ \- the solid circles shown in both
panels correspond to the optimal IB obtained using the larger $\sigma$ value.
By increasing the local atomic resolution, the IB can decrease by up to 50%
and thus enhance the resolution of SOAP for the prediction of several order
parameters.

Figure 4: Convergence of the IB between the SOAP space and individual order
parameters with $\sigma=0.25$ Å for both bulk (left) and surface (right) . The
solid circles represent the optimized IB obtained using $\sigma=1.0$ Å . The
rate of convergence of the IB for different variables are different however,
with a minimum of about 10-20 SOAP vector components, the minimum IB is
reached. Using $\sigma=0.25$ Å significantly increases the information
contained in the SOAP space about the order parameters.

In similar spirit to the analysis presented earlier in Figure 3, the IB
between SOAP and order parameters identifies chemical coordinates such as
coordination number, $d_{ice}$, $d_{5}$ and $\rho_{voro}$, whose chemical
information is well characterized by SOAP. Furthermore, although different
order parameters plateau to the optimal IB with different rates, one only
needs a total of 10-20 components of the full SOAP power spectrum to predict
these order parameters. These trends are found in both bulk and surface liquid
water situations.

Perhaps what is more surprising is that SOAP does not seem to be able to
predict accurately the number of hydrogen bonds, LSI and $q_{tet}$. On the one
hand, this suggests that the SOAP features may not be complete in terms of
their information content of the underlying chemical system. However, many of
these order parameters are constructed based on strict geometrical cutoffs.
For example; along distances or angles or number of neighbours which is then
reflected in the IB.

To illustrate this effect more clearly, we show in Figure 5 a set of water
environments which compares and contrasts distances computed between SOAP,
$q_{tet}$ and hydrogen bonding. We begin with the top panel comparing SOAP and
$q_{tet}$. Within an $r_{cut}$ of 3.7 Å, there are two interstitial oxygen
atoms (purple arrow). Since the definition of $q_{tet}$ only looks at the
first 4 nearest oxygens, the two environments are flagged as non tetrahedral
(left) and highly tetrahedral (right). Due to the restriction of focusing on
the nearest 4 oxygens, the $q_{tet}$ parameter picks up an angle that clearly
deviates from tetrahedrality. In fact, when one restricts the SOAP computation
to only the first four neighbours, the SOAP space is completely predictive of
$q_{tet}$ (see SI Figure 10).

Figure 5: Snapshot of two sets of environments $\chi$, $\chi^{\prime}$ and
$\chi^{\prime\prime}$, $\chi^{\prime\prime\prime}$ which are nearest
neighbours in SOAP space but are distant in $q_{tet}$ (top panel) and
$N_{H.B.}$ (bottom panel) space. Since the definition of $q_{tet}$ focuses on
the first four neighbours of a central atom within the first solvation shell,
the two environments are flagged as not tetrahedral (top left) and tetrahedral
(top right), while the SOAP space predicts the two environments to be similar.
For $N_{H.B.}$, its strict angular cut-off picks up slight angle changes
($\sim$ 2∘) which labels the two environments as having different number of
hydrogen bonds. However, these are clearly two environments that are more
similar than different.

In similar spirit to the analysis presented for $q_{tet}$, the bottom panel
shows two environments that are again close in SOAP space, but far in terms of
hydrogen bonding. Specifically, the bottom left panel shows an environment
that accepts and donates two hydrogen bonds (N${}_{H.B.}=4$) while the bottom
right shows a small change in the local geometry where the angle used for the
hydrogen bonding criterion changes from 29∘ to 31∘ resulting in a defect that
now accepts two but donates only one hydrogen bond (N${}_{H.B.}=3$). In
hydrogen-bonding space, these configurations are topologically different but
in SOAP space, these are, unsurprisingly very similar.

Figure 6: Convergence of the IB between SOAP and selected order parameters, as
a function of the number of SOAP vector components. The compression is done
for all the layers of the density profile depicted in the supplementary
material. These results confirm previous reports showing that most of the
significant structural changes in the water environments occurs in the topmost
layer.

###  3.3 Predictions Within Order Parameters

Our analysis shows that there is a complex interplay of different order
parameters and that they need to be used in combination in order to predict
SOAP similarity both in the bulk and at the air-water interface. In the field
of aqueous science where the structure and dynamics of liquid water is studied
in terms of different local structures, it is quite common to synonymously
associate the different order parameters such as $d_{5}$, LSI and $q_{tet}$ .
However, these order parameters are not equivalent.

Figure 7: Information Imbalance plane 27 for the space of order parameters,
showing the IB obtained between different pairs of order parameters. In the
Bulk (left) the lowest IB obtained is $\sim$ 0.7 which is between $C.N.$ and
$d_{5}$. The rest of the variables are all not informative of each other. In
the Surface (right), the lowest IB obtained $\sim$ 0.4 is also between $C.N.$
and $d_{5}$. In general, most of the variables are uninformative of each other
both in the Bulk and on the Surface.

Figure 7 shows the IB between all pairs of order parameters for both bulk and
surface environments. For bulk environments, we observe that the lowest
imbalance obtained is approximately 0.7; between the coordination number and
$d_{5}$. Essentially all the other order parameters are not informative of
each other consistent with previous work from our group19. Moving to the
surface environments marginally improves the predictability for the
coordination number and $d_{5}$ pair. All in all, it is clear that the
different order parameters used to characterized local water structure contain
very little information about each other. For example, considering the IB
between tetrahedrality (qtet) and local density ($\rho_{voro}$) which is
closer to 1, shows that they provide different information about the local
environment. Thus, an environment that is more tetrahedral may not have a
lower local density and vice-versa.

###  3.4 Diagnostics for Water Structure

We have used the IB approach to examine the coupling between ML-based local
atomic descriptors and chemistry-inspired order parameters. We then explored
whether this could be used as a diagnostic tool to study the length-scale of
the perturbations induced by the prescence of the surface.

Earlier, we compared the IB for bulk and surface water environments, where the
latter focused on interfacial waters that reside only within $\sim$ 3.0 Å of
the WCI. Examining the relative density of water with respect to the WCI (see
SI Figure 1), one observes that the correlations albeit weaker, extend up to 1
nanometer from the surface. The extent of the thickness of this interface as
probed by both surface sensitive vibrational spectroscopy experiments and
simulations, and how it is manifested in terms of local structure, continues
to be a topic of active research. 42, 62, 63.

Since the IB appears to provide a sensitive measure of the surface water
environments, we performed the compression of the full SOAP space for selected
order parameters separately for different layers of water with respect to the
WCI (see SI Figure 1 for an illustration of how the layers are defined). The
left and right panels of Figure 6 shows the IB between SOAP and the
coordination number and $\rho_{voro}$ respectively, as a function of the
number of SOAP components. Our results show that the IB in the second and
third layer are essentially indistinguishable from the bulk. This shows that
there is essentially only a very thin layer of water ($\sim$ 3.0 Å near the
surface) whose local structural properties are significantly different,
consistent with previous studies 42.

##  4 Conclusions

In this work, we have used a recently developed statistical test (IB) to
examine the relationships between the SOAP descriptors and chemically inspired
order parameters that are used to characterize local water structure. We focus
our analysis on how these relationships evolve as one moves from bulk water to
the hydrophobic surface of water.

Examining the IB between the full SOAP space and order parameters shows that
these two classes of variables do not contain the same information. While the
chemical information encoded in the SOAP descriptors can be improved by
reducing the Gaussian smearing, the IB between the order parameters and the
SOAP descriptors obtained is far from optimal. The predictability improves
marginally for water environments sampled at the interface. Given that the
SOAP descriptors are designed to be generic and do not probe specific chemical
details, this observation may not be too surprising. The consistency between
SOAP and the order parameters could be improved by including features that
probe directly the fluctuations of the hydrogen bond network.

Order parameters like the ones we have investigated here to probe the local
structure of water, are used ubiquitously for analyzing molecular dynamics
simulations of liquid water. In particular, these are often used to assign
whether the water structure arises from a low or high density local
environment64, 51, 19. However, our analysis suggests that most of these order
parameters are somewhat independent of each other and cannot be used
synonymously. Furthermore, the SOAP descriptors seem to be probing the local
environment of water molecules in a much less specific manner. One may argue
that order parameters such as the hydrogen-bonding criterion are designed to
be very sensitive to subtle changes in local topology which cannot be captured
by SOAP. We believe that our results suggest that sensitivity analysis of
geometric criteria used in chemistry-inspired order parameters would be
appropriate.

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##  5 Supplementary Information

Figure S1: Snapshot of MD simulaton box (Left)-The water molecules are color-
coded based on their distance from the WCI. Number density profile of water
molecules as a function of the distance from the WCI (Right), the colors
correspond to the distinct structural layers that occur and have been labelled
(L1-L4) for Surface-Bulk respectively. Figure S2:
$\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ for $\sigma=0.25$
using ($n_{max}$,$l_{max}$) = (10,8) (Red Symbols) and ($n_{max}$,$l_{max}$) =
(10,6) (Green Symbols). For fixed $n_{max}$, the predictability does not
change with an increase in $l_{max}$ Figure S3:
$\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ for $\sigma=0.25$
using ($n_{max}$,$l_{max}$) = (12,6) (Red Symbols) and ($n_{max}$,$l_{max}$) =
(10,6) (Green Symbols). For fixed $l_{max}$, the predictability does not
change with an increase in $n_{max}$ Figure S4:
$\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ for $\sigma=0.25$
using ($n_{max}$,$l_{max}$) = (12,8) (Red Symbols) and ($n_{max}$,$l_{max}$) =
(10,6) (Green Symbols). Changing both $n_{max}$ and $l_{max}$ does not
significantly change the predictabilities Figure S5:
$\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ for $\sigma=0.25$
and $r_{cut}$ = 3.7 (Red), $r_{cut}$ = 4.5 (Blue) and $r_{cut}$ = 5.5 (Green).
In general, an $r_{cut}$ of 5.5 makes the SOAP consistently less informative
about the chemical variables (although very slightly). The main thing to
notice is that an $r_{cut}$ of 3.7 is able to capture as much as SOAP can, all
the local fluctuations contained in the Order Parameter. Figure S6: Evolution
of $\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ as a function
of the layer definitions showed in SI figure 1 for $\sigma=1.0$ Å (Left) and
$\sigma=0.25$ Å (Right). There is a general decrease in the IB as we go from a
larger to a smaller $\sigma$. This also confirms that much of the structural
changes that occur is at the topmost layer (L1) Figure S7: $\Delta(\text{Order
Parameters}\longrightarrow\text{SOAP}$) for the TIP3P and TIP4P/2005 water
model for the Surface layer. The optimum set of 5-variables selected are
consistent across the two models with only a difference in one selected
variable ($\text{q}_{tet}$ for the TIP4P/2005 and $\text{S}_{k}$ for TIP3P).
Figure S8: $\Delta(\text{Order Parameters}\longrightarrow\text{SOAP}$) for the
TIP3P and TIP4P/2005 water model for the Bulk layer. The results are
consistent across the two water models. Figure S9: Convergence of
$\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ as a function of
the number of SOAP components for the TIP3P water model. The solid circles
represent the optimized IB obtained for the TIP4P/2005 model for the specified
color coded chemical variable. These results show very small differences in
the IB across the two models and hence are consistent. Figure S10: Information
Imbalance between SOAP and $\text{q}_{tet}$ for SOAP descriptors computed with
6,5 and 4 nearest neighbours. As the number of neighbours used to compute the
SOAP descriptors decreases, the ability of SOAP to predict $\text{q}_{tet}$
increases. With 4 nearest neighbours, the SOAP space is completely predictive
of $\text{q}_{tet}$


$\operatorname{Ext}_{G_{\alpha}}^{\bullet}(\mathcal{L}_{\alpha},\mathcal{L})$.

There is a non-degenerate paring as follows

(7.6) $\begin{split}\langle{-},{-}\rangle:\,K^{0}(R)\times
K_{0}(R)&\to\mathbb{Z}[q,q^{-1}]\\\ (\mathbb{P}\ ,\
M)&\mapsto\operatorname{Dim}\hom_{R}(\mathbb{P},M)\end{split}$

where
$\operatorname{Dim}\hom(\mathbb{P},M)=\sum\limits_{n\in\mathbb{Z}}\operatorname{dim}\operatorname{Hom}_{R}(\mathbb{P},M[n])q^{n}$

######  Lemma 7.6.

Let $\alpha\in Q^{+}$,
$\mathcal{L}_{a}\in\operatorname{add}({\mathcal{P}_{\alpha}})$, and
$\mathbb{P}_{a}=\operatorname{Ext}_{G_{\alpha}}^{\bullet}(\mathcal{L}_{\alpha},{\mathcal{L}_{a}})$.
If we decompose $\mathcal{L}_{a}$ as

$\mathcal{L}_{a}=\bigoplus_{\mu\in\mathcal{P}_{\alpha}}L_{a}(\mu)\boxtimes\operatorname{IC}(\mu)$

then we have
$\operatorname{Dim}\hom(\mathbb{P}_{a},L(\mu))=\operatorname{Dim}L_{a}(\mu)$
where
$\operatorname{Dim}L_{a}(\mu)=\sum\limits_{n\in\mathbb{Z}}\operatorname{dim}L_{a}(\mu)_{n}q^{n}$

###### Proof.

First we assume that $\mathbb{P}_{a}$ is indecomposable projective modules.
For any $\mu^{\prime}\in\operatorname{KP}({\alpha})$, we have

$\operatorname{Hom}_{R}(\mathbb{P}_{\mu^{\prime}},L(\mu))=\delta_{\mu^{\prime},\mu}$

Since
$\mathbb{P}_{a}\cong\displaystyle\bigoplus_{\mu\in\mathcal{P}}\mathbb{P}_{\mu}^{\operatorname{Dim}L_{a}(\mu)}$,
then

$\operatorname{Hom}_{R}(\mathbb{P}_{a},L(\mu))=\operatorname{Hom}_{R}(\mathbb{P}_{\mu}^{\operatorname{Dim}L_{a}(\mu)},L(\mu))=\operatorname{Dim}L_{a}(\mu)$

∎

Next, we define the product on $K_{0}(R)$.

######  Definition 7.7.

For two modules $M,N\in R-\operatorname{gmod}$, we define their product
$M\circ N$ so that

(7.7) $\langle\mathbb{P},M\circ
N\rangle=\langle\operatorname{Res}\mathbb{P},M\boxtimes N\rangle\quad\text{for
any projective modules $\mathbb{P}\in R-\operatorname{proj}$.}$

Since this pairing is non-degenerate, we get $M\circ N$ is well-defined.

######  Remark 7.8.

The above definition coincides with the Induction Functor given in [KL09,
Section 3.1] by [KL09, Propositional 3.3]. Let ${\alpha+\beta}=\gamma$ be a
partition of $\gamma$. There exists a canonical embedding

$R_{\alpha}\otimes R_{\beta}\to R_{\gamma}$

We denote by $e_{{\alpha,\beta}}$ the image of
$\sum\limits_{\mathbf{i}\in\langle I\rangle_{\alpha};\mathbf{j}\in\langle
I\rangle_{\beta}}e_{\mathbf{i}}e_{\mathbf{j}}$. The induction of two modules
$M,N$ is given by $M\circ
N=R_{\gamma}e_{{\alpha,\beta}}\otimes_{R_{\alpha}\otimes R_{\beta}}M\boxtimes
N$. For any $\mathbf{i}\in\langle I\rangle_{\gamma}$ we have

$\displaystyle e_{\mathbf{i}}R_{\gamma}e_{{\alpha,\beta}}$
$\displaystyle=\hom(\mathbb{P}_{\mathbf{i}},\mathbb{P}_{\alpha,\beta})$
$\displaystyle=\operatorname{Ext}_{G_{\gamma}}^{\bullet}(\mathcal{L}_{\mathbf{i}},\mathcal{L}_{\alpha}\star\mathcal{L}_{\beta})\quad$
by isomorphism (7.4)
$\displaystyle=\operatorname{Ext}_{G_{\gamma}}^{\bullet}(\operatorname{Res}_{{\alpha,\beta}}\mathcal{L}_{\mathbf{i}},\mathcal{L}_{\alpha}\boxtimes\mathcal{L}_{\beta})$
by the adjoint property of $(\operatorname{Res},\star)$
$\displaystyle=\hom(\operatorname{Res}_{{\alpha,\beta}}\mathbb{P}_{\mathbf{i}},R_{\alpha}\otimes
R_{\beta})$ by isomorphism (7.4)

Here $\hom(A,B)=\oplus_{n\in\mathbb{Z}}\operatorname{Hom}_{R_{Q}}(A,B[1])$.
Hence, we have

$\displaystyle\hom(\mathbb{P}_{\mathbf{i}},M\circ N)$
$\displaystyle=e_{\mathbf{i}}R_{\gamma}e_{{\alpha,\beta}}\otimes_{R_{\alpha}\otimes
R_{\beta}}(M\boxtimes N)$
$\displaystyle=\hom(\operatorname{Res}_{{\alpha,\beta}}\mathbb{P}_{\mathbf{i}},R_{\alpha}\otimes
R_{\beta})\otimes_{R_{\alpha}\otimes R_{\beta}}(M\boxtimes N)$
$\displaystyle=\hom(\operatorname{Res}_{{\alpha,\beta}}\mathbb{P}_{\mathbf{i}},M\boxtimes
N)$

Let us consider the functor
$\operatorname{Res}_{{\alpha,\beta}}(M)\overset{\operatorname{\scriptstyle
def.}}{=}e_{{\alpha,\beta}}M$. Since

$\hom(\mathbb{P}_{\mathbf{j}}\boxtimes\mathbb{P}_{\mathbf{k}},\operatorname{Res}_{{\alpha,\beta}}(M))=e_{\mathbf{j}}e_{\mathbf{k}}M=\hom(\mathbb{P}_{\mathbf{j}}\star\mathbb{P}_{\mathbf{k}},M)=\hom(\mathbb{P}_{\mathbf{j}\mathbf{k}},M)$

then for any two projective modules $\mathbb{P}_{1},\mathbb{P}_{2}$, we obtain

(7.8)
$\hom(\mathbb{P}_{1}\boxtimes\mathbb{P}_{2},\operatorname{Res}_{{\alpha,\beta}}(M))=\hom(\mathbb{P}_{1}\star\mathbb{P}_{2},M)$

###  7.3. Two product of simple modules

We will give an interpretation of the Jordan-Holder filtration of the
Induction of two simple modules over $R_{Q}$ using the Decomposition (6.7).

######  Lemma 7.9.

For $\gamma\in Q^{+}$ and a given $\mathcal{L}_{a}\in\mathcal{Q}_{\gamma}$,
let $\mathbb{P}_{a}$ be its corresponding projective module. Suppose that

$\operatorname{Res}_{{\alpha,\beta}}^{\gamma}(\mathcal{L}_{a})=\sum\limits_{\mu\in\operatorname{KP}({\alpha}),\nu\in\operatorname{KP}({\beta})}J_{a}({\mu,\nu})(\operatorname{IC}_{\mu}\boxtimes\operatorname{IC}_{\nu})$

Then we get

$\langle\mathbb{P}_{a},\,L(\mu)\circ
L(\nu)\rangle=\operatorname{Dim}J_{a}(\mu,\nu)$

###### Proof.

Recall $\langle\mathbb{P}_{a},\,L(\mu)\circ
L(\nu)\rangle=\langle\operatorname{Res}\mathbb{P}_{a},L(\mu)\boxtimes
L(\nu)\rangle$. By Definition (7.4), we get

$\displaystyle\langle\operatorname{Res}\mathbb{P}_{a},L(\kappa)\boxtimes
L(\nu)\rangle$ $\displaystyle=$
$\displaystyle\left\langle\sum\limits_{\mu^{\prime}\in\operatorname{KP}({\alpha}),\nu^{\prime}\in\operatorname{KP}({\beta})}J_{a}(\mu^{\prime},\nu^{\prime})\boxtimes(\mathbb{P}_{\mu^{\prime}}\otimes\mathbb{P}_{\nu^{\prime}}),L(\mu)\boxtimes
L(\nu)\right\rangle$ $\displaystyle=$ $\displaystyle J_{a}(\mu,\nu)$

∎

Since $L(\mu)\circ L(\nu)$ is finite dimensional, it admits a simple
filtration. We will show that the multiplicity of $L(\lambda)$ in this
filtration, denoted by $[L(\mu)\circ L(\nu)):L(\lambda)]$, is equal to
$\operatorname{Dim}\hom_{R}(\mathbb{P}_{\lambda},L(\mu)\circ L(\nu))$. Namely,

(7.9) $\operatorname{Dim}\hom_{R}(\mathbb{P}_{\lambda},L(\mu)\circ
L(\nu))=[L(\mu)\circ L(\nu)):L(\lambda)]$

We show this by induction on the length of this filtration. When $L(\mu)\circ
L(\nu)$ is a simple module, there is nothing to prove. Suppose that this holds
when the length of the filtration of modules $M$ over $R_{Q}$ is less than the
length of the filtration of $L(\mu)\circ L(\nu)$. Consider the following short
exact sequence:

$0\to M\to L(\mu)\circ L(\nu)\to N\to 0$

and applying the functor $\hom_{R}(\mathbb{P}_{\lambda},-)$, we obtain the
following short exact sequence:

$0\to\hom_{R}(\mathbb{P}_{\lambda},M)\to\hom_{R}(\mathbb{P}_{\lambda},L(\mu)\circ
L(\nu))\to\hom_{R}(\mathbb{P}_{\lambda},N)\to 0$

This means

$\operatorname{Dim}\hom_{R}(\mathbb{P}_{\lambda},L(\mu)\circ
L(\nu))=\operatorname{Dim}\hom_{R}(\mathbb{P}_{\lambda},M)+\operatorname{Dim}\hom_{R}(\mathbb{P}_{\lambda},N)$

On the other hands, the multiplicity of $L(\lambda)$ in $L(\mu)\circ L(\nu)$
is equal to

$[L(\mu)\circ L(\nu):L(\lambda)]=[M:L(\lambda)]+[N:L(\lambda)]$

Our assumption implies that
$[M:L(\lambda)]=\operatorname{Dim}\hom_{R}(\mathbb{P}_{\lambda},M)$ and
$[N:L(\lambda)]=\operatorname{Dim}\hom_{R}(\mathbb{P}_{\lambda},N)$ and then
$\hom_{R}(\mathbb{P}_{\lambda},L(\mu)\circ L(\nu))=[L({\mu})\circ
L({\nu}):L(\lambda)]$.

Using the above lemma, we get

######  Theorem 7.10.

As above assumption, we have

$[L(\kappa)\circ
L(\nu):L(\lambda)]=[\operatorname{Res}_{{\alpha,\beta}}^{\gamma}\operatorname{IC}(\lambda):\operatorname{IC}(\kappa)\boxtimes\operatorname{IC}(\nu)]=K_{\lambda}({\mu,\nu})$

where
$[\operatorname{Res}_{{\alpha,\beta}}^{\gamma}\operatorname{IC}(\lambda):\operatorname{IC}(\kappa)\boxtimes\operatorname{IC}(\nu)]$
refers to the multiplicity of
$\operatorname{IC}(\kappa)\boxtimes\operatorname{IC}(\nu)$ in
$\operatorname{Res}_{{\alpha,\beta}}^{\gamma}\operatorname{IC}(\lambda)$ as
Lemma 7.9. In particular, we have $K_{\lambda}({\mu,\nu})\neq 0$ only if
$\lambda\leq\mu\oplus\nu$, and the top degree of $K_{\lambda}({\mu,\nu})$ is
less than and equal to

$2e_{\lambda}({\mu,\nu})+d_{\lambda}({\mu,\nu})$

If $({\mu,\nu})\in\operatorname{ext}_{\alpha,\beta}^{ger}(\lambda)$, then

$K_{\lambda}({\mu,\nu})=q^{\epsilon(v_{1},\alpha,v_{2},\beta)}\operatorname{P}_{q}({{\operatorname{Gr}}_{v_{1}-v_{\mu}}(CK(M_{\mu}))\times{\operatorname{Gr}}_{v_{2}-v_{\nu}}(CK(M_{\nu}))})$

where $e_{\lambda}({\mu,\nu})$ is the dimension of
$\mathcal{E}xt_{\lambda}({\mu},{\nu})$, (see 4.4).

###### Proof.

Since $[L(\kappa)\circ
L(\nu)):L(\lambda)]=\operatorname{Dim}\hom_{R}(\mathbb{P}_{\lambda},L(\kappa)\circ
L(\nu))$. By the above Lemma and Theorem 6.10, we get required results ∎

######  Proposition 7.11.

Suppose that $Q$ is a quiver without loops, (not necessary a Dynkin quiver).
Let ${\alpha+\beta}=\gamma$ be a partition of $\gamma\in Q^{+}$,
${\mu,\nu}\in\operatorname{KP}({\alpha})\times\operatorname{KP}({\beta})$, and
$\lambda\in\operatorname{KP}({\gamma})$. We have

$[\operatorname{Res}_{{\alpha,\beta}}L(\lambda):L(\mu)\boxtimes
L(\nu)]=[\operatorname{IC}(\mu)\star\operatorname{IC}(\nu):\operatorname{IC}(\lambda)]=J_{\lambda}(\kappa,\nu)$

where
$[\operatorname{IC}(\mu)\star\operatorname{IC}(\nu):\operatorname{IC}(\lambda)]$
refers to the multiplicity of $\operatorname{IC}(\lambda)$ in
$\operatorname{IC}(\mu)\star\operatorname{IC}(\nu)$.

###### Proof.

Recall $[\operatorname{Res}_{{\alpha,\beta}}L(\lambda):L(\mu)\boxtimes
L(\nu)]=\operatorname{Dim}\hom(\mathbb{P}_{\mu}\boxtimes\mathbb{P}_{\nu},\operatorname{Res}_{{\alpha,\beta}}L(\lambda))$
and

$\hom(\mathbb{P}_{\mu}\boxtimes\mathbb{P}_{\nu},\operatorname{Res}_{{\alpha,\beta}}L(\lambda))=\hom(\mathbb{P}_{\mu}\star\mathbb{P}_{\nu},L(\lambda))$

Let us decompose
$\mathbb{P}_{\mu}\star\mathbb{P}_{\nu}=\bigoplus_{\lambda^{\prime}\in\operatorname{KP}({\gamma})}J_{\lambda^{\prime}}({\mu,\nu})\mathbb{P}_{\lambda^{\prime}}$.
It follows that
$\hom(\mathbb{P}_{\mu}\star\mathbb{P}_{\nu},L(\lambda))=J_{\lambda}({\mu,\nu})$.
The definition 7.4 implies that
$[\operatorname{IC}(\mu)\star\operatorname{IC}(\nu):\operatorname{IC}(\lambda)]=J_{\lambda}({\mu,\nu})$,
we finish our proof. ∎

######  Corollary 7.12.

Under above assumptions, $\operatorname{Res}_{{\alpha,\beta}}L(\lambda)\neq 0$
only if there exists a pair ${\mu,\nu}$ in $\operatorname{KP}({\alpha})$ and
$\operatorname{KP}({\beta})$ such that $\lambda\geq\mu*\nu$.

###### Proof.

By the above Proposition and Theorem 6.7, we obtain our conclusion. Because we
just consider the support of perverse sheaves $\operatorname{IC}(\lambda)$,
the conclusion of Theorem 6.7 holds for any quivers $Q$ without loops. ∎

######  Remark 7.13.

This Corollary is closely related to the notion _semicuspidal_ modules
$L(\alpha)$ where $\alpha$ is a positive root. Suppose
$\operatorname{Res}_{{\mu,\nu}}L(\alpha)\neq 0$, then there exists a pair
$({\mu,\nu})$ such that $\alpha\geq\mu*\nu$. On the other hand, $\alpha$ is
the minimal Konstant partition in $\operatorname{KP}({\alpha})$. It follows
that $\alpha=\mu*\nu$. Let $M_{\alpha}$ is indecomposable representation
associated with $\alpha$. If there exists a pair $({\mu,\nu})$ such that

$0\to M_{\nu}\to M_{\alpha}\to M_{\mu}\to 0$

It follows that $[M_{\mu},M_{\nu}]^{1}\neq 0$. On the other hand, any summands
in $M_{\nu}$ is greater than $\alpha$, as there is an injective map
$M_{\nu}\to M_{\alpha}$ by (2.5). Similarly, we have any summands in $M_{\mu}$
is less than $\alpha$. Therefore, $\operatorname{Res}_{{\mu,\nu}}L(\alpha)\neq
0$ only if $\mu_{i}<\alpha$ and $\nu_{j}>\alpha$ for any $\mu_{i}\in\mu$ and
$\nu_{j}\in\nu$.

######  Corollary 7.14.

Give two simple modules $L(\mu)$ and $L(\nu)$ such that
$\mu\in\operatorname{KP}({\alpha})$ and $\nu\in\operatorname{KP}({\beta})$,
let us assume that the head of $L(\mu)\circ L(\nu)$ is a simple module, which
is denoted by $L(\mu)\triangledown L(\nu)$. Denote by $\mu\triangledown\nu$
the Konstant partition associated with this head. Denote by $L(\mu)\triangle
L(\nu)$ the soc of $L(\mu)\circ L(\nu)$. We have

(7.10)
$\mu\oplus\nu\geq\mu\triangledown\nu\geq\nu*\mu;\qquad\mu\oplus\nu\geq\mu\triangle\nu\geq\mu*\nu$

###### Proof.

Let us consider the case $L(\mu\triangle\nu)\to L(\mu)\circ L(\nu)$. It
follows by adjoint functor that
$\operatorname{Res}_{\beta,\alpha}L(\mu\triangle\nu)\to L(\nu)\boxtimes
L(\mu)$. It implies that $J_{\mu\triangle\nu}(\nu,\mu)\neq 0$ and
$K_{\lambda}({\mu,\nu})\neq 0$. By Corollary 7.12 and Theorem 7.10, we have
$\mu\oplus\nu\geq\mu\triangle\nu\geq\mu*\nu$.

In the case $L(\mu)\circ L(\nu)\to L(\mu\triangledown\nu)\to 0$. Applying the
functor $(-)^{*}$, we obtain

$0\to L(\mu\triangledown\nu)\to q^{({\alpha,\beta})}L({\nu})\circ L({\mu})$

By the above conclusion, we have

$\mu\oplus\nu\geq\mu\triangledown\nu\geq\nu*\mu$

∎

####  7.3.1. Support pairs

In this section, we give the notion of support pair. This will give a
necessary condition for some conjectures introduced in [LM22, Conjecture 1.3]
and [KR11, Probelem 7.6]. However, we should remark that this condition is not
sufficient to answer these conjectures. Because this notion just focus on the
supports of the perverse sheaves on representation spaces $E_{\alpha}$, (see
Lemma 3.4). This will omit some information about the decomposition (6.7). One
of the main reason is that the closure of orbits in $E_{\alpha}$ is singular
in general. The singularity makes this approach more difficult than we expect.
Therefore, we limit our attention to the support of perverse sheaves
$\operatorname{IC}(\lambda)$.

######  Definition 7.15.

We call a pair
$({\mu,\nu})\in\operatorname{KP}({\alpha})\times\operatorname{KP}({\beta})$
for a _support pair_ if and only if for any $\lambda<\mu\oplus\nu$ we have
$(\mu,\nu)\notin\operatorname{ext}_{\alpha,\beta}^{ger}(\lambda)$. This notion
is inspired by Proposition 6.10

For [KR11, Probelem 7.6], we have the following theorem.

######  Theorem 7.16.

Let $L(\mu)$ and $L(\nu)$ be two simple modules of $R_{Q}$ where $\mu,\nu$ are
their Konstant partitions, respectively. If $L(\mu)\circ L(\nu)$ is a simple
module, then $({\mu,\nu})$ is a support pair. Namely, for any non-travial
extesnion $\lambda\in\operatorname{ext}({\mu,\nu})$, we have

$[M_{\nu},M_{\mu}\oplus M_{\nu}]>[M_{\nu},M_{\lambda}]$

and

$[M_{\mu},M_{\mu}\oplus M_{\nu}]>[M_{\mu},M_{\lambda}]$

###### Proof.

It is easy to see that $L(\mu)\circ L(\nu)$ is a simple module if and only if
$K_{\lambda}({\mu,\nu})=0$ for all $\lambda<\mu\oplus\nu$. By Proposition
6.10, we have
$({\mu,\nu})\notin\operatorname{ext}_{\alpha,\beta}^{ger}(\lambda)$.
Otherwise,

$K_{\lambda}({\mu,\nu})=q^{\epsilon(v_{1},\alpha,v_{2},\beta)}\operatorname{P}_{q}({{\operatorname{Gr}}_{v_{1}-v_{\mu}}(CK(M_{\mu}))\times{\operatorname{Gr}}_{v_{2}-v_{\nu}}(CK(M_{\nu}))})\neq
0$

By the definitation of support pair, we have that

$[M_{\nu},M_{\mu}\oplus M_{\nu}]>[M_{\nu},M_{\lambda}]$

Since $L(\mu)\circ L(\nu)$ is a simple module if and only if $L(\nu)\circ
L(\mu)$ is a simple module, we see that

$[M_{\mu},M_{\mu}\oplus M_{\nu}]>[M_{\mu},M_{\lambda}]$

∎

Another useful application of this theorem is related to [VV03, Conjecture
4.3].

######  Remark 7.17.

Recall the [VV03, Conjecture 4.3] or [BZ92]: for a pair of dual canonical
bases $(\mathbf{b}^{*}(\mu),\mathbf{b}^{*}(\nu))$

$\mathbf{b}^{*}(\mu)\star\mathbf{b}^{*}(\nu)\in q^{v}\mathbf{B}^{*}$ if and
only if
$\mathbf{b}^{*}(\mu)\star\mathbf{b}^{*}(\nu)=q^{\epsilon(v_{1},\alpha,v_{2},\beta)}\mathbf{b}^{*}(\mu\oplus\nu)$.

The above Theorem provides a proof of this conjecture. Because the simple
modules $L(\mu),L(\nu)$ correspond to the dual canonical bases
$\mathbf{b}^{*}(\mu),\mathbf{b}^{*}(\nu)$ under the isomorphism
$K_{0}(R_{Q})\cong U_{q}(\operatorname{\mathfrak{n}})^{*}$ (see [KL09]) and
$K_{\mu\oplus\nu}({\mu,\nu})=q^{\epsilon(v_{1},\alpha,v_{2},\beta)}$ by
Theorem 7.10, since $v_{1}=v_{\mu}$ and $v_{2}=v_{\nu}$ when
$M_{\lambda}=M_{\mu}\oplus M_{\nu}$ by Remark 5.8.

Here is another application of our results.

######  Theorem 7.18.

Let ${\alpha+\beta}=\gamma$ be a partition of $\gamma\in Q^{+}$ and
$({\mu,\nu})\in\operatorname{KP}({\alpha})\times\operatorname{KP}({\beta})$.
If $({\mu,\nu})\in\operatorname{ext}_{\alpha,\beta}^{ger}(\mu*\nu)$ then
$q^{n}L(\mu*\nu)$ is the socle of $L(\mu)\circ L(\nu)$ for some integer $n$.

###### Proof.

Let $\lambda=\mu*\nu$, we have
$\mathcal{E}xt_{\lambda}({\mu},{\nu})=\operatorname{Hom}_{\Omega}(U,W)$. It
implies that $2e_{\lambda}({\mu,\nu})$ is maximal with respect to any
extension
$M_{\lambda^{\prime}}\in\operatorname{Ext}_{Q}^{1}({M_{\mu}},{M_{\nu}})$. If
$({\mu,\nu})\in\operatorname{ext}_{{\alpha,\beta}}^{ger}(\mu*\nu)$, then
$K_{\lambda}({\mu,\nu})\neq 0$ with the top degree
$2e_{\lambda}({\mu,\nu})+d_{\lambda}({\mu,\nu})$ by Theorem 6.10.

For others $\lambda^{\prime}>\mu*\nu$, we have the top degree of
$K_{\lambda^{\prime}}({\mu,\nu})$ is less than or equal to

$2e_{\lambda^{\prime}}({\mu,\nu})+d_{\lambda^{\prime}}({\mu,\nu})<2e_{\lambda}({\mu,\nu})+d_{\lambda}({\mu,\nu})$

This means the top degree of $K_{\mu*\nu}({\mu,\nu})$ is maximal for all
Knostant partitions $\lambda^{\prime}\in\operatorname{ext}({\mu,\nu})$. On the
other hand, by Corollary 7.14 the Konstant partitions $\mu\triangle\nu$ of the
socle of $L({\mu})\circ L({\nu})$ satisfy
$\mu\triangle\nu\in\operatorname{ext}({\mu,\nu})$. By [McN14, Lemma 7.5],
$K_{\mu\triangle\nu}({\mu,\nu})$ admits the maximal top degree for all
$K_{\lambda^{\prime}}({\mu,\nu})\neq 0$. This leads to that
$\mu\triangle\nu=\mu*\nu$. ∎

######  Remark 7.19.

This theorem gives a new viewpoint to [LM20, Conjecture 1.3]. We remark that
this couldn’t answer that conjecture. Comparing the conditions in 7.18 and
7.16, it is easy to see that a support pair $({\mu,\nu})$ always satisfies the
condition in 7.18.

Thanks to Lapid, he provides some useful examples as follows.

######  Example 7.20.

We remark that if $L(\mu*\nu)$ is a submodule of $L({\mu})\circ L({\nu})$ the
condition
$({\mu,\nu})\in\operatorname{ext}_{{\alpha,\beta}}^{\operatorname{min}{}}(\mu*\nu)$
doesn’t hold in general. The counter-example is given by E. Lapid: In type
$A_{3}$, let us consider the Konstant partitions
$\mu=[2,1]=[2,2]+[1,1],\nu=[3,2]=[3,3]+[2,2]$. Namely, they are both
semisimple representations of $A_{3}$. It is easy to see that
$\operatorname{Ch}L(\mu)=[2,1]$ and $\operatorname{Ch}L(\nu)=[3,2]$. It
implies by shuffle product that

$\displaystyle\operatorname{Ch}L(\mu)\circ\operatorname{Ch}L(\nu)=[2,1,3,2]+[2,3,1,2]+[3,2,2,1]$

By [LM22, Example 5.2], the socle of $L({\mu})\circ L({\nu})$ is
$L([2,1,3,2])$. The representation associated with this segment is the direct
sum of simple representation $S_{2}\oplus S_{1}$ and $S_{3}\oplus S_{2}$. On
the other hand,
$\operatorname{ext}_{\alpha,\beta}^{ger}([2,1]*[3,2])=[2,3][1,2]$ by Example
4.10. Therefore, $[2,1][3,2]$ is not in
$\operatorname{ext}_{\alpha,\beta}^{ger}([2,1]*[3,2])$.

Meanwhile, following [LM22, Example 5.2], let us consider the following
example

######  Example 7.21.

Let $\lambda=\sum_{i=1}^{r}[a_{i},b_{i}]$ be a multisegment in
$\operatorname{KP}({\gamma})$ such that $a_{i}<a_{i+1}$ for all $i\in[1,r-1]$
and $b_{i}<b_{i+1}$ for all $i\in[1,r-1]$. For each $i$ let $t_{i}$ such that
$a_{i}\leq t_{i}\leq b_{i}+1$, $t_{i}<t_{i+1}$. Let

$\mu=\sum_{i=1}^{r}[a_{i},t_{i}]\quad\nu=\sum_{i=1}^{r}[t_{i}+1,b_{i}]$

We will show that $\lambda=\mu*\nu$ and $({\mu,\nu})$ is the generic pair of
$\lambda$. However, the following identity

(7.11) $[M_{\nu},M_{\mu}\oplus M_{\nu}]=[M_{\nu},M_{\lambda}]$

doesn’t hold in general. The counter example is given in 4.10. Therefore, the
conclusion of Theorem 7.18 is not good enough. But this is the first attempt
to prove [LM22, Conjecture 5.1].

###### Proof.

Suppose that
$\mu\in\operatorname{KP}({\alpha}),\nu\in\operatorname{KP}({\beta})$. We
decompose $V=W\oplus U$ and $V^{i}=W^{i}\oplus U^{i}$ for $i\in[1,r]$ such
that $\operatorname{dim}V_{i}=[a_{i},b_{i}]$,
$\underline{\operatorname{dim}}W=\mu$, and
$\underline{\operatorname{dim}}U=\nu$.

We will show that $\mu*\nu=\lambda$. Consider the following short exact
sequence

$0\to M_{\nu}\stackrel{{\scriptstyle
f}}{{\rightarrow}}M_{\lambda^{\prime}}\stackrel{{\scriptstyle
g}}{{\rightarrow}}M_{\mu}\to 0$

Since the head of $M_{\mu}$ coincides with $\oplus_{i=1}^{r}S_{a_{i}}$, we get
$\oplus_{i=1}^{r}S_{a_{i}}\subset\operatorname{hd}M_{\lambda^{\prime}}$.
Similarly, we have
$\oplus_{i=1}^{r}S_{b_{i}}\subset\operatorname{soc}M_{\lambda^{\prime}}$. If
there exists other $c\in I$ such that
$S_{c}\in\operatorname{hd}M_{\lambda^{\prime}}$, let us consider

$r_{c-1,c}(\lambda^{\prime})=\sum_{[c-1,c]\subset[d,e]}d_{\lambda^{\prime}}[d,e]$

where $d_{\lambda^{\prime}}[d,e]$ refers to the multiplicity of the segment
$[d,e]$ in $\lambda^{\prime}$. It is easy to see that the rank
$r_{c-1,c}(\lambda^{\prime})$ is less than $\alpha_{c}$, as
$\sum_{[c]\subset[d,e]}d_{\lambda^{\prime}}[d,e]=\alpha_{c}$ and there is a
segment $[c,l]$ in $\lambda^{\prime}$ by our assumption. However,
$r_{c-1,c}(\lambda)=\alpha_{c}$. It implies that $\lambda^{\prime}>\lambda$
using (2.11).

Similarly, if there exists other $c\in I$ such that
$S_{c}\in\operatorname{soc}M_{\lambda^{\prime}}$, we obtain
$r_{c,c+1}(\lambda^{\prime})<\alpha_{c}=r_{c,c+1}(\lambda)$. Thus, we have
$\lambda^{\prime}=\sum_{k=1}^{r}[a_{k},b^{\prime}_{k}]$ such that
$\\{b^{\prime}_{k}\\}_{k\in[1,r]}=\\{b_{k}\\}_{k\in[1,r]}$.

We will show that $b_{k}=b^{\prime}_{k}$ for any $k$. Suppose that $k$ is the
minimal element with the condition $b^{\prime}_{k}\neq b_{k}$ segment. For
simplicity, we assume that $k=1$, otherwise, we subtract the segments
$[a_{i},b_{i}]$ for all $i<k$. Since $b_{j}=b^{\prime}_{1}>b_{1}$ for some
$j>1$, we have $[a_{1},b_{1}]\subset[a_{1},b_{j}]$. Denote by $M^{\prime}$ the
representation corresponding to $[a_{1},b_{j}]$. Let us consider the
restriction $M^{\prime}$ to $U$ denoted by $M^{\prime}_{U}$, which is a
subrepresentation of $M_{\nu}$. As its soc is $S_{b_{j}}$, it follows that
$M^{\prime}_{U}=M[s+1,b_{j}]\subset M[t_{j}+1,b_{j}]$. On the other hand,
$M^{\prime}_{W}=M[a_{1},s]\subset M[a_{1},t_{1}]$. It implies that $t_{j}\leq
s\leq t_{1}$, which leads to a contradiction $t_{j}>t_{1}$. Therefore, we have
$\lambda=\mu*\nu$.

We will show that $({\mu,\nu})$ is a generic pair of $\lambda$. Suppose that
$(\mu^{\prime},\nu)$ be a pair of multisegments such that $\mu^{\prime}<\mu$
and

$0\to M_{\nu}\to M_{\lambda}\to M_{\mu^{\prime}}\to 0$

We have
$\operatorname{hd}M_{\mu^{\prime}}\subset\operatorname{hd}M_{\lambda}=\oplus_{i=1}^{r}S_{a_{i}}$.
This means

$\mu^{\prime}=([a_{i},v_{i}]_{i\in[1,s]})$

where $s\leq r$. On the other hand, $\mu^{\prime}$ can be obtained from $\mu$
by a sequence of extensions of 2.9. It is easy to see that the end points in
the multisegments of the extension as in 2.9 are the same with that of the
original multisegments. For instance, for the segments $([a,b],[c,d])$ such
that $a\leq c\leq b\leq d$, the multisegments of their extension are
$([a,d],[c,b])$. This implies that the set

$\\{v_{i}\\}_{i\in[1,s]}\subset\\{t_{j}\\}_{j\in[1,r]}$

For simplicity, we assume that $t_{i}+1\neq a_{i}$. Otherwise, we consider the
sub multisegment $\lambda^{\prime}$ obtained by deleting those $[a_{i},b_{i}]$
such that $t_{i}+1=a_{i}$.

Let $k$ be the minimal number $i$ satisfying the relation $v_{i}\neq t_{i}$.
Namely, we have $v_{i}=t_{i}$ for all $i<k$ and $v_{k}=t_{l}>t_{k}$ for some
$l>k$. For any pair $[a_{k},v_{k}],[t_{j}+1,b_{j}]$, the segments of their
extension have the same end points with $[a_{k},v_{k}],[t_{j}+1,b_{j}]$ by 2.9
unless $v_{k}=t_{l}$. But the segments in $\lambda$ implies that there exits
no extension of $[a_{k},v_{k}]$ and $[t_{l},b_{l}]$. Otherwise, we get the
segment $[a_{k},b_{l}]\in\lambda$. The relation $b_{l}\neq b_{k}$ leads to a
contradiction. Therefore, the segment $[a_{k},b_{k}]\in\lambda$ is given by
the unique extension of $[a_{k},v_{k}],[t_{k}+1,b_{k}]$. By a product, we
obtain the segment $[t_{k}+1,v_{k}]$.

Then the point $t_{k}+1$ and $v_{k}=t_{l}$ will appear in the segments in
$\lambda$ by 2.9. This means $t_{k}+1,t_{l}\in\\{a_{i},b_{i}\\}_{i\in[1,r]}$.
Suppose that $t_{k}+1=a_{s}$ for some $s\in[1,r]$, then the segment containing
the end point $t_{k}+1$ is of the form $[t_{k}+1,b_{s}]$. The segment
$[a_{s},v_{s}]$ gives the head of $M_{\lambda}$ with two copy of simple
representation $S_{a_{s}}$. It is a contradiction. If that $t_{k}+1=b_{s}$ for
some $s\in[1,r]$, then the segment containing the end point $t_{k}+1$ is of
the form $[a_{s},t_{k}+1]$. We see that the socle of $M_{\lambda}$ has two
copy of simple representation $S_{a_{s}}$, which leads to a contradiction.
Therefore, the condition $v_{k}>t_{k}$ leads to a contradiction. We prove that
$v_{k}=t_{k}$ for all $k\in[1,r]$. That is $\mu^{\prime}=\mu$.

Similarly, one can show that $\nu$ is the generic subrepresentation of
$M_{\lambda}$ by $M_{\mu}$. Therefore, we have that $({\mu,\nu})$ is the
generic extension of $\lambda$.
∎

If $[M_{\mu},M_{\nu}]^{1}=0$, it is easy to see that
$({\mu,\nu})\in\operatorname{ext}_{\alpha,\beta}^{\operatorname{min}}(\mu*\nu)$,
as $\mu*\nu=\mu\oplus\nu$. This give another proof of [LM22, Lemma 5.4].

####  7.3.2. In the case of rigid representations

In this section, we will consider the rigid representations of $Q$.

######  Corollary 7.22.

Suppose $\mu$ and $\nu$ are minimal elements in $\operatorname{KP}({\alpha})$
(resp:$\operatorname{KP}({\beta})$). In other words, $M_{\mu}$ and $M_{\nu}$
are rigid modules. We have that $L({\mu})\circ L({\nu})$ is a simple module if
and only if $[M_{\mu},M_{\nu}]^{1}=[M_{\nu},M_{\mu}]^{1}=0$ In other words
$M_{\mu}\oplus M_{\nu}$ is a rigid module.

###### Proof.

If $L({\mu})\circ L({\nu})$ is a simple module over $R_{Q}$, then
$({\mu,\nu})$ is a support pair by Theorem 7.16. That means for any
$\lambda<\mu\oplus\nu$
$({\mu,\nu})\notin\operatorname{ext}_{\alpha,\beta}^{ger}(\lambda)$. But
$({\mu,\nu})$ is the unique minimal pair in
$\operatorname{KP}({\alpha})\times\operatorname{KP}({\beta})$. It implies that
there is no Konstant partition $\lambda$ satisfying $\lambda<\mu\oplus\nu$.
Namely, $M_{\mu}\oplus M_{\nu}$ is a rigid representation of $Q$.

Suppose that $M_{\mu}\oplus M_{\nu}$ is a rigid representation of $Q$. By
Proposition 7.10, if simple module $L(\lambda)$ in the Jordan-Holder
filtration of $L({\mu})\circ L({\nu})$, then $\lambda\leq\mu\oplus\nu$. Since
$\mu\oplus\nu$ is the unique minimal Konstant partition in
$\operatorname{KP}({\gamma})$, the only simple module in the Jordan Holder
filtration of $L({\mu})\circ L({\nu})$ is $L(\mu\oplus\nu)$. ∎

Next we will consider the case $[M_{\mu},M_{\nu}]^{1}=1$.

######  Proposition 7.23.

Let us assume that $[M_{\mu},M_{\nu}]^{1}=1$. We have the following short
exact sequence

$0\to q^{n_{2}}L(\mu*\nu)\to L({\mu})\circ L({\nu})\to
q^{n_{1}}L(\mu\oplus\nu)\to 0$

###### Proof.

By Theorem 7.16, we have $L({\mu})\circ L({\nu})$ is not a simple module. By
Proposition 7.10, if $[L({\mu})\circ L({\nu}):L(\lambda)]\neq 0$ then
$M_{\lambda}\in\operatorname{Ext}_{Q}^{1}({M_{\mu}},{M_{\nu}})$. Since
$[M_{\mu},M_{\nu}]^{1}=1$, It implies that there are exactly two Konstant
partitions $\mu\oplus\nu$ and $\mu*\nu$ satisfying $[L({\mu})\circ
L({\nu}):L(\lambda)]\neq 0$. Since
$({\mu,\nu})\in\operatorname{ext}_{\alpha,\beta}^{ger}(\lambda)$, it follows
by Theorem 7.18 that $q^{n_{2}}L(\mu*\nu)$ is the socle of $L({\mu})\circ
L({\nu})$. The cokernal of this embedding gives rise to the simple module
$q^{n_{1}}L(\mu\oplus\nu)$.
∎

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