# Developing a Ranking Problem Library (RPLIB) from a data-oriented
perspective ††thanks: Citation: Authors. Title. Pages…. DOI:000000/11111.

Paul E. Anderson, Brandon Tat, Charlie Ward
Department of Computer Science and Software Engineering
California Polytechnic State University
San Luis Obispo, CA, USA
{pander14, bjtat<EMAIL_ADDRESS>
&Amy N. Langville, Kathryn E. Pedings-Behling
Department of Mathematics
College of Charleston
Charleston, SC 29401, USA
{langvillea<EMAIL_ADDRESS>

###### Abstract

We present an improved library for the ranking problem called RPLIB. RPLIB
includes the following data and features. (1) Real and artificial datasets of
both pairwise data (i.e., information about the ranking of pairs of items) and
feature data (i.e., a vector of features about each item to be ranked). These
datasets range in size (e.g., from small $n=10$ item datasets to large
datasets with hundred of items), application (e.g., from sports to economic
data), and source (e.g. real versus artificially generated to have particular
structures). (2) RPLIB contains code for the most common ranking algorithms
such as the linear ordering optimization method and the Massey method. (3)
RPLIB also has the ability for users to contribute their own data, code, and
algorithms. Each RPLIB dataset has an associated .JSON model card of
additional information such as the number and set of optimal rankings, the
optimal objective value, and corresponding figures.

_K_ eywords ranking problem library $\cdot$ linear ordering problem $\cdot$
integer programming $\cdot$ linear programming $\cdot$ rankability $\cdot$
artificial data $\cdot$ Massey rankings $\cdot$ Colley rankings

##  1 Introduction

Constructing a ranking of items from a set of pairwise comparisons is a common
problem with applications in a wide range of fields including sports,
economics, social science, and many others. A ranking problem can be defined
as a dataset of items to be ranked. Such a dataset could include features of
the items or pairwise comparisons between items. The current approach to
producing rankings often focuses on selecting the single ranking best
supported by the observed data but fails to communicate the ranking’s
uncertainty. In prior work [1, 2, 3, 4, 5], we introduced the idea of the
rankability of a ranking dataset. We tied rankability to the presence, number,
and properties of multiple optimal solutions and thereby quantified a
dataset’s inherent ability to produce a meaningful ranking of its items.

Often progress in a particular field has been sparked by public datasets and
libraries, e.g., ImageNet[6], the Linear Ordering Library[7], and kaggle.com/.
For example, TSPLIB [8] provides researchers of the traveling salesman problem
(TSP) a unified place and problem instances by which to compare their
algorithm’s speed, scalability, and solutions and to compete for titles such
as the largest TSP solved or fastest runtime. RPLIB, our Rankability Problem
Library, was motivated by such libraries.

We hope RPLIB will spark similar research for the ranking community and its
instances will serve as comparative benchmarks. By describing RPLIB here, we
also aim to encourage the ranking community to attack some of the open
questions we pose in Section 5.

##  2 Related work

RPLIB builds on the older existing ranking library LOLIB, the Linear Ordering
Library [7]. LOLIB is an excellent repository of ranking data, but its
datasets are static and in need of update. For example, an LOLIB text file
contains just three pieces of information: the problem size, the optimal
objective value for the linear ordering optimization, and one optimal ranking.
In prior work [3, 1], we discovered that 98% of the 50 instances in the LOLIB
IO folder [9] actually have more than one optimal ranking, yet just one is
stored in the LOLIB text file. Unlike LOLIB, RPLIB algorithms produce and the
library stores, in its novel JSON model card (Section 3.4.1), all (or many if
the user terminates the algorithm early) alternate optimal rankings associated
with an instance. This is a novel contribution of RPLIB. Figure 1 summarizes
several ways in which RPLIB extends LOLIB.

Another new contribution in this paper, in Section 5, is the extension of the
definition of rankability from three measures to four measures. We use RPLIB
instances and algorithms to demonstrate the additional benefits of this fourth
rankability measure.

Figure 1: A comparison of the features available in LOLIB versus those in
RPLIB.

##  3 RPLIB

The RPLIB contains data, algorithms, code, and web applications for user
contributions. These are described in the following subsections.

###  3.1 Datasets

RPLIB has three types of unprocessed data: (1) _static real data_ ranging from
college basketball scores, features associated with U.S. colleges, organized
and standardized versions of LOLIB instances, which include economic input-
output instances as well as special artificial instances; (2) _dynamic
artificial data_ that creates rankings with specific properties; and (3)
_user-contributed data_.

The RPLIB website contains additional details beyond those in this paper. See
https://igards.github.io/RPLib/.

####  3.1.1 Static Real Data

RPLIB contains datasets from several real ranking applications.

  * •

Game data from seasons of U.S. college basketball, including the March Madness
tournament.

  * •

economic input-output matrices from Japan’s economic sectors for 1995 and 2005
[10].

  * •

Economic input-output matrices that were originally hosted within the linear
ordering libraries, LOLIB and XLOLIB. These datasets have been updated and
organized in RPLIB.

  * •

Special collections from LOLIB and XLOLIB. These datasets were also cleaned
and organized when moving them to RPLIB.

  * •

US News & World Report data on features associated with liberal arts colleges.

This collection can be filtered, searched, and exported for research purposes.
For example, to find within RPLIB’s hundreds of instances, instances from
LOLIB where the problem size $n$ is between $30\leq n\leq 35$, use the filter
command as shown in Figure 2.

Figure 2: This screenshot shows a powerful feature of RPLIB, the ability to
filter the hundreds of problem instances to find instances satisfying certain
conditions. This example finds all instances from the LOLIB collection that
have $30<n<35$.

To find all instances pertaining to March Madness basketball that contain
greater than 1000 optimal rankings, use the filter command as shown in Figure
3.

Figure 3: This screenshot shows all March Madness instances that contain
greater than 1000 optimal rankings.

####  3.1.2 Dynamic Artificial Data

RPLIB also has the ability for users to _generate_ artificial instances that
have desired structure. An RPLIB user has the ability to tune various
parameters, such as the size of the instance or the amount of noise added to a
matrix. These artificial datasets are useful for theoretical studies,
particularly for the open questions posed in Section 5. Clicking on the
“Artificial Structured Datasets" tab on the left menu of the RPLIB homepage
opens a Google Colaboratory Notebook.

Figure 4 below is screenshot from the RPLIB Google Colaboratory Notebook. In
this example, the command emptyplusnoise(5,20,2,4) creates a $5\times 5$ empty
matrix where 20% of the data is replaced with random entries between the
lowerbound of 2 and the upperbound of 4.

Figure 4: The screenshot above shows how to create an artificial dataset in
the Colab Notebook section of RPLIB. Specifically, this creates a $5\times 5$
empty matrix where 20% of the data is replaced with random entries between the
lowerbound of 2 and the upperbound of 4.

Currently the following structures of artificial data are available. These
artificial datasets were first described in our prior work on rankability [2].

  * •

empty matrix (i.e., matrix of all zeros)

  * •

empty matrix plus a matrix with random noise

  * •

a completely connected matrix

  * •

a completely connected matrix minus a matrix with random noise

  * •

perfect ranking matrix in hillside form plus random noise

  * •

perfect ranking matrix in dominance form plus random noise

  * •

cyclic matrix

  * •

matrix structured so that it has $c!$ multiple optimal rankings with
variations in the top, middle, or bottom of the ranking. See Section 4.

  * •

ability to turn weighted matrices into unweighted matrices

  * •

ability to add or remove a percentage of entries in the matrix (i.e., edges in
the associated graph)

  * •

ability to generate a $\b{D}$ matrix by simulating individual games with a
probability of upset

####  3.1.3 User-contributed Data

RPLIB makes it easy for users to contribute their own data, either real static
data or artificial structured data. There are two ways to make a contribution.
(1) An RPLIB user can follow the “Contribute Dataset" link on the RPLIB
homepage, which opens a form that prompts the user for their name and email,
the name and description of the dataset, the type of data, and the data
itself. Contributions will be reviewed by the RPLIB team prior to being added
to the library. (2) Alternatively, RPLIB users can create artificial datasets
via RPLIB’s Colab tool and then follow the link within that tool to directly
submit their generated data. See Figure 5.

Ranking research has continued for decades, whereas rankability research has
been recent. We invite researchers from these communities to add methods,
data, and code to RPLIB. For example, rankability researchers from the U.S.
and Europe have created interesting artificial datasets and new methods [5,
11].

Figure 5: There are two avenues for user to submit datasets. The left of the
figure shows the submission link directly on the RPLIB website. The top of the
figure shows a submission link within the Colab Notebook where users can
create and submit their own artificial datasets. The remainder of the figure
is a screenshot of a portion of the submission form.

###  3.2 Dataset Processing

In this section, we use a few examples to explain the Processed Data section
of RPLIB. Figure 6 contains screenshots taken directly from RPLIB and
highlights the difference between unprocessed and processed data. This example
is for the 2002 season of March Madness basketball. Screenshot A is the
_unprocessed_ data. Each row contains data for one game that season, showing
the point score information. For example, row 1 indicates that Arizona scored
71 points, beating Maryland who scored 67 points. Screenshot B shows a
powerful feature of RPLIB–the ability to search and filter data. In this
example, we filtered this unprocessed instance to find all games between
Cincinnati and Charlotte during that 2002 season. Contrast this with
Screenshot C, which shows a subset of the (alphabetically ordered) _processed_
data, a square matrix of pairwise comparisons. An empty cell means the two
teams have not played each other. Non-empty cell $(i,j)$ contains the number
of times team $i$ beat team $j$ in that matchup.

Figure 6: These screenshots are taken directly from RPLIB. Screenshot A is the
_unprocessed_ data for the 2002 March Madness basketball season. Screenshot B
shows a powerful feature of RPLIB–the ability to search and filter data. In
this example, we filtered the unprocessed data to find all instances of games
between Cincinnati and Charlotte. Screenshot C is a subset of the
(alphabetically ordered) _processed_ data, which creates a square matrix of
pairwise comparisons. Empty cells mean the teams have not played each other.
Non-empty cell $(i,j)$ is the number of times team $i$ beat team $j$ in that
matchup.

Our next example is the US News & World Report data, which is used to rank
U.S. colleges. RPLIB contains a subset of this data, specifically Liberal Arts
colleges. Figure 7 shows both the unprocessed and processed data. Figure 7A is
the _unprocessed_ data, which are data associated with various features (e.g.,
graduation rate, retention rate) for each college. Once again, the data is
_processed_ to create a square matrix of pairwise comparisons, as shown in
Figure 7B.

Figure 7: These screenshots are taken directly from RPLIB. Screenshot A is the
_unprocessed_ data for 2002 US News and World Report Liberal Arts College
data. Screenshot B shows a powerful feature of RPLIB–the ability to search and
filter data. In this example, we filtered the unprocessed data and compared
the data from Amherst and Bowdoin Colleges. Screenshot C is a subset of the
(alphabetically ordered) _processed_ data, which creates a square matrix of
pairwise comparisons. Entry $(i,j)$ is the number of features out of the six
total where college $i$ outperformed college $j$.

We invite the community to add new transformations that convert unprocessed
data to processed data. This can be done with a standard GitHub pull request.
Once verified, such transformations can be run on all datasets through RPLIB’s
infrastructure for automated analysis.

###  3.3 RPLIB Methods

The library contains implementations of two classes of ranking methods:

  1. 1.

Optimization: The linear ordering method [12] and its hillside variant [13, 2,
3, 4, 14] are available. These methods output an optimal value, a collection
of multiple optimal rankings (if any exist), the diameter of the set of
multiple optimal rankings found, the solutions nearest to and farthest from
the centroid of the optimal face, a matrix $\b{X}^{*}$ [3] containing rank-
ordered information about pairs of items, and a measure indicating the
location of the indecision among the set of optimal rankings.

  2. 2.

Linear Algebra: The Massey [15] and Colley [16] linear systems are available.
These methods output the rating vector, its associated ranking vector, and a
matrix $\b{Y}^{*}$ similar to the $\b{X}^{*}$ mentioned above, where
$\b{Y}^{*}(i,j)$ gives information about the certainty that item $i$ is ranked
above item $j$. The rating vector can be used to create a pseudo-optimal set
of rankings.

Users are able to download RPLIB code in order to run the optimization and
linear algebra methods on their own data with their own machines.
Alternatively, users can choose to upload their data using the Contribute
Dataset option on RPLIB. The RPLIB team will process the data for contribution
to the website. Depending on user feedback, future versions of RPLIB may
include additional ranking methods, such as the Elo [17, 18] and learning to
rank [19] methods.

###  3.4 RPLIB Outputs

####  3.4.1 RPLIB Model Card

Because ranking is a common task in machine learning, we adopt the standard
documentation from that field, namely storing and visualizing the output as a
_model card_ [20]. An RPLIB instance contains a .JSON file and a Google Colab
Notebook that generates images associated with that instance. We intentionally
chose JSON as the format for RPLIB data. JSON (JavaScript Object Notation)
format is favored for its readability, nesting features, interoperability, and
library support in modern programming languages. Because tools that convert
JSON to other languages are common, our choice of JSON increases the usability
of RPLIB. The JSON file contains some (or all) of the following items, which
are displayed in our RPLIB tool as a model card. The Colab Notebook with
images is described in the next section.

  * •

Dominance data. This can be either a square matrix $\b{D}$ of dominance
relations between all pairs of items or a rectangular matrix of features for
each item.

  * •

Problem size $n$, i.e., the number of items to be ranked.

  * •

Optimal objective value. As described in [2], many ranking and rankability
methods optimize an objective function.

  * •

Optimal ranking(s).

  * •

Diameter of the set of optimal rankings, i.e., the distance (e.g., Kendall
tau) between the two farthest optimal rankings.

  * •

Farthest (by Kendall tau) pair of optimal rankings.

  * •

Closest (by Kendall tau) pair of distinct optimal rankings.

  * •

Ranking closest to the centroid of all optimal rankings.

  * •

Ranking farthest from the centroid of all optimal rankings.

  * •

full set $P$ of all optimal rankings for some instances with small $n$.

  * •

partial set $P$ of all known optimal rankings for larger instances.

Figure 8: A JSON Model Card. In this instance with dataset_id 363, there are
1,095 optimal rankings with an objective value of 204. Clicking on a JSON item
that has an arrow, such as D, expands to show more information. D is a 65 x 65
matrix. The first row corresponds to the team labeled Gonzaga and this can be
further expanded to reveal the data for the 65 entries on this row of the D
matrix. Similarly, expanding the centroid_solution displays a vector of 65
elements, showing that the team with ID 0 is in rank position 6, team with ID
1 is in rank position 55, and so on.

####  3.4.2 RPLIB Figure Outputs

Each RPLIB instance can be loaded in a Google Colab Notebook (see the top left
of Figure 9) and associated methods run, which generate associated measures
(top right of Fig. 9), and figures, such as a pixel plot of nonzero elements
in a reordered $\b{X}(\b{r},\b{r})^{*}$ matrix and spaghetti plots that
visually compare two rankings (see the bottom of Figure 9).

Figure 9: A Google Colab Notebook provides the ability for users to generate
and save any measures and images associated with a data instance. In this
example, for the instance with ID 636, we show the pixel plots
$\b{X}(\b{r},\b{r})^{*}$ matrix and a spaghetti plot between the farthest pair
of optimal rankings.

##  4 Example Use of RPLIB: refining the definition of rankability

In prior work, Anderson et al. define rankability $r$ as a function of three
measures, $k$, the distance of $\b{D}$ from a perfectly rankable data matrix,
$|P|$, the number of optimal rankings, and $\tau$, the diameter (i.e., the
distance between the two farthest rankings) of the set of optimal rankings [1,
2, 3, 4, 14]. Mathematically, they define $r=f(k,|P|,\tau)$ and show that
rankability $r$ is correlated with predictability on the ranking task of
predicting outcomes in the March Madness college basketball tournament. In
other words, March Madness outcomes are straightforward and follow the
rankings in more rankable years and are wild for less rankable years. Figure
10 shows the $\b{X}^{*}$ matrices for two years of March Madness, 2002 and
2008. This figure plots the fractional elements in $\b{X}^{*}$, which measure
the locations of disagreements among the optimal rankings in the set $P$ of
optimal rankings. Notice that for 2002, there is little disagreement about the
teams that belong at the top of the ranking, while there is much more
disagreement about the bottom of the ranking. The 2008 year shows more
disagreement in the top of the ranking, certainty in the middle of the
ranking, and some disagreement in the bottom. With respect to March Madness,
we expect the 2008 year to have some upsets in later rounds of the tournament,
when two top teams face each other. In this section, _we quantify the
uncertainty in locations of the ranking by creating a fourth rankability
measure, called $\beta$, which we define and describe in the remainder of this
section._

Figure 10: $\b{X}^{*}$ matrices for two years of March Madness, 2002 (left)
and 2008 (right).

RPLIB provides a standardized approach for generating and archiving structured
artificial data. We demonstrate the importance of archiving structured
artificial data for the research community and the utility of RPLIB by
expanding the definition of rankability from three measures to four.
Mathematically, we test whether the following expanded definition of
rankability is useful: $r=f(k,|P|,\tau,\beta)$, where $\beta$ is a measure of
the location of indecision in multiple optimal rankings. In most ranking
applications, one is most interested in the top-$k$ part of the ranking.

Thus, we use RPLIB to generate four artificial matrices from the Special
class, $\b{D}_{1}$, $\b{D}_{2}$, $\b{D}_{3}$, and $\b{D}_{4}$, designed to
have particular structure. Users have the ability in the RPLIB Colab Notebook
to create matrices in the Special class that have a desired number of multiple
optimal solutions. For example, when the user selects beginning and ending row
indices of 6 and 10, respectively, this function creates a dominance matrix
that has $(10-6+1)!=5!=120$ multiple optimal rankings. In this way, we
generated the four dominance matrices $\b{D}_{1}$, $\b{D}_{2}$, $\b{D}_{3}$,
and $\b{D}_{4}$ and their corresponding $\b{X}^{*}$ matrices shown in Figure
11.

Figure 11: These four $\b{D}$ matrices (left) and their corresponding
$\b{X}^{*}$ matrices (right) have the same $k$, $p=|P|$, and $\tau$ scores,
but different $\beta$ scores.

Ranking researchers prefer indecision, if it occurs, to appear toward the
bottom of the ranking. This distinction is not captured by the existing
rankability measures $k$, $p$, and $\tau$. Thus, we propose a fourth
rankability measure $\beta$, defined in Figure 12, which uses the number and
location of fractional entries of the $X^{*}$ matrix from the linear ordering
model to penalize fractional entries that are further away from the diagonal
and lower in the ranking. Errors, ambiguity, or indecision in the bottom of
rankings are less concerning than the same thing at the top of the ranking;
the weighted definition of $\beta$ takes this into account.

Figure 12: Definition of $\beta$ for a matrix of size $n$

This new rankability measure works as intended. Remember that lower scores are
better. See the $\beta$ scores for $\b{D}_{1}$, $\b{D}_{2}$, $\b{D}_{3}$, and
$\b{D}_{4}$ in Figure 11. See Figure 10 for the $\beta$ scores for the 2002
and 2008 years of March Madness. Thus, $\beta$ is capturing something that the
other measures do not.

As shown in Figure 11, the three prior metrics ($k$, $p$, $\tau$) do not
capture the location of the multiple optimal solutions within the rankings
(i.e., they do not capture position specific ambiguity in the rankings). Our
proposed metric $\beta$ does take the position specific information into
account.

##  5 Open Questions

Our hope is that RPLIB will facilitate the study of open questions in the
field of ranking. A few open questions follow.

  * •

_Instance Difficulty._ What makes an instance of the ranking problem
difficult, i.e., which instances require a long time to find one (or all)
optimal ranking(s)? Are there properties of the input matrix that indicate the
difficulty of a ranking instance?

  * •

_Inverse Problem._ Another open ranking problem is the inverse problem.
Suppose you are given an optimal objective value and set of optimal rankings.
Does there exist an input matrix associated with this output data? Under what
conditions is the input matrix unique?

  * •

_Rankability Score._ Anderson et al. [1, 2, 3] define the related ranking
problem, the rankability problem. We hope RPLIB will advance open questions in
that new field. For example, Anderson et al. defined rankability as a function
of several measures such as the four measures described in Section 4. How to
combine these measures to create a scalar rankability score is an open
question.

  * •

_Improving Rankability._ What data can be collected to improve the rankability
of a given dataset?

##  6 Future Work

In version 2.0 of RPLIB, we plan an interactive tool that enables users to
analyze their own data using algorithms from the library. We also plan an
interactive tool for one particular dataset, the US News and Report data on
rankings of colleges from their features. We envision a tool that parents,
students, and guidance counselors can use to personalize rankings of colleges
for an individual student based on their preference and weights of features.

##  7 Conclusion

RPLIB is a library for researchers of the ranking (or linear ordering)
problem. The library is a searchable database of hundreds of instances of
ranking problems from both real and artificial data with algorithms, code, and
an interactive notebook. We invite the ranking community to use the library,
give feedback for future versions, and contribute work.

## Acknowledgements

We thank California Polytechnic State University students Marisa Aquilina,
Ethan Goldfarb, and Jackson Waschura for their contributions to RPLIB.

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J0337-0502 | 0.0135 | $~{}7.06^{+0.24}_{-0.21}$ | $-0.32^{+0.07}_{-0.11}$ | $7.23\pm 0.04$ |  $\cdots$ | $-1.80\pm 0.01$ | $-3.82\pm 0.02$ | $-2.31\pm 0.01$
J0405-3648 | 0.0028 | $~{}6.61^{+0.28}_{-0.28}$ | $-1.81^{+0.31}_{-0.27}$ | $6.99\pm 0.07$ |  $\cdots$ | $-1.45\pm 0.07$ |  $\cdots$ | $-2.19\pm 0.07$
J0808+3948 | 0.0912 | $~{}9.12^{+0.30}_{-0.17}$ | $~{}1.26^{+0.18}_{-0.25}$ | $8.77\pm 0.05$ | $-0.49\pm 0.05$ | $-1.63\pm 0.12$ |  $\cdots$ | $-2.77\pm 0.14$
J0823+2806 | 0.0472 | $~{}9.38^{+0.33}_{-0.19}$ | $~{}1.48^{+0.15}_{-0.32}$ | $8.25\pm 0.01$ | $-0.53\pm 0.02$ | $-1.70\pm 0.02$ | $-3.61\pm 0.05$ | $-2.37\pm 0.02$
J0926+4427 | 0.1807 | $~{}8.76^{+0.30}_{-0.26}$ | $~{}1.03^{+0.13}_{-0.13}$ | $8.05\pm 0.09$ | $-0.56\pm 0.12$ | $-1.66\pm 0.12$ |  $\cdots$ | $-2.40\pm 0.12$
J0934+5514 | 0.0025 | $~{}6.27^{+0.15}_{-0.20}$ | $-1.52^{+0.09}_{-0.07}$ | $7.09\pm 0.01$ | $-0.64\pm 0.01$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J0938+5428 | 0.1021 | $~{}9.15^{+0.18}_{-0.29}$ | $~{}1.05^{+0.20}_{-0.17}$ | $8.22\pm 0.02$ | $-0.59\pm 0.03$ | $-1.63\pm 0.03$ |  $\cdots$ | $-2.39\pm 0.02$
J0940+2935 | 0.0017 | $~{}6.71^{+0.23}_{-0.40}$ | $-2.01^{+0.42}_{-0.37}$ | $7.99\pm 0.09$ | $-0.73\pm 0.12$ | $-1.83\pm 0.10$ |  $\cdots$ | $-2.73\pm 0.10$
J0942+3547 | 0.0149 | $~{}7.56^{+0.21}_{-0.29}$ | $-0.76^{+0.19}_{-0.12}$ | $8.00\pm 0.09$ | $-0.59\pm 0.02$ | $-1.63\pm 0.02$ |  $\cdots$ | $-2.32\pm 0.02$
J0944+3442 | 0.0048 | $~{}6.83^{+0.44}_{-0.25}$ | $-0.78^{+0.19}_{-0.16}$ | $7.66\pm 0.22$ | $-0.57\pm 0.14$ | $-1.39\pm 0.13$ |  $\cdots$ | $-2.34\pm 0.13$
J0944-0038 | 0.0200 | $~{}8.19^{+0.40}_{-0.23}$ | $-0.01^{+0.28}_{-0.65}$ | $7.82\pm 0.01$ | $-0.70\pm 0.01$ | $-1.66\pm 0.02$ |  $\cdots$ | $-2.29\pm 0.02$
J1016+3754 | 0.0039 | $~{}6.72^{+0.27}_{-0.22}$ | $-1.17^{+0.18}_{-0.18}$ | $7.57\pm 0.04$ | $-0.68\pm 0.02$ | $-1.72\pm 0.02$ |  $\cdots$ | $-2.34\pm 0.02$
J1024+0524 | 0.0332 | $~{}7.89^{+0.37}_{-0.24}$ | $~{}0.21^{+0.14}_{-0.12}$ | $7.89\pm 0.01$ | $-0.67\pm 0.01$ | $-1.76\pm 0.02$ |  $\cdots$ | $-2.38\pm 0.02$
J1025+3622 | 0.1265 | $~{}8.87^{+0.25}_{-0.27}$ | $~{}1.04^{+0.14}_{-0.18}$ | $8.19\pm 0.02$ | $-0.62\pm 0.03$ | $-1.79\pm 0.04$ |  $\cdots$ | $-2.52\pm 0.03$
J1044+0353 | 0.0129 | $~{}6.80^{+0.41}_{-0.26}$ | $-0.59^{+0.11}_{-0.14}$ | $7.55\pm 0.02$ | $-0.70\pm 0.01$ | $-2.12\pm 0.01$ | $-3.93\pm 0.01$ | $-2.54\pm 0.01$
J1105+4444 | 0.0215 | $~{}8.98^{+0.29}_{-0.24}$ | $~{}0.69^{+0.28}_{-0.22}$ | $8.28\pm 0.01$ | $-0.66\pm 0.02$ | $-1.81\pm 0.02$ | $-3.66\pm 0.04$ | $-2.48\pm 0.02$
J1112+5503 | 0.1316 | $~{}9.59^{+0.33}_{-0.19}$ | $~{}1.60^{+0.20}_{-0.25}$ | $8.02\pm 0.05$ | $-0.62\pm 0.07$ | $-1.65\pm 0.11$ |  $\cdots$ | $-2.34\pm 0.08$
J1119+5130 | 0.0045 | $~{}6.77^{+0.15}_{-0.28}$ | $-1.58^{+0.21}_{-0.12}$ | $7.58\pm 0.08$ | $-0.65\pm 0.03$ | $-1.72\pm 0.04$ |  $\cdots$ | $-2.50\pm 0.05$
J1129+2034 | 0.0047 | $~{}8.09^{+0.37}_{-0.27}$ | $-0.37^{+0.38}_{-0.56}$ | $8.31\pm 0.01$ | $-0.67\pm 0.02$ | $-1.76\pm 0.02$ |  $\cdots$ | $-2.49\pm 0.02$
J1132+1411 | 0.0176 | $~{}7.31^{+0.23}_{-0.26}$ | $-1.07^{+0.27}_{-0.35}$ | $8.20\pm 0.01$ | $-0.58\pm 0.02$ | $-1.70\pm 0.02$ | $-3.48\pm 0.03$ | $-2.41\pm 0.02$
J1132+5722 | 0.0050 | $~{}8.68^{+0.28}_{-0.19}$ | $~{}0.44^{+0.24}_{-0.27}$ | $7.34\pm 0.02$ | $-0.62\pm 0.04$ | $-1.48\pm 0.06$ |  $\cdots$ | $-2.26\pm 0.06$
J1144+4012 | 0.1269 | $~{}9.89^{+0.18}_{-0.29}$ | $~{}1.51^{+0.20}_{-0.29}$ | $8.65\pm 0.08$ | $-0.45\pm 0.15$ |  $\cdots$ |  $\cdots$ | $-2.73\pm 0.18$
J1148+2546 | 0.0451 | $~{}8.14^{+0.34}_{-0.24}$ | $~{}0.53^{+0.17}_{-0.14}$ | $8.09\pm 0.02$ | $-0.72\pm 0.01$ | $-1.81\pm 0.01$ |  $\cdots$ | $-2.47\pm 0.01$
J1150+1501 | 0.0024 | $~{}6.84^{+0.28}_{-0.30}$ | $-1.33^{+0.29}_{-0.23}$ | $8.15\pm 0.02$ | $-0.70\pm 0.01$ | $-1.71\pm 0.01$ | $-3.63\pm 0.02$ | $-2.36\pm 0.01$
J1157+3220 | 0.0110 | $~{}9.04^{+0.32}_{-0.18}$ | $~{}0.97^{+0.21}_{-0.42}$ | $8.44\pm 0.03$ | $-0.69\pm 0.04$ | $-1.76\pm 0.04$ | $-3.55\pm 0.06$ | $-2.50\pm 0.04$
J1200+1343 | 0.0668 | $~{}8.12^{+0.47}_{-0.42}$ | $~{}0.75^{+0.20}_{-0.16}$ | $8.11\pm 0.01$ | $-0.56\pm 0.03$ | $-1.58\pm 0.02$ |  $\cdots$ | $-2.28\pm 0.02$
J1225+6109 | 0.0023 | $~{}7.12^{+0.34}_{-0.24}$ | $-1.08^{+0.26}_{-0.26}$ | $8.02\pm 0.06$ | $-0.71\pm 0.01$ | $-1.82\pm 0.01$ | $-3.70\pm 0.04$ | $-2.43\pm 0.02$
J1253-0312 | 0.0227 | $~{}7.65^{+0.51}_{-0.23}$ | $~{}0.56^{+0.15}_{-0.15}$ | $7.98\pm 0.01$ | $-0.63\pm 0.01$ | $-1.45\pm 0.01$ | $-3.28\pm 0.01$ | $-2.24\pm 0.01$
J1314+3452 | 0.0029 | $~{}7.56^{+0.30}_{-0.21}$ | $-0.67^{+0.23}_{-0.55}$ | $8.27\pm 0.23$ | $-0.67\pm 0.01$ | $-1.79\pm 0.01$ | $-3.85\pm 0.03$ | $-2.48\pm 0.01$
J1323-0132 | 0.0225 | $~{}6.31^{+0.26}_{-0.10}$ | $-0.72^{+0.08}_{-0.09}$ | $7.72\pm 0.02$ | $-0.64\pm 0.01$ | $-1.73\pm 0.01$ |  $\cdots$ | $-2.23\pm 0.01$
J1359+5726 | 0.0338 | $~{}8.41^{+0.31}_{-0.26}$ | $~{}0.42^{+0.20}_{-0.14}$ | $8.05\pm 0.01$ | $-0.64\pm 0.02$ | $-1.78\pm 0.02$ |  $\cdots$ | $-2.44\pm 0.02$
J1416+1223 | 0.1232 | $~{}9.59^{+0.32}_{-0.26}$ | $~{}1.57^{+0.21}_{-0.25}$ | $8.15\pm 0.03$ | $-0.51\pm 0.06$ | $-1.68\pm 0.10$ |  $\cdots$ | $-2.47\pm 0.08$
J1418+2102 | 0.0086 | $~{}6.22^{+0.49}_{-0.35}$ | $-1.13^{+0.15}_{-0.16}$ | $7.50\pm 0.03$ | $-0.64\pm 0.01$ | $-1.91\pm 0.01$ | $-3.72\pm 0.01$ | $-2.41\pm 0.01$
J1428+1653 | 0.1817 | $~{}9.56^{+0.15}_{-0.23}$ | $~{}1.22^{+0.26}_{-0.19}$ | $8.28\pm 0.07$ | $-0.58\pm 0.08$ | $-1.79\pm 0.09$ |  $\cdots$ | $-2.44\pm 0.08$
J1429+0643 | 0.1735 | $~{}8.80^{+0.35}_{-0.21}$ | $~{}1.42^{+0.11}_{-0.17}$ | $7.95\pm 0.02$ | $-0.55\pm 0.02$ | $-1.70\pm 0.03$ |  $\cdots$ | $-2.24\pm 0.02$
J1448-0110 | 0.0274 | $~{}7.61^{+0.41}_{-0.24}$ | $~{}0.39^{+0.13}_{-0.14}$ | $8.04\pm 0.02$ | $-0.65\pm 0.02$ | $-1.61\pm 0.01$ | $-3.55\pm 0.03$ | $-2.34\pm 0.01$
J1521+0759 | 0.0943 | $~{}9.00^{+0.29}_{-0.30}$ | $~{}0.95^{+0.16}_{-0.17}$ | $8.66\pm 0.05$ | $-0.37\pm 0.14$ | $-1.62\pm 0.31$ |  $\cdots$ | $-2.68\pm 0.15$
J1525+0757 | 0.0758 | $10.10^{+0.28}_{-0.42}$ | $~{}1.00^{+0.69}_{-0.24}$ | $8.76\pm 0.05$ | $-0.36\pm 0.08$ | $-1.68\pm 0.31$ |  $\cdots$ | $-2.66\pm 0.14$
J1545+0858 | 0.0377 | $~{}7.52^{+0.43}_{-0.26}$ | $~{}0.37^{+0.13}_{-0.17}$ | $7.75\pm 0.02$ | $-0.65\pm 0.00$ | $-1.69\pm 0.01$ |  $\cdots$ | $-2.31\pm 0.01$
J1612+0817 | 0.1491 | $~{}9.78^{+0.28}_{-0.26}$ | $~{}1.58^{+0.28}_{-0.24}$ | $8.48\pm 0.19$ | $-0.43\pm 0.06$ | $-1.78\pm 0.21$ |  $\cdots$ | $-2.49\pm 0.10$

Note. — Columns 2-4: Redshift, stellar mass and SFR taken from Paper I,
respectively. Column 5: metallicity, and columns 6-9: the Ne/O, S/O, Cl/O and
Ar/O abundance ratios.


# Investigating the dependence of collective dynamics on n/p asymmetry for
light nuclei

R. T. deSouza<EMAIL_ADDRESS>Varinderjit Singh  S. Hudan  Department of
Chemistry and Center for Exploration of Energy and Matter, Indiana University
2401 Milo B. Sampson Lane, Bloomington, Indiana 47408, USA  Z. Lin  Department
of Physics and Center for Exploration of Energy and Matter, Indiana University
2401 Milo B. Sampson Lane, Bloomington, Indiana 47408 USA  Department of
Physics, Arizona State University,
450 E. Tyler Mall, Tempe, AZ 85287-1504 USA  C. J. Horowitz  Department of
Physics and Center for Exploration of Energy and Matter, Indiana University
2401 Milo B. Sampson Lane, Bloomington, Indiana 47408, USA

###### Abstract

The dynamics present in the fusion of neutron-rich nuclei is explored through
the comparison of experimental cross-sections at above-barrier energies with
measurements of the interaction cross-section at relativistic energies. The
increase of fusion dynamics with increasing neutron excess is clearly
demonstrated. Experimental cross-sections are compared with the predictions of
a Sao Paulo model using relativistic mean field density distributions and the
impact of different interactions is explored.

###### pacs:

21.60.Jz, 26.60.Gj, 25.60.Pj, 25.70.Jj

Nuclei are extremely interesting quantal systems, which despite a limited
number of constituent particles, manifest collective dynamics. This collective
dynamics is observed in many forms including the giant dipole resonance
Bertrand (1976); Kobayashi  _et al._ (1989), shape coexistence Heyde and Wood
(2011), and fission Meitner and Frisch (1939); Vandenbosch and Huizenga
(1973). Although typically associated with the structure and reactions of mid-
mass and heavy nuclei, collectivity for very light nuclei has recently been
reported Morse  _et al._ (2018). Nuclear fission and nuclear fusion provide
examples in which collective degrees of freedom undergo substantial change as
the reaction proceeds. Of particular interest is the role of collectivity for
neutron-rich nuclei as for these nuclei the dependence of the dynamics on the
asymmetry between the neutron and proton densities can be probed. Fusion
reactions provide a powerful means to assess the response of neutron-rich
nuclei to perturbation. As fusion involves the interplay of the repulsive
Coulomb and attractive nuclear potentials, by examining fusion for an isotopic
chain one probes the neutron density distribution and how that density
distribution evolves as the two nuclei approach and overlap Varinderjit Singh
_et al._ (2017); Vadas  _et al._ (2018). In the following manuscript we
propose a novel perspective for investigating the role of collective dynamics
in fusion. Moreover, using this new perspective we elucidate the dependence of
the fusion dynamics on n/p asymmetry for the first time including the
indication that for light nuclei fusion dynamics is enhanced with increasing
neutron number.

Measurement of the interaction cross-section, $\sigma_{Int.}$ in high energy
collisions is an effective means to invesitgate the spatial extent of the
matter distribution Ozawa  _et al._ (1996). The interaction cross-section in
these measurements is simply defined as the total nuclear reaction cross-
section resulting in a change of either the atomic number (Z) or mass number
(A) of the projectile. Systematic comparison of these cross-sections for
lithium isotopes revealed the halo nature of 11Li Tanihata  _et al._ (1985a,
b). Presented in Fig. 1 are the interaction cross-sections of carbon isotopes
with a carbon target. Measurements for A$\geq$12 were made at E/A $\sim$ 900
MeV at GSI-Darmstadt and utilized the high-resolution fragment separator FRS
to resolve the reaction products Ozawa  _et al._ (2001a, b). These data,
represented as solid symbols in Fig. 1, are supplemented by the results of
earlier measurements at the LBL Bevalac indicated by open symbols Ozawa  _et
al._ (1996). The overall trend observed is an approximately linear increase in
$\sigma_{Int.}$ with neutron excess, (N-Z). At the high incident energy that
these experiments were conducted at, one expects the sudden approximation to
be valid. Hence, the measured interaction cross-section, $\sigma_{Int.}$
provides a direct measure of the extent of the matter distribution. Through
comparison with a Glauber model, the rms matter radii of these nuclides has
been extracted Kanungo  _et al._ (2016).

Closer examination of Fig. 1 provides an indication of the impact of shell
structure on $\sigma_{Int.}$. The dependence of $\sigma_{Int.}$ on neutron
excess for 12$\leq$A$\leq$14 is weak as is the dependence for
16$\leq$A$\leq$18\. Between 14C and 16C one observes a jump in $\sigma_{Int.}$
from a value of $\sim$850 mb to $\sim$1050 mb. This increase reflects the
completion of the 1p${}_{\frac{1}{2}}$ with N=8 and the population of the
1d${}_{\frac{5}{2}}$ shell. This observation is significant as it indicates
that the shell structure of the neutron-rich isotopes is accessible through
measurement of $\sigma_{Int.}$ for an isotopic chain.

Figure 1:  Comparison of the interaction cross-section $\sigma_{Int.}$ for
various carbon isotopes with the prediction of the average fusion cross-
section at above-barrier energies using a RMF-SP model. See text for details.
Figure 2:  Neutron density distributions predicted by the RMF model with the
FSUGOLD interaction for carbon isotopes.

Juxtaposed with the measured interaction cross-sections in Fig. 1 are the
results of calculations of the fusion cross-section at energies just above the
fusion barrier. These calculations were performed at energies of ECM = 1-2
MeV/A and utilize the Sao Paulo model for calculating the fusion cross-section
Gasques  _et al._ (2004). In this model the density distributions of the
colliding nuclei are assumed to be frozen during the fusion process thus the
calculated cross-sections reflect the size of the colliding nuclei. Moreover,
use of a common target nucleus allows one to assess the change in the size of
the projectile nucleus with increasing neutron excess. The density
distributions used in the Sao Paulo calculations were determined using a
relativistic mean field (RMF) model Serot and Walecka (1986); Ring (1996). In
order to investigate the sensitivity of the calculated cross-sections to the
interaction used in the RMF calculations, the RMF calculations were performed
using two sets of interactions FSUGOLD and NL3 Shen  _et al._ (2011). In
contrast to the widely used NL3 interaction, the FSUGOLD corresponds to a
softer interaction Fattoyev  _et al._ (2010). To facilitate the comparison of
the fusion cross-sections predicted by the RMF-SP model with $\sigma_{Int.}$
we have calculated the average fusion cross-section over the interval 14
MeV$\leq$ECM$\leq$17 MeV and designate this quantity
$<$$\sigma_{fusion}$$>$${}_{14}^{17}$. Although the calculations with the
FSUGOLD interaction manifest a consistently larger fusion cross-section,
nonetheless both RMF-SP calculations exhibit a similar dependence on neutron
number. For N$<$Z the value of $<$$\sigma_{fusion}$$>$${}_{14}^{17}$ is
approximately constant while for N$>$Z it increases approximately linearly
with (N-Z). The slope of the predicted cross-sections for the two interactions
shown is to first order the same indicating that while the absolute size of
the nucleus depends on the interaction used in the RMF model, the increase in
size with increasing N is relatively insensitive to the interaction utilized.
Moreover for N$>$Z, the slope of the predicted above-barrier fusion cross-
section is very similar to that for $\sigma_{Int.}$. This similarity of the
two slopes arises from the fact that the $\sigma_{Int.}$ measures the size of
the nucleus and the RMF-SP with the frozen density distributions is
intrinsically related to the same quantity. As such, the quantity
$\sigma_{Int.}$ provides a key reference from which to examine fusion
dynamics.

Figure 3:  Proton density distributions predicted by the RMF model with the
FSUGOLD interaction for carbon isotopes.

In Fig. 2 the density distributions for neutrons predicted by the RMF model
for the various carbon isotopes are displayed. The density distributions for
N$\neq$Z are compared with that of 12C for reference. As expected, the tail of
the neutron density distribution extends further for the more neutron-rich the
isotope. The value of calculating these neutron density distributions for an
isotopic chain lies in the ability to examine the systematic dependence on
neutron number. The evolution of the nuclear size on neutron number may have
different sensitivity to the model uncertainties as compared to the absolute
size. Presented in Fig. 3 are the density distributions for protons predicted
by the RMF model for the different isotopes of carbon. While the distributions
are all quite close as might be expected, as the isotope becomes more neutron-
rich the tail of the proton distribution extends slightly further out. This
change in the proton distribution is due to the attractive nuclear force of
the valence neutrons. The dependence of the charge radii on neutron number for
the carbon isotopic chain has recently been determined through measurement of
the charge changing cross-section Kanungo  _et al._ (2016).

Figure 4:  Fusion excitation functions for 10-15C + 12c. Experimental data are
compared with the results of a RMF-SP model using FSUGOLD (left column) and
NL3 (right column). The fusion excitation function for 13C (not shown) is
comparable to that of 14C Almaraz-Calderon  _et al._ (2015); Carnelli (2014).

To investigate the evolution of fusion dynamics with increasing neutron number
we examine the fusion cross-section for AC+ 12C at near barrier energies.
Using a novel active target approach the fusion excitation functions for these
reactions was measured by the ANL group Carnelli  _et al._ (2014). This active
target approach is particularly well suited to studying reactions with low
intensity beams and allowed measurement of the fusion excitation function with
beam intensities as low as 500 ions/s. Depicted in Fig. 4 (both columns) are
the fusion cross-section data for 10-15C. The data for 10,15C have been taken
from Carnelli  _et al._ (2015). Using the same approach though at higher beam
intensity, 12,14C cross-sections were also measured Almaraz-Calderon  _et al._
(2015); Carnelli (2014) and are shown in Fig. 4. The fusion excitation
functions observed for 12,13C + 12C are in good agreement with those published
in the literature Kovar  _et al._ (1979); Dayras  _et al._ (1976). The
measured excitation functions manifest the expected dependence indicative of a
barrier driven process. The experimental data are compared with the results of
the RMF-SP model with the FSUGOLD and NL3 interactions depicted in the left
and right columns respectively. While in the case of 10C the models
overpredict the experimental results, in the remainder of the cases the
agreement is reasonable. Close comparison of the left and right columns
indicates that the calculations with FSUGOLD consistently predict larger
cross-sections than those with NL3, consistent with the observation in Fig. 1.
It is noteworthy though that this increase in the cross-section for FSUGOLD as
compared to NL3 is typical of the entire above-barrier regime. In Fig. 4 the
blue bar indicates the value of the average cross-section as well as the
energy interval over which the average was calculated. For 12-15C the average
cross-section is clearly representative of the above-barrier cross-section. In
the case of 10C, however there is significant variation in the measured cross-
sections and the average cross-section calculated is more sensitive to the
choice of energy interval.

In Fig. 5 the average above-barrier fusion cross-sections for the carbon
isotopic chain are compared with $\sigma_{Int.}$. One observes that for 12C
the fusion cross-section and $\sigma_{Int.}$ are essentially the same. For
N$>$Z however, the fusion cross-section depends more strongly on neutron
excess than $\sigma_{Int.}$ does. Since the dependence of $\sigma_{Int.}$ on
increasing neutron number indicates the inherent growth in the size of the
neutron density distribution with increasing neutron number, the cross-section
in the case of fusion, above that of $\sigma_{Int.}$ reflects the impact of
dynamics in the fusion process. The quantity
($<$$\sigma_{fusion}$$>$${}_{14}^{17}$-$\sigma_{Int.}$) can be viewed as a
measure of the fusion dynamics. Moreover, the increase in this quantity,
dictated by the larger slope for fusion as compared to $\sigma_{Int.}$,
indicates that this dynamics evolves with increasing neutron excess.

It is also instructive to compare the behavior of the fusion data with the
results of the RMF-SP model. The experimental data indicates a stronger
dependence on neutron excess than the RMF-SP model independent of the
interaction chosen. This comparison thus also indicates an increased role for
dynamics with increasing neutron excess. Thus, the increased role of fusion
dynamics with neutron excess is realized in two independent ways.
Juxtaposition of the $<$$\sigma_{fusion}$$>$${}_{14}^{17}$ with
$\sigma_{Int.}$, namely a data to data comparison, indicates enhanced fusion
dynamics with increasing neutron excess. This result is supported by the
comparison of $<$$\sigma_{fusion}$$>$${}_{14}^{17}$ with the cross-sections
predicted by the RMF-SP model.

Although the span of neutron excess for the fusion data presented is presently
limited, the new generation of radioactive beam facilities allows one to
extend these measurements to even more neutron-rich carbon isotopes.
Measurement of fusion with beams of 16,17C and possibly 18,19C at FRIB FRIB
is envisoned. Similar measurement for the oxygen isotopic chain extending
nearly to the neutron-drip line is also possible.

Examination of the fusion cross-section at above-barrier energies for an
isotopic chain is a powerful tool. Comparison of fusion cross-sections just
above the barrier with the interaction cross-section, $\sigma_{Int.}$, at high
energies where the sudden approximation is valid allows extraction of not just
the fusion dynamics but the dependence of the dynamics on neutron excess.
Investigating this dynamics for the most neutron-rich nuclei accessible could
provide valuable insight into the dynamics of extremely asymmetric nuclear
matter.

Figure 5:  Comparison of the average fusion cross-section
$<$$\sigma_{fusion}$$>$${}_{14}^{17}$ with the interaction cross-section
$\sigma_{Int.}$ for various carbon isotopes. Also shown are the predictions of
the average fusion cross-section using a RMF-SP model.

###### Acknowledgements.

This work was supported by the U.S. Department of Energy under Grant No. DE-
FG02-88ER-40404 (Indiana University). CJH is supported in part by U.S. DOE
grants DE-FG02-87ER40365 and DE-SC0018083. ZL gratefully acknowledges support
from National Science Foundation under PHY-1613708 (Arizona State University).

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# On the $\mathrm{GL}(n)$-module structure of Lie nilpotent associative
relatively free algebras

Elitza Hristova  Institute of Mathematics and Informatics, Bulgarian Academy
of Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria
<EMAIL_ADDRESS>
###### Abstract.

Let $K\left\langle X\right\rangle$ denote the free associative algebra
generated by a set $X=\\{x_{1},\dots,x_{n}\\}$ over a field $K$ of
characteristic $0$. Let $I_{p}$, for $p\geq 2$, denote the two-sided ideal in
$K\left\langle X\right\rangle$ generated by all commutators of the form
$[u_{1},\dots,u_{p}]$, where $u_{1},\dots,u_{p}\in K\left\langle
X\right\rangle$. We discuss the $\mathrm{GL}(n,K)$-module structure of the
quotient $K\left\langle X\right\rangle/I_{p+1}$ for all $p\geq 1$ under the
standard diagonal action. We give a bound on the values of partitions
$\lambda$ such that the irreducible $\mathrm{GL}(n,K)$-module $V_{\lambda}$
appears in the decomposition of $K\left\langle X\right\rangle/I_{p+1}$ as a
$\mathrm{GL}(n,K)$-module. As an application, we take $K={\mathbb{C}}$ and we
consider the algebra of invariants $({\mathbb{C}}\left\langle
X\right\rangle/I_{p+1})^{G}$ for $G=\mathrm{SL}(n,{\mathbb{C}})$,
$\mathrm{O}(n,{\mathbb{C}})$, $\mathrm{SO}(n,{\mathbb{C}})$, or
$\mathrm{Sp}(2s,{\mathbb{C}})$ (for $n=2s$). By a theorem of Domokos and
Drensky, $({\mathbb{C}}\left\langle X\right\rangle/I_{p+1})^{G}$ is finitely
generated. We give an upper bound on the degree of generators of
$({\mathbb{C}}\left\langle X\right\rangle/I_{p+1})^{G}$ in a minimal
generating set. In a similar way, we consider also the algebra of invariants
$({\mathbb{C}}\left\langle X\right\rangle/I_{p+1})^{G}$, where
$G=\mathrm{UT}(n,{\mathbb{C}})$, and give an upper bound on the degree of
generators in a minimal generating set. These results provide useful
information about the invariants in ${\mathbb{C}}\left\langle
X\right\rangle^{G}$ from the point of view of Classical Invariant Theory. In
particular, for all $G$ as above we give a criterion when a $G$-invariant of
${\mathbb{C}}\left\langle X\right\rangle$ belongs to $I_{p}$.

###### Key words and phrases:

Lie nilpotent associative relatively free algebras, product of commutators,
$\mathrm{GL}(n)$-module structure, invariant theory, Hilbert series.

###### 2010 Mathematics Subject Classification:

13A50; 16R10; 20G05

Partially supported by Grant KP-06 N32/1 of the Bulgarian National Science
Fund.

##  1\. Introduction

Let $V$ be a finite dimensional vector space over a field $K$ of
characteristic zero and let $X=\\{x_{1},\dots,x_{n}\\}$ be a basis for $V$. We
consider the free associative algebra $K\left\langle X\right\rangle$ generated
by $X$, which can be naturally identified with the tensor algebra $T(V)$ of
$V$. We define inductively left-normed long commutators in $K\left\langle
X\right\rangle$ in the usual way: $[u_{1},u_{2}]=u_{1}u_{2}-u_{2}u_{1}$ and
for $i\geq 3$, we have $[u_{1},\dots,u_{i}]=[[u_{1},\dots,u_{i-1}],u_{i}]$,
where $u_{1},\dots,u_{i}$ are arbitrary elements from $K\left\langle
X\right\rangle$. The commutator $[u_{1},\dots,u_{i}]$ is called a commutator
of length $i$ or simply an $i$-commutator. Furthermore, we call a commutator
pure if it is of the form $[x_{i_{1}},\dots,x_{i_{j}}]$ for some
$x_{i_{1}},\dots,x_{i_{j}}\in X$.

The lower central series of $K\left\langle X\right\rangle$ is the descending
filtration

$L_{1}\supseteq L_{2}\supseteq\dots\supseteq L_{p}\supseteq\cdots,$

defined inductively by $L_{1}=K\left\langle X\right\rangle$ and
$L_{p}=[L_{p-1},K\left\langle X\right\rangle]$ for $p\geq 2$. Then $L_{p}$ is
the Lie ideal in $K\left\langle X\right\rangle$ generated by all commutators
of length $p$. For $p\geq 2$, let $I_{p}=K\left\langle X\right\rangle\cdot
L_{p}$ denote the two-sided associative ideal in $K\left\langle
X\right\rangle$ generated by $L_{p}$. In other words, $I_{p}$ is the T-ideal
generated by the commutator $[x_{1},\dots,x_{p}]$. Following the traditions in
the theory of groups and Lie algebras we denote by $\mathfrak{N}_{p}$ the
variety of Lie nilpotent of class $\leq p$ associative algebras (i.e.,
$\mathfrak{N}_{p}$ is the variety of associative algebras satisfying the
polynomial identity $[x_{1},\dots,x_{p+1}]=0$). The algebra $K\left\langle
X\right\rangle/I_{p+1}$ is the relatively free algebra of rank $n$ in the
variety $\mathfrak{N}_{p}$. We denote it in the standard way, i.e.,
$F_{n}(\mathfrak{N}_{p})=K\left\langle X\right\rangle/I_{p+1}$. When $p=1$,
$F_{n}(\mathfrak{N}_{1})$ is equal to the symmetric algebra $S(V)$ of $V$.
When $p=2$, $F_{n}(\mathfrak{N}_{2})$ is the relatively free algebra of rank
$n$ in the variety generated by the Grassmann algebra.

The group $\mathrm{GL}(n):=\mathrm{GL}(V,K)$ acts on the vector space $V$ with
basis $X$ and this action is extended in a natural way to the diagonal action
of $\mathrm{GL}(n)$ on $K\left\langle X\right\rangle$. Since for any $p$,
$I_{p}$ is a $\mathrm{GL}(n)$-submodule of $K\left\langle X\right\rangle$, the
algebra $F_{n}(\mathfrak{N}_{p})$ is also a $\mathrm{GL}(n)$-module. The
$\mathrm{GL}(n)$-module structure of $F_{n}(\mathfrak{N}_{p})$ is known for
$p=1,2,3,4$ (see [18] for $p=3$ and [17] for $p=4$). Our first goal is to give
some results on the $\mathrm{GL}(n)$-module structure of
$F_{n}(\mathfrak{N}_{p})$ for any $p$. More precisely, let
$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{n})\in{\mathbb{Z}}^{n}_{\geq
0}$ with $\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{n}\geq 0$ be a non-
negative integer partition and let $V_{\lambda}$ denote the irreducible
$\mathrm{GL}(n)$-module corresponding to the partition $\lambda$. In Section
3, we give a bound on the values of $\lambda$ such that $V_{\lambda}$ appears
in the decomposition of $F_{n}(\mathfrak{N}_{p})$ as a
$\mathrm{GL}(n)$-module. The main results in this direction are Theorem 3.3
and Corollary 3.4. The proofs are based on several known results on inclusions
of products of commutators, which are recalled in Section 2, and one new
result on inclusions of products of commutators, which is stated in Section 3.
In the end of Section 3, we consider separately the case $n=3$. As an
application of our results on the $\mathrm{GL}(n)$-module structure of
$F_{n}(\mathfrak{N}_{p})$, in Section 4 we set $K={\mathbb{C}}$ and consider
the algebra of invariants $F_{n}(\mathfrak{N}_{p})^{G}$ where $G$ is one of
the classical complex groups $\mathrm{SL}(n):=\mathrm{SL}(V,{\mathbb{C}})$,
$\mathrm{O}(n):=\mathrm{O}(V,{\mathbb{C}})$,
$\mathrm{SO}(n):=\mathrm{SO}(V,{\mathbb{C}})$, and
$\mathrm{Sp}(2s):=\mathrm{Sp}(V,{\mathbb{C}})$ (the last in the case $n=2s$).
By a theorem of Domokos and Drensky ([6], [8]), $F_{n}(\mathfrak{N}_{p})^{G}$
is finitely generated. In Section 4, we give an upper bound on the degree of
the generators of $F_{n}(\mathfrak{N}_{p})^{G}$ in a minimal generating set
(Corollary 4.4). The advantage of our upper bound is that it is very explicit.
We also give a general form for the Hilbert series
$H(F_{n}(\mathfrak{N}_{p})^{G},t)$, where $G$ is again one of
$\mathrm{SL}(n)$, $\mathrm{O}(n)$, $\mathrm{SO}(n)$, and $\mathrm{Sp}(2s)$,
and show that $F_{n}(\mathfrak{N}_{p})^{\mathrm{SL}(n)}$ and
$F_{n}(\mathfrak{N}_{p})^{\mathrm{Sp}(2s)}$ are finite-dimensional algebras.
In a similar way we consider also the algebra of invariants
$F_{n}(\mathfrak{N}_{p})^{G}$, where
$G=\mathrm{UT}(n):=\mathrm{UT}(n,{\mathbb{C}})$ is the unitriangular group,
i.e., the subgroup of $\mathrm{GL}(n)$ consisting of upper triangular matrices
with $1$’s on the diagonal. It is known (see [7]) that
$F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)}$ is finitely generated for any
$p\geq 1$. In Section 4 again, we give an upper bound on the degree of the
generators in a minimal generating set (Corollary 4.7). The results from
Section 4 are then translated in the language of Classical Invariant Theory in
Section 5. In particular, for each $G$ as above, we give a criterion when a
$G$-invariant of $T(V)$ belongs to $I_{p}$.

##  2\. Preliminaries

In this section we summarize several results on inclusions of products of
commutators, which will be extensively used in the next sections. We state all
results for a field of characteristic $0$, even though most of the results
hold for more general fields.

The oldest result in this direction is the following lemma due to Jennings
from 1947.

######  Lemma 2.1 ([14]).

Let $p\geq 3$. Then the product of any two $p$-commutators is an element of
$I_{p+1}$.

###### Proof.

Consider the product $[c_{1},x][c_{2},y]$ where $c_{1}$ and $c_{2}$ are
arbitrary $p-1$-commutators (i.e., $c_{1}$ and $c_{2}$ belong to $L_{p-1}$)
and $x,y\in K\left\langle X\right\rangle$. We will show that
$[c_{1},x][c_{2},y]$ can be expressed using elements of $I_{p+1}$. We consider
the expression

$\displaystyle[c_{2},c_{1}y,x]=-[[c_{1}y,c_{2}],x]=-[c_{1}[y,c_{2}],x]-[[c_{1},c_{2}]y,x]$
$\displaystyle=c_{1}[c_{2},y,x]+[c_{1},x][c_{2},y]-[c_{1},c_{2}][y,x]-[c_{1},c_{2},x]y.$

Hence,

(1)
$\displaystyle[c_{1},x][c_{2},y]=[c_{2},c_{1}y,x]+[c_{1},c_{2}][y,x]+[c_{1},c_{2},x]y-c_{1}[c_{2},y,x].$

Since $[c_{1},c_{2}]\in L_{2p-2}\subset I_{2p-2}$ and for $p\geq 3$,
$I_{2p-2}\subseteq I_{p+1}$ the lemma follows. ∎

Variations of the above proof lead to the following variations of the lemma of
Jennings:

######  Lemma 2.2.

Let $p\geq 3$ and let $3\leq r\leq p$. The product of an $r$-commutator and a
$p$-commutator is an element of $I_{p+1}$.

###### Proof.

Let $c_{1}$ be an $r-1$-commutator and let $c_{2}$ be a $p-1$-commutator. Let
$x$,$y$ be arbitrary elements of $K\left\langle X\right\rangle$. From the
proof of Lemma 2.1 we know that

$\displaystyle[c_{1},x][c_{2},y]=[c_{2},c_{1}y,x]+[c_{1},c_{2}][y,x]+[c_{1},c_{2},x]y-c_{1}[c_{2},y,x].$

Furthermore, $[c_{1},c_{2}]\in L_{p+r-2}\subset I_{p+r-2}$. For $p\geq 3$ and
$3\leq r\leq p$, we have $I_{p+r-2}\subseteq I_{p+1}$, hence the lemma
follows. ∎

######  Lemma 2.3.

Let $p\geq 2$ and let $2\leq r\leq p$. Let $y$ be an arbitrary element of
$K\left\langle X\right\rangle$. Let $[c_{1},y]\in L_{r}$ and $[c_{2},y]\in
L_{p}$. Then,

$[c_{1},y][c_{2},y]\in I_{p+1}.$

In particular, for $x,y,z_{1},\dots,z_{p-1}\in K\left\langle X\right\rangle$
we have

$[x,y][z_{1},\dots,z_{p-1},y]\in I_{p+1}.$

###### Proof.

From Equation(1) we obtain that

$\displaystyle[c_{1},y][c_{2},y]=[c_{2},c_{1}y,y]+[c_{1},c_{2}][y,y]+[c_{1},c_{2},y]y-c_{1}[c_{2},y,y],$

which proves the statement. ∎

The next result improves considerably Lemma 2.1 and Lemma 2.2 (but not Lemma
2.3). It was first proved by Latyshev in 1965 and then was independently
rediscovered by Gupta and Levin in 1983.

######  Theorem 2.4 ([15], [12]).

For all $m_{1}$ and $m_{2}$

$I_{m_{1}}I_{m_{2}}\subseteq I_{m_{1}+m_{2}-2}.$

When one of the integers $m_{1}$ and $m_{2}$ is odd, an even stronger
inclusion holds. The following theorem was proved several times with different
restrictions on the characteristic of $K$. Here we cite only two of the
sources. The most general result (i.e., for most general fields $K$) was
proved in [5].

######  Theorem 2.5 ([1], [5]).

If $m_{1}$ or $m_{2}$ is odd then

$I_{m_{1}}I_{m_{2}}\subseteq I_{m_{1}+m_{2}-1}.$

The proof of Theorem 2.5 given in [1] is based on the following statement,
which is of interest of its own.

######  Theorem 2.6 ([1]).

$[L_{m_{1}},I_{m_{2}}]\subseteq L_{m_{1}+m_{2}},$

whenever $m_{2}$ is odd.

The next statement is derived from Theorems 2.4 and 2.5 and can be found in
[3] and [5].

######  Theorem 2.7.

For every $l\geq 2$ and for every choice of positive integers
$m_{1},\dots,m_{l}$ one of the following holds:

  * (i)

If $k$ indices among $m_{1},\dots,m_{l}$ are odd with $0\leq k<l$ then

$I_{m_{1}}\cdots I_{m_{l}}\subseteq I_{m_{1}+\dots+m_{l}-2l+k+2}.$

The number $m_{1}+\dots+m_{l}-2l+k+2$ is always even.

  * (ii)

If all indices are odd then

$I_{m_{1}}\cdots I_{m_{l}}\subseteq I_{m_{1}+\dots+m_{l}-l+1}.$

The number $m_{1}+\dots+m_{l}-l+1$ is always odd.

In the end of this section, let us fix $n=3$, i.e., let us consider the free
associative algebra $K\left\langle x_{1},x_{2},x_{3}\right\rangle$. Then the
following result holds:

######  Theorem 2.8 ([16]).

Consider the algebra $K\left\langle x_{1},x_{2},x_{3}\right\rangle$. Then

$I_{m_{1}}I_{m_{2}}\subseteq I_{m_{1}+m_{2}-1},$

for all integers $m_{1},m_{2}\geq 2$.

It is worth pointing out, that the restriction $n=3$ in Theorem 2.8 is
essential. It was proved in [4] that in general for even integers $m_{1}$ and
$m_{2}$ we have

$I_{m_{1}}I_{m_{2}}\nsubseteq I_{m_{1}+m_{2}-1}.$

##  3\. The $\mathrm{GL}(n)$-module structure of $F_{n}(\mathfrak{N}_{p})$

In this section we discuss the $\mathrm{GL}(n)$-module structure of
$F_{n}(\mathfrak{N}_{p})$ for any $p\geq 1$. First we introduce some
notations. Let $B_{n}$ denote the subalgebra of $K\left\langle X\right\rangle$
generated by all pure commutators $[x_{i_{1}},\dots,x_{i_{r}}]$ with $r\geq
2$. The elements of $B_{n}$ are called proper polynomials. Clearly, $B_{n}$ is
a $\mathrm{GL}(n)$-submodule of $K\left\langle X\right\rangle$. Let
$\varphi_{p}:K\left\langle X\right\rangle\rightarrow F_{n}(\mathfrak{N}_{p})$
denote the natural surjective $K$-algebra homomorphism. Then
$\varphi_{p}(B_{n})$ is a $\mathrm{GL}(n)$-submodule of
$F_{n}(\mathfrak{N}_{p})$. By a theorem of Drensky ([9]), for any $p\geq 1$ we
have the following decomposition as $\mathrm{GL}(n)$-modules:

(2) $\displaystyle
F_{n}(\mathfrak{N}_{p})\cong_{\mathrm{GL}(n)}S(V)\otimes\varphi_{p}(B_{n}),$

where again $S(V)$ denotes the symmetric algebra of $V$.

We start by proving the following lemma, which improves considerably Lemma 2.3
from Section 2.

######  Lemma 3.1.

Let $m_{1}$ and $m_{2}$ be even integers. Let $[c_{1},y]\in L_{m_{1}}$ and
$[c_{2},y]\in L_{m_{2}}$, where $y$ is an arbitrary element from
$K\left\langle X\right\rangle$. Then,

$[c_{1},y][c_{2},y]\in I_{m_{1}+m_{2}-1}.$

###### Proof.

From Equation (1) we have that

$\displaystyle[c_{1},y][c_{2},y]=[c_{2},c_{1}y,y]+[c_{1},c_{2}][y,y]+[c_{1},c_{2},y]y-c_{1}[c_{2},y,y].$

We need to show that all non-zero terms on the right side of the above
equation belong to $I_{m_{1}+m_{2}-1}$. This clearly holds for
$[c_{1},c_{2},y]y$. Consider the term $c_{1}[c_{2},y,y]$. $c_{1}$ is a
commutator of odd length $m_{1}-1$, whereas $[c_{2},y,y]$ is a commutator of
odd length $m_{2}+1$. By Theorem 2.5, $c_{1}[c_{2},y,y]\in I_{m_{1}+m_{2}-1}$.
It remains to consider the term $[c_{2},c_{1}y,y]$. We have that $c_{2}\in
L_{m_{2}-1}$ and $c_{1}y\in I_{m_{1}-1}$. By Theorem 2.6,

$[c_{2},c_{1}y]\in L_{m_{1}+m_{2}-2}.$

This completes the proof.

∎

######  Corollary 3.2.

Let $m_{1},\dots,m_{l}$ be even integers, let $y\in K\left\langle
X\right\rangle$, and let $[c_{i},y]\in L_{m_{i}}$ for $i=1,\dots,l$. Then

$[c_{1},y]\cdots[c_{l},y]\in I_{m_{1}+\dots+m_{l}-l+1}.$

###### Proof.

If $l$ is even, for $k=1,\dots,\frac{l}{2}$, Lemma 3.1 implies that
$[c_{2k-1},y][c_{2k},y]\in I_{m_{2k-1}+m_{2k}-1}$. Moreover, the number
$m_{2k-1}+m_{2k}-1$ is odd. Let us denote it by $j_{k}$. Therefore, by Theorem
2.7(ii) we obtain that

$([c_{1},y][c_{2},y])\cdots([c_{l-1},y][c_{l},y])\in I_{j_{1}}\cdots
I_{j_{l/2}}\subseteq I_{j_{1}+\dots+j_{l/2}-l/2+1}=I_{m_{1}+\dots+m_{l}-l+1}.$

Next, if $l$ is odd, we can apply the above considerations for $l-1$. We
obtain that

$[c_{1},y]\cdots[c_{l-1},y]\in I_{m_{1}+\dots+m_{l-1}-l+2}.$

Furthermore, $m_{1}+\dots+m_{l-1}-l+2$ is an odd number. Hence, by Theorem
2.5,

$[c_{1},y]\cdots[c_{l-1},y][c_{l},y]\in
I_{m_{1}+\dots+m_{l-1}-l+2}I_{m_{l}}\subseteq I_{m_{1}+\dots+m_{l}-l+1}.$

∎

We now give the following estimate for the $\mathrm{GL}(n)$-module structure
of $\varphi_{p}(B_{n})$ for any $p\geq 1$.

######  Theorem 3.3.

Let $p\geq 1$. Suppose that

$\varphi_{p}(B_{n})\cong_{\mathrm{GL}(n)}\bigoplus_{\lambda}k_{\lambda}V_{\lambda},$

where again $V_{\lambda}$ denotes the irreducible $\mathrm{GL}(n)$-module
corresponding to the non-negative integer partition
$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{n})$. Then $\lambda_{1}\leq
p-1$.

###### Proof.

For $p=1$ the statement is trivially true. Therefore, we give a proof for the
case $p\geq 2$. The algebra $K\left\langle X\right\rangle$ has a
${\mathbb{Z}}_{\geq 0}^{n}$-grading given by the weight space decomposition as
a $\mathrm{GL}(n)$-module. More precisely,

$K\left\langle
X\right\rangle=\bigoplus_{\mu=(\mu_{1},\dots,\mu_{n})\in{\mathbb{Z}}_{\geq
0}^{n}}W(\mu),$

where $W(\mu)$ is the weight space corresponding to the weight $\mu$. This
means that, if $g=\mathrm{diag}(\xi_{1},\dots,\xi_{n})\in\mathrm{GL}(n)$ and
$w\in W(\mu)$, then

$g(w)=\xi_{1}^{\mu_{1}}\cdots\xi_{n}^{\mu_{n}}w.$

Therefore, a basis for $W(\mu)$ is given by the monomial
$x_{1}^{\mu_{1}}\cdots x_{n}^{\mu_{n}}$ and all its distinct permutations.

Consequently, $B_{n}$ and $\varphi_{p}(B_{n})$ also have ${\mathbb{Z}}_{\geq
0}^{n}$-gradings given respectively by

$B_{n}=\bigoplus_{\mu=(\mu_{1},\dots,\mu_{n})\in{\mathbb{Z}}_{\geq
0}^{n}}W(\mu)\cap B_{n};$
$\varphi_{p}(B_{n})=\bigoplus_{\mu=(\mu_{1},\dots,\mu_{n})\in{\mathbb{Z}}_{\geq
0}^{n}}\varphi_{p}(W(\mu)\cap B_{n}).$

Notice that $\varphi_{p}(W(\mu)\cap B_{n})$ consists of sums of products of
pure commutators such that the variable $x_{i}$ appears in each product
exactly $\mu_{i}$ times for all $i=1,\dots,n$.

Consider now a partition
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{n})$ such that
$V_{\lambda}$ appears in the decomposition of $\varphi_{p}(B_{n})$. Therefore,
$\varphi_{p}(W(\lambda)\cap B_{n})\neq\emptyset$ and hence
$\varphi_{p}(B_{n})$ contains an element $w$ such that $w$ is a sum of
products of pure commutators and the variable $x_{1}$ appears in each product
exactly $\lambda_{1}$ times. We will estimate the maximal possible number of
appearances of $x_{1}$ in an element of $\varphi_{p}(B_{n})$. For brevity, let
us denote this maximal number by $N$.

Let $u$ be a product of pure commutators, such that the number of appearances
of $x_{1}$ in $u$ is equal to $N$. We notice the following property, which was
also considered in [3]. If $u\in I_{k}$ for some $2\leq k\leq p$ and if we
permute the commutators which participate in $u$ then we obtain again a
product of pure commutators which belongs to $I_{k}$. This follows from the
fact that for all $i,j\geq 1$, $[L_{i},L_{j}]\subseteq L_{i+j}$. Therefore,
without loss of generality, we may consider that $u=u_{1}u_{2}u_{3}$, such
that the following conditions are fulfilled: $u_{1}$ is a product of $l_{1}$
pure commutators of lengths respectively $m_{1},\dots,m_{l_{1}}$, such that
each $m_{i}$ is odd; $u_{2}$ is a product of $l_{2}$ pure commutators of
lengths respectively $n_{1},\dots,n_{l_{2}}$, such that all $n_{i}$ are even
and the last position in each commutator is equal to $x_{1}$; $u_{3}$ is a
product of $l_{3}$ pure commutators of lengths respectively
$r_{1},\dots,r_{l_{3}}$, such that all $r_{i}$ are even and the last position
in each commutator is different from $x_{1}$. Let us denote the maximal number
of appearances of $x_{1}$ in $u_{i}$ by $N_{i}$, for $i=1,2,3$. We need to
estimate $N_{1}$, $N_{2}$, and $N_{3}$.

By Theorem 2.7 (ii), $u_{1}\in I_{m_{1}+\dots+m_{l_{1}}-l_{1}+1}$. We set
$p_{1}=m_{1}+\dots+m_{l_{1}}-l_{1}+1$. The maximal number of appearances of
$x_{1}$ in $u_{1}$ is $N_{1}=m_{1}+\dots+m_{l_{1}}-l_{1}$ (since in a pure
$m_{i}$-commutator the variable $x_{1}$ can appear at most $m_{i}-1$ times).
Therefore, $N_{1}=p_{1}-1$.

Next, we consider the element $u_{2}$. By Corollary 3.2, $u_{2}\in
I_{n_{1}+\dots+n_{l_{2}}-{l_{2}}+1}$. Similarly as above, we set
$p_{2}=n_{1}+\dots+n_{l_{2}}-{l_{2}}+1$ and we obtain that $N_{2}=p_{2}-1$.

Finally, we take $u_{3}$. By Theorem 2.7 (i), we have that $u_{3}\in
I_{r_{1}+\dots+r_{l_{3}}-2l_{3}+2}$. We set
$p_{3}=r_{1}+\dots+r_{l_{3}}-2l_{3}+2$. Since in each of the commutators in
$u_{3}$ the last position is different from $x_{1}$ we have that the maximal
number of appearances of $x_{1}$ in $u_{3}$ is
$N_{3}=r_{1}+\dots+r_{l_{3}}-2l_{3}$. Hence, $N_{3}=p_{3}-2$.

In short, the above implies that

$u=u_{1}u_{2}u_{3}\in I_{p_{1}}I_{p_{2}}I_{p_{3}}\subseteq I_{p}.$

Notice that $p_{1}$ is always an odd number, whereas $p_{3}$ is always an even
number. Therefore, Theorem 2.5 implies that

$I_{p_{1}}I_{p_{2}}\subseteq I_{p_{1}+p_{2}-1}.$

Hence, by Theorem 2.4 we obtain that

$I_{p_{1}}I_{p_{2}}I_{p_{3}}\subseteq I_{p_{1}+p_{2}-1}I_{p_{3}}\subseteq
I_{p_{1}+p_{2}+p_{3}-3}\subseteq I_{p}.$

Therefore, $p_{1}+p_{2}+p_{3}-3\leq p$. Thus,
$N=N_{1}+N_{2}+N_{3}=p_{1}-1+p_{2}-1+p_{3}-2\leq p-1$. This proves the
statement.

∎

Theorem 3.3 together with Equation (2) lead to the following corollary.

######  Corollary 3.4.

Let $p\geq 1$. Suppose that

$F_{n}(\mathfrak{N}_{p})\cong_{\mathrm{GL}(n)}\bigoplus_{\lambda}m_{\lambda}V_{\lambda},$

where again $V_{\lambda}$ denotes the irreducible $\mathrm{GL}(n)$-module
corresponding to the non-negative integer partition
$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{n})$. Then $\lambda_{2}\leq
p-1$.

###### Proof.

For $p=1$ the statement is again trivially true. For $p\geq 2$, the proof is a
standard task on branching rules. We apply Pieri’s branching rule (also known
as Young’s rule) to Equation (2) and then we use Theorem 3.3. One may read the
description of Pieri’s rule in e.g. [11]. ∎

We notice that the upper bound on $\lambda_{1}$ which is obtained in Theorem
3.3 is exact for all known cases. More precisely, using the decomposition of
$\varphi_{p}(B_{n})$ over $\mathrm{GL}(n)$, which is known for $p=2,3,4$, we
see that for $p=2$, $\lambda_{1}\leq 1=p-1$ and for $p=3$, $\lambda_{1}\leq
2=p-1$. For $p=4$, we have that $\lambda_{1}\leq 3=p-1$.

For $p=5$, Theorem 3.3 gives that $\lambda_{1}\leq 4$.

In the end of the section, we would like to consider separately the case
$n=3$, i.e., we take the free associative algebra $K\left\langle
x_{1},x_{2},x_{3}\right\rangle$ and the relatively free algebra
$F_{3}(\mathfrak{N}_{p})$ of rank $3$ in the variety $\mathfrak{N}_{p}$. Then,
we have the stronger theorem on inclusions of products of commutators (Theorem
2.8) which leads to a much shorter proof of Theorem 3.3. For completeness of
the exposition, we present this proof below.

Shorter proof of Theorem 3.3 for $n=3$.

Consider a partition $\lambda$ such that $V_{\lambda}$ appears in the
decomposition of $\varphi_{p}(B_{3})$ as a $\mathrm{GL}(3)$-module. Therefore,
as in the general proof of Theorem 3.3, $\varphi_{p}(B_{3})$ contains an
element $u$ such that $u$ is a product of pure commutators and the variable
$x_{1}$ appears in $u$ exactly $\lambda_{1}$ times. We will again estimate the
maximal number of appearances of the variable $x_{1}$ in $u$. By Theorem 2.8,
for all integers $m_{1},\dots,m_{l}\geq 2$

$I_{m_{1}}I_{m_{2}}\cdots I_{m_{l}}\subseteq I_{m_{1}+m_{2}+\dots+m_{l}-l+1}.$

Hence, $u$ is at most a product of an $m_{1}$-commutator, $m_{2}$-commutator,
$\dots$, and $m_{l}$-commutator such that $m_{1}+\dots+m_{l}-l+1=p$. Hence
$m_{1}+\dots+m_{l}=p+l-1$. The number of appearances of $x_{1}$ in such a
product of pure commutators $u$ is at most $p-1$. This proves the statement.

##  4\. The algebra of invariants $F_{n}(\mathfrak{N}_{p})^{G}$ for
$G=\mathrm{SL}(n)$, $\mathrm{O}(n)$, $\mathrm{SO}(n)$, $\mathrm{Sp}(2s)$, and
$\mathrm{UT}(n)$

In this section we set $K={\mathbb{C}}$. Using results from [2] and [10], we
will obtain some bounds on the generators of the algebra of invariants
$F_{n}(\mathfrak{N}_{p})^{G}$ for $G=\mathrm{SL}(n)$, $\mathrm{O}(n)$,
$\mathrm{SO}(n)$, $\mathrm{Sp}(2s)$, and $\mathrm{UT}(n)$.

######  Proposition 4.1.

Let $p\geq 1$ and let $G=\mathrm{SL}(n)$ or $G=\mathrm{Sp}(2s)$ (for $n=2s$).
Then, the algebra of invariants $F_{n}(\mathfrak{N}_{p})^{G}$ is finite-
dimensional. More precisely, the Hilbert series
$H(F_{n}(\mathfrak{N}_{p})^{G},t)$ is a polynomial and it holds that

$\displaystyle\deg H(F_{n}(\mathfrak{N}_{p})^{G},t)\leq n(p-1),$

where in the case $G=\mathrm{Sp}(2s)$ we have that $n=2s$.

###### Proof.

We denote by $F_{n}^{(i)}(\mathfrak{N}_{p})$ the homogeneous component of
total degree $i$ in $F_{n}(\mathfrak{N}_{p})$. Then we can write the
decomposition of $F_{n}(\mathfrak{N}_{p})$ as a $\mathrm{GL}(n)$-module in the
following way:

$F_{n}(\mathfrak{N}_{p})=\bigoplus_{i\geq
0}F_{n}^{(i)}(\mathfrak{N}_{p})\cong_{\mathrm{GL}(n)}\bigoplus_{i\geq
0}\bigoplus_{\lambda}m_{i,\lambda}V_{\lambda},$

where $|\lambda|=i$ and, by Corollary 3.4, $\lambda_{2}\leq p-1$.

Consider first the case when $G=\mathrm{Sp}(2s)$. Theorem 2.2 from [10]
implies that

$H(F_{2s}(\mathfrak{N}_{p})^{\mathrm{Sp}(2s)},t)=\sum_{i\geq
0}\left(\sum_{\lambda}m_{i,\lambda}\right)t^{i},$

where the second sum runs over partitions $\lambda$ with
$\lambda_{1}=\lambda_{2}$, $\dots$, $\lambda_{2s-1}=\lambda_{2s}$ such that
$|\lambda|=i$ and $\lambda_{2}\leq p-1$. The largest partition $\lambda$ which
satisfies these conditions is
$\lambda_{1}=\lambda_{2}=\dots=\lambda_{2s-1}=\lambda_{2s}=p-1$. Therefore,
the highest power of $t$ which can occur in the Hilbert series
$H(F_{2s}(\mathfrak{N}_{p})^{\mathrm{Sp}(2s)},t)$ is
$i=\lambda_{1}+\dots+\lambda_{2s}=2s(p-1)$. This proves the statement for
$G=\mathrm{Sp}(2s)$.

Next, let $G=\mathrm{SL}(n)$. The results from [2] together with Corollary 3.4
imply that

$H(F_{n}(\mathfrak{N}_{p})^{\mathrm{SL}(n)},t)=\sum_{i\geq
0}\left(\sum_{\lambda}m_{i,\lambda}\right)t^{i},$

where the second sum runs over partitions $\lambda$ with
$\lambda_{1}=\lambda_{2}=\dots=\lambda_{n}$ such that $|\lambda|=i$ and
$\lambda_{2}\leq p-1$. Hence, the largest partition $\lambda$ which satisfies
these conditions is again $\lambda_{1}=\lambda_{2}=\dots=\lambda_{n}=p-1$.
Therefore, the highest power of $t$ which can occur in the Hilbert series
$H(F_{n}(\mathfrak{N}_{p})^{\mathrm{SL}(n)},t)$ is again $i=n(p-1)$. This
completes the proof. ∎

######  Proposition 4.2.

Let $p\geq 1$ and let $G=\mathrm{O}(n)$. Then,

$H(F_{n}(\mathfrak{N}_{p})^{\mathrm{O}(n)},t)=\frac{p(t)}{1-t^{2}},$

where $p(t)$ is a polynomial such that

$\displaystyle\deg p(t)\leq\left\\{\begin{array}[]{ll}n(p-1)&\text{if
}p-1\text{ is even}\\\ n(p-2)&\text{if }p-1\text{ is odd}.\end{array}\right.$

###### Proof.

We proceed as in the proof of the previous proposition. We write the
decomposition of $F_{n}(\mathfrak{N}_{p})$ as a $\mathrm{GL}(n)$-module in the
following way:

$F_{n}(\mathfrak{N}_{p})=\bigoplus_{i\geq
0}F_{n}^{(i)}(\mathfrak{N}_{p})\cong_{\mathrm{GL}(n)}\bigoplus_{i\geq
0}\bigoplus_{\lambda}m_{i,\lambda}V_{\lambda},$

where again $|\lambda|=i$ and $\lambda_{2}\leq p-1$. Then Theorem 2.2 from
[10] implies that

$H(F_{n}(\mathfrak{N}_{p})^{\mathrm{O}(n)},t)=\sum_{i\geq
0}\left(\sum_{\lambda}m_{i,\lambda}\right)t^{i},$

where the second sum runs over even partitions $\lambda$ such that
$|\lambda|=i$ and $\lambda_{2}\leq p-1$.

If a partition
$\lambda^{\prime}=(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\dots,\lambda_{n}^{\prime})$
satisfies the above conditions and appears in the decomposition of
$F_{n}(\mathfrak{N}_{p})^{\mathrm{O}(n)}$, then due to Equation (2) all
partitions of the form
$(\lambda_{1}^{\prime}+2k,\lambda_{2}^{\prime},\dots,\lambda_{n}^{\prime})$
for $k=1,2,\dots$ also appear in the decomposition of
$F_{n}(\mathfrak{N}_{p})^{\mathrm{O}(n)}$. Thus, the following term appears in
$H(F_{n}(\mathfrak{N}_{p})^{\mathrm{O}(n)},t)$:

$t^{|\lambda^{\prime}|}(1+t^{2}+t^{4}+\dots)=\frac{t^{|\lambda^{\prime}|}}{1-t^{2}}.$

We can write all terms in $H(F_{n}(\mathfrak{N}_{p})^{\mathrm{O}(n)},t)$ in
this way for a suitable $\lambda^{\prime}$. Theorem 3.3 implies that when
$p-1$ is even, the largest suitable $\lambda^{\prime}$ that satisfies the
above conditions is $\lambda^{\prime}=(p-1,p-1,\dots,p-1)$. Similarly, when
$p-1$ is odd, the largest suitable $\lambda^{\prime}$ that satisfies the above
conditions is $\lambda^{\prime}=(p-2,p-2,\dots,p-2)$. This proves the
statement. ∎

In the same way we can prove the following proposition.

######  Proposition 4.3.

Let $p\geq 1$ and let $G=\mathrm{SO}(n)$. Then,

$H(F_{n}(\mathfrak{N}_{p})^{\mathrm{SO}(n)},t)=\frac{p(t)}{1-t^{2}},$

where $p(t)$ is a polynomial such that

$\displaystyle\deg p(t)\leq n(p-1).$

By a theorem of Domokos and Drensky ([6]), the algebra of invariants
$F_{n}(\mathfrak{N}_{p})^{G}$ is finitely generated for any reductive subgroup
$G$ of $\mathrm{GL}(n)$. Using the above results we can give an explicit upper
bound for the degree of the generators in a minimal generating set of
$F_{n}(\mathfrak{N}_{p})^{G}$ for $G=\mathrm{SL}(n)$, $\mathrm{O}(n)$,
$\mathrm{SO}(n)$, or $\mathrm{Sp}(2s)$. First, we introduce the following
notation. Given a graded algebra $A=\bigoplus_{i\geq 0}A^{i},$ we denote by
$\beta(A)$ the minimal non-negative integer $i$ such that $A$ is generated by
homogeneous elements of degree at most $i$. We write $\beta(A)=\infty$ if
there is no such $i$.

######  Corollary 4.4.

Let $G$ be one of $\mathrm{SL}(n)$ or $\mathrm{Sp}(2s)$. Then

$\displaystyle\beta(F_{n}(\mathfrak{N}_{p})^{G})\leq n(p-1),\text{ for all
}p\geq 1.$

For $G=\mathrm{O}(n)$ we have that

$\displaystyle\beta(F_{n}(\mathfrak{N}_{p})^{\mathrm{O}(n)})\leq\left\\{\begin{array}[]{ll}n(p-1)&\text{if
}p-1\text{ is even and }p>1\\\ n(p-2)&\text{if }p-1\text{ is odd and }p>2\\\
2&\text{if }p=1,2.\end{array}\right.$

For $G=\mathrm{SO}(n)$ we have that

$\displaystyle\beta(F_{n}(\mathfrak{N}_{p})^{\mathrm{SO}(n)})\leq\left\\{\begin{array}[]{ll}n(p-1)&\text{if
}p>1\\\ 2&\text{if }p=1.\end{array}\right.$

###### Proof.

By Proposition 4.1, the statement is clear for $G=\mathrm{SL}(n)$ and
$G=\mathrm{Sp}(2s)$. Hence, we consider the case of $G$ being one of
$\mathrm{O}(n)$ and $\mathrm{SO}(n)$. By Propositions 4.2 and 4.3

$H(F_{n}(\mathfrak{N}_{p})^{G},t)=\frac{p(t)}{1-t^{2}}=p(t)(1+t^{2}+t^{4}+\dots+t^{2k}+\dots).$

$F_{n}(\mathfrak{N}_{p})^{G}$ has one free generator of degree $2$, namely
$u=x_{1}^{2}+\dots+x_{n}^{2}$. Therefore, if $\\{f_{1},\dots,f_{l}\\}$ is a
basis for the homogeneous part $\bigoplus_{i=0}^{\deg
p(t)}F_{n}^{(i)}(\mathfrak{N}_{p})^{G}$ then $\\{f_{1},\dots,f_{l},u\\}$ is a
generating set (not necessarily minimal) for the whole
$F_{n}(\mathfrak{N}_{p})^{G}$. This proves the statement. ∎

######  Remark 4.5.

In [8], Domokos and Drensky give a general bound for
$\beta(F_{n}(\mathfrak{N}_{p})^{G})$ for any reductive group $G$ as a special
case of a more general statement. The advantage of the bound that we find in
Corollary 4.4 is that it is very explicit and can be used in explicit
computations.

In a similar way we can consider the algebra of invariants
$F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)}$, where
$\mathrm{UT}(n):=\mathrm{UT}(n,{\mathbb{C}})$ denotes the unitriangular group,
i.e., the subgroup of $\mathrm{GL}(n)$ consisting of upper triangular matrices
with $1$’s on the diagonal. $\mathrm{UT}(n)$ is a maximal unipotent subgroup
of $\mathrm{GL}(n)$ and of $\mathrm{SL}(n)$. Using results from [2] we prove
the following proposition.

######  Proposition 4.6.

Let $p\geq 1$ and let $G=\mathrm{UT}(n)$. Then,

$H(F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)},t)=\frac{p(t)}{1-t},$

where $p(t)$ is a polynomial such that

$\displaystyle\deg p(t)\leq n(p-1).$

###### Proof.

We proceed as before. We write the decomposition of $F_{n}(\mathfrak{N}_{p})$
as a $\mathrm{GL}(n)$-module in the following way:

$F_{n}(\mathfrak{N}_{p})=\bigoplus_{i\geq
0}F_{n}^{(i)}(\mathfrak{N}_{p})\cong_{\mathrm{GL}(n)}\bigoplus_{i\geq
0}\bigoplus_{\lambda}m_{i,\lambda}V_{\lambda},$

where again $|\lambda|=i$ and $\lambda_{2}\leq p-1$. Each module $V_{\lambda}$
has a one-dimensional $\mathrm{UT}(n)$-invariant subspace generated by any
highest weight vector. Therefore (see also [2]),

$H(F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)},t)=\sum_{i\geq
0}\left(\sum_{\lambda}m_{i,\lambda}\right)t^{i},$

where the second sum runs over partitions $\lambda$ such that $|\lambda|=i$
and $\lambda_{2}\leq p-1$.

Then, as in the proof of Proposition 4.2, we notice the following. If a
partition
$\lambda^{\prime}=(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\dots,\lambda_{n}^{\prime})$
satisfies the above conditions and appears in the decomposition of
$F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)}$, then due to Equation (2) all
partitions of the form
$(\lambda_{1}^{\prime}+k,\lambda_{2}^{\prime},\dots,\lambda_{n}^{\prime})$ for
$k=1,2,\dots$ also appear in the decomposition of
$F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)}$. Thus, the following term appears
in $H(F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)},t)$:

$t^{|\lambda^{\prime}|}(1+t+t^{2}+t^{3}+\dots)=\frac{t^{|\lambda^{\prime}|}}{1-t}.$

We can write all terms in $H(F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)},t)$ in
this way for a suitable $\lambda^{\prime}$. Theorem 3.3 implies that the
largest suitable $\lambda^{\prime}$ that satisfies the above conditions is
$\lambda^{\prime}=(p-1,p-1,\dots,p-1)$. This proves the statement. ∎

The algebra $F_{n}(\mathfrak{N}_{p})$ is left Noetherian, hence Theorem 7.2
from [7] implies that the algebra of invariants
$F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)}$ is finitely generated. Using
Proposition 4.6 we can give an upper bound on the degree of the generators of
$F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)}$ in a minimal generating set.

######  Corollary 4.7.

For the algebra of invariants $F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)}$ it
holds that

$\displaystyle\beta(F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)})\leq\left\\{\begin{array}[]{ll}n(p-1)&\text{if
}p>1\\\ 1&\text{if }p=1.\end{array}\right.$

###### Proof.

We notice that $x_{1}$ is a free generator of
$F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)}$ of degree $1$. Furthermore, by
Proposition 4.6

$H(F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)},t)=\frac{p(t)}{1-t}=p(t)(1+t+t^{2}+t^{3}+\dots).$

Therefore, as in the proof of Corollary 4.4, if $\\{f_{1},\dots,f_{l}\\}$ is a
basis for the homogeneous part $\bigoplus_{i=0}^{\deg
p(t)}F_{n}^{(i)}(\mathfrak{N}_{p})^{\mathrm{UT}(n)}$ then
$\\{f_{1},\dots,f_{l},x_{1}\\}$ is a generating set (not necessarily minimal)
for the whole $F_{n}(\mathfrak{N}_{p})^{\mathrm{UT}(n)}$. This proves the
statement. ∎

######  Remark 4.8.

Notice that for $p=2$, using results from [10], we can compute explicitly
$\beta(F_{n}(\mathfrak{N}_{p})^{G})$ for $G=\mathrm{SL}(n)$, $\mathrm{O}(n)$,
$\mathrm{SO}(n)$, $\mathrm{Sp}(2s)$, and $\mathrm{UT}(n)$. We see that the
bounds obtained in Corollary 4.4 and 4.7 for $p=2$ are exact when
$G=\mathrm{SL}(n)$, $\mathrm{O}(n)$, $\mathrm{SO}(n)$, and $\mathrm{UT}(n)$.
Namely,
$\beta(F_{n}(\mathfrak{N}_{2})^{\mathrm{SL}(n)})=\beta(F_{n}(\mathfrak{N}_{2})^{\mathrm{SO}(n)})=\beta(F_{n}(\mathfrak{N}_{2})^{\mathrm{UT}(n)})=n=n(p-1)$
and $\beta(F_{n}(\mathfrak{N}_{2})^{\mathrm{O}(n)})=2$. For
$G=\mathrm{Sp}(2s)$, we obtain
$\beta(F_{2s}(\mathfrak{N}_{2})^{\mathrm{Sp}(2s)})=2<2s(p-1)$ for $s>1$, i.e.,
when $G=\mathrm{Sp}(2s)$ the bound from Corollary 4.4 is not exact for $p=2$.

##  5\. Applications to Classical Invariant Theory

In this section we will reformulate Corollaries 4.4 and 4.7 in the language of
Classical Invariant Theory. We set again $K={\mathbb{C}}$. First we recall a
classical result due to Weyl involving the algebra of invariants
${\mathbb{C}}\left\langle X\right\rangle^{G}=T(V)^{G}$, where $G$ is one of
$\mathrm{O}(n)$ or $\mathrm{Sp}(2s)$. In order to state this result, we follow
the notations and formulations in [13]. Let
$\left\langle\cdot,\cdot\right\rangle$ denote the non-degenerate symmetric (in
the case of $G=\mathrm{O}(n)$) or skew symmetric (in the case of
$G=\mathrm{Sp}(2s))$ form on $V$ preserved by the action of $G$. Using the
natural identification of $V^{*}$ with $V$ we obtain that the form
$\left\langle\cdot,\cdot\right\rangle$ is an invariant for $G$ in $V\otimes
V=V^{\otimes 2}$. Following [13], we denote this invariant by $\theta_{2}$.
For $k=2j$ we may choose an isomorphism

$i_{k}:V^{\otimes k}\cong\otimes^{j}V^{\otimes 2}.$

Specifying $i_{k}$ amounts to pairing off the factors of $V^{\otimes k}$.
Then, in $\otimes^{j}V^{\otimes 2}$ we have the invariant
$\otimes^{j}\theta_{2}$. Hence, in $V^{\otimes k}$ we have the invariant
$\theta_{k}=i^{-1}(\otimes^{j}\theta_{2})$. The symmetric group $S_{k}$ acts
on $V^{\otimes k}$ by permuting the factors. Then the following result of Weyl
holds:

######  Theorem 5.1 ([19],[13]).

Let $G$ be one of $\mathrm{O}(n)$ or $\mathrm{Sp}(2s)$. If $k$ is odd, there
are no $G$-invariants in $V^{\otimes k}$. If $k$ is even, the translates of
$\theta_{k}$ by $S_{k}$ span the $G$-invariants in $V^{\otimes k}$.

An analogous result can be found in the literature for $G=\mathrm{SL}(n)$.

Propositions 4.1, 4.2, and 4.3 and Corollaries 4.4 and 4.7 lead to the
following further characterization of the $G$-invariants in $T(V)$. We include
also the cases $G=\mathrm{SL}(n)$, $\mathrm{SO}(n)$ and $\mathrm{UT}(n)$ in
our considerations.

######  Theorem 5.2.

Let $G=\mathrm{SL}(n)$ or $G=\mathrm{Sp}(2s)$ (for $n=2s$). For any $p\geq 1$,
if $f$ is a $G$-invariant in $T(V)$ of degree greater than $n(p-1)$ then $f\in
I_{p+1}$.

######  Theorem 5.3.

Let $G=\mathrm{O}(n)$.

  * (i)

If $f$ is a $G$-invariant in $T(V)$ of degree greater than $2$ then either
$f\in I_{3}$ or $f$ is equal to a power of $\theta_{2}$.

  * (ii)

For any $p\geq 3$ such that $p-1$ is an even number, if $f$ is a $G$-invariant
in $T(V)$ of degree greater than $n(p-1)$ then either $f\in I_{p+1}$ or $f$
can be expressed by invariants of smaller degree.

  * (iii)

For any $p\geq 3$ such that $p-1$ is an odd number, if $f$ is a $G$-invariant
in $T(V)$ of degree greater than $n(p-2)$ then either $f\in I_{p+1}$ or $f$
can be expressed by invariants of smaller degree.

######  Theorem 5.4.

Let $G=\mathrm{SO}(n)$.

  * (i)

If $f$ is a $G$-invariant in $T(V)$ of degree greater than $2$ then either
$f\in I_{2}$ or $f$ can be expressed by invariants of smaller degree.

  * (ii)

For any $p\geq 2$, if $f$ is a $G$-invariant in $T(V)$ of degree greater than
$n(p-1)$ then either $f\in I_{p+1}$ or $f$ can be expressed by invariants of
smaller degree.

######  Theorem 5.5.

Let $G=\mathrm{UT}(n)$.

  * (i)

If $f$ is a $G$-invariant in $T(V)$ of degree greater than $1$ then either
$f\in I_{2}$ or $f$ can be expressed by invariants of smaller degree.

  * (ii)

For any $p\geq 2$, if $f$ is a $G$-invariant in $T(V)$ of degree greater than
$n(p-1)$ then either $f\in I_{p+1}$ or $f$ can be expressed by invariants of
smaller degree.

## Acknowledgements

I am very grateful to Vesselin Drensky for the productive discussions that we
had during my work on this text.

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