# The Gaze and Mouse Signal as additional Source for User Fingerprints in
Browser Applications

Wolfgang Fuhl<EMAIL_ADDRESS>0000-0001-7128-298X ,  Nikolai
Iraj Sanamrad<EMAIL_ADDRESS>and  Enkelejda
Kasneci<EMAIL_ADDRESS>University TübingenSand
14TübingenBaden WürttembergGermany72076

(2018)

###### Abstract.

In this work we inspect different data sources for browser fingerprints. We
show which disadvantages and limitations browser statistics have and how this
can be avoided with other data sources. Since human visual behavior is a rich
source of information and also contains person specific information, it is a
valuable source for browser fingerprints. However, human gaze acquisition in
the browser also has disadvantages, such as inaccuracies via webcam and the
restriction that the user must first allow access to the camera. However, it
is also known that the mouse movements and the human gaze correlate and
therefore, the mouse movements can be used instead of the gaze signal. In our
evaluation we show the influence of all possible combinations of the three
information sources for user recognition and describe our simple approach in
detail. The data and the Matlab code can be downloaded here
https://atreus.informatik.uni-
tuebingen.de/seafile/d/8e2ab8c3fdd444e1a135/?p=%2FThe%20Gaze%20and%20Mouse%20Signal%20as%20additional%20Source%20...&mode=list.

Browser Fingerprint, Fingerprint, Browser Watermarking, Watermarking, Mouse,
Gaze, Eye Tracking, Person Reidentification

††copyright: acmcopyright††journalyear: 2018††doi:
10.1145/1122445.1122456††conference: ETRA 2021: ACM Symposium on Eye Tracking
Research & Applications; May 25-29, 2021; Stuttgart, Germany††booktitle: ETRA
2021: ACM Symposium on Eye Tracking Research & Applications, May 25-29, 2021,
Stuttgart, Germany††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Security
and privacy Browser security††ccs: General and reference Empirical
studies††ccs: Information systems Personalization††ccs: Computing
methodologies Cross-validation Figure 1. The data sources for browser
fingerprinting.

##  1\. Introduction

User identification plays a crucial role in many industrial sectors in which
eye tracking (Fuhl, 2019) did not play a big role so far. In its original form
it is used to protect data and access to networks or premises (Lee et al.,
2010; Choubey and Choubey, 2013; Fuhl et al., 2020a). Today, there is a
growing need for user identification, especially in the online environment,
which includes both personalized advertising (Tucker, 2014) and product
placement (Shamdasani et al., 2001; Fossen and Schweidel, 2019), but also
online banking (Lee et al., 2010) or external access to corporate networks
(Cole, 2011). For security-critical applications such as external access to
company networks or online banking, user IDs and passwords have become widely
accepted. When using security-critical functionalities, additional security
prompts such as a generated PIN or SMS prompts are added. In online
advertising, as well as product placement, companies try to identify a person
without accessing critical personal data. This is guaranteed in the modern
world by so-called cookies (Juels et al., 2006) since those have to be
activated by the user, stateless approches only use browser statistics (Juels
et al., 2006). A disadvantage of this method is that the statistics can be
used to identify a computer very effectively, but in the case of computers
with multiple users all of them are treated as the same person. For the
password and user recognition procedure there are also disadvantages. For
example if the identification and password is known by an attacker, the
attacker can gain access.

In this work we analyze new data sources, like the eye signal and mouse
movements. The basic idea is that a person can be identified by means of gaze
signals (Fuhl et al., 2020b) or human visual behavior. This has been shown
several times (Holland and Komogortsev, 2011; Fuhl et al., 2019a; Fuhl et al.,
2020a). Since the gaze signal can only be computed with a webcam in a browser,
it requires the user to activate and allow the acces to the webcam.
Additionally, the quality of the camera as well as the different lighting
conditions influence the accuracy of the gaze signal (Papoutsaki et al.,
2016). Further scientific work has already been done on the correlation of
mouse movements and the human eye signal (Liebling and Dumais, 2014; Guo and
Agichtein, 2010; Navalpakkam et al., 2013). It has been found that when
clicking or the end point of a mouse movement almost always corresponds to the
eye position (Liebling and Dumais, 2014; Guo and Agichtein, 2010; Navalpakkam
et al., 2013). With this information, the technique of webcam based eye
tracking has changed in that the mouse information is used to calibrate the
eye tracker (Papoutsaki et al., 2016). Another advantage of mouse movements is
that this information is freely accessible in the browser and does not have to
be activated manually by the user like the camera. We show in this thesis that
the mouse information is sufficient to identify a user, which is also
scientifically based on the fact that visual behavior is user-specific
(Holland and Komogortsev, 2011; Fuhl et al., 2019a; Fuhl et al., 2020a) and
that mouse movements in the browser correlate with visual behavior (Liebling
and Dumais, 2014; Guo and Agichtein, 2010; Navalpakkam et al., 2013).

The application of these data sources in the industrial environment is
enormous. For example, it enables continuous user validation for online
banking and external access to corporate networks. It would not be enough to
have only the password and the user ID, one would also have to be able to
emulate the behavior of the correct human user. For user-specific advertising
and product placement, it is also possible to differentiate between users on a
shared computer and identification of the same user on different computers.

  1. (1)

First work dealing with gaze behavior and mouse movements for a browser based
fingerprint. Since browser statistics make it impossible to distinguish
between two different users on the same PC, new data sources are needed. We
show that this is possible with the visual behavior as well as with the mouse
movements.

  2. (2)

Using mouse movements as fingerprints could prove to be very interesting in
practice, because this information can be easily retrieved. We show how much
information this data source provides and compare it with the webcam based
gaze signal and browser statistics. We also show the effectiveness of using
combined data sources.

  3. (3)

We compare the traditional browser statistics with new data sources such as
the gaze behavior and mouse movements of the user. Since it has already been
shown that human visual behavior can be effectively used to identify
individuals (Holland and Komogortsev, 2011; Fuhl et al., 2019a; Fuhl et al.,
2020a) and mouse movements correlate with visual behavior (Liebling and
Dumais, 2014; Guo and Agichtein, 2010; Navalpakkam et al., 2013). Thus it is
logical and justifiable why mouse movements can also be effectively used as
fingerprints.

  4. (4)

In our evaluation we only use resource-saving machine learning methods (Fuhl
et al., 2020d; Fuhl and Kasneci, 2020a, c, b; Fuhl, 2020) without validation
methods (Fuhl and Kasneci, 2019; Fuhl et al., 2020f) that can be executed even
inexpensively in the browser of the user. This extends the field of
application of the used data sources as well as the presented simple approach
for direct use.

##  2\. Related work

In this section, we discuss the state of the art regarding browser-based user
identification. The first work which has dealt with browser-based user
identification is (Mayer, 2009). It analyzed and used statistics about the
browser configuration, version and installed extensions. In (Eckersley, 2010)
the approach was proven in a larger study and thus it was shown that the
digital fingerprint can be effectively used for user identification via
statistics. Further studies (Alaca and Van Oorschot, 2016; Englehardt and
Narayanan, 2016; Laperdrix et al., 2020; Kobusińska et al., 2018) dealt with
extensions of the statistical features and their quality for user recognition.
In (Alaca and Van Oorschot, 2016; Kobusińska et al., 2018) an analysis of the
stability of the individual characteristics was also carried out. (Laperdrix
et al., 2020; Kobusińska et al., 2018) examined different browsers and also
analyzed security settings that can prevent fingerprinting. There was also a
long term study which dealt with the creation of a unique fingerprint over
years (Gómez-Boix et al., 2018). Cross-browser fingerprinting was covered in
(Cao et al., 2017) using both operating system and hardware features. A
further extension of these approaches is the use of hashing algorithms to make
the calculation and identification more effective (Gabryel et al., 2020).

Applications for browser-based fingerprints are described in the literature as
user tracking (Eckersley, 2010; Englehardt and Narayanan, 2016), abuse
prevention (Vastel et al., 2020), and authentication (Alaca and Van Oorschot,
2016) in many different contexts. For example, security companies use the
fingerprint to detect bots or abnormal behavior on web pages (mis, 2020b, a).
In (Vastel et al., 2020) it is also shown that fingerprinting can be used to
easily block scripts that collect data from web pages, but the authors also
show that this protection can be easily circumvented. A fingerprint for mobile
devices, which was calculated on all hardware and installed software, is
described in (Bursztein et al., 2016). This makes it possible to distinguish
between the real device and a simulated environment of the same device and
thus block network traffic in case of a simulated device.

Literature that deals with abuse prevention is mostly in the context of
advertising or e-commerce. This concerns click fraud or credit card payments.
Two new inference techniques were presented in (Nagaraja and Shah, 2019). The
first technique recognizes click patterns within an advertising network and
thus can prevent click fraud. In the second technique, bait clicks are
injected and resulting conspicuous patterns are detected. (Renjith, 2018)
deals with credit card fraud. Here, cheap goods or services are sought that
have never been shipped or performed. The authors use different features and
machine learning algorithms to detect this type of fraud.

There is also already some work in the field of deep neural networks for fraud
detection. In (Zhang et al., 2019) a deep neural network was presented, which
analyses the data for similar behaviour. This allows fraud cases, which are
repeated and follow the same procedures, to be detected and traced. This
technique also helps to protect against known fraud, because the behavior is
conspicuous for the system. Several interconnected neural networks have also
been used to detect intrusion into computer networks (Ludwig, 2019). Here
several deep neural networks are used to monitor network communication. These
networks detect patterns in the communication which do not correspond to the
norm and warn early in case of a possible intrusion. Deep Boltzmann machines
were used for fraud detection in biometric systems (de Souza et al., 2019).
For this purpose, features from the deep layers of the network were used, as
these have proven to be more robust against attempts of fraud. Another use of
deep neural networks for fingerprint calculation is described in
(Salakhutdinov and Hinton, 2009). Here, auto-encoders are used to calculate a
hash of a document. Similar documents produce a similar hash. This technique
can also be applied to browser statistics to obtain a fingerprint of a user.

While click patterns(Nagaraja and Shah, 2019) or drawing symbols (Syukri et
al., 1998) have already been used to calculate a fingerprint, we are not aware
of any work that deals with mouse movement behavior for the recognition of
persons via the web browser. In addition, we use the gaze behavior or the scan
path (Fuhl et al., 2019a, b) of the test person, which to our knowledge has
also not been used for the generation of fingerprints until today. In our
work, we compare mouse movements and gaze behavior with conventional browser
statistics and also perform analyses regarding combinatorial approaches.

##  3\. Method & Data recording

Figure 2. The initial calibration process. The dots disappear after the
subject clicked them with the mouse.

The data was recorded on two PCs with two different browsers each. On each PC,
six recordings per person were made, each of which had a minimum length of
five minutes and could be of any length. On each computer there were two
browsers, each of which was used for three recordings. The choice of websites
was limited to six and selected with buttons on the top and bottom of the web
page. In addition all sub-pages could be reached as well as other sites could
be visited through internal links. This limitation was due to the need of an
iframe which allowed us to keep the recording software running constantly. We
have also chosen to use a fixed selection of pages that are the same for all
subjects, as it is guaranteed that the subjects receive the same stimulus.
This reduces the influence of completely different pages on the gaze and mouse
data. Before each recording the eye tracker software WebGazer (Papoutsaki et
al., 2016) was calibrated. Figure 2 shows our calibration screen. During the
calibration the subject had to gaze at each calibration point and click on
each point two times. This information was given to WebGazer (Papoutsaki et
al., 2016) as calibration coordinates. After the calibration the recording
started. In total, six people performed the study, which brings the total
number of images to 72 ($2browser*2computers*3images*6people=$72).

The collected data is the eye signal encoded as heatmap, the mouse movements
as heatmap and the browser statistics. For the heatmap we have quantized the
data into a $10\times$10 grid, which is valid for both the eye movement (Fuhl
et al., 2020e; Fuhl et al., 2018h; Fuhl et al., 2018a, b; Fuhl and Kasneci,
2018) and the mouse movement. In addition we normalized the sum of the heatmap
to one. The collected browser statistics are standard values like webdriver,
webgl, header, language, device meomory, etc. according to the FingerprintJS
(mis, 2020c). To store the data online we used a local Apache server
(Wolfgarten, 2004) with a MySQL database (Greenspan and Bulger, 2001), to
which the data was sent via Ajax (Garrett et al., 2005) using Javascript
(Goodman, 2007).

Figure 3. The gaze, mouse, and absolute difference for three users. Each row
corresponds to one subject.

Figure 3 shows the normalized gaze and mouse movement data. Each row
corresponds to a separate image. Comparing the mouse and gaze data, a clear
difference can be seen where both signals have the main focus relatively
central. Since we used an iframe for our recordings, we could not use tabs in
the browser. Scrolling was also done mainly over the mouse wheel and the
scrollbar was a bit inside the screen. In the third column in Figure 3, which
represents the absolute difference, it can be seen that the signals are
clearly different. Nevertheless the signals correlate with each other, which
is of course also due to the fact that the heatmap is a quantization and is
invariant to time.

Figure 4. Correlation between the mouse and the gaze data over all samples as
whisker plot.

Figure 4 shows the distribution as a whisker plot of the correlation
coefficient over all recorded data between the gaze and mouse signal. The blue
box represents the 75% confidence interval and the red line represents the
median. Since all values of the correlation coefficient are above 0.5, it can
be assumed that there is a very strong relation between the gaze signal and
the mouse signal in our recordings. As already mentioned, this is reinforced
by the heatmap quantization, the normalization, and the removal of the time
dependency in the heatmap.

##  4\. Evaluation

Figure 5. The confusion matrices with the browser statistics as well as for
the gaze and mouse data as input separately. On the bottom right of each
confusion matrix, the overall accuracy can be seen.

For all our evaluations, we performed a 50% to 50% split between training and
validation data. We made sure that every user is present in the training data
as well as every computer and browser. This has been done because it is
necessary to have seen the user at least once to recognize him. To make the
comparison to the browser statistics fair, we made sure that every computer
and browser is in the training data as well. As machine learning method we
have chosen bagged decision trees with the standard parameters of Matlab
2020b. The only setting we have made is the number of trees to be trained,
which we have set to 50 for all evaluations. We did not perform any data
augmentation or other preprocessing. We have chosen this approach because it
is the easiest to reproduce. In addition, this work is not about the best
possible results but about the proof of concept of using the mouse as well as
gaze data to create a digital fingerprint. The script and data can be viewed
in the supplementary material and tested together with Matlab.

Figure 5 shows the results for the use of the individual data sources (browser
statistics, mouse, and gaze) separately. The results are displayed as a
confusion matrix to view each class separately. Each confusion matrix also has
the overall accuracy in the lower right corner. The matrix on the left side
was only evaluated with the browser statistics as input. The overall accuracy
is exactly at the chance level (16.66%). This shows that the browser
statistics, in the case of computers with multiple users, cannot be used
effectively to distinguish between the different users. This is of course
logical, because for example two users on the same computer in the same
browser have the same statistical values. The middle confusion matrix in
Figure 5 shows the results achieved with the mouse heatmap. The overall
accuracy of 69.4% is significantly above the chance level of 16.66%. Thus it
is clear that the mouse data contains information about the user. Furthermore,
this data can be used to distinguish between users who have used the same
computer and the same browser. The right matrix in Figure 5 shows the results
of the gaze data. These results exceed the results of the mouse data with
80.6% accuracy. This also means that the gaze data can be used to
differentiate between users on the same computer and this even better than any
other data source.

Figure 6. The confusion matrices for the combinational evaluations. The left
most confusion matrix is the browser and mouse data as input. In the central
plot the results for the browser and gaze data as input are shown. For the
right confusion matrix we combined the gaze and mouse heatmap. On the bottom
right of each confusion matrix, the overall accuracy can be seen.

Figure 6 shows the results for the combinatorial use of the individual data
(browser statistics, mouse, and gaze). Like the individual results, the
combinatorial evaluations are also displayed as a confusion matrix. The left
matrix shows the results of the combination of browser statistics and the
mouse heatmap. This combination is slightly worse than using the mouse heatmap
alone (69.4 to 63.9%). This is mainly due to the fact that in our study the
users are equally distributed over all computers and browsers. Normally, that
is, that every user has his own computer, except for a few, the browser
statistics alone would be very effective and the combination of mouse and
browser statistics would be much better. In the middle matrix of Figure 6 the
results of the combination of gaze and browser statistics are shown. Here a
slight improvement can be seen (80.6% to 83.3%). This is not very much but the
same applies as for the combination of mouse and browser statistics. Usually
the browser statistics is very effective, because most of the users have their
own computer, so this combination would be much more effective. The last and
right matrix in Figure 6 shows the results of the mouse and gaze heatmaps.
Here is only a small improvement to the gaze heatmaps alone (2.7%). One reason
for this is that the two heatmaps have a very similar content and therefore
correlate strongly (Figure 4). In addition, larger data volumes of
significantly more than six users would have to be included in order to be
able to finally evaluate this. In general it can be assumed that this
combination is not very effective.

Figure 7. The confusion matrix with the combination of all data sources which
are the browser statistics, the gaze heatmap, and the mouse heatmap. On the
bottom right, the overall accuracy can be seen.

Figure 7 shows the results for the combinatorial use of all data together
(browser statistics, mouse, and gaze). As in all previous evaluations, we have
used a confusion matrix. The overall result of all data as input is as good as
the result when using the gaze data alone (80.6 to 80.6). The individual
values of the correct and incorrect classifications in the matrix differ
slightly but the overall result shows no improvement. Of course, as for all
combinatorial analysis with the browser data, the browser data usually works
very well and the result is certainly much better, if there are only a few
users which share a computer. Also the combination of the gaze and mouse
heatmap may not be optimal, because these data correlate too strongly.
However, the gaze heatmap can be replaced with the mouse heatmap. The mouse
data have the clear advantage that they can always be retrieved and do not
require calibration.

##  5\. Conclusion

In this work we have done a small study and analyzed both gaze and mouse data
to use them for a digital fingerprint. In our evaluation it is clearly shown
that these signals can be used individually as well as in combination for
fingerprinting. It also shows that in the case of computers used by multiple
users, the browser statistic fails and can no longer distinguish the persons.
With our data we can confirm, as in previous work, that the gaze and mouse
signals are dependent on each other.

The gaze signal scored best in our evaluation to distinguish the users, but it
is hardly usable in practice. This is due to the fact that the camera has to
be activated by the user and the eye tracker software has to be calibrated in
an initial step. This would only be possible in security-critical applications
such as online banking, where the customer is willing to protect his account.

The mouse signal is a signal that can always be queried without the user
having to allow it. Also, no calibration is necessary and the signal itself is
very precise under all external conditions. Since this signal can also be
effectively used for digital fingerprinting and works well in the case of
multiple users on the same computer, the authors believe that it is highly
interesting for real-world applications. The mouse signal can of course be
used for many applications in which a person should be recognized. One example
is an additional security layer for online banking. If a user’s mouse movement
data were additionally analyzed and compared by the bank, the misuse of a
password and user ID that has become public could be prevented.

A limitation of this work is the small number of test persons in the study.
This is due to the current Corona regulations. Therefore, we can only prove a
possible effectiveness of the mouse as well as the eye signal, so this work
can be considered as proof of concept. Future research in this area should in
a first step create a larger study in which significantly more test persons
and devices are used.

Further planned improvements are integrating modern pupil detektion algorithms
(Fuhl et al., 2015b; Fuhl et al., 2016d, f, 2017e, c, a; Fuhl et al., 2016e,
2017a, 2018e; Fuhl et al., 2020c, 2017c; Fuhl et al., 2018d), extracting the
information of the eyelids (Fuhl et al., 2016b, 2017d; Fuhl et al., 2017b,
2019c, 2019d; Fuhl et al., 2018c), and using saliency metrics too (Fuhl et
al., 2018g; Geisler et al., 2017). We also want to improve the approach using
different visualization techniques for debugging as well as improving the grid
based heatmap generation with dynamic AOIs (Kübler et al., 2015; Fuhl et al.,
2018f, 2015a).

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A well-known representation-theoretic model for the transformed Macdonald polynomial $\Htil_\mu(Z;t,q)$, where $\mu$ is an integer partition, is given by the Garsia-Haiman module $\Hcal_\mu$. We study the $\frac{n!}{k}$ conjecture of Bergeron and Garsia, which concerns the behavior of certain $k$-tuples of Garsia-Haiman modules under intersection. In the special case that $\mu$ has hook shape, we use a basis for $\Hcal_\mu$ due to Adin, Remmel, and Roichman to resolve the $\frac{n!}{2}$ conjecture by constructing an explicit basis for the intersection of two Garsia-Haiman modules.

§ INTRODUCTION

The Macdonald polynomials $\Htil_\mu(Z;t,q)$ have become a central object of study in algebra, combinatorics, and geometry since their introduction by Macdonald in [15]. Of particular import is the expansion of $\Htil_\mu(Z;t,q)$ in the Schur basis:

\[
\Htil_\mu(Z;t,q) = \sum_\lambda \Ktil_{\lambda \mu}(t,q) s_\lambda(Z).
\]

The (since resolved) Macdonald positivity conjecture asserts that $\Ktil_{\lambda \mu}(t,q)$ is a polynomial in $t$ and $q$ with nonnegative integer coefficients. This conjecture suggests that $\Htil_\mu(Z;t,q)$ should have a representation-theoretic interpretation as the character of a doubly graded symmetric group module, and indeed such an interpretation is given by the Garsia-Haiman module $\Hcal_\mu$:

The transformed Macdonald polynomial $\Htil_\mu(Z;t,q)$ equals the bigraded Frobenius series of the Garsia-Haiman module $\Hcal_\mu$.

In fact, Garsia and Haiman were able to show that in order to establish this connection, it would be enough to prove that $\dim(\Hcal_\mu) = n!$. Thus the problem was reduced to this $n!$ conjecture, which was later resolved by Haiman ([14]).

This paper studies the related $\frac{n!}{k}$ conjecture of Bergeron and Garsia, which makes a similar assertion about the dimension of the intersection of certain $k$-tuples of Garsia-Haiman modules:

Let $\lambda$ be an integer partition of $n+1$, and let $\mu^{(1)}, \ldots, \mu^{(k)}$ be partitions of $n$ each obtained from $\lambda$ by removing a removable cell from the Young diagram of $\lambda$. Then,

\[
\dim\left( \bigcap_{i=1}^k \Hcal_{\mu^{(i)}} \right) = \frac{n!}{k}.
\]

We resolve the $\frac{n!}{k}$ conjecture in the special case that $\lambda$ has hook shape — the only nontrivial case is $k=2$ since a hook has at most two removable cells, hence the title. The rest of the paper is organized as follows: after outlining some necessary prerequisites in Section <ref>, we introduce the Garsia-Haiman module $\Hcal_\mu$ in Section <ref>. In particular, we define a basis for $\Hcal_\mu$ when $\mu$ has hook shape; each basis element is indexed by a standard filling of $\mu$, and is given by a monomial which encodes particular inversions in the standard filling. This basis is simply a restatement of the $k$-th Artin basis of [1], using the language of standard fillings rather than permutations.

Equipped with this basis, we proceed in Section <ref> to construct an explicit basis for the intersection of two Garsia-Haiman modules with hook shape by defining a bijective map between particular subsets of standard fillings which leaves the corresponding basis element unchanged. Our bijection makes use of two “Foata-like" maps which preserve particular inversions in the first row and column of a standard filling of hook shape.

§ MACDONALD POLYNOMIALS

We begin by cataloguing some necessary terminology. An integer partition of $n \in \NN$ is a nonincreasing sequence of positive integers $\mu = (\mu_1 \ge \mu_2 \ge \cdots \ge \mu_k)$ such that $\sum_i \mu_i = n$, abbreviated $\mu \vdash n$. We say that an integer partition $\mu$ has hook shape if $\mu = (a, 1, \ldots, 1)$ for some $a \ge 1$. The Young diagram of $\mu$ a subset of $\NN \times \NN$ with $\mu_i$ left-justified cells in row $i$. We opt for French (coordinate) notation, so that the partition $\mu = (4,3,3)$ has Young diagram

\[
\ytableausetup{aligntableaux=bottom, boxsize=1.2em}
\begin{ytableau}
~ & ~ & ~ \\
~ & ~ & ~ \\
~ & ~ & ~ & ~
\end{ytableau}.
\]

By a slight abuse of notation we identify $\mu$ with its Young diagram; e.g. when we refer to “the cells of $\mu$" we really mean “the cells of the Young diagram of $\mu$".

The length of a partition $\mu$, denoted $\ell(\mu)$, is the number of nonzero rows of $\mu$, and the conjugate of $\mu$, denoted $\mu'$, is the partition obtained from $\mu$ by transposing rows and columns. We say a cell $c$ in $\mu$ is removable if there are no cells immediately above or directly right of $c$; in other words, $c$ is removable if the diagram $\mu - \{ c \}$ still has partition shape. For instance, the cells at the ends of rows $1$ and $3$ in the above diagram are removable, but the cell at the end of row $2$ is not.

A filling of $\mu \vdash n$ is a map $S : \mu \to \NN$ — i.e. an assignment of positive integers to the cells of $\mu$ — and a standard filling of $\mu$ is a bijective map $S : \mu \to \{ 1, 2, \ldots, n \}$. For instance, the six standard fillings of $\mu = (2,1)$ are:

\[
\ytableausetup{aligntableaux=bottom, boxsize=1.2em}
\begin{array}{cccccc}
\begin{ytableau}
3 \\
1 & 2
\end{ytableau},
\begin{ytableau}
2 \\
1 & 3
\end{ytableau},
\begin{ytableau}
3 \\
2 & 1
\end{ytableau},
\begin{ytableau}
1 \\
2 & 3
\end{ytableau},
\begin{ytableau}
2 \\
3 & 1
\end{ytableau},
\begin{ytableau}
1 \\
3 & 2
\end{ytableau}.
\end{array}
\]

For a filling $S$, we let $S_{i,j}$ denote the entry in row $i$, column $j$ of $S$.

Let $Z = z_1, z_2, \ldots$ denote a countably infinite set of variables. The symmetric group $\Sym_n$ is the group of permutations of $[n] = \{ 1, 2, \ldots, n \}$, and we say that a function $f \in \QQ[Z]$ is symmetric if it is unchanged under the $\Sym_n$-action given by permuting indices, for any $n \in \NN$. Let $\Lambda \subset \QQ[Z]$ denote the subring of symmetric functions. Many bases for $\Lambda$ are known, but perhaps the most famous (and most useful from a representation-theoretic point of view) is the basis of Schur functions $\{ s_\mu \}$, where $\mu$ ranges over the collection of all integer partitions.

Macdonald ([15]) defined an exceptional new basis for the ring $\Lambda_{(t,q)}$ of symmetric functions with coefficients in the field $\QQ(t.q)$ which simultaneously generalizes the Schur functions, Hall-Littlewood polynomials, and Jack symmetric functions, among others. Macdonald's polynomials were not given by an explicit formula — rather, they were characterized by particular triangularity and orthogonality relations with respect to a $(t,q)$-deformation of the Hall inner product — but a certain transformed version $\Htil_\mu(Z;t,q)$ of the Macdonald polynomial has an elegant combinatorial description, proved in [12]. For any integer partition $\mu$, $\Htil_\mu(Z;t,q)$ may be described as a $(t,q)$-weighted sum over fillings of $\mu$, where the $t$- and $q$-weights are given by combinatorial statistics on fillings which were originally defined by Haglund ([11]). See [11] or [12] for the precise definition of these statistics.

Moreover, $\Htil_\mu(Z;t,q)$ has an elegant expansion in the Schur basis:

\begin{equation}\label{eq:schur-exp}
\Htil_\mu(Z;t,q) = \sum_\lambda \Ktil_{\lambda \mu}(t,q) s_\lambda(Z).
\end{equation}

The coefficients $\Ktil_{\lambda \mu}(t,q)$ are a priori rational functions in $t$ and $q$, but Macdonald conjectured that $\Ktil_{\lambda \mu}(t,q) \in \NN[t,q]$; this Macdonald positivity conjecture was resolved in [14], by realizing $\Ktil_{\lambda \mu}(t,q)$ as the doubly graded character multiplicities of a certain doubly graded $\Sym_n$-module, which we describe in the following section.

§ GARSIA-HAIMAN MODULES

Garsia and Haiman ([9]) proposed the following representation-theoretic interpretation for $\Htil_\mu(Z;t,q)$. Consider the polynomial ring $\QQ[X,Y] = \QQ[x_1, \ldots, x_n ; y_1, \ldots, y_n]$ along with the diagonal $\Sym_n$-action given by permuting the $X$- and $Y$-variables simultaneously and identically; that is,

\[
\sigma \cdot f(x_1, \ldots, x_n ; y_1, \ldots, y_n) = f(x_{\sigma(1)}, \ldots, x_{\sigma(n)} ; y_{\sigma(1)}, \ldots, y_{\sigma(n)})
\]

for any $f(X,Y) \in \QQ[X,Y]$ and $\sigma \in \Sym_n$. For $\mu \vdash n$, define the polynomial

\[
\Delta_\mu = \det(x_i^{p_j-1}y_i^{q_j-1})_{i,j=1}^n \in \QQ[X,Y],
\]

where $\{ (p_1,q_1), \ldots, (p_n, q_n) \}$ encode the coordinates of the cells of $\mu$, taken in any order. For instance, we label the cells of $\mu = (3,2)$ as follows:

\[
\ytableausetup{aligntableaux=bottom, boxsize=2em}
\begin{ytableau}
\scriptstyle{(2,1)} & \scriptstyle{(2,2)} \\
\scriptstyle{(1,1)} & \scriptstyle{(1,2)} & \scriptstyle{(1,3)}
\end{ytableau}.
\]

Then $\Delta_\mu \neq 0$ since the above biexponents are distinct, and $\Delta_\mu$ is an $\Sym_n$-alternating polynomial which is doubly homogeneous of $X$-degree $b(\mu) = \sum_{i=1}^{\ell(\mu)} (i-1) \mu_i$ and of $Y$-degree $b(\mu') = \sum_{j=1}^{\ell(\mu')} (j-1) \mu'_j$. Let

\[
\ii_\mu = \{ f \in \QQ[X,Y] : f(\tfrac{\partial}{\partial x_1}, \ldots, \tfrac{\partial}{\partial x_n}; \tfrac{\partial}{\partial y_1}, \ldots, \tfrac{\partial}{\partial y_n}) \Delta_\mu = 0 \}
\]

denote the ideal of polynomials whose corresponding differential operator annihilates $\Delta_\mu$. Note that, since $\Delta_\mu$ is doubly homogeneous and $\Sym_n$-alternating, $\ii_\mu$ is an $\Sym_n$-invariant, doubly homogeneous ideal.

The Garsia-Haiman module is the quotient ring

\[
\Hcal_\mu = \QQ[X,Y] / \ii_\mu
\]

viewed as a doubly graded $\Sym_n$-module, with the $\Sym_n$-action induced by the diagonal action on $\QQ[X,Y]$.

Since $\Hcal_\mu$ is doubly graded, we have a direct sum decomposition

\[
\Hcal_\mu = \bigoplus_{i=1}^{b(\mu)} \bigoplus_{j=1}^{b(\mu')} (\Hcal_\mu)_{i,j},
\]

where $(\Hcal_\mu)_{i,j}$ denotes the submodule spanned by elements of total $X$-degree $i$ and total $Y$-degree $j$.

Garsia and Haiman conjectured that the Frobenius series of $\Hcal_\mu$ (defined below) is precisely $\Htil_\mu(Z;t,q)$. Due to known identities involving the usual Kostka numbers $K_{\lambda \mu}$, it is necessary that $\Hcal_\mu$ afford a doubly graded version of the regular representation of $\Sym_n$, and in particular that the dimension of $\Hcal_\mu$ is $n!$, for this identity to hold. Strikingly, this apparently weaker condition is sufficient to establish the desired result:

If the dimension of $\Hcal_\mu$ is $n!$, then the bigraded Frobenius series of $\Hcal_\mu$ given by

\[
Frob_{\Hcal_\mu}(Z;t,q) = \sum_{i,j} t^iq^j ch((\Hcal_\mu)_{i,j}),
\]

where $ch$ is the map which sends the irreducible $\Sym_n$-representation $S^\lambda$ to the Schur function $s_\lambda$, equals the transformed Macdonald polynomial $\Htil_\mu(Z;t,q)$.

Thus the problem was reduced to this comparatively simpler $n!$ conjecture, which was later resolved by Haiman:

The dimension of $\Hcal_\mu$ is $n!$.

This in turn proves the Macdonald positivity conjecture, since it realizes $\Ktil_{\lambda \mu}(t,q)$ as the doubly graded multiplicity of $S^\lambda$ in $\Hcal_\mu$. The proof of the $n!$ conjecture in [14], however, relies on deep algebro-geometric results, and it remains an open problem to construct an explicit basis for $\Hcal_\mu$ in general.

Many bases for $\Hcal_\mu$ are known when $\mu$ has hook shape, however (see [1],[2],[3],[4],[10],[16]). We opt to work with the $k$-th Artin basis of [1], which was also known to Garsia and Haiman in <cit.>. Using the following terminology, we reframe this basis in a form which is more amenable to our later constructions.

Let $SF(\mu)$ denote the set of standard fillings of $\mu$. Given $S \in SF(a,1^\ell)$, define a row inversion to be a pair $S_{1,i} > S_{1,j}$ where $i < j$. Similarly, define a column inversion to be a pair $S_{j,1} > S_{i,1}$ for $i < j$ (note the swapped indices). Denote the collection of row (resp. column) inversions in $S$ by

\begin{align*}
\mathsf{rowInv}(S) &= \{ (t,r) : t > r, t \text{ left of } r \text{ in row $1$ of } S \}, \\
\mathsf{colInv}(S) &= \{ (d,c) : d > c, d \text{ above } c \text{ in column $1$ of } S \}.
\end{align*}

Given $S \in SF(a,1^\ell)$, define the polynomial $\varphi_S(X,Y) \in \QQ[X,Y]$ by

\[
\varphi_S(X,Y) = \prod_{(d,c) \in \mathsf{colInv}(S)} x_d \prod_{(t,r) \in \mathsf{rowInv}(S)} y_r
\]

(cf. [1], Definition 1.6).

The following standard filling of $\mu = (5,1^4)$:

\[
S =
\ytableausetup{centertableaux, boxsize = 1.2em}
\begin{ytableau}
4 \\
1 \\
9 \\
7 \\
5 & 6 & 3 & 2 & 8
\end{ytableau}
\]

\begin{align*}
\mathsf{rowInv}(S) &= \{ (5,\mathbf{3}), (5,\mathbf{2}), (6,\mathbf{3}), (6,\mathbf{2}), (3,\mathbf{2}) \}, \\
\mathsf{colInv}(S) &= \{ (\mathbf{4},1), (\mathbf{9},7), (\mathbf{9},5), (\mathbf{7},5) \},
\end{align*}

so $\varphi_S(X,Y) = x_4x_7x_9^2y_2^3y_3^2$.

The set

\[
\{ \varphi_S(X,Y) : S \in SF(a,1^\ell) \}
\]

forms a basis for the Garsia-Haiman module $\Hcal_{(a,1^\ell)}$.

The theory of orbit harmonics developed by Garsia and Haiman ([8]) lays the groundwork for a new, direct proof that $\{ \varphi_S \}$ forms a basis for $\Hcal_\mu$, which we sketch here. Let $\alpha_1, \ldots, \alpha_n, \beta_1, \ldots, \beta_n$ be distinct rational numbers. To any standard filling $S$ we associate an orbit point $p_S = (\alpha_{i_1}, \ldots, \alpha_{i_n}; \beta_{j_1}, \ldots, \beta_{j_n})$, and a polynomial

\[
\psi_S = \prod_{(d,c) \in \mathsf{colInv}(S)} (x_d - \alpha_{\mathsf{row}(c)}) \prod_{(t,r) \in \mathsf{rowInv}(S)} (y_r - \beta_{\mathsf{col}(t)}),
\]

so that $\varphi_S$ is the leading term of $\psi_S$. Garsia and Haiman proved that if

* the matrix $(\psi_S(p_T))$ is nonsingular, and

* the Hilbert series of $\{ \psi_S \}$ is symmetric,

then the leading terms $\{ \varphi_S \}$ form a basis for $\Hcal_\mu$. There exists a total ordering on standard fillings under which $(\psi_S(p_T))$ is upper-triangular with nonzero diagonal entries (hence nonsingular), and the symmetry of the Hilbert series is evidenced by the fact that $\varphi_S(t,q) = t^{n(\mu)}q^{n(\mu')} \varphi_{\widetilde{S}}(t^{-1},q^{-1})$, where $\widetilde{S}$ is obtained from $S$ by replacing the entry $i$ with $n-i+1$.

§ A PROOF OF THE $\FRAC{N!}{2}$ CONJECTURE FOR HOOK SHAPES

Now equipped with an explicit basis for $\Hcal_{(a,1^\ell)}$, we can study the remarkable intersection properties of these modules which were conjectured in [5]. In full generality, the $\frac{n!}{k}$ conjecture is:

Let $\lambda \vdash n+1$, and let $\mu^{(1)}, \ldots, \mu^{(k)}$ be partitions of $n$ each be obtained from $\lambda$ by removing a removable cell from the Young diagram of $\lambda$. Then,

\[
\dim\left( \bigcap_{i=1}^k \Hcal_{\mu^{(i)}} \right) = \frac{n!}{k}.
\]

We prove the $\frac{n!}{k}$ conjecture when $\lambda$ has hook shape; in particular, the only nontrivial case is $k=2$, since a hook shape only has two removable cells. These ideas are also pursued in [6], where the authors obtain a plethystic formula for the Frobenius series of the intersection of two Garsia-Haiman modules with hook shape. For the rest of the paper, assume $\lambda = (a,1^\ell)$ is a partition of $n+1 := a+\ell$ with $a \ge 2$ and $\ell \ge 1$ — the conjecture is trivial otherwise — and let $\mu = (a,1^{\ell-1}), \rho = (a-1,1^\ell)$ be partitions of $n$ obtained from $\lambda$ by removing a cell from the end of the first column, and from the end of the first row, respectively.

For partitions $\mu, \rho$ as defined above,

\[
\dim (\Hcal_\mu \cap \Hcal_\rho) = \frac{n!}{2}.
\]

The rest of Section <ref> is devoted to a proof of Theorem <ref>.

As above, let $SF(\mu)$ (resp. $SF(\rho)$) denote the set of standard fillings of $\mu$ (resp. $\rho$). Define a map $\mathsf{bump} : SF(\mu) \to SF(\rho)$, where $\mathsf{bump}(S)$ is the filling of $\rho$ obtained from $S$ by moving each entry in the first column up one row, and then pushing each remaining entry in the first row to the left one column; that is,

\[
\mathsf{bump} \left(
\ytableausetup{centertableaux, boxsize=2.3em}
\begin{ytableau}
v_{\ell} \\
\vdots \\
v_2 \\
v_1 & u_1 & u_2 & \cdots & u_{a-1}
\end{ytableau}
\right)
\begin{ytableau}
v_\ell \\
\vdots \\
v_2 \\
v_1 \\
u_1 & u_2 & \cdots & u_{a-1}
\end{ytableau}.
\]

Now consider the subsets

\begin{align*}
SF_<(\mu) &= \{ S \in SF(\mu) : S_{1,1} < S_{1,a} \}, \\
SF_<(\rho) &= \{ T \in SF(\rho) : T_{\ell+1,1} < T_{1,1} \},
\end{align*}

where $S_{i,j}$ denotes the entry in row $i$, column $j$ of $S$.

We have $\# SF_<(\mu) = \# SF_<(\rho) = \frac{n!}{2}$, and we claim furthermore that

\[
\{ \varphi_S : S \in SF_<(\mu) \} = \{ \varphi_T : T \in SF_<(\rho) \}.
\]

To prove this assertion, we will define a bijective map $\theta : SF_<(\mu) \rightarrow SF_<(\rho)$ such that $\varphi_S = \varphi_{\theta(S)}$ for any $S \in SF_<(\mu)$.

There are two main steps in defining the map $\theta$:

* Given a filling $S \in SF_<(\mu)$, consider $\mathsf{bump}(S) \in SF(\rho)$. This new filling is potentially missing some row inversions in $S$ which were created by $u = S_{1,1}$, which now resides in the second row, so we must rearrange what remains of the first row to reintroduce the necessary inversions.

* After rearranging the first row there is a new entry $v$ in row $1$, column $1$, so we may have introduced some column inversions which were not there before, and we must rearrange the first column to remove these new inversions.

To this end, we define a map $\mathsf{arm}_u$ which reintroduces the row inversions removed in the first step, and a map $\mathsf{leg}_v$ which removes the column inversions introduced in the second. In what follows, let

\[
\Wcal = \{ w_1w_2 \cdots w_m : m, w_i \in \NN \}
\]

denote the set of words of finite length in the alphabet $\NN$.

For a fixed $u \in \NN$, consider the set of words

\[
\Acal_u = \{ w_1w_2\cdots w_m : m \in \NN, w_i \in \NN - \{ u \}, w_m > u \}.
\]

Define a map $\mathsf{arm}_u: \Acal_u \to \Wcal$ as follows: given $w = w_1 w_2 \cdots w_m \in \Acal_u$, let $b_1, b_2, \ldots, b_j$ denote the indices for which $w_{b_i} < u$. If there are no such indices, then define $\mathsf{arm}_u(w) = w$. Otherwise,

* Draw a vertical bar immediately to the left of $w_{b_i}$ if either $w_{b_i - 1} > u$, or if $i = 1$ and $b_1 = 1$.

* Within each newly created block which contains at least one of the $w_{b_i}$'s, move the leftmost entry which is $> u$ to the front of the block, immediately right of the (leftmost) vertical bar.

Define the resulting word to be $\mathsf{arm}_u(w)$. This map is well-defined on $\Acal_u$, since the last letter in $w \in \Acal_u$ is required to be $> u$; that is, we will always have an entry $>u$ to shuffle within each block.

For $w = 49263187 \in \Acal_5$, draw vertical bars to the left of $4$, $2$, and $3$, and within each block shuffle the leftmost number $>5$ to the front:

\[
\begin{tikzpicture}
\draw[thick] (0.3,-0.3) -- (0.3,0.3);
\draw (0.3,-0.2) node (B1) {};
\draw (0.5,0) node[darkred] {$4$};
%\draw (0.5,0.5) node {${b_1}$};
\draw (1,0) node (C1) {$9$};
\draw[thick] (1.3,-0.3) -- (1.3,0.3);
\draw (1.3,-0.2) node (B2) {};
\draw (1.5,0) node[darkred] {$2$};
%\draw (1.5,0.5) node {${b_2}$};
\draw (2,0) node (C2) {$6$};
\draw[thick] (2.3,-0.3) -- (2.3,0.3);
\draw (2.3,-0.2) node (B3) {};
\draw (2.5,0) node[darkred] {$3$};
%\draw (2.5,0.5) node {${b_3}$};
\draw (3,0) node[darkred] {$1$};
%\draw (3,0.5) node {${b_4}$};
\draw (3.5,0) node (C3) {$8$};
\draw (4,0) node {$7$};
\draw [->,darkgreen] (C1.south) to [out=235,in=305] (B1.south east);
\draw [->,darkgreen] (C2.south) to [out=235,in=305] (B2.south east);
\draw [->,darkgreen] (C3.south) to [out=235,in=305] (B3.south east);

\draw (5.25,0) node {$\longmapsto$};

\draw[thick] (6.3,-0.3) -- (6.3,0.3);
\draw (6.5,0) node {$9$};
\draw (7,0) node {$4$};
\draw[thick] (7.3,-0.3) -- (7.3,0.3);
\draw (7.5,0) node {$6$};
\draw (8,0) node {$2$};
\draw[thick] (8.3,-0.3) -- (8.3,0.3);
\draw (8.5,0) node {$8$};
\draw (9,0) node {$3$};
\draw (9.5,0) node {$1$};
\draw (10,0) node {$7$};
\end{tikzpicture}
\]

So, $\mathsf{arm}_5(49263187) = 94628317$.

For a fixed $v \in \NN$, consider the set of words

\[
\Lcal_v = \{ w_1w_2\cdots w_m : m \in \NN, w_i \in \NN - \{ v \}, w_1 < v \}.
\]

Define $\mathsf{leg}_v : \Lcal_v \to \Wcal$ as follows: given $w = w_1 w_2 \cdots w_m \in \Lcal_v$, let $c_1, c_2, \ldots, c_k$ denote the indices for which $w_{c_i} > v$. If there are no such indices, then define $\mathsf{leg}_v(w) = w$. Otherwise,

* Draw a vertical bar immediately to the right of $w_{c_i}$ if either $w_{c_i + 1} < v$, or if $i = k$ and $c_k = m$.

* Within each newly created block which contains at least one of the $w_{c_i}$'s, move the rightmost entry which is $< v$ to the end of the block, immediately left of the (rightmost) vertical bar.

Define the resulting word to be $\mathsf{leg}_v(w)$. This map is well-defined on $\Lcal_v$, since the first letter in $w \in \Lcal_v$ is required to be $< v$; that is, we will always have an entry $< v$ to shuffle within each block.

For $w = 48731926 \in \Lcal_5$, draw vertical bars to the right of $7, 9$, and $6$, and within each block shuffle the rightmost number $< 5$ to the end:

\[
\begin{tikzpicture}
\draw (0.5,0) node (C1) {$4$};
\draw (1,0) node[darkred] {$8$};
%\draw (1,0.5) node {${c_1}$};
\draw (1.5,0) node[darkred] {$7$};
\draw[thick] (1.7,-0.3) -- (1.7,0.3);
\draw (1.7,-0.2) node (B1) {};
%\draw (1.5,0.5) node {${c_2}$};
\draw (2,0) node {$3$};
\draw (2.5,0) node (C2) {$1$};
\draw (3,0) node[darkred] {$9$};
\draw[thick] (3.2,-0.3) -- (3.2,0.3);
\draw (3.2,-0.2) node (B2) {};
%\draw (3,0.5) node {${c_3}$};
\draw (3.5,0) node (C3) {$2$};
\draw (4,0) node[darkred] {$6$};
%\draw (4,0.5) node {${c_4}$};
\draw[thick] (4.2,-0.3) -- (4.2,0.3);
\draw (4.2,-0.2) node (B3) {};
\draw [->,darkgreen] (C1.south) to [out=305,in=235] (B1.south west);
\draw [->,darkgreen] (C2.south) to [out=305,in=235] (B2.south west);
\draw [->,darkgreen] (C3.south) to [out=305,in=235] (B3.south west);

\draw (5.25,0) node {$\longmapsto$};

\draw (6 + 0.5,0) node {$8$};
\draw (6 + 1,0) node {$7$};
\draw (6 + 1.5,0) node {$4$};
\draw[thick] (6 + 1.7,-0.3) -- (6 + 1.7,0.3);
\draw (6 + 2,0) node {$3$};
\draw (6 + 2.5,0) node {$9$};
\draw (6 + 3,0) node {$1$};
\draw[thick] (6 + 3.2,-0.3) -- (6 + 3.2,0.3);
\draw (6 + 3.5,0) node {$6$};
\draw (6 + 4,0) node {$2$};
\draw[thick] (6 + 4.2,-0.3) -- (6 + 4.2,0.3);
\end{tikzpicture}
\]

So, $\mathsf{leg}_5(48731926) = 87439162$.

Note the similarity of $\mathsf{arm}_u$ and $\mathsf{leg}_v$ to the maps $\gamma_x$ introduced by Foata ([7]), which he used to bijectively prove that the major index and inversion number have the same distribution over $\Sym_n$.

Using the maps $\mathsf{arm}_u$ and $\mathsf{leg}_v$ defined above, we define $\theta : SF_<(\mu) \to SF_<(\nu)$ as follows. Given $S \in SF_<(\mu)$, let $u = S_{1,1}$, and let $v$ be the leftmost entry in the first row of $S$ which is $> u$. Then define

\[
\theta(S) = \mathsf{leg}_v \circ \mathsf{arm}_u \circ \mathsf{bump}(S),
\]

where $\mathsf{arm}_u$ acts only on the first row of $\mathsf{bump}(S)$ — that is, we replace the first row of $\mathsf{bump}(S)$ with $\mathsf{arm}_u(S_{1,2} S_{1,3} \cdots S_{1,a})$ — and $\mathsf{leg}_v$ acts on the first column of $\mathsf{arm}_u(\mathsf{bump}(S))$, strictly above the first row, so that these entries are replaced by $\mathsf{leg}_v(S_{1,1} S_{2,1} \cdots S_{\ell,1})$, entered from bottom to top.

Since $S_{1,a} > u$ and $u < v$ for any $S \in SF_<(\mu)$, each step in the composition is defined; thus $\theta$ is well-defined. Furthermore we have $\theta(S)_{\ell + 1,1} < v = \theta(S)_{1,1}$ by construction, so that $\theta(S) \in SF_<(\rho)$.

Let $\mu = (5,1^4)$ and $\rho = (4,1^5)$, and let

\[
S =
\ytableausetup{centertableaux, boxsize = 1.2em}
\begin{ytableau}
9 \\
1 \\
7 \\
4 \\
5 & 6 & 3 & 2 & 8
\end{ytableau} \in SF_<(\mu).
\]

Then in the language of the above definition, we have $u = 5, v = 6$, so that

\begin{align*}
\theta(S) &= \mathsf{leg}_6 \circ \mathsf{arm}_5 \circ \mathsf{bump}(S) = \mathsf{leg}_6 \circ \mathsf{arm}_5\left(
\begin{ytableau}
9 \\
1 \\
7 \\
4 \\
5  \\
6 & {\color{darkred} 3} & {\color{darkred} 2} & {\color{darkgreen} 8}
\end{ytableau}
\right) \\
&= \mathsf{leg}_6\left(
\begin{ytableau}
{\color{darkred} 9} \\
{\color{darkgreen} 1} \\
{\color{darkred} 7} \\
{\color{darkgreen} 4} \\
5  \\
6 & {\color{darkgreen} 8} & 3 & 2
\end{ytableau}
\right) =
\begin{ytableau}
{\color{darkgreen} 1} \\
9 \\
{\color{darkgreen} 4} \\
7 \\
5  \\
6 & 8 & 3 & 2
\end{ytableau} \in SF_<(\rho).
\end{align*}

Note that $\varphi_S = \varphi_{\theta(S)} = x_7^2x_9^4y_2^3y_3^2$.

For partitions $\mu = (a,1^{\ell - 1})$ and $\rho = (a-1, 1^\ell)$, the map $\theta : SF_<(\mu) \to SF_<(\rho)$ defined above satisfies $\varphi_S = \varphi_{\theta(S)}$ for every $S \in SF_<(\mu)$.

We begin by comparing the first rows of $S$ and $\theta(S)$ to demonstrate that

\begin{equation}\label{eq:row-invs}
\{ r : (t,r) \in \mathsf{rowInv}(S) \text{ for some } t \} = \{ r' : (t',r') \in \mathsf{rowInv}(\theta(S)) \text{ for some } t' \}.
\end{equation}

Since the smaller entries of the row inversions in $S$ determine the $y$-terms in $\varphi_S$ (resp. $\theta(S)$), it will follow that $\varphi_S$ and $\varphi_{\theta(S)}$ have identical $y$-terms.

First, each row inversion in $\mathsf{bump}(S)$ is also a row inversion in $S$, but $S$ may have additional row inversions created by $u = S_{1,1}$ which are absent in $\mathsf{bump}(S)$. Let $r_1, r_2, \ldots, r_p$ denote the elements in the first row of $S$ which are $< u$, so that

\[
\mathsf{rowInv}(S) = \mathsf{rowInv}(\mathsf{bump}(S)) \cup \{ (u, r_1), (u, r_2), \ldots, (u, r_p) \}.
\]

We claim that performing $\mathsf{arm}_u$ on $\mathsf{bump}(S)$ introduces exactly one new row inversion involving each of $r_1, r_2, \ldots, r_p$, and does not create any other new row inversions, nor alter any existing ones.

Indeed, in performing the “blocking" procedure of $\mathsf{arm}_u$, each of the $r_i$'s will be contained in a unique block; in each of these blocks we move some number $z > u > r_i$ to the left of each $r_i$ contained in the block, so that exactly one new row inversion is created with each of $r_1, r_2, \ldots, r_p$ in $\mathsf{arm}_u(\mathsf{bump}((S)))$. Furthermore, since $z$ is chosen to be the leftmost entry in each block which is $> u$,

(i) the only new row inversions created within a block are those created with some $r_i$, and

(ii) no existing row inversions are removed, since $z$ does not move to the left of a number larger than itself.

We only shuffle numbers within blocks which contain at least one of the $r_i$'s, and performing $\mathsf{leg}_v$ on the resulting filling does not affect the first row, so it follows that (<ref>) holds.

Now we compare the first columns of $S$ and $\theta(S)$ to demonstrate that

\begin{equation}\label{eq:col-invs}
\{ d : (d,c) \in \mathsf{colInv}(S) \text{ for some } c \} = \{ d' : (d',c') \in \mathsf{colInv}(\theta(S)) \text{ for some } c' \}.
\end{equation}

Since the larger entries of the column inversions in $S$ determine the $x$-terms in $\varphi_S$ (resp. $\theta(S)$), it will follow that $\varphi_S$ and $\varphi_{\theta(S)}$ have identical $x$-terms.

First, note that $v = \mathsf{arm}_u(\mathsf{bump}(S))_{1,1}$ by construction. Then every column inversion in $S$ is also a column inversion in $\mathsf{arm}_u(\mathsf{bump}(S))$, except that $\mathsf{arm}_u(\mathsf{bump}(S))$ may have additional column inversions created by $v$ which are not present in $S$. Let $d_1, d_2, \ldots, d_q$ denote the elements in the first column of $\mathsf{arm}_u(\mathsf{bump}(S))$ which are $> v$, so that

\[
\mathsf{colInv}(S) = \mathsf{colInv}(\mathsf{arm}_u(\mathsf{bump}(S))) - \{ (d_1, v), (d_2, v), \ldots, (d_q, v) \}.
\]

We claim that performing $\mathsf{leg}_v$ on $\mathsf{arm}_u(\mathsf{bump}(S))$ removes exactly one column inversion involving each of $d_1, d_2, \ldots, d_q$. When we perform the “blocking" procedure required by $\mathsf{leg}_v$ in the first column of $\mathsf{arm}_u(\mathsf{bump}(S))$, each of the $d_j$'s is contained in a unique block, and within each block some number $t < v < d_j$ is shuffled above each $d_j$ in the block, so that exactly one column inversion involving each of $d_1, d_2, \ldots, d_q$ is removed. Furthermore, since $t$ is chosen to be the the topmost number $< v$ in its block,

(i) the only column inversions removed within a block are those involving some $d_j$, and

(ii) no new column inversions are created, since $t$ does not move above any number smaller than itself.

Thus (<ref>) holds, and we conclude that $\varphi_S = \varphi_{\theta(S)}$.

The map $\theta : SF_<(\mu) \to SF_<(\rho)$ is a bijection.

First, we claim that $\theta$ is injective. Indeed, if $\theta(S) = \theta(T)$ for $S, T \in SF_<(\mu)$, then $\varphi_{\theta(S)} = \varphi_{\theta(T)}$. But then by Theorem <ref>, we obtain $\varphi_S = \varphi_T$, i.e. $S = T$. Thus $\theta$ is injective, and since $\# SF_<(\mu) = \# SF_<(\rho) = \frac{n!}{2}$, the result follows.

We have $\dim (\Hcal_\mu \cap \Hcal_\rho) \ge \frac{n!}{2}$.

We have exhibited that $\varphi_S \in \Hcal_\mu \cap \Hcal_\rho$ for any $S \in SF_<(\mu)$ (equivalently $SF_<(\rho)$), and $\{ \varphi_S : S \in SF_<(\mu) \}$ is linearly independent, so it follows that

\[
\dim (\Hcal_\mu \cap \Hcal_\rho) \ge \# \{ \varphi_S : S \in SF_<(\mu) \} = \frac{n!}{2}.
\]

In fact, we claim that $\{ \varphi_S : S \in SF_<(\mu) \}$ (or equivalently $\{ \varphi_T : T \in SF_<(\rho) \}$) forms a basis for $\Hcal_\mu \cap \Hcal_\rho$, aided by the following lemma.

For partitions $\mu = (a,1^{\ell-1}), \rho = (a-1,1^\ell)$, let $SF_>(\mu) = SF(\mu) - SF_<(\mu)$ and $SF_>(\rho) = SF(\rho) - SF_<(\rho)$. Then

* for any $S \in SF_>(\mu)$, we have $\varphi_S \neq \varphi_T$ for all $T \in SF(\rho)$, and

* for any $T \in SF_>(\rho)$, we have $\varphi_T \neq \varphi_S$ for all $S \in SF(\mu)$.

For the first assertion, let $S \in SF_>(\mu)$. We then have that $S_{1,1} > S_{1,a}$. Let $r = S_{1,a}$, and let $t_1, t_2, \ldots, t_k$ denote the entries in row $1$ of $S$ which are $< S_{1,1}$, ordered in such a way that $t_1, t_2, \ldots, t_j$ are the entries which are also $> r$. Then $S_{1,1}$ forms a row inversion with each of $t_1, t_2, \ldots, t_k$, and $t_1, t_2, \ldots, t_j$ each form a row inversion with $r$. Since there are then $a-k-1$ remaining terms in row $1$ which are $\ge S_{1,1}$ (and thus form row inversions with $r$), the following term divides $\varphi_{S}$:

\begin{equation}\label{eq:bad-term}
m_S(Y) = y_r^{a-k-1+j} y_{t_1} y_{t_2} \cdots y_{t_k}.
\end{equation}

We now claim that $m_S(Y)$ does not divide $\varphi_T$ for any filling $T \in SF(\rho)$. If it did, then each of $r, t_1, t_2, \ldots, t_k$ must appear in the first row of $T$. Furthermore, since $t_1, t_2, \ldots, t_j$ are the only of these elements which could be the larger entry in a row inversion involving $r$, there must be at least $a-k-1$ other entries in row $1$ which are $> r$ in order for $\varphi_T$ to have a $y_r$-degree of $a-k-1+j$. There are thus at least $a$ distinct entries in the first row of $T$, a contradiction since $\rho = (a-1,1^\ell)$. Thus $\varphi_S \neq \varphi_T$ for all $T \in SF(\rho)$.

The second assertion follows from a completely analogous argument, where we instead show that for any $T \in SF_>(\rho)$, one can infer from $\varphi_T$ that there must be $\ell + 1$ distinct entries in the first column of $T$.

For $\mu = (a, 1^{\ell - 1}), \rho = (a-1,1^\ell)$, we have $\dim(\Hcal_\mu \cap \Hcal_\rho) \le \frac{n!}{2}$.

We demonstrate that no nontrivial linear combination of $\{ \varphi_S : S \in SF_>(\mu) \}$, or of $\{ \varphi_T : T \in SF_>(\rho) \}$, lies in $\Hcal_\mu \cap \Hcal_\rho$. First, assume that

\[
\sum_{S \in SF_>(\mu)} c_S \varphi_S \in \Hcal_\mu \cap \Hcal_\rho
\]

for some $c_S \in \QQ$. Then, there exist constants $d_T \in \QQ$ so that

\[
\sum_{S \in SF_>(\mu)} c_S \varphi_S = \sum_{T \in SF(\rho)} d_T \varphi_T.
\]

Pick any $\widetilde{S} \in SF_>(\mu)$ and write

\[
c_{\widetilde{S}} \varphi_{\widetilde{S}} = \sum_{T \in SF(\rho)} d_T \varphi_T - \sum_{S \in SF_>(\mu) - \{ \widetilde{S} \}} c_S \varphi_S.
\]

By Lemma <ref>, and using the fact that $\{ \varphi_S : S \in SF(\mu) \}$ and $\{ \varphi_T : T \in SF(\rho) \}$ are linearly independent sets of monomials, the monomial $m_{\widetilde{S}}(Y)$ defined in (<ref>) must divide each term on the right-hand side which has a nonzero coefficient. (More precisely, none of the terms within a given sum can cancel because they are linearly independent, and none of the terms between the sums can cancel because $\varphi_T$ cannot contain the same monomial $m_S(Y)$ as $\varphi_S$ for any $S \in SF_>(\mu)$.)

But, again applying Lemma <ref>, $m_{\widetilde{S}}(Y)$ cannot divide $\varphi_T$ for any $T \in SF(\rho)$, forcing $d_T = 0$ for all $T$. Thus the above equation reduces to

\[
c_{\widetilde{S}} \varphi_{\widetilde{S}} = - \sum_{S \in SF_>(\mu) - \{ \widetilde{S} \}} c_S \varphi_S,
\]

forcing $c_S = 0$ for all $S$, since $\{ \varphi_S : S \in SF_>(\mu) \}$ is linearly independent. Thus

\[
\sum_{S \in SF_>(\mu)} c_S \varphi_S \in \Hcal_\mu \cap \Hcal_\rho \Rightarrow c_S = 0 \text{ for all } S.
\]

An identical argument shows that

\[
\sum_{T \in SF_>(\rho)} d_T \varphi_T \in \Hcal_\mu \cap \Hcal_\rho \Rightarrow d_T = 0 \text{ for all } T
\]

by appealing to the second assertion in Lemma <ref>. Thus no nonzero element of $\Hcal_\mu \cap \Hcal_\rho$ can be written as a linear combination of $\{ \varphi_S : S \in SF_>(\mu) \}$ or of $\{ \varphi_T : T \in SF_>(\rho) \}$, proving that $\dim(\Hcal_\mu \cap \Hcal_\rho) \le \frac{n!}{2}$.

Combining Theorems <ref> and <ref> proves Theorem <ref>.

Dually, one may define the Garsia-Haiman module as

\[
D_\mu = \text{span}\{  f(\tfrac{\partial}{\partial x_1}, \ldots, \tfrac{\partial}{\partial x_n}; \tfrac{\partial}{\partial y_1}, \ldots, \tfrac{\partial}{\partial y_n}) \Delta_\mu : f \in \QQ[X,Y] \},
\]

the span of all partial derivatives of all orders of $\Delta_\mu$; the map $\Hcal_\mu \to D_\mu$ given by $f(X,Y) \mapsto f(\frac{\partial}{\partial X}, \frac{\partial}{\partial Y})\Delta_\mu$ is an isomorphism of doubly graded $\Sym_n$-modules (see [13], Proposition 3.4).

It is worth pointing out that the original $\frac{n!}{k}$ conjecture in [5] is stated in terms of $D_\mu$, rather than $\Hcal_\mu$. Our basis for $\Hcal_\mu \cap \Hcal_\rho$ does not immediately yield a basis for $D_\mu \cap D_\rho$ in the sense that

\[
\{ \varphi_S(\tfrac{\partial}{\partial X},\tfrac{\partial}{\partial Y}) \Delta_\mu : S \in SF_<(\mu) \} \neq \{ \varphi_T(\tfrac{\partial}{\partial X},\tfrac{\partial}{\partial Y}) \Delta_\rho : T \in SF_<(\rho) \}.
\]

In fact, experimental evidence suggests that these sets are disjoint.

As the above isomorphism is bidegree-complementing, the naive/optimistic conjecture would be that the complements $SF_>(\mu)$ and $SF_>(\rho)$ index a basis for $D_\mu \cap D_\rho$. However, it is not true in general that $\varphi_S(\frac{\partial}{\partial X},\frac{\partial}{\partial Y})\Delta_\mu \in D_\rho$ for arbitrary $S \in SF_>(\mu)$ (and vice-versa), either. Thus it appears to be nontrivial to construct an explicit basis for $D_\mu \cap D_\rho$ in the present paradigm.

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Ron M. Adin, Jeffrey B. Remmel, and Yuval Roichman.
The combinatorics of the Garsia-Haiman modules for hook shapes.
Electron. J. Combin., 15(1):Research Paper 38, 42, 2008.

[2]
E. E. Allen.
Bitableaux bases for some Garsia-Haiman modules and other related
Electron. J. Combin., 9(1):Research Paper 36, 59, 2002.

[3]
Sami Assaf and Adriano Garsia.
A kicking basis for the two column Garsia-Haiman modules.
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