# Neural network analysis of neutron and X-ray reflectivity data:
Incorporating prior knowledge for tackling the phase problem

Valentin Munteanua Vladimir Starostina Alessandro Grecoa Linus Pithana
Alexander Gerlacha Alexander Hinderhofera Stefan Kowarikb Frank Schreibera [a]
Institute of Applied Physics, University of Tübingen, Auf der Morgenstelle 10,
72076 Tübingen, Germany  [b] Department of Physical Chemistry, University of
Graz, Heinrichstraße 28, 8010 Graz, Austria

###### Abstract

Due to the lack of phase information, determining the physical parameters of
multilayer thin films from measured neutron and X-ray reflectivity curves is,
on a fundamental level, an underdetermined inverse problem. This so-called
phase problem poses limitations on standard neural networks, constraining the
range and number of considered parameters in previous machine learning
solutions. To overcome this, we present an approach that utilizes prior
knowledge to regularize the training process over larger parameter spaces. We
demonstrate the effectiveness of our method in various scenarios, including
multilayer structures with box model parameterization and a physics-inspired
special parameterization of the scattering length density profile for a
multilayer structure. By leveraging the input of prior knowledge, we can
improve the training dynamics and address the underdetermined (”ill-posed”)
nature of the problem. In contrast to previous methods, our approach scales
favorably when increasing the complexity of the inverse problem, working
properly even for a 5-layer multilayer model and an N-layer periodic
multilayer model with up to 17 open parameters.

##  I Introduction

X-ray and neutron reflectometry (XRR and NR) are well-established and
indispensable experimental techniques commonly used to investigate the
scattering length density (SLD) profile along the direction perpendicular to
the surface of a sample such as thin films and multilayers [1, 2, 3, 4, 5].
The most common way of modeling the SLD profile of a measured sample is via
the box model parameterization, where the physical parameters of interest are
the thickness, the roughness, and the constant SLD of each layer in a
multilayer structure. Still, more complex parameterizations of the SLD profile
can be employed based on preexisting physical knowledge or intuition about the
investigated structure. XRR and NR have been extensively used in both in situ
and ex situ studies of a large variety of systems, such as liquid and solid
thin films [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], layers of polymers [17,
18], lipids [19, 20, 21, 22], self-assembled monolayers [23, 24] and organic
solar cells [25, 26, 27]. In addition, polarized NR[28] can be used to study
the magnetic properties of thin films, but it is important to realize that
these successful uses of reflectometry usually involve some form of
complementary information in the data analysis, such as typical densities or
reasonable intervals for the film thickness, provided by the experimentalist.

Reflectometry can be counted among the techniques affected by the phase
problem. In the absence of complementary information, the lack of phase
information introduces a degree of ambiguity when trying to reconstruct the
SLD profile of the investigated sample from the measured reflectivity curve,
i.e. the inverse problem is, on a fundamental level, underdetermined (”ill-
posed”). This intrinsic ambiguity of the scattering method is further
magnified by additional ambiguity due to the limited experimental accuracy,
such as instrument noise, measurement artifacts, and the finite number of
measured points over the domain of the momentum transfer vector $q_{z}$.

In recent years, machine learning has emerged as an alternative to classical
methods of analyzing surface scattering data [29], being attractive due to its
very fast prediction times and the ability to be incorporated into the
operating pipelines of large-scale measurement facilities. In particular, fast
machine learning-based solutions are ideal for enabling an experimental
feedback loop during reflectometry measurements performed in real time [30,
31]. While many machine learning approaches dedicated to reflectivity exist
[32, 33, 34, 35, 36, 37, 38, 39], most of them do not address the inherent
ambiguity of the reflectivity data directly, instead training neural networks
over specific parameter domains where the ambiguity is not prominent enough to
prevent convergence. Such networks lack the flexibility to be used in more
general scenarios, typically requiring to be retrained on new parameter
domains for each use case. Specifically, Ref. [34], while successful within
its target, only addresses the case of a layer grown on top of a
silicon/silicon oxide substrate.

In our method proposed here, we enhance the regularization of the inverse
problem by including prior boundaries for the parameters as supplementary
inputs to the neural network. This allows the network to be trained over wide
parameter domains while the regression is being conducted over small enough
subdomains defined by the prior bounds to mitigate ambiguity. During
inference, the output of the network is defined not only by the measured
reflectivity curve but also by the prior experimental knowledge for the
considered physical scenario. Different choices of prior bounds can lead to
the recovery of distinct solution branches from the larger parameter domain.

Another limitation of existing approaches is that they require a specific
discretization of the reflectivity curves, as imposed by common network
architectures. While experimental curves can be interpolated to the required
discretization, interpolation is prone to introduce unphysical artifacts, for
example around deep minima of Kiessig fringes in NR / XRR curves. To address
this limitation, we introduce the use of a neural operator for processing
reflectivity curves with variable discretizations (number of points and $q$
ranges).

In the following, we first discuss the theoretical concepts necessary to
understand our approach in Section II, after which we present the technical
details of the implementation in Section III. Finally, we demonstrate the
results of our method for different parameterizations of the SLD profile (two-
layer box model, five-layer box model, physics-informed model with $N$
repeating monolayer units) on both simulated and experimental reflectivity
curves.

##  II Theoretical considerations

###  II.1 The phase problem

The phase problem is a ubiquitous issue for experimental techniques utilizing
the interference of waves $Ae^{-i\omega t}$ as a means of probing the physical
properties of materials, caused by the fact that detectors cannot record the
phase of the signal, but only its intensity $\left|A\right|^{2}$ [40]. This
loss of information introduces a degree of ambiguity when trying to
reconstruct the physical quantities of interest from the measured signal. It
is well-known that neutron and X-ray reflectometry are among the techniques
affected by the phase problem[41], which precludes an analytical solution via
the Gel’fand-Levitan-Marchenko[42] inverse scattering equation. Some
approaches for experimentally tackling the phase problem of neutron and X-ray
reflectometry have been developed, such as the reference layer method [43] and
the Lloyd mirage technique [44], but they have certain practical limitations.
Of particular interest is the use of magnetic reference layers [45] in
polarized neutron reflectometry where the SLD of the reference layer depends
on the polarization of the incident neutrons. Three distinct measurements for
up, down, and non-polarized neutron beams then allow for the determination of
the phase. Also for NR, the amount of structural detail extracted from
reflectivity data can be maximized by flexibly varying the SLD of specific
structures within the sample via isomorphic isotropic substitution [46], such
as the replacement of protium with deuterium (selective deuteration) in the
hydrocarbon chains of lipid bilayers [47]. Alternatively, the degree of
ambiguity can be diminished when measuring a series of reflectivity curves for
a sample with evolving structure (e.g. during film deposition). Notably, such
theoretical ambiguity is in practice further accentuated by experimental
sources of error (noise, finite instrumental resolution, artifacts) and the
discrete nature of the measurement process, the scattered intensity being
recorded over a finite range of the momentum transfer $q_{z}$ at a finite
number of points.

A consequence of the phase problem is the underdetermined (”ill-posed”) nature
of the parameter recovery from the measured data, different SLD profiles
corresponding to equivalent reflectivity curves. When taking into account
multiple scattering at the interfaces, as described by Parratt’s recursive
formalism [48] or Abelès transfer-matrix method [49], some information about
the phase can be recovered in the low $q_{z}$ region [50], although the
difference between reflectivity curves can still become vanishingly small
[51], especially when compounded with systematic measurement errors of the
total reflection edge.

In the following, we briefly summarize a theoretical derivation of how such
SLD profiles can be identified in the mathematically simpler kinematical
approximation, which neglects multiple scattering at the sample interfaces so
that the calculation remains analytical. In the kinematical approximation, the
scattered intensity is proportional to the square of the total scattering
amplitude:

$R(q_{z})=\frac{R_{\text{F}}(q_{z})}{\rho_{\text{s}}^{2}}\left|\int_{-\infty}^{+\infty}\frac{d\rho(z)}{dz}e^{iq_{z}z}dz\right|^{2}=\frac{R_{\text{F}}(q_{z})}{\rho_{\text{s}}^{2}}\left|\mathrm{FT}\left(\frac{d\rho(z)}{dz}\right)\right|^{2},$
(1)

where $R_{\text{F}}(q_{z})$ denotes the Fresnel reflectivity, $\rho(z)$ is the
SLD profile along the perpendicular direction $z$ to the sample and
$\rho_{\text{s}}$ is the SLD of the substrate.

For the commonly used box model with interfacial roughness parametrized via
Névot-Croce factor [52], an SLD profile can be written as a sum of the error
functions:

$\rho(z)=\rho_{\mathrm{ambient}}+\sum_{i=1}^{N}\Delta\rho_{i}\cdot\mathrm{erf}\left(\frac{z-z_{i}}{\sqrt{2}\sigma_{i}}\right),$
(2)

where $\mathrm{erf}(z)$ is the error function, $N$ is the number of
interfaces, $z_{i}$ the position of the $i$th interface, $\sigma_{i}$ is the
roughness of the $i$th interface and $\Delta\rho_{i}$ is the SLD difference
between interfaces $i$ and $i+1$.

By substituting Equation (2) into Equation (1), we can explicitly calculate
the Fourier transform:

$\left|\mathrm{FT}\left(\frac{d\rho(z)}{dz}\right)\right|^{2}=\left|\int_{-\infty}^{+\infty}\sum_{i=1}^{N}\Delta\rho_{i}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(z-z_{i})^{2}}{2\sigma^{2}}}e^{iq_{z}z}dz\right|^{2}=\left|\sum_{i=1}^{N}\Delta\rho_{i}e^{iq_{z}z_{i}-q_{z}^{2}\sigma_{i}^{2}/2}dz\right|^{2}$
(3)

Finally, we obtain a decomposition of the scattered intensity into the sum of
a constant term and several sinusoidal components with amplitudes
$\Delta\rho_{i}\Delta\rho_{j}e^{-q_{z}^{2}(\sigma_{i}^{2}+\sigma_{j}^{2})/2}$
and frequencies $\Delta z_{ij}$:

$\displaystyle\frac{R_{\text{box}}(q_{z})}{R_{\text{F}}(q_{z})}\rho_{\text{s}}^{2}$
$\displaystyle=\sum_{i=1}^{N}\Delta\rho_{i}^{2}e^{-q_{z}^{2}\sigma_{i}^{2}}+2\sum_{i=2}^{N}\sum_{j=1}^{i-1}\Delta\rho_{i}\Delta\rho_{j}e^{-q_{z}^{2}(\sigma_{i}^{2}+\sigma_{j}^{2})/2}\cos(q_{z}\Delta
z_{ij}).$ (4)

By solving for all combinations of $\Delta\rho_{i}$ and $z_{i}$ with the same
scattered intensity, one can identify classes of the theoretical identical
solutions in the kinematical approximation, such as profiles with mirrored
derivatives as reported in earlier studies [51, 53, 54] (shown in Figure 1).
Notably, restricting the shape of the considered SLD profiles to the box model
with a fixed number of layers already imposes a strong regularization on the
space of admissible solutions.

Figure 1: (a) Two SLD profiles with mirrored derivatives (b) The derivatives
of the two SLD profiles (c) The corresponding reflectivity curves for the two
SLD profiles are almost identical (they would be exactly identical in the
kinematical approximation), despite the fact that the SLD profiles in (a) are
very dissimilar.

###  II.2 Solving ill-posed inverse problems using neural networks

If a forward process $f(\mathbf{x})=\mathbf{y}$ maps the hidden parameters
$\mathbf{x}$ describing a physical system to an observable signal
$\mathbf{y}$, the inverse problem can be described as retrieving the physical
parameters from a possibly noise-corrupted measurement of the signal [55].
When the inverse problem $f^{-1}(\textbf{y})=\textbf{x}$ represents a one-to-
one mapping, neural networks can be straightforwardly trained to approximate
the inverse function. In contrast, a many-to-one mapping of the forward
process leads to an underdetermined, ill-posed inverse problem, and the
corresponding one-to-many inverse mapping $f^{-1}(\textbf{y})$ cannot be
approximated via regression. Attempting to train a neural network as a point
estimator over the domain containing the non-uniqueness would lead to
incorrect predictions corresponding to an average of distinct solution
branches. Different machine learning approaches exist for addressing ill-posed
inverse problems [56, 57, 58], which generally depend on the specific
application.

The present study proposes a novel approach for tackling ill-posed inverse
problems, which takes advantage of the a priori knowledge of the
experimentalist. This approach is inspired by conventional reflectometry
analysis and combines the high speed of neural networks with the flexibility
of conventional fitting procedures.

Typically, some physical parameters of the studied samples are known to the
experimentalists with narrow uncertainty ranges, such as densities of the used
materials, thicknesses of the deposited layers, etc. In conventional fitting
procedures, this information is used to set upper and lower bounds on each
sample parameter within which solutions are allowed, the bounds being narrower
for layer SLDs and wider for layer thicknesses and roughnesses. Our
implementation allows us to provide such prior knowledge as input to the
neural network in the form of the two prior bounds for each parameter, the
regression problem being constrained to the local domain defined by the prior
bounds, thus avoiding the non-uniqueness associated with the full parameter
domain. As long as the prior bounds are narrow enough not to include multiple
solution branches, the inverse mapping is well-determined and can be
approximated by a neural network. In practice, even a small amount of prior
knowledge can be enough to rule out ambiguity in reflectometry analysis. We
note that narrow priors do not guarantee a single solution, as multiple
solutions might occur close to each other. However, our general assumption
shared by experimentalists who analyze reflectometry data via conventional
fitting is that for a large set of cases, the prior information is sufficient
to isolate a single solution.

Our method can be envisioned as simultaneously learning the inverse problem on
all the possible subdomains (or a subset of subdomains) of the full parameter
domain, using a single neural network. Thus, some parallels can be drawn to
approaches such as [59] where a Beta Variational Autoencoder ($\beta$-VAE) is
trained simultaneously for all values of the $\beta$ coefficient using a
single neural network (a task which would typically require retraining for
each $\beta$ value).

###  II.3 Discretization-invariant learning

Neural networks can only learn mappings between finite-dimensional vector
spaces, some architectures such as the multilayer perceptron requiring a fixed
discretization (range and resolution) of the input. A new paradigm is
represented by neural operators, which can learn mappings between infinite-
dimensional function spaces, allowing discretization-invariant learning [60,
61]. For an input function $v_{0}(x)$ defined over a domain $D$, a neural
operator is constructed as a series of transformations $v_{t}\mapsto v_{t+1}$:

$v_{t+1}(x)=\sigma\left(Wv_{t}(x)+(\mathfrak{K}v_{t})(x)\right)$

where $W$ is a linear transformation, $\sigma$ is a non-linear activation
function and $\mathfrak{K}$ is a non-local integral operator with learnable
kernel $\mathfrak{k}$:

$(\mathfrak{K}v_{t})(x)=\int_{y\in D}\mathfrak{k}(x,y)v_{t}(y)dy$

The Fourier Neural Operator (FNO) [62] represents an efficient and expressive
implementation of a neural operator which imposes
$\mathfrak{k}(x,y)=\mathfrak{k}(x-y)$ and leverages the convolution theorem.
It parameterizes the kernel operator directly in Fourier space as:

$(\mathfrak{K}v_{t})(x)=\mathfrak{F}^{-1}\left(T\left(\mathfrak{F}v_{t}\right)\right)(x)$

where $\mathfrak{F}$ and $\mathfrak{F}^{-1}$ represent the direct and,
respectively, the inverse discrete Fourier transform and $T$ is a learnable
linear transformation.

Neural operators have been predominantly used in the fields of differential
equation solving and physics-informed learning [63, 64]. Here, we use a neural
operator to learn a vector embedding for reflectivity curves with variable
discretizations ($q$ ranges and numbers of points) for our regression inverse
problem. Such an approach is beneficial as it confers a higher degree of
flexibility to the trained model, enabling the use of the full measured signal
without relying on interpolation.

##  III Design and implementation of our method

###  III.1 Training methodology and neural network architecture

To enable the regularization of the inverse problem by confining the
regression within a sample-dependent local domain, we generate ground truth
values of the parameters for training in the following manner. Firstly, for
each parameter, we obtain the center ($c$) and the width ($w$) of the local
domain, the center being uniformly sampled from the global domain (i.e. the
parameter range) and the width being uniformly sampled from a predefined width
range for that parameter. Secondly, the ground truth values of the parameters
are obtained by uniform sampling within the local domain $[c-w/2,c+w/2]$
defined by the previously sampled values, the two prior bounds being the edges
of this local domain, $c-w/2$ and $c+w/2$.

At each training step, we simulate a new batch of reflectivity curves from the
sampled ground truth parameters using a fast, GPU-accelerated Pytorch [65]
implementation of the Abelès transfer-matrix method. As a preprocessing step,
we set intensities below 10-10 which cannot be recorded in most experimental
scenarios (although there are exceptions) to this chosen minimum threshold, we
add noise to the curve, and we apply a logarithmic transformation followed by
a linear rescaling. The prior bounds are normalized with respect to the
corresponding parameter ranges and the ground truth parameters are normalized
with respect to the local domain they were sampled from, such that all the
inputs and outputs to the neural network are in the range [$-$1, 1].

As shown in Figure 2, an embedding of the reflectivity curve, together with
the prior bounds are input to a fully connected neural network (also known as
multilayer perceptron or MLP [66]), the loss being computed as the mean-
squared error between the neural network output and the ground truth
parameters (normalized with respect to the prior bounds). Notably, obtaining
the final prediction requires reversing the normalization of the neural
network output with respect to the prior bounds. The architecture of our
model, as well as the sampling procedure, are shown in Figure 2 (parameter
scaling omitted for simplicity). The MLP consists of a sequence of
$n_{\text{blocks}}$=6 residual blocks inspired by the ResNet[67] architecture
design, each block containing 2 hidden layers of width
dim${}_{\text{hidden}}$=1024 neurons. The use of skip connections has the role
of facilitating gradient propagation and preventing singularities [68]. Batch
normalization [69] is known to improve the convergence of neural networks, so
we use it to normalize the intermediate features before activation. Since the
type of activation function can have a significant impact on the performance
of a neural network, we explored the use of several popular activation
functions (ReLU[70], GELU[71], SELU[72], Mish[73]), choosing GELU as the
default.

We trained our models using the AdamW[74] optimizer, a version of Adam[75]
with decoupled weight decay regularization. The weight decay coefficient is
kept at the default value of 0.01 . The initial learning rate of 0.0001 was
decreased by a factor of 10 on plateau of the loss. We used the largest batch
size that fits in our GPU memory (4096) to ensure stable gradients.

Figure 2: Neural network architecture and steps involved in our method.
Firstly, the ground truth parameters and the prior bounds are sampled using
the described procedure. An embedding of the reflectivity curves together with
the prior bounds are provided as input to the MLP. The MLP consists of
residual blocks with batch normalization (BN), nonlinear activation (chosen to
be GELU) and linear layers.

###  III.2 Embedding networks

Going beyond previous publications, where a reflectivity curve is directly
provided as input to the MLP, we first use a network that produces a latent
embedding of the reflectivity curve which is subsequently fed to the MLP.

When training a model on reflectivity curves with fixed discretization, we use
a 1D CNN [76] as the embedding network, which is parameter-efficient and
better leverages the sequential characteristics of the data. While less
popular than their 2D counterparts, 1D CNNs have been successfully used in a
variety of tasks involving 1D signals such as automatic speech recognition
[77] and time-series prediction [78]. The 1D CNN, as shown in Figure 3a,
consists of a sequence of convolutional layers with kernel size 3, stride 2
and padding 1, the dimension of the signal being (approximately) halved after
each layer. At the same time, the number of channels is doubled after each
layer starting from 32 up to ch${}_{\text{out}}$=512. An adaptive average
pooling layer with output size dim${}_{\text{avpool}}$=8 ensures a fixed input
size (ch${}_{\text{out}}$*dim${}_{\text{avpool}}$) for a linear layer which
produces the final embedding with dimension dim${}_{\text{emb, CNN}}$ = 128.
While the adaptive average pooling allows obtaining a fixed size embedding
from curves with variable discretizations, CNNs do not enable discretization-
invariant learning, as shown in previous studies [62].

When training a model on reflectivity curves with variable discretizations, we
employ a Fourier Neural Operator (FNO) as the embedding network (3b), as
theoretically motivated in the previous section on discretization-invariant
learning. In this scenario the minimum and maximum values of $q$, as well as
the number of points in the curve are also uniformly sampled for each batch
from the considered ranges. The input to the FNO is the reflectivity curve
together with the corresponding $q$ values (concatenated along the channel
axis). After the input is raised to a higher channel space
ch${}_{\text{FNO}}$=128 by a pointwise linear operation, a sequence of
$n_{\text{FNO}}$=5 spectral blocks are applied which implement the kernel
operator in Fourier space as illustrated in Figure 3b. The number of Fourier
modes kept after performing the discrete Fourier transform is a hyperparameter
set at $n_{\text{modes}}$=16. Finally, a mean pooling over the input dimension
followed by a linear layer produces the final embedding with dimension
dim${}_{\text{emb, FNO}}$ = 256.

Figure 3: Architecture of the embedding networks (a) 1D CNN, consisting of
convolutions with kernel size 3, stride 2 and padding 1 (b) FNO, consisting of
spectral blocks which implement the neural operator kernel in Fourier space.

##  IV Results

We trained neural networks according to the previously described approach for
different parameterizations of the thin film SLD profile. Each of the
following subsections elaborates on a network trained for data with a
different parameterization: either a different number of layers (two and five)
for the box model or a physics-informed special parameterization of a
multilayer structure with repeating monolayers. We use the 1D CNN as the
default embedding network. In the last subsection the FNO is used as the
embedding network for data with variable discretization. We evaluate
performance metrics on statistically significant batches of simulated data and
we illustrate the applicability of our method on experimental reflectivity
data when available.

###  IV.1 Two-layer box model

This subsection shows the performance evaluation for a neural network trained
on data corresponding to the two-layer parameterization of the box model, the
total number of physical parameters predicted by the neural network being 8 (3
parameters per layer (thickness, roughness, SLD) plus 2 additional parameters
for the substrate (roughness and SLD)). The $q$ range of the simulated curves
is [0.02, 0.15] Å-1 which aligns with the range of the test experimental data,
and the resolution is 128 points. Various types of noise were applied to the
simulated reflectivity curves in order to provide robustness to experimental
artifacts, namely Poisson noise, $q$-position noise, curve shifting and curve
scaling based on previous investigations [33].

The chosen parameter ranges are [0, 500] Å for the thicknesses, [0, 60] Å for
the roughnesses, and [$-$25, 25] $10^{-6}$Å-2 for the SLDs. Notably, the
negative values of the SLD include specific cases of NR. The ranges of the
prior bound widths are [0.01, 500] Å for the thicknesses, [0.01, 60] Å for the
roughnesses and [0.01, 4] $10^{-6}$Å-2 for the SLDs. While the prior bound
widths for thicknesses and roughnesses span the whole domain of these
parameter types, the maximum prior bound width for the SLDs was reduced, since
this type of parameter is associated with the highest amount of prior
experimental knowledge. Figure 4 illustrates examples of input simulated
reflectivity curves for this model together with the neural network
predictions (top row) as well as the ground truth and predicted SLD profiles
(bottom row). Figure 5 shows the box plots of the absolute errors for each of
the 8 predicted parameters ($a.$ thicknesses, $b.$ roughnesses, $c.$ SLDs),
computed over a batch of 4096 simulated curves, the ground truth parameters
and prior bounds being generated as in the training procedure. We can see from
the box plots that the prediction errors for thicknesses, roughnesses and SLDs
are quite low considering the large parameter ranges used for training.

It is important to understand how the performance of the model depends on the
input prior bounds, as this informs users how much prior knowledge they should
provide for an expected prediction quality. To evaluate such a dependence, we
sample batches of ground truth parameters and prior bounds such that the prior
bound widths are fixed for each batch. The prior bound widths are varied by
multiplying the maximum bound width of one parameter type at a time (while for
the other parameter types keeping the maximum bound width) by a scalar value
in the range [0, 1]. As shown in Figure 6, we observe that the absolute errors
of the thicknesses, roughnesses and SLDs decrease when the relative prior
bound width for the corresponding parameter type is decreased, as expected. A
relative bound width of 0 represents the trivial case when the prior bounds
define the ground truth parameters exactly.

Figure 4: (a)-(c) Examples of input simulated reflectivity curves (blue
markers) and the corresponding neural network predictions (red line) for the
2-layer model. (d)-(f) Ground truth (blue) and predicted (red) SLD profiles
corresponding to the reflectivity curves in the top row. SLD profiles
corresponding to the minimum (green) and maximum (orange) prior bounds used
for the prediction are also shown as dotted lines. Figure 5: Boxplots of the
absolute errors of each predicted parameter of the 2-layer model, computed
over a batch of 4096 simulated curves, with the prior bounds being uniformly
sampled. The horizontal red line denotes the median, the black dot denotes the
mean and the lower and upper extremities of the box plot denote the
25${}^{\text{th}}$ percentile and the 75${}^{\text{th}}$ percentile
respectively. Figure 6: Dependence of the mean absolute error of each
parameter type ($a.$ thickness, $b.$ roughness, $c.$ SLD) for the 2-layer
model as a function of the prior bound width.

We demonstrate the applicability of our method for analyzing experimental
reflectivity data using XRR curves from a previously published dataset [79],
containing in situ measurements performed during the deposition of a layer of
organic material of diindenoperylene (DIP) or pentacene (PEN), on top of a
thin silicon oxide layer sitting on a silicon substrate, together with the
ground truth manual fits of the parameters. We are able to leverage the a
priori experimental knowledge about this dataset by setting narrow prior
bounds around the known values of the SLD for the substrate (Si) and the
bottom layer (SiO2). In previous approaches [34] using this dataset, all the
parameters of the substrate and SiO2 layer were kept at predefined constant
values during training and only the thickness, roughness and SLD of the top
layer were predicted. Our approach is capable of also tackling this specific
use case while being trained for more general cases. Figure 7 shows input
experimental curves together with the neural network predictions (top row), as
well as the corresponding SLD profiles (bottom row).

Figure 7: (a)-(c) Examples of input experimental reflectivity curves (blue
markers) and the corresponding neural network predictions (red line). (d)-(f)
Ground truth (blue) and predicted (red) SLD profiles corresponding to the
reflectivity curves in the top row. SLD profiles corresponding to the minimum
(green) and maximum (orange) prior bounds used for the prediction are also
shown as dotted lines.

###  IV.2 Five-layer box model

Increasing the number of layers in the box model parameterization increases
the difficulty of training a neural network for the inverse problem since the
reflectivity curves become more complex and also the degree of non-uniqueness
is increased. Nevertheless, we demonstrate that our method can still be
successfully applied to models with an increased number of layers, namely a
five-layer model, the total number of physical parameters predicted by the
neural network being 17 (3 parameters per layer (thickness, roughness, SLD)
plus 2 additional parameters for the substrate (roughness and SLD)). We
increase the $q$ range of the simulated curves to [0.02, 0.3] Å-1 and the
resolution to 256 points. The chosen parameter ranges are [0, 300] Å for the
thicknesses, [0, 60] Å for the roughnesses, and [0, 25] $10^{-6}$Å-2 for the
SLDs. The ranges of the prior bound widths are [0.01, 300] Å for the
thicknesses, [0.01, 60] Å for the roughnesses, and [0.01, 4] $10^{-6}$Å-2 for
the SLDs. Figure 8 shows examples of input simulated curves and predictions
for the 5-layer model, together with the corresponding SLD profiles. The
absolute errors between the ground truth and the predicted parameters are
displayed in Figure 9. We observe that the prediction errors do not vary much
depending on the specific layer in the model. While, as expected, the
prediction errors for the 5-layer model are higher than the prediction errors
for the 2-layer model (the difference being more pronounced for roughnesses
and less pronounced for SLDs), the performance of our model is still very good
on this more challenging use case.

Figure 8: (a)-(c) Examples of input simulated reflectivity curves (blue
markers) and the corresponding neural network predictions (red line) for the
5-layer model. (d)-(f) Ground truth (blue) and predicted (red) SLD profiles
corresponding to the reflectivity curves in the top row. SLD profiles
corresponding to the minimum (green) and maximum (orange) prior bounds used
for the prediction are also shown as dotted lines. Figure 9: Boxplots of the
absolute errors of each predicted parameter of the 5-layer model, computed
over a batch of 4096 simulated curves, with the prior bounds being uniformly
sampled. The horizontal red line denotes the median, the black dot denotes the
mean and the lower and upper extremities of the box plot denote the
25${}^{\text{th}}$ percentile and the 75${}^{\text{th}}$ percentile
respectively.

###  IV.3 Complex multilayer model

In this subsection, instead of the box model parameterization, we employ a
physics-informed parameterization of a complex multilayer structure, as
illustrated in Figure 10. The physical scenario is the following: on top of a
silicon/silicon oxide substrate we consider a thin film composed of repeating
identical monolayers (grey curve in Figure 10), each monolayer consisting of
two boxes with distinct SLDs. A sigmoid envelope modulating the SLD profile of
the monolayers defines the film thickness and the roughness at the top
interface (green curve in Figure 10). A second sigmoid envelope can be used to
modulate the amplitude of the monolayer SLDs as a function of the displacement
from the position of the first sigmoid (red curve in Figure 10). These two
sigmoids allow one to model a thin film that is coherently ordered up to a
certain coherent thickness and gets incoherently ordered or amorphous toward
the top of the film. Such a scenario is sometimes encountered when Kiessig and
Laue fringes show different periods. In addition, a layer between the
substrate and the multilayer is introduced to account for the interface
structure, which does not necessarily have to be identical to the multilayer
period. This so-called ”phase layer” (i.e. a layer that strongly influences
the relative scattering phase between the substrate and the multilayer) is
important as the relative phase between strong substrate reflection and a
multilayer Bragg reflection can lead to very different shapes of the curve
around the Bragg reflection for constructive or destructive interference. The
17 parameters describing the model together with their training ranges are
displayed in Table 1. For experimental XRR curves of DIP monolayers, measured
at a laboratory X-ray source, we leverage prior knowledge about this system
(layer spacing, substrate SLDs, approximate SLD values for the two boxes in
the monolayer) to set suitable prior bounds. Figure 11 shows input
experimental curves together with the neural network predictions for the model
with physics-informed parameterization. Again the prediction quality is good,
demonstrating the potential of the proposed method to fit experimental data in
increasingly complex scenarios such as multilayer structures featuring Bragg
peaks. It is worth noting that in this case the gradient descent polishing
procedure introduced in previous studies [34] can further improve the fit and
it can be performed both with the 17 introduced parameters and with the full
set of box model parameters, allowing the final solution to potentially evolve
beyond the designed parameterization.

Figure 10: Physics-informed parameterization of the SLD profile for a thin film consisting of repeating identical monolayers on top of a substrate. The grey curve shows the base SLD profile of the monolayers, the green curve shows the SLD profile with surface roughness and the red curve shows the modulation of the SLD amplitude. The blue curve represents the final SLD profile. Table 1: Parameters of the model with physics-informed parameterization of the SLD profile, together with the parameter ranges and the ranges of the prior bound widths used for training. Some of the parameters are relative with respect to the monolayer thickness. Parameter | Parameter range | Prior bound width range
---|---|---
monolayer thickness | [10, 20] Å | [0.1 , 10] Å
relative roughness of the monolayer interfaces | [0, 0.3] | [0.1, 0.3]
SLD of the first box in the monolayer | [0, 20] $10^{-6}\text{\AA}^{-2}$ | [0.1, 5] $10^{-6}\text{\AA}^{-2}$
SLD difference between the second and the first box in the monolayer | [-10, 10] $10^{-6}\text{\AA}^{-2}$ | [0.1, 5] $10^{-6}\text{\AA}^{-2}$
fraction of the monolayer thickness belonging to the first box | [0.01, 0.99] | [0.01, 1]
roughness of the silicon substrate | [0, 10] Å | [0.01, 10] Å
SLD of the silicon substrate | [19, 21] $10^{-6}\text{\AA}^{-2}$ | [0.01, 2] $10^{-6}\text{\AA}^{-2}$
thickness of the silicon oxide layer | [0, 10] Å | [0.01, 10] Å
roughness of the silicon oxide layer | [0, 10] Å | [0.01, 10] Å
SLD of the silicon oxide layer | [17, 19] $10^{-6}\text{\AA}^{-2}$ | [0.01, 2] $10^{-6}\text{\AA}^{-2}$
SLD of the phase layer | [0, 25] $10^{-6}\text{\AA}^{-2}$ | [0.01, 25] $10^{-6}\text{\AA}^{-2}$
relative thickness of the phase layer | [0, 1] | [0.01, 1]
relative roughness of the phase layer | [0, 1] | [0.01, 1]
relative position of the first sigmoid (total film thickness) | [0, 25] | [0.1, 25]
relative width of the first sigmoid | [0, 5] | [0.1, 5]
relative position of the second sigmoid (coherently ordered film thickness) | [-10, 10] | [0.1, 20]
relative width of the second sigmoid | [0, 20] | [0.1, 20]
Figure 11: (a)-(b) Examples of input experimental reflectivity curves of DIP
monolayers grown on top of a silicon/silicon oxide substrate (blue markers)
with corresponding neural network predictions (red lines) for the model with
physics-informed parameterization. (c)-(d) Predicted SLD profiles
corresponding to the reflectivity curves in the top row.

###  IV.4 Model with Fourier Neural Operator embedding network

In this subsection, we show results obtained when using the FNO as the
embedding network instead of the 1D CNN used in the previous sections. The
number of points in the simulated curves are in the range [128, 256], the
minimum value of $q$ is in the range [0.01, 0.03] Å-1 and the maximum value of
$q$ is in the range [0.15, 0.4] Å-1. We consider a 2-layer box model with
parameter ranges [0, 300] Å for the thicknesses, [0, 60] Å for the
roughnesses, and [0, 25] $10^{-6}$Å-2 for the SLDs. The ranges of the prior
bound widths are [0.01, 300] Å for the thicknesses, [0.01, 60] Å for the
roughnesses and [0.01, 4] $10^{-6}$Å-2 for the SLDs. Due to increased memory
demands the batch size is decreased to 1024. Figure 12 shows input curves with
variable number of points and $q$ ranges, together with the neural network
predictions (top row), as well as the corresponding SLD profiles (bottom row).
By using a FNO as the embedding network, our approach is successfully extended
to curves with variable discretizations. A disadvantage in this scenario is
that the network takes longer to converge.

Figure 12: (a)-(c) Examples of input simulated reflectivity curves with
variable discretizations (blue markers) and the corresponding neural network
predictions (red line) for a 2-layer model with FNO embedding network. (d)-(f)
Ground truth (blue) and predicted (red) SLD profiles corresponding to the
reflectivity curves in the top row. SLD profiles corresponding to the minimum
(green) and maximum (orange) prior bounds used for the prediction are also
shown as dotted lines.

##  V Conclusions

In this study, we address the phase problem as the primary obstacle in machine
learning-based approaches for extracting information from X-ray reflectivity
(XRR) and neutron reflectivity (NR) data. The lack of phase information
induces non-uniqueness in the space of possible solutions, the inverse problem
becoming increasingly underdetermined the larger the considered parameter
space. This prevents the successful training of neural networks for complex
multilayer structures, with many free parameters. Therefore, previous
solutions were limited to relatively simple layer structures with few
parameters. To tackle this issue, we propose a procedure that enables training
a neural network on a continuous range of smaller subspaces of a large
parameter space. Our method allows users to incorporate prior experimental
knowledge by specifying upper and lower bounds for each parameter during
inference. This approach overcomes limitations in existing methods, as it
allows training networks with a larger number of parameters and expanded
parameter ranges, while still enabling proper convergence. Additionally, we
introduce the use of a neural operator to process reflectivity curves with
varying discretizations. We validate the effectiveness of our approach by
training networks using different physical models: two-layer and five-layer
box model parameterizations, as well as a specialized parameterization for
repeating identical monolayers. In contrast to previous work, our approach
scales favourably when increasing the complexity of the inverse problem,
giving good predictions even for the challenging 5-layer multilayer model. We
note that the proposed approach can be adapted to tackle other inverse
problems in science affected by the non-uniqueness issue.

##  VI Author contributions

V.M. trained neural networks, performed all the tests, and implemented the FNO
technique. V.S. introduced the prior-aware ML approach, designed the neural
network, and implemented the training procedure. S.K. and V.S. designed and
implemented multilayer parameterization. A.H. performed the experiments.
A.Gr., L.P., A.G., A.H., S.K., and F.S. provided expertise in surface
scattering. All the authors contributed to writing the paper, analysis, and
interpretation of the results. A.H. and F.S. supervised the research process.

##  VII Acknowledgements

This research is part of a project (VIPR, 05D23VT1) funded by the German
Federal Ministry for Science and Education (BMBF). This work is partly
supported by the consortium DAPHNE4NFDI in the context of the work of the NFDI
e.V., funded by the German Research Foundation (DFG). We acknowledge DESY
(Hamburg, Germany), a member of the Helmholtz Association, for the provision
of experimental facilities.

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# Nichols algebras over classical Weyl groups

Zhengtang Tan a, Weicai Wu b, Shouchuan Zhang c
$a$. College of Engineering and Design, Hunan Normal University, Changsha
410081, P.R. China
$b$. College of Mathematics, Hunan Institute of Science and Technology,
Yueyang 414006, P.R. China
$c$. Department of Mathematics, Hunan University, Changsha 410082, P.R. China
Emails<EMAIL_ADDRESS>(WCW)

###### Abstract

We show that except in several cases conjugacy classes of classical Weyl
groups $W(B_{n})$ and $W(D_{n})$ are of type D. We prove that except in three
cases Nichols algebras of irreducible Yetter-Drinfeld (YD in short )modules
over the classical Weyl groups are infinite dimensional.

2010 Mathematics Subject Classification: 16W30, 22E60, 11F23

Keywords: Rack, classical Weyl groups, Nichols algebras.

##  1 Introduction

Nichols algebras play a fundamental role in the classification of finite-
dimensional complex pointed Hopf algebras by means of the lifting method
developed by Andruskiewitsch and Schneider [AS02]. In this context, given a
group $G$, an important step to classify all finite-dimensional complex
pointed Hopf algebras $H$ with group-like $G(H)=G$ is to determine all the
pairs $(\mathcal{O},\rho)$ such that the associated Nichols algebra
$\mathfrak{B}(\mathcal{O},\rho)$ is finite-dimensional. Here the pairs
$(\mathcal{O},\rho)$ are such that $\mathcal{O}$ runs over all conjugacy
classes of $G$ and $\rho$ runs over all irreducible representations of the
centralizer of $g$ in $G$, with $g\in\mathcal{O}$ fixed. In general, this is a
difficult task since Nichols algebra is defined by generators and relations.
In practice, it is often useful to discard those pairs such that
$\dim\mathfrak{B}(\mathcal{O},\rho)=\infty$. There are properties of the
conjugacy class $\mathcal{O}$ that imply that
$\dim\mathfrak{B}(\mathcal{O},\rho)=\infty$ for any $\rho$, one of which is
the property of being of type D. This is useful since it reduces the
computations to operations inside the group and avoids hard calculations of
generators and relations of the corresponding Nichols algebra.

This work contributes to the classification of finite-dimensional Hopf
algebras over an algebraically closed field of characteristic $0$. This
problem was posed by Kaplansky in 1975. The lifting method of Andruskiewitsch
and Schneider describes a way to classify finite-dimensional complex Hopf
algebras. In [AS10] they obtained the classification over the finite abelian
groups whose order is relative prime with 210. Nichols algebras of braided
vector spaces $(\mathbb{C}X,c_{q})$, where $X$ is a rack and $q$ is a
2-cocycle in $X$, were studied in [AG03]. It was shown [AFGV08, AFZ09, AZ07]
that Nichols algebras $\mathfrak{B}({\mathcal{O}}_{\sigma},\rho)$ over
symmetry groups are infinite dimensional, except in a number of remarkable
cases corresponding to ${\mathcal{O}}_{\sigma}$. Two of the present authors
[ZZ12] showed that except in three cases Nichols algebras of irreducible
Yetter-Drinfeld (YD) modules over classical Weyl groups
$A\rtimes\mathbb{S}_{n}$ supported by $\mathbb{S}_{n}$ are infinite
dimensional. However, the classification has not been completed for Nichols
algebras over general classic Weyl groups $W(B_{n})$ and $W(D_{n})$.

Note that $\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$ is isomorphic to Weyl
groups $W(B_{n})$ and $W(C_{n})$ of $B_{n}$ and $C_{n}$ for $n>2$. If
$A=\\{a\in\mathbb{Z}_{2}^{n}\mid\ a=({a_{1}},{a_{2}},\cdots,{a_{n}})$ $\hbox{
with }a_{1}+a_{2}+\cdots+a_{n}\equiv 0\ ({\rm mod}\ 2)\\}$, then
$A\rtimes\mathbb{S}_{n}$ is isomorphic to Weyl group $W(D_{n})$ of $D_{n}$ for
$n>3$. Obviously, when $A=\\{a\in{\mathbb{Z}}_{2}^{n}\mid\
a=({a_{1}},{a_{2}},\cdots,{a_{n}})$ $\hbox{ with all }a_{i}\equiv 0\ ({\rm
mod}\ 2)\\}$, $A\rtimes\mathbb{S}_{n}$ is isomorphic to Weyl group of
$A_{n-1}$ for $n>1$. Note that $\mathbb{S}_{n}$ acts on $A$ as follows: for
any $a\in A$ with $a=({a_{1}},{a_{2}},\cdots,{a_{n}})$ and
$\sigma,\tau\in\mathbb{S}_{n}$, $\sigma\cdot
a:=({a_{\sigma^{-1}(1)}},{a_{\sigma^{-1}(2)}},\cdots,{a_{\sigma^{-1}(n)}}).$

It is clear that

$\displaystyle(a,\sigma)^{-1}=(-(a_{\sigma(1)},a_{\sigma(2)},\cdots,a_{\sigma(n)}),\sigma^{-1})=(-\sigma^{-1}(a),\sigma^{-1}),$
$\displaystyle(b,\tau)(a,\sigma)(b,\tau)^{-1}=(b+\tau(a)-\tau\sigma\tau^{-1}(b),\tau\sigma\tau^{-1}).$
$\displaystyle a\sigma=(a,\sigma).$

Without specification, $K_{n}:=\\{a\in\mathbb{Z}_{n}\mid
a_{1}+a_{2}+\cdots+a_{n}=0\\}$ and $K_{n}\rtimes\mathbb{S}_{n}$ is a subgroup
of $\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$. Let $\mathbb{W}_{n}$ denote
$K_{n}\rtimes\mathbb{S}_{n}$ or $\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$
throughout this paper. When we say type D we mean a rack or conjugacy class of
type D and not a Weyl group of type ${\rm D}_{n}$.

In this paper we prove that except in several cases conjugacy classes of
classical Weyl groups $\mathbb{W}_{n}$ are of type D, and except in three
cases Nichols algebras of irreducible YD modules over the classical Weyl
groups are infinite dimensional.

The work is organized as follows. In Section Preliminaries and conventions we
provide some preliminaries and set our notations. In Section 2 we determine
when the conjugacy classes of juxtapositions of two elements are of type D. In
Section 3 we examine the rack and conjugacy classes of $\mathbb{W}_{n}$. In
Section 4 we classify Nichols algebras of irreducible YD modules over the
classical Weyl groups.

## Preliminaries and conventions

Let $\mathbf{k}$ be an algebraically closed field of characteristic zero and
$(V,C)$ a braided vector space, i.e. $V$ is a vector space over $\mathbf{k}$
and $C\in{\rm Aut}(V\otimes V)$ is a solution of the braid equation $({\rm
id}\otimes C_{V,V})(C_{V,V}\otimes{\rm id})({\rm id}\otimes
C_{V,V})=(C_{V,V}\otimes{\rm id})({\rm id}\otimes C_{V,V})(C_{V,V}\otimes{\rm
id})$. $C_{i,i+1}:={\rm id}^{\otimes i-1}\otimes C_{V,V}\otimes{\rm
id}^{\otimes n-i-1}$. Let $S_{m}\in{\rm End}_{\mathbf{k}}(V^{\otimes m})$ and
$S_{1,j}\in{\rm End}_{\mathbf{k}}(V^{\otimes(j+1)})$ denote the maps
$S_{m}=\prod\limits_{j=1}^{m-1}({\rm id}^{\bigotimes m-j-1}\bigotimes
S_{1,j})$ , $S_{1,j}={\rm
id}+C_{12}^{-1}+C_{12}^{-1}C_{23}^{-1}+\cdots+C_{12}^{-1}C_{23}^{-1}\cdots
C_{j,j+1}^{-1}$ (in leg notation) for $m\geq 2$ and $j\in\mathbb{N}$. Then the
subspace $S=\bigoplus\limits_{m=2}^{\infty}{\rm ker}S_{m}$ of the tensor
$T(V)=\bigoplus\limits_{m=0}^{\infty}V^{\otimes m}$ is a two-sided ideal, and
algebra $\mathfrak{B}(V)=T(V)/S$ is termed the Nichols algebra associated to
$(V,C)$.

For $s\in G$ and $(\rho,V)\in\widehat{G^{s}}$, here is a precise description
of the YD module $M({\mathcal{O}}_{s},\rho)$, introduced in [Gr00]. Let
$t_{1}=s,t_{2},\cdots,t_{m}$ be a numeration of ${\mathcal{O}}_{s}$, which is
a conjugacy class containing $s$, and let $g_{i}\in G$ such that $g_{i}\rhd
s:=g_{i}sg_{i}^{-1}=t_{i}$ for all $1\leq i\leq m$. Then
$M({\mathcal{O}}_{s},\rho)=\oplus_{1\leq i\leq m}g_{i}\otimes V$. Let
$g_{i}v:=g_{i}\otimes v\in M({\mathcal{O}}_{s},\rho)$, $1\leq i\leq m$, $v\in
V$. If $v\in V$ and $1\leq i\leq m$, then the action of $h\in G$ and the
coaction are given by

$\displaystyle\delta(g_{i}v)=t_{i}\otimes g_{i}v,\qquad
h\cdot(g_{i}v)=g_{j}(\nu_{i}(h)\cdot v),$ (1)

where $hg_{i}=g_{j}\nu_{i}(h)$, for unique $1\leq j\leq m$ and $\nu_{i}(h)\in
G^{s}$. The explicit formula for the braiding is then given by

$C(g_{i}v\otimes g_{j}w)=t_{i}\cdot(g_{j}w)\otimes
g_{i}v=g_{j^{\prime}}(\nu_{j}(t_{i})\cdot w)\otimes g_{i}v$ (2)

for any $1\leq i,j\leq m$, $v,w\in V$, where
$t_{i}g_{j}=g_{j^{\prime}}\nu_{j}(t_{i})$ for unique $j^{\prime}$, $1\leq
j^{\prime}\leq m$ and $\nu_{j}(t_{i})\in G^{s}$. Let
$\mathfrak{B}({\mathcal{O}}_{s},\rho)$ denote
$\mathfrak{B}(M({\mathcal{O}}_{s},\rho))$. $M({\mathcal{O}}_{s},\rho)$ is a
simple YD module (see [DPR91, Ci97, AZ07]).

We briefly recall the definition and main properties of racks; see [AG03] for
details, more information and bibliographical references. A rack is a pair
$(X,\rhd)$, where $X$ is a non-empty set and $\rhd:X\times X\rightarrow X$ is
an operation such that $x\rhd x=x$, $x\rhd(y\rhd z)=(x\rhd y)\rhd(x\rhd z)$
and $\phi_{x}$ is invertible for any $x,y,z\in X$, where $\phi_{x}$ is a map
from $X$ to $X$ sending $y$ to $x\rhd y$ for any $x,y\in X.$ For example,
$({\mathcal{O}}_{s}^{G},\rhd)$ is a rack with $x\rhd y:=xyx^{-1}.$

If $R$ and $S$ are two subracks of $X$ with $R\cup S=X$, $R\cap S=\emptyset$,
$x\rhd y\in S$, $y\rhd x\in R$, for any $x\in R,y\in S,$ then $R\cup S$ is
called a decomposition of subracks of $X$. Furthermore, if there exist $a\in
R$, $b\in S$ such that ${\rm sq}(a,b):=a\rhd(b\rhd(a\rhd b))\not=b$, then $X$
is called to be of type D. Notice that if a rack $Y$ contains a subrack $X$ of
type ${\rm D}$, then $Y$ is also of type ${\rm D}$ ( see [AFGV08]).

##  2 Conjugacy classes of juxtapositions

In this section we determine when the conjugacy classes of juxtapositions of
two elements are of type D.

######  Lemma 2.1.

Let $G=\mathbb{W}_{n}$ and $(a,\tau),(b,\mu)\in G$. Let $(c,\lambda)$ denote
$(a,\tau)\rhd((b,\mu)\rhd((a,\tau)\rhd(b,\mu)))$.

(i)  Then

$\displaystyle c$ $\displaystyle=$
$\displaystyle(a+\tau\cdot[b+\mu\cdot(a+\tau\cdot b+(\tau\rhd\mu)\cdot
a)+(\mu\rhd(\tau\rhd\mu))\cdot b]$ (3) $\displaystyle+$
$\displaystyle(\tau\rhd(\mu\rhd(\tau\rhd\mu)))\cdot a.$

(ii) If $\tau$ and $\mu$ are commutative, then

$\displaystyle c=a+\tau\mu\cdot a+\tau\mu^{2}\cdot a+\mu\cdot a+\tau\cdot
b+\tau^{2}\mu\cdot b+\tau\mu\cdot b.$ (4)

(iii) If $\tau$ and $\mu$ are commutative, then ${\rm
sq}((a,\tau),(b,\mu))=(b,\mu)$ if and only if

$\displaystyle a+\tau\mu\cdot a+\tau\mu^{2}\cdot a+\mu\cdot a=b+\tau\cdot
b+\tau^{2}\mu\cdot b+\tau\mu\cdot b.$ (5)

(iv) If $\tau$ and $\mu$ are commutative with $(b,\mu)=\xi\rhd(a,\tau)$ and
$\xi\cdot a=a$, then $c=a+\tau\mu^{2}\cdot a+\mu\cdot a+\tau\cdot
a+\tau^{2}\mu\cdot a$.

(v) If $\tau$ and $\xi$ are commutative with $\tau^{2}=1$ and
$(b,\mu)=\xi\rhd(a,\tau)$ and $\xi\cdot a=a$, then $c=a$.

Proof. It is clear. $\Box$

If the lengths of independent cycles of $\pi$ and $\tau$ are different, then
they are called mutually orthogonal, written as $\pi\bot\tau.$ If the lengths
of independent sign cycles of $(a,\pi)$ and $(b,\tau)$ are different, then
they are called mutually orthogonal, written as $a\pi\bot b\tau.$ Obviously,
$a\pi\bot b\tau$ if and only if $\pi\bot\tau.$ If ${\rm ord(\pi)}$ and ${\rm
ord(\tau)}$ are coprime and one of the two elements does not have any fixed
point, then $\pi\bot\tau.$ Notice that if $\pi\bot\tau$, then
$\mathbb{S}_{m+n}^{\pi\\#\tau}=\mathbb{S}_{n}^{\pi}\\#\mathbb{S}_{m}^{\tau}$.

For any $a\pi\in\mathbb{W}_{n}$ and $b\tau\in\mathbb{W}_{m}$, define
$a\pi\\#b\tau\in\mathbb{W}_{m+n}$ as follows:

$(a\\#b)_{i}:=\left\\{\begin{array}[]{ll}a_{i}&\hbox{when }i\leq n\\\
b_{i-n}&\hbox{when }i>n\end{array}\right.,$
$(\pi\\#\tau)(i):=\left\\{\begin{array}[]{ll}\pi(i)&\hbox{when }i\leq n\\\
\tau(i-n)+n&\hbox{when }i>n\end{array}\right.$ and
$a\pi\\#b\tau:=(a\\#b,\pi\\#\tau)$. Obviously
$a\pi\\#b\tau\in\mathbb{W}_{m+n}$ and it is called a juxtaposition of $a\pi$
and $b\tau$ (see [AZ07]). Let $\overrightarrow{\nu_{n,m}}$ be a map from
$\mathbb{W}_{n}$ to $\mathbb{W}_{m+n}$ by sending $a\pi$ to
$\overrightarrow{\nu_{n,m}}(a\pi):=a\pi\\#1_{\mathbb{W}_{m}}$; let
$\overleftarrow{\nu_{n,m}}$ be a map from $\mathbb{W}_{m}$ to
$\mathbb{W}_{m+n}$ by sending $b\tau$ to
$\overleftarrow{\nu_{n,m}}(b\tau):=1_{\mathbb{W}_{n}}\\#b\tau$.

######  Lemma 2.2.

Assume $a\pi\bot b\tau$ with $a\pi,a^{\prime}\pi^{\prime}\in\mathbb{W}_{n}$
and $b\tau,b^{\prime}\tau^{\prime}\in\mathbb{W}_{m}$. Then

(i) $(a\pi\\#b\tau)(a^{\prime}\pi^{\prime}\\#b^{\prime}\tau^{\prime})=(a\pi
a^{\prime}\pi^{\prime}\\#b\tau b^{\prime}\tau^{\prime})$.

(ii)
$a\pi\\#b\tau=\overrightarrow{\nu_{n,m}}(a\pi)\overleftarrow{\nu_{n,m}}(b\tau)=\overleftarrow{\nu_{n,m}}(b\tau)\overrightarrow{\nu_{n,m}}(a\pi)$.

(iii)
$\mathbb{W}_{m+n}^{a\pi\\#b\tau}=\mathbb{W}_{n}^{a\pi}\\#\mathbb{W}_{m}^{b\tau}=\overrightarrow{\nu_{n,m}}(\mathbb{W}_{n}^{a\pi})\overleftarrow{\nu_{n,m}}(\mathbb{W}_{m}^{b\tau})$
as directed products.

(iv) For any $\rho\in\widehat{\mathbb{W}_{m+n}^{a\pi\\#b\tau}}$, there exist
$\rho_{1}\in\widehat{\mathbb{W}_{n}^{a\pi}}$,
$\rho_{2}\in\widehat{\mathbb{W}_{m}^{b\tau}}$ such that
$\rho=\rho_{1}\otimes\rho_{2}$.

(v)
$(a\pi\\#b\tau)\rhd(a^{\prime}\pi^{\prime}\\#b^{\prime}\tau^{\prime})=(a\pi\rhd
a^{\prime}\pi^{\prime})\\#(b\tau\rhd b^{\prime}\tau^{\prime})$.

(vi)
$\mathcal{O}_{a\pi\\#b\tau}^{\mathbb{W}_{m+n}}=\mathcal{O}_{a\pi}^{\mathbb{W}_{n}}\\#\mathcal{O}_{b\tau}^{\mathbb{W}_{m}}$.

Proof. (i), (ii) and (v) are clear.

(iii) By [AZ07, Section 2.2],
$\mathbb{S}_{m+n}^{\pi\\#\tau}=\mathbb{S}_{n}^{\pi}\\#\mathbb{S}_{m}^{\tau}.$
Obviously,
$\mathbb{W}_{n}^{a\pi}\\#\mathbb{W}_{m}^{b\tau}\subseteq\mathbb{W}_{m+n}^{a\pi\\#b\tau}$.
For any $c\xi\in\mathbb{W}_{m+n}^{a\pi\\#b\tau},$ then
$\xi\in\mathbb{S}_{m+n}^{\pi\\#\tau}$ and there exist
$\mu\in\mathbb{S}_{n}^{\pi}$ and $\lambda\in\mathbb{S}_{m}^{\tau}$ such that
$\xi=\mu\\#\lambda$. Consequently, $c\xi=d\mu\\#f\lambda.$ Considering
$c\xi(a\pi\\#b\tau)=(a\pi\\#b\tau)c\xi$, we have
$d\mu\in\mathbb{W}_{n}^{a\pi}$ and $f\lambda\in\mathbb{W}_{m}^{b\tau}$. This
completes the proof.

(iv) It follows from (iii).

(vi) By (v),
$\mathcal{O}_{a\pi}^{\mathbb{W}_{n}}\\#\mathcal{O}_{b\tau}^{\mathbb{W}_{m}}\subseteq\mathcal{O}_{a\pi\\#b\tau}^{\mathbb{W}_{m+n}}.$
Consequently (vi) follows from (iii). $\Box$

######  Remark 2.3.

(i), (ii) and (v) above still hold when $a\pi$ and $b\tau$ are not mutually
orthogonal.

######  Theorem 2.4.

If $\mathcal{O}_{a\tau}$ is of type D, then $\mathcal{O}_{a\tau\\#b\mu}$ is
also of type D.

Proof. Let $X=R\cup S$ be a subrack decomposition of $\mathcal{O}_{a\tau}$ and
of type D. It is clear that $X\\#b\mu=R\\#b\mu\cup S\\#b\mu$ is a subrack
decomposition of $\mathcal{O}_{a\tau\\#b\mu}$ and of type D. $\Box$

######  Lemma 2.5.

Assume $a\pi\bot b\tau$. Let
$\rho=\rho_{1}\otimes\rho_{2}\in\widehat{\mathbb{W}_{m+n}^{a\pi\\#b\tau}}$,
$\rho_{1}\in\widehat{\mathbb{W}_{n}^{a\pi}}$,
$\rho_{2}\in\widehat{\mathbb{W}_{m}^{b\tau}}$, $a\pi\in\mathbb{W}_{n}$ and
$b\tau\in\mathbb{W}_{m}$ with $q_{a\pi,a\pi}{\rm id}=\rho_{1}(a\pi)$ and
$q_{b\tau,b\tau}{\rm id}=\rho_{2}(b\tau)$. If
$\dim\mathfrak{B}({\mathcal{O}}_{a\pi\\#b\tau}^{\mathbb{W}_{n+m}},\rho)<\infty$,
then

(i) $M({\mathcal{O}}_{a\pi}^{\mathbb{W}_{n}},\rho_{1})$ is isomorphic to a YD
submodule of $M({\mathcal{O}}_{a\pi\\#b\tau}^{\mathbb{W}_{n+m}},\rho)$ over
$\mathbb{W}_{n}$ when $q_{b\tau,b\tau}=1$; hence
$\dim\mathfrak{B}({\mathcal{O}}_{a\pi}^{\mathbb{W}_{n}},\rho_{1})<\infty$.

(ii) $q_{a\pi,a\pi}q_{b\tau,b\tau}=-1$.

(iii) $q_{b\tau,b\tau}=1$ and $q_{a\pi,a\pi}=-1$ when ${\rm ord}(b\tau)\leq 2$
and ${\rm ord}(q_{a\pi,a\pi})\not=1$.

(iv) $q_{b\tau,b\tau}=1$ and $q_{a\pi,a\pi}=-1$ when ${\rm ord}(a\pi)$ and
${\rm ord}(b\tau)$ are coprime and ${\rm ord}(b\tau)$ is odd.

Proof. (i) The proof is similar to the one in [AZ07, Section 2.2]. Indeed, Let
$t_{1}=\pi,t_{2},\cdots,t_{\mid{\mathcal{O}}_{\pi}\mid}$ be a numeration of
${\mathcal{O}}_{\pi}$, and let $g_{i}\in G$ such that $g_{i}\rhd\pi:=g_{i}\pi
g_{i}^{-1}=t_{i}$ for all $1\leq i\leq\mid{\mathcal{O}}_{\pi}\mid$. Let
$s_{1}=\tau,s_{2},\cdots,s_{\mid{\mathcal{O}}_{\tau}\mid}$ be a numeration of
${\mathcal{O}}_{\tau}$, and let $h_{i}\in G$ such that
$h_{i}\rhd\tau:=h_{i}\tau h_{i}^{-1}=s_{i}$ for all $1\leq
i\leq\mid{\mathcal{O}}_{\tau}\mid$. Thus $\\{t_{i}\\#s_{j}\mid 1\leq
i\leq\mid{\mathcal{O}}_{\pi}\mid,1\leq j\ \leq\
\mid{\mathcal{O}}_{\tau}\mid\\}={\mathcal{O}}_{\pi\\#\tau}$. Let $V$ and $W$
be representation spaces of $\rho_{1}$ and $\rho_{2}$, respectively. Let
$0\not=w_{0}\in W$. Define a map $\psi$ from
$M({\mathcal{O}}_{\pi}^{{\mathbb{W}}_{n}},\rho_{1})$ to
$M({\mathcal{O}}_{\pi\\#\tau}^{{\mathbb{W}}_{m}\\#{\mathbb{W}}_{n}},\rho_{1}\otimes\rho_{2})$
by sending $g_{i}v$ to $(g_{i}\\#h_{1})v\otimes w_{0}$ for any $v\in V,$
$1\leq i\leq\mid{\mathcal{O}}_{\pi}\mid.$ It is clear that $\psi$ is
injective. Now we show that $\psi$ is a homomorphism of braided vector spaces.
For any $v,v^{\prime}\in V,$ we have that

$\displaystyle(\psi\otimes\psi)(C(g_{i}v\otimes g_{j}v^{\prime}))$
$\displaystyle=$ $\displaystyle(\psi\otimes\psi)(g_{i\rhd
j}\rho_{1}(\nu_{j}(t_{i}))(v^{\prime})\otimes g_{i}v)$ $\displaystyle=$
$\displaystyle(g_{i\rhd j}\\#h_{1}\rho_{1}(\nu_{j}(t_{i})(v^{\prime})\otimes
w_{0})\otimes(g_{i}\\#h_{1}v\otimes w_{0})\ \ \ \hbox{and }$ $\displaystyle
C(\psi(g_{i}v)\otimes\psi(g_{j}v^{\prime})$ $\displaystyle=$ $\displaystyle
C((g_{i}\\#h_{1}v\otimes w_{0})\otimes(g_{j}\\#h_{1}v^{\prime}\otimes w_{0}))$
$\displaystyle=$ $\displaystyle(g_{i\rhd
j}\\#h_{1}(\rho_{1}\otimes\rho_{2})(\nu_{j,1}(t_{i}\\#s_{1}))(v^{\prime}\otimes
w_{0}))\otimes(g_{i}\\#h_{1}v\otimes w_{0}).$

It is clear that $\nu_{j,1}(t_{i}\\#h_{1})=\nu_{j}(t_{i})\\#h_{1}$. Therefore
$\psi$ is a homomorphism of braided vector spaces.

(ii) and (iii) are clear.

(iv) Obviously, ${\rm ord}(q_{a\pi,a\pi})\mid{\rm ord}(a\pi)$ and ${\rm
ord}(q_{b\tau,b\tau})\mid{\rm ord}(b\tau)$. Therefore, ${\rm
ord}(q_{b\tau,b\tau})$ is odd. The least common factor $({\rm
ord}(q_{a\pi,a\pi}),{\rm ord}(q_{b\tau,b\tau}))=1$ since $({\rm
ord}{(a\pi}),{\rm ord}({b\tau}))=1$. By Part (ii), ${\rm
ord}(q_{a\pi,a\pi}){\rm ord}(q_{b\tau,b\tau})=2$. Consequently, ${\rm
ord}(q_{a\pi,a\pi})=2$ and ${\rm ord}(q_{b\tau,b\tau})=1$. $\Box$

##  3 Conjugacy classes of $\mathbb{W}_{n}$

In this section we prove that except in several cases conjugacy classes of
classical Weyl groups $\mathbb{W}_{n}$ are of type D.

######  Lemma 3.1.

Let $p$ be odd with $p\geq 5$ and $a\tau\in\mathbb{W}_{n}$ with $\tau=(1\
2\cdots\ p)$. Then $\mathcal{O}_{a\tau}^{\mathbb{W}_{n}}$ is of type D.

Proof. (i) Assume that $a\tau$ is a negative cycle (defined in [ZZ12, Appendix
]) and $b=(1,0,\cdots,0,a_{p+1},a_{p+2},\cdots,a_{n})$. Thus $a\tau$ and
$b\tau$ are conjugate. We assume $a=(1,1,\cdots,1,$
$a_{p+1},a_{p+2},\cdots,a_{n})$ without lost generality. Obviously, the right
hand side of (5) is non-vanishing for $\mu=\tau^{2}$, i.e. ${\rm
sq}(a\tau,b\tau^{2})\not=b\tau^{2}.$ Let
$R:=\mathbb{Z}_{2}^{n}\rtimes\tau\cap\mathcal{O}_{a\tau}^{\mathbb{W}_{n}}$ and
$S:=\mathbb{Z}_{2}^{n}\rtimes\tau^{2}\cap\mathcal{O}_{a\tau}^{\mathbb{W}_{n}}$.
It is clear that $R\cup S$ is a subrack decomposition of
$\mathcal{O}_{a\tau}^{\mathbb{W}_{n}}$, hence it is of type D; notice that
$\tau$ and $\tau^{2}$ are conjugate in $\mathbb{S}_{n}.$

(ii) Assume that $a\tau$ is a positive cycle. Let
$b=(1,0,0,1,0,a_{p+1},a_{p+2},\cdots,a_{n})$ when $p=5;$
$b=(1,1,0,\cdots,0,a_{p+1},a_{p+2},\cdots,a_{n})$ when $p>5.$ Thus $0\tau$,
$a\tau$ and $b\tau$ are conjugate. We assume $a=0$ without lost generality. It
is clear that the right hand side of (5) is not equal to $0$. Consequently,
$R\cup S$ is a subrack decomposition and is of type D as Part (i).

$\Box$

######  Lemma 3.2.

If $\sigma$ is of type ($3^{2}$), then
$\mathcal{O}_{a\sigma}^{\mathbb{W}_{6}}$ is of type ${\rm D}$ for all
$a\in\mathbb{Z}_{2}^{6}.$

Proof. Let $\pi=(1\ 2\ 3)$, $\xi=(4\ 5\ 6)$, $\tau=(1\ 2\ 3)(4\ 5\ 6)$ and
$\mu=(1\ 2\ 3)^{2}(4\ 5\ 6)$. We have that the $4$-th, $5$-th and $6$-th
components of (5) are

$\displaystyle(a_{6}+a_{5},a_{4}+a_{6},a_{5}+a_{4})=(b_{6}+b_{5},b_{4}+b_{6},b_{5}+b_{4}).$
(6)

By (5), (i) If $a=(1,1,\cdots,1)$ and $b=(1,0,0,1,0,0)$, then (6) does not
hold.

(ii) If $a=0$ and $b=(0,0,0,1,1,0)$, then (6) does not hold.

(iii) If $a=(1,0,0,0,0,0)$ and $b=(1,0,0,1,1,0)$, then (6) does not hold.

(iv) If $a=(0,0,0,1,0,0)$ and $b=(0,0,0,0,1,0)$, then (6) does not hold. Let
$R:=\mathbb{Z}_{2}^{6}\rtimes\tau\cap\mathcal{O}_{a\sigma}^{\mathbb{W}_{6}}$
and
$S:=\mathbb{Z}_{2}^{6}\rtimes\mu\cap\mathcal{O}_{a\sigma}^{\mathbb{W}_{6}}$.
It is clear that $R\cup S$ is a subrack decomposition of
$\mathcal{O}_{a\sigma}^{\mathbb{W}_{6}}$. Consequently
$\mathcal{O}_{a\sigma}^{\mathbb{W}_{6}}$ is of type D. $\Box$

######  Lemma 3.3.

If $\sigma$ is of type ($2^{2},3^{1}$), then
$\mathcal{O}_{a\sigma}^{\mathbb{W}_{7}}$ is of type ${\rm D}$ for all
$a\in\mathbb{Z}_{2}^{7}.$

Proof. Let $\pi=(5\ 6\ 7)$, $\xi=(1\ 2)(3\ 4)$, $\lambda=(1\ 3)(2\ 4)$,
$\tau=\pi\xi$ and $\mu=\pi\lambda$. If $a=(a_{1},a_{2},a_{3},a_{4},0,0,0)$,
let $b=(b_{1},b_{2},b_{3},b_{4},1,1,0)$. If
$a=(a_{1},a_{2},a_{3},a_{4},1,1,1)$, let $b=(b_{1},b_{2},b_{3},b_{4},1,0,0)$.
Then the $5$-th, $6$-th and $7$-th components of (5) are
$(a_{6}+a_{7},a_{7}+a_{5},a_{5}+a_{6})=(b_{6}+b_{7},b_{7}+b_{5},b_{5}+b_{6})$,
respectively. Consequently, (5) does not hold.

Let
$R:=\mathbb{Z}_{2}^{7}\rtimes\tau\cap\mathcal{O}_{a\sigma}^{\mathbb{W}_{7}}$
and
$S:=\mathbb{Z}_{2}^{7}\rtimes\mu\cap\mathcal{O}_{a\sigma}^{\mathbb{W}_{7}}$.
It is clear that $R\cup S$ is a subrack decomposition of
$\mathcal{O}_{a\sigma}^{\mathbb{W}_{7}}$. Consequently
$\mathcal{O}_{a\sigma}^{\mathbb{W}_{7}}$ is of type $D.$ $\Box$

######  Example 3.4.

If $\tau$ and $\mu$ are of type $(2^{2})$, then $a\tau$ and $b\mu$ are square
commutative when they are conjugate to each other or
$(-1)^{\sum_{i}^{4}a_{i}}=(-1)^{\sum_{i}^{4}b_{i}}$.

Proof. Let $\tau=(1\ 2)(3\ 4)$. Then $\mu=(1\ 3)(2\ 4)$ or $\mu=\tau.$ It is
clear that equation (5) becomes
$a_{1}+a_{2}+a_{3}+a_{4}=b_{1}+b_{2}+b_{3}+b_{4}.$ $\Box$

######  Lemma 3.5.

(i) Assume that $\tau$ and $\mu$ are conjugate with ${\rm
sq}(\tau,\mu)\not=\mu$ in $\mathbb{S}_{n}$ and $\tau(n)=\mu(n)=n$. If
$a\in\mathbb{Z}_{2}^{n}$ and there exists $i$ such that $a_{i}\not=a_{n}$ with
$\tau(i)=\mu(i)=i$, then $\mathcal{O}_{a\tau}$ is of type $D.$

(ii) Assume that $n>4$ and $a\tau\in\mathbb{W}_{n}$ with type $(1^{n-2},2)$ of
$\tau.$ If there exist $i,j$ such that $\tau(i)=i$ and $\tau(j)=j$ with
$a_{i}\not=a_{j}$, then $\mathcal{O}_{a\tau}$ is of type D.

(iii) Assume that $n>5$ and $a\tau\in\mathbb{W}_{n}$ with type $(1^{n-3},3)$
of $\tau.$ If there exist $i,j$ such that $\tau(i)=i$ and $\tau(j)=j$ with
$a_{i}\not=a_{j}$, then $\mathcal{O}_{a\tau}$ is of type D.

Proof. (i) Let
$R:=\\{d\xi\in\mathcal{O}_{a\tau}^{\mathbb{W}_{n}}\mid\xi(n)=n;d_{n}=0\\}$ and
$S:=\\{d\xi\in\mathcal{O}_{a\tau}^{\mathbb{W}_{n}}\mid\xi(n)=n;d_{n}=1\\}$.
Obviously, $R\cup S$ is a subrack decomposition. Let $\xi\in\mathbb{S}_{n}$
with $\xi(i)=i$ and $\xi(n)=n$ such that $\xi\rhd\tau=\mu.$ By simple
computation we have $(i,n)\xi\rhd(a\tau)=b\mu$ with $b_{n}=a_{i}.$
Consequently, $a\tau$ and $b\mu$ are in the same set of $R$ and $S$, which
implies that $R\cup S$ is of type ${\rm D}$.

(ii) It is clear that ${\rm sq}(\tau,\mu)\not=\mu$ with $\tau:=(1,2)$ and
$\mu:=(2,3)$. Applying Part (i) we complete the proof.

(iii) It is clear that ${\rm sq}(\tau,\mu)\not=\mu.$ with $\tau=(1,2,3)$ and
$\mu=(2,4,3)$. Applying Part (i) we complete the proof. $\Box$

######  Lemma 3.6.

Let $\sigma=(a,\tau)\in\mathbb{W}_{n}$. If
$\mathcal{O}_{\tau}^{\mathbb{S}_{n}}$ is of type D, then so is
$\mathcal{O}_{(a,\tau)}^{\mathbb{W}_{n}}$.

Proof. Let $X=S\cup T$ be a subrack decomposition of
$\mathcal{O}_{\tau}^{\mathbb{S}_{n}}$ and $s\in S,t\in T$ such that

$\displaystyle s\rhd(t\rhd(s\rhd t))\not=t.$ (7)

Let $h\rhd\tau=s$ and $g\rhd\tau=t$ with $h,g\in{\mathbb{S}_{n}}$. It is clear
${\rm sq}((h\cdot a,s),(g\cdot a,t))\not=(g\cdot a,t)$ since (7); $(h\cdot
a,s)=h\rhd(a,\tau)$ and $(g\cdot a,t)=g\rhd(a,\tau)$.

$(<{\mathbb{S}_{n}}\cdot a>,X)$ is a subrack, where $<{\mathbb{S}_{n}}\cdot
a>$ is the subgroup generated by subset ${\mathbb{S}_{n}}\cdot a$ of
$\mathbb{Z}_{2}^{n}$. In fact, for any $h,g\in{\mathbb{S}_{n}},\xi,\mu\in X,$
we have

$\displaystyle(h\cdot a,\xi)\rhd(g\cdot a,\mu)=((h+\xi g+\xi\mu\xi^{-1}h)\cdot
a,\xi\rhd\mu)\in(<{\mathbb{S}_{n}}\cdot a>,X).$

Thus $(<{\mathbb{S}_{n}}\cdot a>,X)$ is a subrack. Consequently
$(<{\mathbb{S}_{n}}\cdot
a>,X)\cap\mathcal{O}_{(a,\tau)}^{\mathbb{W}_{n}}=(<{\mathbb{S}_{n}}\cdot
a>,S)\cap\mathcal{O}_{(a,\tau)}^{\mathbb{W}_{n}}\cup(<{\mathbb{S}_{n}}\cdot
a>,T)\cap\mathcal{O}_{(a,\tau)}^{\mathbb{W}_{n}}$ is a subrack decomposition
of $\mathcal{O}_{(a,\tau)}^{\mathbb{W}_{n}}$ and is of type D. $\Box$

######  Theorem 3.7.

Let $G=\mathbb{W}_{n}$ with $n>4$. Let $\tau\in\mathbb{S}_{n}$ be of type
$(1^{\lambda_{1}},2^{\lambda_{2}},\dots,n^{\lambda_{n}})$ and
$a\in\mathbb{Z}_{2}^{n}$ with $\sigma=(a,\tau)\in G$ and $\tau\not=1$. If
$\mathcal{O}_{\sigma}^{G}$ is not of type D, then the type of $\tau$ belongs
to one in the following list.

  1. (i)

$(2,3)$; $(2^{3});$

  2. (ii)

$(2^{4});$ $(1,2^{2}),(1^{2},3),(1^{2},2^{2});$

  3. (iii)

$(1^{n-2},2)$ and $(1^{n-3},3)$ $(n>5)$ with $a_{i}=a_{j}$ when $\tau(i)=i$
and $\tau(j)=j$.

Proof. It follows from Lemma 3.3, Lemma 3.5, Lemma 3.6, Lemma 3.1, Lemma 3.2
and [AFGV08, Theorem 4.1]. $\Box$

##  4 Nichols algebras of irreducible YD module over $\mathbb{W}_{n}$

In this section we show that except in three cases Nichols algebras of
irreducible YD modules over classical Weyl groups $\mathbb{W}_{n}$ are
infinite dimensional.

Let ${\rm supp}M:=\\{g\in G\mid M_{g}\not=0\\}$ for $G$-comodule $(M,\delta)$,
where $M_{g}:=\\{x\in M\mid\delta(x)=g\otimes x\\}$.

We shall use the following facts:

######  Theorem 4.1.

([HS08, Cor. 8.4]) Let $n\in\mathbb{N}$, $n\geq 3$, and assume that
$G=\mathbb{S}_{n}$ is the symmetric group. Let $U$ be a YD module over $G$. If
$\mathfrak{B}(U)$ is finite dimensional, then $U$ is an irreducible YD module
over $G$.

######  Theorem 4.2.

( [AFGV08, Th. 1.1]) Let $m\geq 5$. Let $\sigma\in\mathbb{S}_{m}$ be of type
$(1^{n_{1}},2^{n_{2}},\dots,m^{n_{m}})$, let $\mathcal{O}_{\sigma}$ be the
conjugacy class of $\sigma$ and let
$\rho=(\rho,V)\in\widehat{{S_{m}}^{\sigma}}$. If
$\dim\mathfrak{B}(\mathcal{O}_{\sigma},\rho)<\infty$, then the type of
$\sigma$ and $\rho$ are in the following list:

  1. (i)

$(1^{n_{1}},2)$, $\rho_{1}={\rm sgn}$ or $\epsilon$, $\rho_{2}={\rm sgn}$.

  2. (ii)

$(2,3)$ in $\mathbb{S}_{5}$, $\rho_{2}={\rm sgn}$,
$\rho_{3}=\overrightarrow{\chi_{0}}$.

  3. (iii)

$(2^{3})$ in $\mathbb{S}_{6}$,
$\rho_{2}=\overrightarrow{\chi_{1}}\otimes\epsilon$ or
$\overrightarrow{\chi_{1}}\otimes{\rm sgn}$.

######  Theorem 4.3.

If $G$ is a finite group and $\mathcal{O}_{\sigma}^{G}$ is of type D, then dim
$\mathfrak{B}(\mathcal{O}_{\sigma}^{G},\rho)=\infty$ for any
$\rho\in\widehat{G^{\sigma}}$.

Proof. It follows from [AFGV08, Theorem 3.6]. $\Box$

######  Theorem 4.4.

Assume $n>4$. Let $\tau\in\mathbb{S}_{n}$ be of type
$(1^{\lambda_{1}},2^{\lambda_{2}},\dots,n^{\lambda_{n}})$ and
$a\in\mathbb{Z}_{2}^{n}$ with $\sigma=(a,\tau)\in\mathbb{W}_{n}$ and
$\tau\not=1$. If dim
$\mathfrak{B}(\mathcal{O}_{\sigma}^{\mathbb{W}_{n}},\rho)<\infty$, then the
type of $\tau$ belongs to one in the following list:

  1. (i)

$(2,3)$; $(2^{3});$

  2. (ii)

$(2^{4});$ $(1,2^{2}),(1^{2},3),(1^{2},2^{2});$

  3. (iii)

$(1^{n-2},2)$ and $(1^{n-3},3)$ $(n>5)$ with $a_{i}=a_{j}$ when $\tau(i)=i$
and $\tau(j)=j$.

Proof. It follows from Theorem 3.7 and Theorem 4.3. $\Box$

Let $a\mu=c\tau\\#d\xi\in\mathbb{W}_{n}$ with $c\tau\in\mathbb{W}_{m}$,
$d\xi\in\mathbb{B}_{n-m}$ and $c\tau\bot d\xi$. Let
$\rho=\rho_{1}\otimes\rho_{2}\in\widehat{\mathbb{W}_{n}^{a\mu}}=\widehat{\mathbb{W}_{m}^{c\tau}}\times\widehat{\mathbb{W}_{n-m}^{d\xi}}$.
If $c=(1,1,\cdots,1)$, then
$\rho_{1}=(\chi_{1}\otimes\mu_{1})\uparrow_{G^{a\mu}_{\chi_{1}}}^{G^{a\mu}}$
with $\chi_{1}\in\widehat{(Z_{2}^{m})^{\tau}}$,
$\mu_{1}\in\widehat{(\mathbb{S}_{m}^{\tau})_{\chi_{1}}}$ (see [ZZ12, provious
Proposition 2.9 in Section 2.5] ), where $G=\mathbb{W}_{n}$. Case $a=0$ and
$a=(1,1,\cdots,1)$ were studied in paper [ZZ12, Theorem 1.1 and Table 1].
Other cases are listed as follows:

######  Corollary 4.5.

Under notation above assume
$\dim\mathfrak{B}({\mathcal{O}}_{a\mu}^{\mathbb{W}_{n}},\rho)<\infty$ with
$d_{1}=d_{2}=\cdots=d_{n-m}$ and $\xi={\rm id}^{\otimes(n-m)}$.

(i) Then $\rho_{1}(c\tau)=-id$ when $\rho_{2}(d\xi)={\rm id}$ and
$\rho_{1}(c\tau)={\rm id}$ when $\rho_{2}(d\xi)=-{\rm id}$.

(ii) Case $\tau=(1\ 2)$, $c=0$ and $d=(1,1,\cdots,1)$. Then
$\rho_{1}(c\tau)=\pm{\rm id}$, $\rho_{2}(d\xi)=\mp{\rm id}$, $\chi_{1}(c)=1$
and $\mu_{1}(\tau)=\pm 1.$

(iii) Case $\tau=(1\ 2)$, $c=(1,1)$ and $d=0$. Then $\rho_{1}(c\tau)=-{\rm
id}$, $\rho_{2}(d\xi)={\rm id}$.

(iv)  Case $\tau=(1\ 2\ 3)$, $c=0$ and $d=(1,1,\cdots,1)$. Then
$\rho_{1}(c\tau)={\rm id}$, $\rho_{2}(d\xi)=-id$, $\chi_{1}(c)=1$ and
$\mu_{1}(\tau)=1.$

(v) Case $\tau=(1\ 2\ 3)$, $c=(1,1,1)$ and $d=0$. Then $\rho_{1}(c\tau)=-id$,
$\rho_{2}(d\xi)={\rm id}$, $\chi_{1}(c)=-1$ and $\mu_{1}(\tau)=1.$

(vi) Case $\tau=(1\ 2)(3\ 4)$, $c=0$ and $d=(1,1)$. Then $\rho_{1}(c\tau)={\rm
id}$, $\rho_{2}(d\xi)=-id$.

(vii) Case $\tau=(1\ 2)(3\ 4)$, $c=(1,0,1,0)$ and $d=(1,1)$. Then
$\rho_{1}(c\tau)=\pm{\rm id}$, $\rho_{2}(d\xi)=\mp{\rm id}$.

(viii) Case $\tau=(1\ 2)(3\ 4)$, $c=(1,0,1,0)$ and $d=0$. Then
$\rho_{1}(c\tau)=-id$, $\rho_{2}(d\xi)={\rm id}$.

(ix) Case $\tau=(1\ 2)(3\ 4)$, $c=(1,0,0,0)$ and $d=(1,1)$. Then
$\rho_{1}(c\tau)=\pm{\rm id}$, $\rho_{2}(d\xi)=\mp{\rm id}$.

(x) Case $\tau=(1\ 2)(3\ 4)$, $c=(1,0,0,0)$ and $d=(0,0)$. Then
$\rho_{1}(c\tau)=-id$, $\rho_{2}(d\xi)={\rm id}$.

Proof. (iv) If $\rho_{2}(d\xi)={\rm id}$, then
$\dim\mathfrak{B}({\mathcal{O}}_{c\tau}^{\mathbb{W}_{3}},\rho_{1})<\infty$ by
Lemma 2.5, which constracts to [ZZ12, Theorem 1.1].

(v) If $\chi_{1}(c)=1$, then there exists a contradiction by [ZZ12,
Proposition 2.4, Theorem 1.1].

The others follow from Lemma 2.5 and [ZZ12, Theorem 1.1]. $\Box$

## Appendix

In this section we give the general FK’ Conjecture and another proof of
Theorem 4.3.

##  Appendix A General FK’ Conjecture.

$\mathfrak{B}({\mathcal{O}}_{\sigma},\rho)$ is finite dimensional when $n\leq
5$ according to [MS, FK99, AZ07]. However, it is open whether
$\mathfrak{B}({\mathcal{O}}_{\sigma},\rho)$ is finite dimensional when $n>5.$
We know the relationship between FK’ Conjecture and Nichols algebra
$\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes{\rm sgn})$ of
transposition over symmetry group(see [MS, Theorem 5.7]). That is, if
$\dim\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes{\rm sgn})=\infty$,
then so is $\dim\mathcal{E}_{n}$.

######  Definition A.1.

(See [FK99, Def. 2.1]) algebra ${\mathcal{E}}_{n}$ is generated by
$\\{x_{ij}\mid 1\leq i<j\leq n\\}$ with definition relations:

(i) $x_{ij}^{2}=0$ for $i<j.$

(ii) $x_{ij}x_{jk}=x_{jk}x_{ik}+x_{ik}x_{ij}$ and
$x_{jk}x_{ij}=x_{ik}x_{jk}+x_{ij}x_{ik},$ for $i<j<k.$

(iii) $x_{ij}x_{kl}=x_{kl}x_{ij}$ for any distinct $i,j,k,l$ and $l,$
$i<j,k<l.$ Equivalently, algebra ${\mathcal{E}}_{n}$ is generated by
$\\{x_{ij}\mid i\not=j,1\leq i,j\leq n\\}$ with definition relations:

(i) $x_{ij}^{2}=0$, $x_{ij}=-x_{ji}$;

(ii) $x_{ij}x_{jk}+x_{jk}x_{ki}+x_{ki}x_{ij}=0;$

(iii) $x_{ij}x_{kl}=x_{kl}x_{ij}$;

for any distinct $i,j,k$ and $l.$

S. Fomin and A.N. Kirillov [FK99, Conjecture 2.2] point out a conjecture “
${\mathcal{E}}_{n}$ is finite dimensional ”. Consequently, we have

######  Problem A.2.

Let $\alpha_{ijk},\beta_{ijk},\gamma_{ij},\lambda_{ijkl}\in\\{1,-1\\}$ for any
distinct $i,j,k$ and $l$ with $1\leq i,jk,l\leq n$. Assume that algebra
$A(\alpha,\beta,\gamma,\lambda)$ is generated by $\\{x_{ij}\mid i\not=j,1\leq
i,j\leq n\\}$ with definition relations:

(i) $x_{ij}^{2}=0$, $x_{ij}=\gamma_{ij}x_{ji}$;

(ii) $x_{ij}x_{jk}+\alpha_{ijk}x_{jk}x_{ki}+\beta_{ijk}x_{ki}x_{ij}=0;$

(iii) $x_{ij}x_{kl}=\lambda_{ijkl}x_{kl}x_{ij}$;

for any distinct $i,j,k$ and $l.$

Then $A(\alpha,\beta,\gamma,\lambda)$ is finite dimensional when $n>5$.

Obviously, $\mathcal{E}_{n}=A(1,1,-1,1),$ i.e.
$\mathcal{E}_{n}=A(\alpha,\beta,\gamma,\lambda)$ with
$\alpha_{i,j,k}=1,\beta_{i,j,k}=1,\gamma_{ij}=-1,\lambda_{i,j,k,l}=1$ for any
distinct $i,j,k$ and $l.$

##  Appendix B Proof of Theorem 4.3.

Proof. (i) There exists subrack $X\subseteq\mathcal{O}_{\sigma}^{G}$. $R\cup
S=X$ is a subrack decomposition of $X$there exists $a\in R$, $b\in S$ such
that ${\rm sq}(a,b):=a\rhd(b\rhd(a\rhd b))\not=b$. Let $H$ be the subgroup
generated by $X$ in $G$. It is clear
${\mathcal{O}}_{\sigma_{1}}^{H}\not={\mathcal{O}}_{\sigma_{2}}^{H}$ for any
$\sigma_{1}\in R,\sigma_{2}\in S.$ In fact, if
${\mathcal{O}}_{\sigma_{1}}^{H}={\mathcal{O}}_{\sigma_{2}}^{H}$, then there
exists $t\in H$, such that $\sigma_{2}=t\sigma_{1}t^{-1}$.
$t=\prod_{i=1}^{n}t_{i}$ whit $t_{i}\in X$ or $t_{i}^{-1}\in X.$ Therefore
$\sigma_{2}\in R$, which is a contradiction.

(ii) By (i), we have ${\mathcal{O}}_{a}^{H}\not={\mathcal{O}}_{b}^{H}$ and
they are not square commutative. By [HS08, Th. 8.6],
$\dim\mathfrak{B}(M({\mathcal{O}}_{a}^{H},\lambda)+M({\mathcal{O}}_{b}^{H}),\xi))=\infty,$
for any $\lambda\in\widehat{H^{a}}$, $\xi\in\widehat{H^{b}}$. Consequently,
dim $\mathfrak{B}(\mathcal{O}_{\sigma}^{G},\rho)=\infty$ for any
$\rho\in\widehat{G^{\sigma}}$ by [AFGV08, Lemma 3.2 ii]. $\Box$

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