# HarperValleyBank: A Domain-Specific Spoken Dialog Corpus

###### Abstract

We introduce HarperValleyBank, a free, public domain spoken dialog corpus. The
data simulate simple consumer banking interactions, containing about 23 hours
of audio from 1,446 human-human conversations between 59 unique speakers. We
selected intents and utterance templates to allow realistic variation while
controlling overall task complexity and limiting vocabulary size to about 700
unique words. We provide audio data along with transcripts and annotations for
speaker identity, caller intent, dialog actions, and emotional valence. The
data size and domain specificity makes for quick transcription experiments
with modern end-to-end neural approaches. Further, we provide baselines for
representation learning, adapting recent work to embed waveforms for
downstream prediction tasks. Our experiments show that tasks using our
annotations are sensitive to both the model choice and corpus size.

##  1 Introduction

Recent innovations in deep learning approaches substantially improved spoken
dialog systems in both academic research and industry applications. Speech
recognition systems now regularly leverage neural network models to achieve
near human performance [1, 2, 3, XXXciteConformer]. Modern systems
increasingly use architecture themes including attention and end-to-end
recurrent neural networks to encode few assumptions and rapidly adapt to new
data [4, 5, 6, 7, 8, 9]. In parallel, approaches to spoken and text-based
dialog systems increasingly leverage neural networks for dialog management and
state representation [10, 11, 12].

We developed the HarperValleyBank corpus for homeworks and projects in
Stanford’s _Spoken Language Processing_ course111cs224s.stanford.edu, as well
as for research on speech recognition combined with other spoken language
tasks. The goals of the corpus are to provide:

  * •

Freely available data for education, research, or commercial development. We
release the data using a Creative Commons license (CC-
BY)222creativecommons.org/licenses/by/4.0.

  * •

Sufficient size and variability to meaningfully evaluate end-to-end neural
approaches for speech transcription.

  * •

Manageable overall size and complexity to enable students to quickly iterate
on experiments without requiring expensive compute hardware for training.

  * •

Annotations for dialog-relevant tasks aside from speech transcription (e.g.
intent, dialog action) to enable multi-task training and representation
transfer.

  * •

Realistic domain-specific, goal-oriented conversations to evaluate
representation transfer approaches across domains in spoken dialog systems.

There is significant recent work in representation learning for domain and
task transfer via embedding models [13, 14, 15, 16], which can be trained
without any supervision and reused for many downstream tasks like predicting
speaker identity [17, 18] and commands [19, 20]. Representation transfer is
important to warm start deep learning based dialog systems on new task
domains. Part of our motivation in designing this corpus is providing a
realistic test case for representation learning approaches to adapt to speech
tasks. See [21] for a review of availaable dialog datasets. We recorded two-
sided phone conversations to simulate customer call center interactions in a
financial services domain. The dataset is representative of human to human
goal-oriented dialogs for consumer banking with a narrowly scoped set of
intents.

In the next sections, we provide more details on the corpus and its
collection, followed by experiments showcasing its applications to automatic
speech recognition and unsupervised representation learning. In Sec. 2, we
discuss basic corpus statistics, caller intents, and the data generation and
annotation process. In Sec. 3, we explore end-to-end neural models with multi-
task objective functions to simultaneously perform speech-to-text
transcription and caller intent prediction. In Sec. 4, we explore using caller
intents and sentiments as downstream objectives to evaluate representation
transfer, and report unsupervised baselines. The full dataset with a PyTorch
implementation reproducing the speech recognition and transfer experiments is
publically available333https://github.com/cricketclub/gridspace-stanford-
harper-valley.

##  2 The HarperValleyBank Corpus

We compile a dataset of recorded audio conversations between an agent and a
customer of a bank. Conversations are goal-oriented, such as ordering a new
checkbook or checking the balance of an account. Fig. 1 shows an example
conversation from the dataset. We collected data using the Gridspace Mixer
platform, where crowd workers are randomly paired for short telephone
conversations. Mixer membership includes hundreds of past and current
professional call center agents who are trained to perform assorted Mixer
tasks in domains including healthcare, telecommunications, and commerce.

###  2.1 Data Collection Procedure

Using the Mixer web platform, a person is randomly assigned the role of agent
or customer and provided a script for the interaction along with a telephone
number to call to start the conversation. Roles are randomly assigned for each
call, so the same worker can appear as both customer and agent in different
conversations. We created a set of conversation goals and scripts for each
interaction using templates intended to capture variety in each intent while
keeping workers’ word choices and the overall interactions fairly simple with
limited vocabulary. We do not control the noise environment or microphones
used by each worker, and there is natural variation across different types of
phones and environments.

When a person calls in, they are paired with the next available conversation
partner. The groups are large enough that many unique pairings occur over the
course of one session. Once the Mixer task is live for the caller, the web
application will change state, informing the caller whether they are acting in
the role of the agent or caller. The instructions, data, and user interface
adapt to the role and provide a rough script. The customer role initiates a
call task by expressing an intent, and the agent role has an interactive web
interface to simulate completing a task. We encouraged callers to use mobile
phones or headsets to encourage a microphone transfer function that is
acoustically representative of a real call center.

During each call, participants have the script for their side of the
conversation in front of them in a web browser. The worker playing the
customer role is given a single intent for the conversation, along with
specific values for relevant slots (e.g. the amount of money to transfer and
the source/target accounts). When playing the agent role, a worker is shown
some simple buttons and menus they must click to perform the requested
operation (e.g. ”check account balance”).

A conversation is deemed successful and considered for the dataset if the
agent correctly executes the task provided to the customer caller. Names and
slot values for different transactions are randomly generated, and we limit
the number of possible names and proper nouns to reduce overall corpus
vocabulary size. The Gridspace Mixer platform handles generating random
templates from a high level specification, all associated telephony operations
to pair callers, and recording audio along with metadata for each interaction.

agent hello this is harper valley national bank my name is jay how can I help
you today caller hi my name is mary davis caller [noise] caller i would like
to schedule an appointment agent yeah sure what day what time caller thursday
one thirty p m agent that’s done anything else caller that’s it agent have a
good one.

Figure 1: Example conversation from corpus.

###  2.2 Data Labelling

Gridspace Mixer trains a subset of its community to perform a wide range of
annotations. For this corpus, Mixers performed three primary labeling tasks:
text transcript, audio quality, and script adherence ratings. Gridspace has
provided the Mixer community with a highly specialized speech labeling tool
called Scriber. Scriber is designed for rapid human transcription and data
labeling. The tool also provides a wide array of convenience and ergonomic
functions, designed to enable efficient labeling of large spoken language
datasets. Every person trained to use Scriber must go through several training
sessions, which requires them to watch training videos and perform well on a
quiz. For dialog actions and emotional valence, labels were instead produced
using a Gridspace API rather than human annotation. As a result, there may be
some noise or bias, but our experiments indicate they are reasonable for a
benchmark task.

The HarperValleyBank corpus was collected over three separate Mixer sessions
and then filtered post-annotation, informed by the script adherence labels and
audio quality labels, to ensure the data was simple and low variance. This
filtering ensures the corpus provides conversational and task-oriented speech
data while regulating for simplicity. The primary target of the cleaning were
conversations where calls dropped or there were other technical issues which
derailed the conversation. Specifically we removed conversations with script
adherence ratings less than 4 and audio quality ratings less than 3.
Furthermore we filtered out conversations which contained some words such as
’frozen’, ’website’, and ’refresh’, which indicated conversation about
technical issues with the task interface. In total we removed 375
conversations.

###  2.3 Corpus Statistics

HarperValleyBank Statistics |
---|---
Hours of audio | 23.7
# of conversations/transcripts | 1,446
# of utterances | 25,730
# of unique words | 735
Mean # of lines per conversation | 17.8
Median # of lines per conversation | 16
Mean # of words per utterance | 4.1
Median # of words per utterance | 4.5
# of unique speakers | 59
# of task classes | 8
# of dialog action classes | 16
# of sentiment classes | 3
Table 1: Basic corpus statistics .

Table 1 shows basic statistics of the HarperValleyBank corpus. The corpus
contains about 23 hours of audio in total, across 1,446 conversations.
Conversations range from 2 to 60 utterances, with an average of 18. Each
utterance roughly corresponds to a single turn in the conversation. Due to
automatic segmentation of utterances, there can be multiple utterances in a
row from a single speaker’s turn. Notably, the corpus has a small vocabulary
of approximately 700 unique words. Many of the most common words in the
vocabulary are domain-specific to customer service e.g. “help”, “thank”, or
“please”. Fig. 2b depicts how vocabulary size scales with dataset size.

(a)

(b)

Figure 2: (a) The empirical distribution over the number of utterances in a
conversation transcript. (b) The number of unique words spoken as a function
of the conversation count.

The Gridspace platform records each side of the conversation separately, and
we release the audio in speaker-separated files encoded as 8kHz per the
original telephony data. We transcribed the utterances via crowd workers with
basic speech transcription training again using the Gridspace Mixer platform.
Workers are not instructed to carefully transcribe word fragments or non-
speech noises. Leveraging crowd workers and transcribing without precise
fragments and non-speech tags has been shown to be a viable approach for
training speech recognition systems [22]. In addition to human transcriptions
of each conversation, we include four additional labels:

Intent. Each conversation has a single intent representing the customer’s goal
in the conversation. An intent can be one of eight categories: order checks,
check balance, replace card, reset password, get branch hours, pay bill,
schedule appointment, transfer money. Fig. 3a shows the distribution of
intents to be roughly balanced. Conversation intents are derived automatically
from the tasks assigned to callers during collection.

Emotional Valence. Utterances are automatically labeled with three sentiment
categories, negative, neutral, and positive. There is a probability estimate
label for each category, generated by the Gridspace Speech API that is trained
on a large corpus of proprietary data from multiple domains. Fig. 3b shows the
distribution of each sentiments across utterances.

Speaker ID. Utterances have a unique ID out of 59 speakers. The number of
utterances per speaker are imbalanced, with most speakers responsible for less
than 50 utterances.

Dialog Action. Every utterance is accompanies by one or more labels
representing a ”conversational mode”. The distribution over actions is
imbalanced with “greeting” being the most frequent and many infrequent actions
combined into the “other” category. The 16 possible actions are: “yes”
response, greeting, response, data confirmation, procedure explanation, data
question, closing, data communication, “bear with me” response,
acknowledgement, data response, filler disfluency, thanks, open question,
problem description, and other. Fig. 3d shows the distribution of dialog
actions for utterances.

(a) Intent

(b) Sentiment

(c) Dialog Actions

Figure 3: Distribution over auxiliary labels. Subfigures (a) & (c) show the
counts for intent and dialog action. Subfigure (d) show boxplots for each of
the three sentiments.

##  3 Spoken Language Understanding

Using the HarperValleyBank corpus, we evaluate three common approaches for
automatic speech recognition: connectionist temporal classification or CTC
[23], Listen-Attend-Spell or LAS [5], and finally, a “multi-task” objective
combining the two previous losses [24], MTL.

In addition to optimizing the speech recognition objective, denoted
$\mathcal{L}_{\textup{asr}}$, we fit four linear layers mapping the encoding
of the audio signal to a prediction of the intent, dialog action, and
sentiment labels. These three auxiliary objectives are optimized jointly with
the speech recognition objective:

$\beta\mathcal{L}_{\textup{asr}}+(1-\beta)\left(\mathcal{L}_{\textup{task}}+\mathcal{L}_{\textup{action}}+\mathcal{L}_{\textup{sent}}\right)$

where $\mathcal{L}_{\textup{task}}$, and $\mathcal{L}_{\textup{sent}}$ are
cross entropy losses, whereas $\mathcal{L}_{\textup{action}}$ comprises a sum
of 16 binary cross entropy losses. The scalar $\beta\in[0,1]$ weights the
recognition and auxiliary objectives.

Model | CER | Action (F1) | Sentiment | Intent
---|---|---|---|---
CTC | 14.43 | 0.3864 | 84.46 | 45.47
LAS | 47.45 | 0.2931 | 72.09 | 34.96
MTL | 38.59 | 0.3222 | 76.12 | 42.28
Table 2: Speech recognition and auxiliary task performance using three modern
end-to-end neural approaches.

###  3.1 Training Details

Audio is preprocessed to 128 Mel-frequency spectrogram features with a
sampling rate of 8kHz, a hop length of 128, and a window size of 256. For CTC,
we encode log-Mel features using a bi-directional LSTM with two layers and 128
hidden dimensions. CTC decoding is done greedily with no language model. In
LAS, the listener network is composed of three stacked pyramid bi-directional
LSTMs with 128 hidden dimensions whereas the speller network is an uni-
directional LSTM with 256 hidden dimensions and a single-headed attention
layer. For MTL, we use the same encoder as LAS but include both the speller
and CTC decoder. In MTL, we can interpret CTC as a secondary objective whose
main role is to regularize the LAS encoder to respect CTC alignments:

$\beta(\alpha\mathcal{L}_{\textup{ctc}}+(1-\alpha)\mathcal{L}_{\textup{las}})+(1-\beta)\left(\mathcal{L}_{\textup{task}}+\mathcal{L}_{\textup{action}}+\mathcal{L}_{\textup{sent}}\right)$

where $\alpha$ is chosen to be 0.7 by grid search. The LAS and MTL objectives
are optimized with teacher forcing with probability 0.5. We train each model
for 200 epochs with Adam [25] using a learning rate of 1e-3, batch size 128,
and gradient clipping.

###  3.2 Results and Analysis

Table 2 shows the performance of CTC, LAS, and MTL on a speaker split test
set. We report CER to measure transcription quality, accuracy for sentiment
and intent prediction, and F1 for dialog action prediction (due to class
imbalance). The highest performing model in each metric is bolded.

Interestingly, we find that CTC outperforms LAS and MTL. We attribute this to
the small size of the HarperValleyBank corpus that encourages expressive
autoregressive models to heavily overfit. In this case, the structure of the
CTC loss is beneficial compared with a model that learns attention. We see
further evidence of this by MTL outperforming LAS, where CTC acts as a
regularizer. LAS can improve a bit by tuning the probability of teacher
forcing (see Fig 4).

(a) CTC CER

(b) LAS CER

(c) Teacher Forcing

Figure 4: Subfigure (a) and (b) show the effect of increased weight on the
test CER error for CTC and LAS models. Subfigure (c) shows the effect of
teacher forcing in training LAS.

##  4 Unsupervised Speech Representations

Unsupervised representation learning seeks to derive useful representations of
speech waveforms without any human annotations. The learned representations
are reused for downstream tasks, such as predicting caller intent or dialog
action. As HarperValleyBank is a small dataset, it is a suitable candidate to
measure the effectiveness of speech representations. In our experiments, we
pretrain representations on the 100 hour split or 960 hour split of
LibriSpeech [17].

Ou baselines use recent ideas from contrastive learning [26, 27, 28, 29, 30,
31, 32, 33] where representations are learned by discriminating between
specific instances in a dataset. In particular, we adapt four algorithms from
computer vision to speech: Instance Discrimination or IR [26], Local
Aggregation or LA [27], Momentum Contrast or MoCo [30], and SimCLR [31]. One
of our contributions is to establish comparable baselines for audio
representation learning. As a close relative, we also evaluate representations
learned using Wav2Vec-1.0 [15] and Wav2Vec-2.0 [16].

Model | Spk. | Intent | Action | Sent.
---|---|---|---|---
Wav2Vec 1.0 (960hr) | 18.2 | 17.1 | 0.0 | 53.7
Wav2Vec 2.0 (100hr) | 22.3 | 19.7 | 0.0 | 54.3
Wav2Vec 2.0 (960hr) | 27.3 | 20.5 | 0.0 | 55.5
IR (100hr) | 99.5 | 99.1 | 0.0 | 51.3
LA (100hr) | 99.5 | 98.8 | 0.0 | 50.5
MoCo (100hr) | 99.6 | 98.9 | 0.0 | 53.2
SimCLR (100hr) | 99.8 | 99.3 | 0.0 | 53.9
IR∗ (100hr) | 99.5 | 84.5 | 0.0 | 51.4
LA∗ (100hr) | 97.5 | 75.8 | 1.4 | 55.1
MoCo∗ (100hr) | 99.1 | 82.6 | 0.0 | 54.0
SimCLR∗ (100hr) | 99.2 | 81.4 | 0.0 | 54.6
IR (960hr) | 99.9 | 99.9 | 17.4 | 66.3
LA (960hr) | 99.9 | 99.9 | 18.4 | 64.6
MoCo (960hr) | 99.9 | 99.9 | 17.3 | 65.5
SimCLR (960hr) | 99.9 | 99.9 | 17.4 | 65.9
IR∗ (960hr) | 99.9 | 86.7 | 17.6 | 64.8
LA∗ (960hr) | 99.9 | 79.8 | 18.0 | 64.6
MoCo∗ (960hr) | 99.5 | 86.1 | 16.2 | 64.3
SimCLR∗ (960hr) | 98.6 | 82.6 | 16.1 | 65.6
Table 3: Performance on speaker identity, intent, dialog action, and emotional
valence. We report F1 score for performance on predicting dialog action. The
superscript (∗) represents using spectral augmentations rather than wavform
augmentations.

###  4.1 Training Details

Given a waveform, we truncate to 150k frames, and compute the log-Mel
spectrogram. Spectrograms are z-scored using training statistics, which we
found to be important for generalization. By default, we augment waveforms by
selecting contiguous crops with a minimum and maximum ratio of 0.08 to 1.0,
along with Gaussian noise with a scale of 1.0. We separately explore first
computing the spectrogram, then applying a time and frequency mask as
augmentation, denoted by the (∗) superscript in Table 3. Regardless, we fit a
ResNet50 [34] to map the spectrogram to a 2048 dimensional embedding.

After representation learning, we measure the quality of an embedding by
linear evaluation [26, 27, 30, 31]. The HarperValleyBank corpus is split into
train (80%) and test (20%) sets by class to ensure both sets have instances of
each class. Thus, each transfer task has its own train test split. Refer to
the public repository for training hyperparameters.

###  4.2 Results and Analysis

Table 3 compares the different unsupervised models: IR, LA, MoCo, and SimCLR
surpass supervised methods (e.g. CTC, LAS, and MTL) in intent prediction,
falling short in dialog action and sentiment prediction. In particular, dialog
action prediction is a surprisingly difficult task for our unsupervised
representations. Whereas supervised methods achieve upwards of 0.3 F1, the
best models in Table 3 achieve only half the score, despite seeing 960 hours
of speech. That being said, adding 860 hours of speech improved the
representations, as shown by overall better performance. Finally, compared to
Wav2Vec, the contrastive objectives find large gains of up to 70%.

##  5 Conclusion

We introduced HarperValleyBank, a new speech corpus of transcribed
conversations between employees and customers in a bank transcaction. The
corpus includes additional labels, including speaker identity, caller intent,
dialog actions, and sentiment. In our experiments, we established baseline
models that showed this corpus to be an interesting challenge for future
algorithms, and a useful educational tool for modern deep learning approaches
to spoken dialog. Our experiments analyzed utterances independently, future
work can explore using the HarperValleyBank corpus in conversation modelling
and its related downstream optimization.

##  6 Acknowledgments

We thank the Gridspace team for their support in data collection and analysis.
Experiments used PyTorch, PyTorch Lightning, and Weights & Biases [35]. We
thank Samuel Kwong and Dan Jurafsky for helpful discussion and feedback.

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smalltableaux

# A combinatorial interpretation of the Bernstein degree
of modules of covariants

William Q. Erickson  William Q. Erickson
Department of Mathematics
Baylor University
One Bear Place #97328
Waco, TX 76798<EMAIL_ADDRESS>and  Markus Hunziker  Markus Hunziker
Department of Mathematics
Baylor University
One Bear Place #97328
Waco, TX 76798<EMAIL_ADDRESS>
###### Abstract.

We introduce sets $\mathcal{R}(\sigma)$ consisting of semistandard tableaux of
shape $\sigma$, subject to a certain restriction on the initial column. These
$\mathcal{R}(\sigma)$ are generalizations of the rational, symplectic, or
orthogonal tableaux (which furnish a weight basis for the irreducible finite-
dimensional representations $U_{\sigma}$ of the classical groups
$H=\operatorname{GL}_{k}$, $\operatorname{Sp}_{2k}$, or
$\operatorname{O}_{k}$, respectively). As our main result, we show that the
cardinality of $\mathcal{R}(\sigma)$ gives the Bernstein degree (i.e.,
multiplicity) of the modules of covariants (of type $U_{\sigma}$) of each
classical group $H$. Via Howe duality, these modules can also be viewed as
$(\mathfrak{g},K)$-modules of unitary highest weight representations of a real
reductive group $G_{\mathbb{R}}$. Notably, our result using
$\mathcal{R}(\sigma)$ is valid regardless of the rank of $H$, whereas the
previous result of Nishiyama–Ochiai–Taniguchi (expressing the Bernstein degree
in terms of $\dim U_{\sigma}$) holds only when $k$ is at most the real rank of
$G_{\mathbb{R}}$. In effect, as $k$ increases beyond this range,
$|\mathcal{R}(\sigma)|$ interpolates between $\dim U_{\sigma}$ and the
dimension of a certain “limiting” $K$-module.

###### Key words and phrases:

Modules of covariants, multiplicity, $(\mathfrak{g},K)$-modules, Bernstein
degree, Howe duality, Stanley decompositions

###### 2020 Mathematics Subject Classification:

Primary 05E10; Secondary 22E47, 17B10

##  1\. Introduction

###  1.1. Overview

Let $H$ be a complex reductive group with finite-dimensional representations
$W$ and $U$. Then $(\mathbb{C}[W]\otimes U)^{H}$ is the space of all
$H$-equivariant polynomial functions $\varphi:W\longrightarrow U$. This space
is called the _module of covariants_ for $W$ of type $U$. It is a module over
the ring of invariants $\mathbb{C}[W]^{H}$ via multiplication of functions. A
fundamental setting in invariant theory is the case where $H$ is a classical
group with defining representation $V$: then we have either
$H=\operatorname{GL}(V)$ acting on $W=V^{*p}\oplus V^{q}$, or
$H=\operatorname{Sp}(V)$ or $\operatorname{O}(V)$ acting on $W=V^{n}$. Upon
fixing a basis for $V$, we clarify its dimension by writing
$H=\operatorname{GL}_{k}$, $\operatorname{Sp}_{2k}$, or
$\operatorname{O}_{k}$. Then the irreducible finite-dimensional
representations of $H$ can be labeled as $U_{\sigma}$, where $\sigma$ belongs
to a certain set of partitions with length restricted by $k$. This parameter
$k$ plays a significant role in our paper.

An important invariant of these modules of covariants (or, in general, any
finitely generated module over a graded ring) is their _multiplicity_ ,
defined as follows. By Weyl’s first fundamental theorem of classical invariant
theory (and by the Hilbert–Serre theorem), the Hilbert series of
$(\mathbb{C}[W]\otimes U_{\sigma})^{H}$ can be written in the form

$\frac{Q(t)}{(1-t^{2})^{d}},$

where $Q(t)$ is a polynomial with integer coefficients such that $Q(1)\neq 0$.
The (positive) integer $Q(1)$ is the _multiplicity_ of $(\mathbb{C}[W]\otimes
U_{\sigma})^{H}$. Because the modules $\smash{(\mathbb{C}[W]\otimes
U_{\sigma})^{H}}$ can be viewed as $(\mathfrak{g},K)$-modules (where
$\mathfrak{g}$ is the complexified Lie algebra of a certain real reductive
group $G_{\mathbb{R}}$; see Section 1.2), it makes sense (following Vogan
[Vogan]) to use the term _Bernstein degree_ (denoted by $\operatorname{Deg}$)
as a synonym for _multiplicity_ in this setting.

A major result of Nishiyama–Ochiai–Taniguchi [NOT]*Thm. 8.6 gives the
following formula for the Bernstein degree, which is valid when $k$ is
sufficiently small (to be clarified in Section 1.2):

(1.1) $\operatorname{Deg}(\mathbb{C}[W]\otimes U_{\sigma})^{H}=\dim
U_{\sigma}\cdot\operatorname{Deg}\mathbb{C}[W]^{H}.$

In the present paper, we generalize this formula so that it holds for all
values of $k$. In particular, our main result (Theorem 2.4) takes the
following form, which is valid regardless of the rank of $H$:

(1.2) $\operatorname{Deg}(\mathbb{C}[W]\otimes
U_{\sigma})^{H}=|\mathcal{R}(\sigma)|\cdot\operatorname{Deg}\mathbb{C}[W]^{H},$

where $\mathcal{R}(\sigma)$ is a certain set of _restricted_ semistandard
tableaux of shape $\sigma$ which we introduce (see Definition 2.1). In fact,
when $k$ is greater than the range considered in [NOT], the Bernstein degree
is precisely the number of these restricted tableaux:

(1.3) $\operatorname{Deg}(\mathbb{C}[W]\otimes
U_{\sigma})^{H}=|\mathcal{R}(\sigma)|.$

We emphasize that the “$k$ sufficiently small” condition was essential to the
approach in [NOT]; we therefore find it especially surprising that there
exists such a formulation (1.3) which is purely combinatorial, and uniform
across the classical families. Moreover, our restricted tableaux have another
curious combinatorial interpretation, in the sense that they interpolate
between the dimensions of two distinguished modules; see (1.8) below.

We should mention here that in the case $H=\operatorname{O}_{k}$, our result
(1.2) still remains a conjecture, due to the difficulty in proving it using
the combinatorics in our previous work [EricksonHunzikerMOC2024]. We have,
however, verified examples by computer, using code that we wrote to compute
generalized BGG resolutions via the combinatorial methods in [EW,
EnrightHunziker04, EHP]. (See Example 2.6.)

###  1.2. Unitary highest weight modules

The modules of covariants $(\mathbb{C}[W]\otimes U_{\sigma})^{H}$ can also be
viewed from the perspective of unitary highest weight representations of real
Lie groups. This is the primary viewpoint taken in [NOT] (where the authors
use the convention of _lowest_ weights), and it provides the context to
clarify our phrase “$k$ sufficiently small” in the discussion above.

Let $G_{\mathbb{R}}/K_{\mathbb{R}}$ be an irreducible Hermitian symmetric
space. Let
$\mathfrak{g}_{\mathbb{R}}=\mathfrak{k}_{\mathbb{R}}\oplus\mathfrak{p}_{\mathbb{R}}$
be the Cartan decomposition of the Lie algebra of $G_{\mathbb{R}}$, which
yields the complexified triangular decomposition

(1.4) $\mathfrak{g}=\mathfrak{p}^{-}\oplus\mathfrak{k}\oplus\mathfrak{p}^{+}.$

The complexified group $K$ acts on $\mathfrak{p}^{+}$, and the closures of the
$K$-orbits in $\mathfrak{p}^{+}$ form a chain of algebraic varieties

(1.5)
$\\{0\\}=\overline{\mathcal{O}}_{0}\subset\overline{\mathcal{O}}_{1}\subset\cdots\subset\overline{\mathcal{O}}_{r}=\mathfrak{p}^{+},$

where $r$ denotes the real rank of $G_{\mathbb{R}}$.

Let $H$ be one of the three classical groups acting on the space $W$,
described in the first paragraph of this paper; recall that the rank of $H$ is
determined by the parameter $k$. We say that $H$ arises in the _dual pair
setting_ , in the following sense; see Tables 1 and 3 for specifics, along
with the appendix. Associated to $H$ is a real group $G_{\mathbb{R}}$, such
that (by Howe duality) the space $\mathbb{C}[W]$ admits a multiplicity-free
decomposition as an $H\times(\mathfrak{g},K)$-module:

(1.6)
$\mathbb{C}[W]\cong\bigoplus_{\sigma\in\Sigma\subseteq\widehat{H}}U_{\sigma}^{*}\otimes\underbrace{(\mathbb{C}[W]\otimes
U_{\sigma})^{H}}_{\cong L_{\lambda(\sigma)}}.$

(See [Howe89, KV].) The direct sum in (1.6) ranges over a certain subset
$\Sigma$ of the set $\widehat{H}$ of finite-dimensional irreducible
representations of $H$, where $U_{\sigma}$ is a model for $\sigma\in\Sigma$.
There is an injective map $\sigma\mapsto\lambda(\sigma)$ from $\Sigma$ to the
set of dominant integral weights of $\mathfrak{k}$, and $L_{\lambda(\sigma)}$
is the simple $\mathfrak{g}$-module with highest weight $\lambda(\sigma)$.
This $L_{\lambda(\sigma)}$ is a _unitary highest weight module_ , in the sense
that it is the $(\mathfrak{g},K)$-module of a unitary highest weight
representation of (the metaplectic double cover of) $G_{\mathbb{R}}$.

It is well known that the associated variety of $L_{\lambda(\sigma)}$ is
$\smash{\overline{\mathcal{O}}_{\min\\{k,r\\}}}$; see [EW]*Thm. 6 or
[BaiHunziker]*Thm. 4.3. The paper [NOT] considered those modules
$L_{\lambda(\sigma)}$ whose associated variety is properly contained in
$\overline{\mathcal{O}}_{r}=\mathfrak{p}^{+}$. Therefore, the result (1.1)
from [NOT] is valid for $k<r$ (actually, it turns out, for $k\leq r$), but not
for $k>r$. Hence the range $k\leq r$ was the precise meaning of our phrase
“$k$ sufficiently small” in (1.1). In this range, (1.1) can be restated as
follows (see [NOT]*Thm. B):

(1.7) $\operatorname{Deg}L_{\lambda(\sigma)}=\dim
U_{\sigma}\cdot\deg\overline{\mathcal{O}}_{k},$

where $\deg$ denotes the degree as an algebraic variety. Likewise, our main
result (1.2), which is valid for all $k$, can also be restated in this
language: namely, in (1.7), replace $\dim U_{\sigma}$ by
$|\mathcal{R}(\sigma)|$, and replace $\overline{\mathcal{O}}_{k}$ by
$\overline{\mathcal{O}}_{\min\\{k,r\\}}$.

###  1.3. Interpretation of restricted tableaux

Depending on the group $H$, our sets $\mathcal{R}(\sigma)$ of restricted
tableaux should be regarded as generalizations of rational tableaux (in the
sense of Stembridge [Stembridge]), or symplectic or orthogonal tableaux (in
the sense of Proctor [ProctorRSK]*p. 30). In fact, when $k\leq r$, the set
$\mathcal{R}(\sigma)$ is precisely the set of these tableaux (up to a constant
shift of the entries). It follows that in the range $k\leq r$, we have
$|\mathcal{R}(\sigma)|=\dim U_{\sigma}$. As $k$ increases, the restriction
defining $\mathcal{R}(\sigma)$ becomes more and more relaxed, until it
vanishes altogether at a certain threshold which we call $s$. (This threshold
$s$ depends on the size of $W$, and stands for _stable range_ 111For us, this
term refers to the fact that the Bernstein degree stabilizes at $\dim
F_{\lambda(\sigma)}$ for all $k\geq s$. The term “stable range” is used in
this context in [HTW]*p. 1609, referring to the fact that
$L_{\lambda(\sigma)}$ is a parabolic Verma module for all $k>s$. We should
point out, however, that the authors of [NOT] use the phrase “stable range” to
describe the opposite extreme $k\leq r$.; see Table 1 for the value of $s$ in
each case.) When $k\geq s$, the space $W$ is said to be _cofree_ , in the
sense of Schwarz [Schwarz], meaning that every module of covariants is free.
The fact that $L_{\lambda(\sigma)}$ is free implies that its Bernstein degree
equals the dimension of the $\mathfrak{k}$-module $F_{\lambda(\sigma)}$ with
highest weight $\lambda(\sigma)$. Equivalently, when $k\geq s$, the set
$\mathcal{R}(\sigma)$ is just a full set of semistandard tableaux without any
restrictions, and we have $|\mathcal{R}(\sigma)|=\dim F_{\lambda(\sigma)}$.
(See (A.1) in the appendix.)

(1.8) Hence as $k$ increases from $r$ to $s$, the cardinality of
$\mathcal{R}(\sigma)$ (besides giving the Bernstein degree of a module of
covariants) interpolates between the dimension of $U_{\sigma}$ and that of
$F_{\lambda(\sigma)}$.

We will focus all of our examples on this somewhat mysterious range $r<k<s$.

To give a specific example of the restriction that defines
$\mathcal{R}(\sigma)$, we look ahead to Definition 2.1 in the case where
$H=\operatorname{Sp}_{2k}$, with $k=5$ and $W=V^{9}$. Given any partition
$\sigma$ with no more than five parts, a tableau $T$ of shape $\sigma$ (with
maximum entry $9$) belongs to $\mathcal{R}(\sigma)$ if and only if it
satisfies the following restriction for each $1\leq i\leq 3$:

$\\#\\{\text{entries in the first column of $T$ which are $\leq 9-2i$}\\}\leq
5-i.$

In other words, in order for a tableau to be in $\mathcal{R}(\sigma)$, the
entries in its first column must satisfy the following: the $3$rd entry must
be $\geq 4$, the $4$th entry must be $\geq 6$, and the $5$th entry must be
$\geq 8$. A brief glance at Definition 2.1 reveals a very similar restriction
for $H=\operatorname{GL}_{k}$ and $\operatorname{O}_{k}$.

The proof of our main result, which we give in Section 4, relies on
combinatorial diagrams called _jellyfish_. We introduced these jellyfish in
[EricksonHunzikerMOC2024], as a way to parametrize Stanley decompositions of
modules of covariants. In that paper, we defined jellyfish only in the range
$k\leq r$, and we used them to recover the result (1.1) from [NOT] via an
elementary combinatorial proof. In the present paper, we extend the definition
of a jellyfish to include all values of $k$ (see Theorem 3.6). In the case
$H=\operatorname{O}_{k}$, we were able in [EricksonHunzikerMOC2024] to define
jellyfish only when the shape $\sigma$ had one column; it is for this reason
that our main result still remains a conjecture for the group
$\operatorname{O}_{k}$.

##  2\. Restricted tableaux and main result

In this section, we define and give examples of the sets $\mathcal{R}(\sigma)$
of restricted tableaux. We then conclude the section by stating our main
result (Theorem 2.4), which interprets the Bernstein degree of modules of
covariants as the cardinality of $\mathcal{R}(\sigma)$.

###  2.1. Partitions and tableaux

A _partition_ is a finite nonincreasing sequence of positive integers. Given a
partition $\pi=(\pi_{1},\ldots,\pi_{m})$, we write $\ell(\pi)\coloneqq m$ for
its _length_. We write $0$ to denote the empty partition, which has length
$0$. The _Young diagram_ associated to a partition $\pi$ is an arrangement of
boxes in left-justified rows, such that the $i$th row from the top contains
$\pi_{i}$ boxes. A _Young tableau_ is a Young diagram in which the boxes are
filled with positive integers (called the _entries_ of the tableau). We say a
tableau is _semistandard_ if the entries are weakly increasing along each row,
and strictly increasing along each column. The _shape_ of a tableau is the
partition corresponding to its underlying Young diagram. We write

$\operatorname{SSYT}(\pi,n)\coloneqq\\{\text{semistandard Young tableaux of
shape $\pi$, with entries $\leq n$}\\}.$

Below, we show an example of a partition $\pi$ along with an element of
$\operatorname{SSYT}(\pi,n)$:

$\ytableausetup{smalltableaux,centertableaux}\pi=(4,3,3,1,1)\quad\leadsto\quad\ytableaushort{1225,234,446,5,6}\in\operatorname{SSYT}(\pi,n),\text{
where $n\geq 6$.}$

Note that if $\ell(\pi)>n$, then we automatically have
$\operatorname{SSYT}(\pi,n)=\varnothing.$ If $\pi=0$ is the empty partition,
then $|{\operatorname{SSYT}(0,n)}|=1$, where the sole element is the empty
tableau.

Since $\pi^{\prime}$ typically denotes the conjugate of $\pi$ (i.e., the
partition whose Young diagram is obtained from that of $\pi$ by reflecting
about the main diagonal), we write $\pi^{\prime}_{1}+\pi^{\prime}_{2}$ to
denote the number of boxes in the first two columns of the Young diagram of
$\pi$. Given a semistandard tableau $T$, we will write $T_{1}$ to denote the
set of entries in its first column (from the left). We write $T_{1}\cup T_{2}$
to denote the multiset of entries in its first two columns. Hence for the
tableau $T$ displayed above, we have $T_{1}\cup T_{2}=\\{1,2,2,3,4,4,5,6\\}$.

When $H=\operatorname{GL}_{k}$, we will also consider finite nonincreasing
sequences $\sigma$ of (possibly nonpositive) integers. Any such sequence can
be expressed uniquely as a pair $\sigma=(\sigma^{+},\sigma^{-})$, where
$\sigma^{+}$ and $\sigma^{-}$ are true partitions: in particular, $\sigma^{+}$
consists of the positive coordinates of $\sigma$, and $\sigma^{-}$ is the
partition obtained by negating and reversing the negative coordinates of
$\sigma$. For example, if $\sigma=(6,3,3,2,0,0,-1,-3,-5)$, then
$\sigma^{+}=(6,3,3,2)$ and $\sigma^{-}=(5,3,1)$.

It is well known that the irreducible finite-dimensional representations
$U_{\sigma}$ of $H$ are indexed by partitions (or pairs thereof) as follows:

(2.1)
$\widehat{H}=\begin{cases}\\{\sigma=(\sigma^{+},\sigma^{-}):\ell(\sigma^{+})+\ell(\sigma^{-})\leq
k\\},&H=\operatorname{GL}_{k},\\\ \\{\sigma:\ell(\sigma)\leq
k\\},&H=\operatorname{Sp}_{2k},\\\
\\{\sigma:\sigma^{\prime}_{1}+\sigma^{\prime}_{2}\leq
k\\},&H=\operatorname{O}_{k}.\end{cases}$

If $H=\operatorname{GL}_{k}$ or $\operatorname{Sp}_{2k}$, then $U_{\sigma}$ is
the irreducible representation of $H$ with highest weight $\sigma$ (in
standard coordinates). For $H=\operatorname{O}_{k}$, which is not connected,
the situation is more subtle; see [GW]*pp. 438–9 for a detailed construction
of the representations $U_{\sigma}$.

###  2.2. Restricted tableaux

$H$ | $\mathbb{C}[W]$ |  $G_{\mathbb{R}}$ | $r$ | $s$ | $\mathcal{S}(\sigma)$
---|---|---|---|---|---
[2pt] $\operatorname{GL}_{k}$ | $\mathbb{C}[V^{*p}\oplus V^{q}]$ |  $\operatorname{U}(p,q)$ | $\min\\{p,q\\}$ | $p+q-1$ | $\phantom{\times}\operatorname{SSYT}(\sigma^{+},q)$
$\times\operatorname{SSYT}(\sigma^{-},p)$ |  |  |  |  |
$\operatorname{Sp}_{2k}$ | $\mathbb{C}[V^{n}]$ |  $\operatorname{O}^{*}(2n)$ | $\lfloor n/2\rfloor$ | $n-1$ | $\operatorname{SSYT}(\sigma,n)$
$\operatorname{O}_{k}$ | $\mathbb{C}[V^{n}]$ |  $\operatorname{Sp}(2n,\mathbb{R})$ | $n$ | $2n-1$ | $\operatorname{SSYT}(\sigma,n)$
Table 1. Data for the three dual pair settings. In each row, $H$ is a complex
classical group acting on $\mathbb{C}[W]$. (Recall that $V$ denotes the
defining representation of $H$.) There is also an action on $\mathbb{C}[W]$ by
$(\mathfrak{g},K)$, where $\mathfrak{g}$ is the complexified Lie algebra of
$G_{\mathbb{R}}$, and $K$ is the complexification of a maximal compact
subgroup (see Table 3). The parameter $r$ is the real rank of
$G_{\mathbb{R}}$. The parameter $s$ is the sharp bound such that $k\geq s$
implies that every module of covariants is free.

For each $\sigma\in\widehat{H}$, we write $\mathcal{S}(\sigma)$ to denote the
full set of semistandard tableaux of shape $\sigma$ given in Table 1. Note
that for $H=\operatorname{GL}_{k}$, since $\sigma=(\sigma^{+},\sigma^{-})$,
each “tableau” $T\in\mathcal{S}(\sigma)$ is actually an ordered pair
$T=(T^{+},T^{-})$. By consulting Table 3 in the appendix, one can see that we
have $\sigma\in\Sigma$ if and only if $\mathcal{S}(\sigma)\neq\varnothing$.
Since $\Sigma$ was the $H$-spectrum of $\mathbb{C}[W]$ in (1.6), this means
that the module of covariants $(\mathbb{C}[W]\otimes U_{\sigma})^{H}$ is
nonzero if and only if $\mathcal{S}(\sigma)$ is nonempty. We now make the
central definition of the paper, namely the subset
$\mathcal{R}(\sigma)\subseteq\mathcal{S}(\sigma)$ given by a certain
restriction on the initial columns.

######  Definition 2.1 (Restricted tableaux).

Let $\sigma\in\widehat{H}$ as in (2.1). We define $\mathcal{R}(\sigma)$ to be
the set of all tableaux $T\in\mathcal{S}(\sigma)$ such that, for all $1\leq
i\leq\min\\{k,\>r,\>s-k\\}$,

$k-i\geq\begin{cases}\\#\\{x\in T^{+}_{1}:x\leq q-i\\}+\\#\\{x\in
T^{-}_{1}:x\leq p-i\\},&H=\operatorname{GL}_{k},\\\ \\#\\{x\in T_{1}:x\leq
n-2i\\},&H=\operatorname{Sp}_{2k},\\\ \\#\\{x\in T_{1}\cup T_{2}:x\leq
n-i\\},&H=\operatorname{O}_{k}.\end{cases}$

######  Remark 2.2.

If the range of $i$-values in Definition 2.1 is empty (meaning that $k\geq
s$), then there are no restrictions and so we have
$\mathcal{R}(\sigma)=\mathcal{S}(\sigma)$. Moreover, depending on the shape
$\sigma$, the first few values of $i$ in Definition 2.1 may yield trivial
restrictions. If we want a _minimal_ system of restrictions, then we can start
at the index $i=k-\ell(\sigma^{+})-\ell(\sigma^{-})$ for
$H=\operatorname{GL}_{k}$, or $i=k-\ell(\sigma)$ for
$H=\operatorname{Sp}_{2k}$, or $i=k-\sigma^{\prime}_{1}-\sigma^{\prime}_{2}$
for $H=\operatorname{O}_{k}$, assuming this $i$ is positive. We also point out
that $\sigma\in\widehat{H}$ if and only if any (and thus _every_) tableau
$T\in\mathcal{S}(\sigma)$ satisfies the restriction in Definition 2.1 at the
index $i=0$.

######  Example 2.3.

  1. (a)

Let $H=\operatorname{GL}_{k}$, where $k=5$, and let $p=3$ and $q=5$. Let
$\sigma=(3,2,1,-2,-2)$, so that $\sigma^{+}=(3,2,1)$ and $\sigma^{-}=(2,2)$.
By Definition 2.1, the set $\mathcal{R}(\sigma)$ consists of those pairs
$(T^{+},T^{-})\in\operatorname{SSYT}(\sigma^{+},5)\times\operatorname{SSYT}(\sigma^{-},3)$
such that

$\displaystyle\\#\\{x\in T^{+}_{1}:x\leq 4\\}+\\#\\{x\in T^{-}_{1}:x\leq 2\\}$
$\displaystyle\leq 4,$ $\displaystyle\\#\\{x\in T^{+}_{1}:x\leq
3\\}+\\#\\{x\in T^{-}_{1}:x\leq 1\\}$ $\displaystyle\leq 3.$

Concretely, $\mathcal{R}(\sigma)$ contains all pairs except those of the
following form:

$\ydiagram{3,2,1}_{\phantom{,}}\ydiagram{2,2,0}$$\ytableaushort{1\none\none,2,3}_{,}\ytableaushort{1\none,2,\none}$$\bigg{(}$$\bigg{)}$120
pairs$\mathcal{R}(\sigma)=\mathcal{S}(\sigma)-\Bigg{\\{}$,$\ydiagram{3,2,1}_{\phantom{,}}\ydiagram{2,2,0}$$\ytableaushort{1\none\none,2,3}_{,}\ytableaushort{1\none,3,\none}$$\bigg{(}$$\bigg{)}$80
pairs,$\ydiagram{3,2,1}_{\phantom{,}}\ydiagram{2,2,0}$$\ytableaushort{1\none\none,2,4}_{,}\ytableaushort{1\none,2,\none}$$\bigg{(}$$\bigg{)}$120
pairs,$\ydiagram{3,2,1}_{\phantom{,}}\ydiagram{2,2,0}$$\ytableaushort{1\none\none,3,4}_{,}\ytableaushort{1\none,2,\none}$$\bigg{(}$$\bigg{)}$105
pairs,$\ydiagram{3,2,1}_{\phantom{,}}\ydiagram{2,2,0}$$\ytableaushort{2\none\none,3,4}_{,}\ytableaushort{1\none,2,\none}$$\bigg{(}$$\bigg{)}$60
pairs$\Bigg{\\}}$.

Therefore since $|\mathcal{S}(\sigma)|=1680$, we have
$|\mathcal{R}(\sigma)|=1680-485=1195$.

  2. (b)

Let $H=\operatorname{Sp}_{2k}$, where $k=4$, and let $n=7$. Let
$\sigma=(4,3,2,2)$. By Definition 2.1, the set $\mathcal{R}(\sigma)$ consists
of those tableaux $T\in\operatorname{SSYT}(\sigma,7)$ such that

$\displaystyle\\{x\in T_{1}:x\leq 5\\}$ $\displaystyle\leq 3,$
$\displaystyle\\{x\in T_{1}:x\leq 3\\}$ $\displaystyle\leq 2.$

Equivalently, the third entry of $T_{1}$ must be at least $4$, and the fourth
entry must be at least $6$. The tableaux violating this restriction are of the
following form:

$\ydiagram{4,3,2,2}$,$\ytableaushort{1\none\none\none,2\none,3,4}$,$2940$$\mathcal{R}(\sigma)=\mathcal{S}(\sigma)-\Bigg{\\{}$$\ydiagram{4,3,2,2}$,$\ytableaushort{1\none\none\none,2\none,3,5}$,$2828$$\ydiagram{4,3,2,2}$,$\ytableaushort{1\none\none\none,2\none,3,6}$,$2429$$\ydiagram{4,3,2,2}$,$\ytableaushort{1\none\none\none,2\none,3,7}$,$1547$$\ydiagram{4,3,2,2}$,$\ytableaushort{1\none\none\none,2\none,4,5}$,$2492$$\ydiagram{4,3,2,2}$,$\ytableaushort{1\none\none\none,3\none,4,5}$,$1820$$\ydiagram{4,3,2,2}$$\ytableaushort{2\none\none\none,3\none,4,5}$$840$$\Bigg{\\}}$.

Since $|\mathcal{S}(\sigma)|=33600$, we have
$|\mathcal{R}(\sigma)|=33600-14896=18704$.

  3. (c)

Let $H=\operatorname{O}_{k}$, where $k=5$, and let $n=4$. Let
$\sigma=(4,3,1)$. By Definition 2.1, the set $\mathcal{R}(\sigma)$ contains
those tableaux $T\in\operatorname{SSYT}(\sigma,4)$ such that

$\displaystyle\\#\\{x\in T_{1}\cup T_{2}:x\leq 3\\}$ $\displaystyle\leq 4,$
$\displaystyle\\#\\{x\in T_{1}\cup T_{2}:x\leq 2\\}$ $\displaystyle\leq 3.$

The tableaux violating this restriction are of the following form:

$\ydiagram{4,3,1}$,$\ytableaushort{11\none\none,22\none,3}$,$20$$\mathcal{R}(\sigma)=\mathcal{S}(\sigma)-\Bigg{\\{}$$\ydiagram{4,3,1}$,$\ytableaushort{11\none\none,22\none,4}$,$20$$\ydiagram{4,3,1}$,$\ytableaushort{11\none\none,23\none,3}$,$16$$\ydiagram{4,3,1}$$\ytableaushort{12\none\none,23\none,3}$$8$$\Bigg{\\}}$.

Since $|\mathcal{S}(\sigma)|=175$, we have $|\mathcal{R}(\sigma)|=175-64=111$.

We now state our main result, which generalizes the result (1.1) of
Nishiyama–Ochiai–Taniguchi [NOT]*Thm. 8.6 for all values of $k$. We will prove
the following theorem in Section 4, after setting up the requisite background
in the next section.

######  Theorem 2.4.

Let $H=\operatorname{GL}_{k}$ or $\operatorname{Sp}_{2k}$, and let
$\sigma\in\widehat{H}$ as in (2.1). In the setting of Table 1, where $k,n,p,q$
are arbitrary positive integers, we have

$\operatorname{Deg}(\mathbb{C}[W]\otimes
U_{\sigma})^{H}=|\mathcal{R}(\sigma)|\cdot\operatorname{Deg}\mathbb{C}[W]^{H},$

where $\mathcal{R}(\sigma)$ is the set of restricted tableaux in Definition
2.1. In particular, if $k\geq r$, then

$\operatorname{Deg}(\mathbb{C}[W]\otimes
U_{\sigma})^{H}=|\mathcal{R}(\sigma)|.$

######  Conjecture 2.5.

Theorem 2.4 also holds for $H=\operatorname{O}_{k}$.

######  Example 2.6.

Using the combinatorial methods developed in [EW, EnrightHunziker04, EHP], we
have written code that computes the resolutions and Hilbert series of the
classical modules of covariants. With this in hand, we revisit Example 2.3 to
show that each case verifies Theorem 2.4 or Conjecture 2.5.

  1. (a)

In the setting of Example 2.3(a), the Hilbert series of
$(\mathbb{C}[V^{*3}\oplus V^{5}]\otimes U_{\sigma})^{\operatorname{GL}_{5}}$
is

$\frac{1680t^{10}-525t^{12}+40t^{14}}{(1-t^{2})^{15}}.$

Hence the Bernstein degree, namely $1680-525+40$, does indeed equal
$|\mathcal{R}(\sigma)|=1195$.

  2. (b)

In the setting of Example 2.3(b), the Hilbert series of
$(\mathbb{C}[V^{7}]\otimes U_{\sigma})^{\operatorname{Sp}_{8}}$ is

$\frac{33600t^{11}-16800t^{13}+2016t^{15}-112t^{17}}{(1-t^{2})^{21}}.$

Hence the Bernstein degree equals
$33600-16800+2016-112=18704=|\mathcal{R}(\sigma)|$.

  3. (c)

In the setting of Example 2.3(c), the Hilbert series of
$(\mathbb{C}[V^{4}]\otimes U_{\sigma})^{\operatorname{O}_{5}}$ is

$\frac{175t^{8}-64t^{10}}{(1-t^{2})^{10}}.$

Hence the Bernstein degree equals $175-64=111=|\mathcal{R}(\sigma)|$.

##  3\. Stanley decompositions via jellyfish diagrams

In this section, we first set down some basic results which will streamline
the proof of Theorem 2.4. Since that theorem is stated only for
$H=\operatorname{GL}_{k}$ and $\operatorname{Sp}_{2k}$, we will treat only
those two groups in this section. We will then extend the definition of a
jellyfish given in [EricksonHunzikerMOC2024] so that it is valid for all
values of $k$.

###  3.1. Fundamental theorems of classical invariant theory

For both $H=\operatorname{GL}_{k}$ and $\operatorname{Sp}_{2k}$, the invariant
ring $\mathbb{C}[W]^{H}$ is generated by certain quadratic contractions which
we call $f_{ij}$, to be defined below. This is the content of Weyl’s first
fundamental theorem (FFT) of classical invariant theory; the second
fundamental theorem (SFT) describes the relations among the $f_{ij}$, via the
vanishing of certain determinants or Pfaffians [Weyl]. We summarize the
relevant details below, following our more expansive treatment in
[EricksonHunzikerMOC2024]*§2.1. The exposition below refers to
$\mathfrak{p}^{-}$ from the triangular decomposition (1.4) of $\mathfrak{g}$,
the complexified Lie algebra of $G_{\mathbb{R}}$.

First let $H=\operatorname{GL}_{k}$, and let
$(v^{*}_{1},\ldots,v^{*}_{p},v_{1},\ldots,v_{q})\in W=V^{*p}\oplus V^{q}$.
Then $\mathbb{C}[W]^{H}$ is generated by the functions

(3.1) $f_{ij}:(v^{*}_{1},\ldots,v^{*}_{p},v_{1},\ldots,v_{q})\longmapsto
v^{*}_{i}(v_{j}),\qquad 1\leq i\leq p,\>1\leq j\leq q.$

We have $S(\mathfrak{p}^{-})\cong\mathbb{C}[z_{ij}:1\leq i\leq p,\>1\leq j\leq
q]$, and (by the SFT) the kernel of the algebra homomorphism

(3.2)
$\displaystyle\begin{split}S(\mathfrak{p}^{-})&\longrightarrow\mathbb{C}[W]^{H},\\\
z_{ij}&\longmapsto f_{ij},\end{split}$

is the ideal generated by the determinants of the $(k+1)$-minors of the matrix
of indeterminates $[z_{ij}]$. It is natural in this context to consider the
indeterminates $z_{ij}$ to be homogeneous of degree 2.

Next, let $H=\operatorname{Sp}_{2k}$, and let $(v_{1},\ldots,v_{n})\in
W=V^{n}$. Then $\mathbb{C}[W]^{H}$ is generated by the functions

(3.3) $f_{ij}:(v_{1},\ldots,v_{n})\longmapsto\omega(v_{i},v_{j}),\qquad 1\leq
i<j\leq n,$

where $\omega$ is the nondegenerate skew-symmetric bilinear form on $V$ which
is preserved by $\operatorname{Sp}_{2k}$. We have
$S(\mathfrak{p}^{-})\cong\mathbb{C}[z_{ij}:1\leq i<j\leq n]$, and (by the SFT)
the kernel of the homomorphism (3.2) is the ideal generated by the
$2(k+1)$-Pfaffians of the alternating matrix of indeterminates $[z_{ij}]$.
Again we view the $z_{ij}$ as homogeneous of degree 2.

###  3.2. Stanley decompositions

Let $M$ be a finitely generated graded $S$-module, where $S$ is a polynomial
ring over $\mathbb{C}$ such that all indeterminates have the same (positive)
degree. Then $M$ admits a certain type of decomposition, not necessarily
unique, called a Stanley decomposition (due to its origin in Stanley’s work
[Stanley82]*Thm. 5.2). Following [BKU]*Def. 2.1, a _Stanley decomposition_ of
$M$ is a finite family $(S_{i},\eta_{i})_{i\in I}$ where $\eta_{i}\in M$ is
homogeneous, and $S_{i}$ is a graded $\mathbb{C}$-algebra retract of $S$ such
that $S_{i}\cap\operatorname{Ann}\eta_{i}=0$, and

(3.4) $M=\bigoplus_{i\in I}S_{i}\eta_{i}$

as a graded vector space. Each component in (3.4) is called a _Stanley space_.
Moreover, the _dimension_ of the $i$th Stanley space is defined to be the
Krull dimension of $S_{i}$. The following well-known result describes how
Stanley spaces of maximal dimension are intimately connected with the
multiplicity of $M$.

######  Lemma 3.1 ([Jahan]*p. 1018).

Let $M$ be a finitely generated graded $S$-module. Then every Stanley
decomposition of $M$ has the same number of Stanley spaces of maximal
dimension. Moreover, this number is the multiplicity (i.e., Bernstein degree)
of $M$.

In this paper, the role of $M$ in (3.4) is played by the module of covariants
$(\mathbb{C}[W]\otimes U_{\sigma})^{H}$, which is a module over the polynomial
ring $S=S(\mathfrak{p}^{-})$ via the homomorphism (3.2). Of course, our $M$ is
also a module over $\mathbb{C}[W]^{H}$ via multiplication of functions, which
is the more natural action in the context of classical invariant theory.
Therefore the role of each $S_{i}$ will be played by a subalgebra of
$\mathbb{C}[W]^{H}$, in particular a polynomial ring generated by certain
$f_{ij}$’s. In our previous work [EricksonHunzikerMOC2024], we introduced
combinatorial objects called _jellyfish_ which parametrize Stanley spaces for
modules of covariants. Because we originally defined jellyfish only in the
range where $k\leq r$, our next task is to generalize the definition to allow
for arbitrary values of $k$.

###  3.3. Jellyfish

Just as in [EricksonHunzikerMOC2024], we will define a jellyfish (of shape
$\sigma$) as an ordered pair $(\mathbf{F},T)$, where $\mathbf{F}$ is a family
of lattice paths, and $T\in\mathcal{S}(\sigma)$ is a tableau depicted as
“tentacles” attached to the endpoints of $\mathbf{F}$. The reader may find it
helpful to consult Figure 1 (page 1) while reading through the details of the
following construction.

We begin with the lattice $\mathbf{L}$ containing one point $(i,j)$ for each
of the generators $f_{ij}$ in (3.1) or (3.3), and we fix a certain
antidiagonal $\mathbf{A}$ and a boundary $\mathbf{B}$, as in Table 2 below:

$H$ | $\mathbf{L}$ |  $\mathbf{A}$ | $\mathbf{B}$
---|---|---|---
[2pt] $\operatorname{GL}_{k}$ | $\left\\{(i,j):\parbox{50.0pt}{$1\leq i\leq p,$\\\ $1\leq j\leq q$}\right\\}$ |  $\\{(i,j):i+j-1=k\\}$ | $\\{(i,j):i=p\text{ or }j=q\\}$
$\operatorname{Sp}_{2k}$ | $\\{(i,j):1\leq i<j\leq n\\}$ |  $\\{(i,j):i+j-2=2k\\}$ | $\\{(i,j):j=n\\}$
Table 2.

We visualize the points $(i,j)$ as matrix coordinates; hence if
$H=\operatorname{GL}_{k}$, then $\mathbf{L}$ is a $p\times q$ grid with
$(1,1)$ in the upper-left, while if $H=\operatorname{Sp}_{2k}$, then
$\mathbf{L}$ is an $(n-1)\times(n-1)$ staircase pattern with $(1,2)$ in the
upper-left. The antidiagonal $\mathbf{A}$ divides $\mathbf{L}$ into a lower
and upper subset, denoted as follows:

(3.5)
$\displaystyle\begin{split}{\downarrow}{\mathbf{A}}&\coloneqq\\{\text{points
lying weakly below the antidiagonal $\mathbf{A}$}\\},\\\
{\uparrow}{\mathbf{A}}&\coloneqq\mathbf{L}-{\downarrow}{\mathbf{A}}.\end{split}$

We observe that the parameters $r$ and $s$ in Table 1 have the following
interpretations in this setting: $r$ is the size of the longest antidiagonal
in $\mathbf{L}$, and $s$ is the size of $\mathbf{B}$.

For us, a _lattice path_ is a path starting on $\mathbf{A}$ and ending on
$\mathbf{B}\cap{\downarrow}{\mathbf{A}}$, consisting of unit steps either down
or to the right. We allow paths that consist of only one point, meaning that
their starting point is their endpoint. The symbol $\mathbf{F}$ will always
denote a family of nonintersecting lattice paths, one path for each point in
$\mathbf{A}$. (To be precise, $\mathbf{F}$ is actually the subset of
$\mathbf{L}$ consisting of those points which lie on the paths in
$\mathbf{F}$.) For visual clarity in our diagrams, we circle the endpoint of
each path in $\mathbf{F}$. We also circle each point in
$\mathbf{B}\cap{\uparrow}{\mathbf{A}}$; in this way, the circled points are
those to which the tentacles (defined below) will be attached. Since
$|\mathbf{B}|=s$, we have

(3.6) $\text{\\# circled
points}=\Big{|}\mathbf{A}\cup(\mathbf{B}\cap{\uparrow}{\mathbf{A}})\Big{|}=\min\\{k,s\\}.$

######  Definition 3.2 (Tentacles).

Let $\sigma\in\widehat{H}$ as in (2.1), and let $T\in\mathcal{S}(\sigma)$ as
in Table 1. Then $T$ can be depicted as a collection of _tentacles_ , in the
following manner:

  1. (a)

Let $H=\operatorname{GL}_{k}$, so that $T=(T^{+},T^{-})$. The tableaux $T^{+}$
and $T^{-}$ are depicted as collections of _vertical_ and _horizontal_
tentacles, respectively. There is one vertical (resp., horizontal) tentacle
for each row of $T^{+}$ (resp., $T^{-}$). Hence there are $\ell(\sigma^{+})$
many vertical tentacles, and $\ell(\sigma^{-})$ many horizontal tentacles. For
each row of $T^{+}$ (resp., $T^{-}$), if its $i$th entry is $x$, then we place
a dot $i$ units below the bottom edge (resp., to the right of the right edge)
of $\mathbf{L}$, aligned with the $x$th column (resp., row) of $\mathbf{L}$,
and then connect all these dots to form a tentacle.

  2. (b)

Let $H=\operatorname{Sp}_{2k}$. For each row of $T$, we construct a tentacle
in the same way as we construct a horizontal tentacle in part (a) above. Note
that each entry $n$ in $T$ is depicted as a dot that is aligned with the $n$th
row of $\mathbf{L}$, which is actually one unit beneath $\mathbf{L}$.

We will call an endpoint of $\mathbf{F}$ _maximal_ if it cannot be moved
farther away from $\mathbf{A}$ (in any family of paths with the same endpoints
as $\mathbf{F}$). We make this precise in the definition below (where by the
“last” antidiagonals we mean the antidiagonals counting from the lower-right
corner of $\mathbf{L}$).

######  Definition 3.3 (Maximal endpoints).

  1. (a)

Let $H=\operatorname{GL}_{k}$. An endpoint $(p,\>q-j)$ along the bottom edge
of ${\downarrow}{\mathbf{A}}$ is _maximal_ if the last $j$ antidiagonals of
$\mathbf{L}$ contain $j$ endpoints.

  2. (b)

Let $G=\operatorname{Sp}_{2k}$. An endpoint $(n-i,\>n)$ is _maximal_ if there
are $\lfloor i/2\rfloor$ endpoints below it.

As an example where $H=\operatorname{GL}_{k}$, in the middle diagram of Figure
1(a), $\mathbf{F}$ has four endpoints along the bottom edge of
${\downarrow}{\mathbf{A}}$. They are all maximal except for the endpoint
$(7,6)=(p,q-3)$, lying three units to the left of the lower-right corner of
$\mathbf{L}$; this endpoint is not maximal because the last three
antidiagonals of $\mathbf{L}$ contain only two endpoints of $\mathbf{F}$. As
an example where $H=\operatorname{Sp}_{2k}$, in the middle diagram of Figure
1(b), $\mathbf{F}$ has three endpoints, of which only the middle endpoint
$(5,9)=(n-4,n)$ is _not_ maximal; this is because there is only one endpoint
below it, rather than $\lfloor 4/2\rfloor=2$.

$\mathbf{A}$$\mathbf{B}$${\uparrow}{\mathbf{A}}$

$\mathbf{F}$

$\sigma=(3,3,3,2,1,0,0,-2,-2,-3)$
$T=\left(\>\ytableaushort{122,334,446,57,6}*[*(blue!20)]{1,1,1,1,1}_{,}\;\ytableaushort{223,34,46,\none,\none}*[*(red!20)]{1,1,1}\>\right)\notin\mathcal{R}(\sigma)$

(a) $H=\operatorname{GL}_{k}$, where $k=10$, $p=7$, and ${q=9}$. Here
$\mathbf{L}$ is a $7\times 9$ grid, where $\mathbf{A}$ is the 10th
antidiagonal (since $k=10$).

$\mathbf{A}$$\mathbf{B}$${\uparrow}{\mathbf{A}}$

$\mathbf{F}$

$\sigma=(3,3,2,2,1)$
$T=\ytableaushort{112,333,45,57,9}*[*(red!20)]{1,1,1,1,1}\notin\mathcal{R}(\sigma)$

(b) $H=\operatorname{Sp}_{2k}$, where $k=5$ and $n=9$. Here $\mathbf{L}$ is an
$8\times 8$ staircase (since $n-1=8$), where $\mathbf{A}$ is the 10th
antidiagonal (since $2k=10$).

Figure 1. Example of a jellyfish, for $H=\operatorname{GL}_{k}$ and
$\operatorname{Sp}_{2k}$. On top, following Table 2, we show the lattice
$\mathbf{L}$, the antidiagonal $\mathbf{A}$, and the boundary $\mathbf{B}$. We
also shade the subset ${\uparrow}{\mathbf{A}}$ defined in (3.5). In the
middle, we show a typical family $\mathbf{F}$, with the endpoints circled. We
also circle the points lying in $\mathbf{B}\cap{\uparrow}{\mathbf{A}}$. On
bottom, for the given $\sigma\in\widehat{H}$ and $T\in\mathcal{S}(\sigma)$, we
depict the jellyfish $(\mathbf{F},T)$ of shape $\sigma$. We highlight the
initial column(s) of $T$, along with the corresponding dots in the tentacles.
To verify that $(\mathbf{F},T)$ is indeed a jellyfish, it is straightforward
to check that the endpoints of $\mathbf{F}$ align with the highlighted dots in
the tentacles according to Definition 3.4. (To illustrate the proof of Lemma
4.1, we have chosen $T\notin\mathcal{R}(\sigma)$.)

######  Definition 3.4 (Jellyfish).

Let $\mathbf{F}$ be a family of $|\mathbf{A}|$ many nonintersecting lattice
paths from $\mathbf{A}$ to $\mathbf{B}\cap{\downarrow}{\mathbf{A}}$. Let
$T\in\mathcal{S}(\sigma)$ for some $\sigma\in\widehat{H}$. We say that the
ordered pair $(\mathbf{F},T)$ is a _jellyfish of shape $\sigma$_ if the
endpoints of $\mathbf{F}$ are compatible with the starting points of the
tentacles in $T$, in the following sense:

  1. (a)

Let $H=\operatorname{GL}_{k}$. To check compatibility, we depict $T$ as a
family of tentacles as in Definition 3.2, and we attach each tentacle (by
drawing a line) to one of the circled points in $\mathbf{B}$. For the
horizontal tentacles, we attach the topmost tentacle to the topmost circle,
and continue in this way from top to bottom. For the vertical tentacles, we
attach the leftmost tentacle to the leftmost circle, and continue in this way
from left to right.

In order for $(\mathbf{F},T)$ to be a jellyfish, the following must be
satisfied. Each endpoint of $\mathbf{F}$ attached to a horizontal tentacle
must be at the same height as that tentacle. All remaining endpoints of
$\mathbf{F}$ must lie along the bottom edge of ${\downarrow}{\mathbf{A}}$. If
an endpoint is maximal, then it can lie weakly to the left of its attached
tentacle (if it has one). If an endpoint is _not_ maximal, then it must be
attached to a tentacle directly beneath it.

  2. (b)

Let $H=\operatorname{Sp}_{2k}$. From top to bottom, we attach each tentacle to
the topmost remaining circled point. If an endpoint of $\mathbf{F}$ is
maximal, then it can lie weakly above its attached tentacle (if it has one).
If an endpoint is _not_ maximal, then it must be attached to a tentacle at the
same height.

######  Remark 3.5.

In [EricksonHunzikerMOC2024], we took an equivalent approach to defining
jellyfish, but from the opposite (and slightly more technical) direction: we
started with $T$, then defined the points at which its tentacles attach to
$\mathbf{B}$, and finally declared that $(\mathbf{F},T)$ is a jellyfish if the
endpoints of $\mathbf{F}$ attach to those tentacles. See Definitions 3.4, 3.6,
and 3.11 in [EricksonHunzikerMOC2024]. From that approach, it is clear that
each $T$ (in fact, just its initial column) determines a unique set of
endpoints, which in turn determines the families $\mathbf{F}$ such that
$(\mathbf{F},T)$ is a jellyfish.

The significance of the jellyfish is explained by the following theorem, which
generalizes Theorems 3.7 and 3.13 in [EricksonHunzikerMOC2024] by allowing
$k\geq r$. We refer the reader to those theorems for a full explanation of the
notation $f_{{\rm cor}(\mathbf{F})}$ and $\varphi_{T}$ below; these are not
relevant to the problem at hand, since in light of Lemma 3.1, our only concern
is the number of Stanley spaces of maximal dimension.

######  Theorem 3.6.

Let $H=\operatorname{GL}_{k}$ or $\operatorname{Sp}_{2k}$ in the setting of
Table 1, where $k,n,p,q$ are arbitrary positive integers. We have a Stanley
decomposition

$(\mathbb{C}[W]\otimes
U_{\sigma})^{H}=\bigoplus_{\mathclap{\begin{subarray}{c}\textup{jellyfish}\\\
(\mathbf{F},T)\\\ \textup{of shape
$\sigma$}\end{subarray}}}\mathbb{C}[f_{ij}:(i,j)\in{\uparrow}{\mathbf{A}}\cup\mathbf{F}]\cdot
f_{{\rm cor}(\mathbf{F})}\cdot\varphi_{T},$

where $f_{ij}\in\mathbb{C}[W]^{H}$ is given in (3.1) or (3.3), and where
$f_{{\rm cor}(\mathbf{F})}\in\mathbb{C}[W]^{H}$ and
$\varphi_{T}\in(\mathbb{C}[W]\otimes U_{\sigma})^{H}$ are given in
[EricksonHunzikerMOC2024]*Thms. 3.7, 3.13.

###### Proof.

Now that we have defined jellyfish for arbitrary values of $k$, this theorem
is proved in exactly the same way as Theorems 3.7 and 3.13 in
[EricksonHunzikerMOC2024]. The only notational difference between those proofs
and the present paper has to do with the starting points of each family
$\mathbf{F}$. In particular, in [EricksonHunzikerMOC2024], we fixed the
starting points along the edge of $\mathbf{L}$ regardless of the value of $k$;
since we stipulated $k\leq r$, the nonintersecting condition on lattice paths
forced $\mathbf{F}$ to contain the first $k$ (for $\operatorname{GL}_{k}$) or
$2k$ (for $\operatorname{Sp}_{2k}$) antidiagonals of $\mathbf{L}$. In the
language of the present paper, these antidiagonals are precisely the points in
${\uparrow}{\mathbf{A}}\cup\mathbf{A}$. Therefore, when $k\leq r$, each
“$\mathbf{F}$” in [EricksonHunzikerMOC2024] is the same as
${\uparrow}{\mathbf{A}}\cup\mathbf{F}$ in this paper. Hence for $k\leq r$, the
present theorem is exactly the same as Theorems 3.7 and 3.13 in
[EricksonHunzikerMOC2024].

Since the proofs in [EricksonHunzikerMOC2024] remain valid in the range $k>r$,
we will simply highlight the combinatorial crux of the proof below. In brief,
the proof relied on showing that no jellyfish contains a _split_. A _split_
was introduced in [Jackson]*Defs. 3.6.15–16 as an arrangement of positive
integers satisfying certain order constraints. (This all follows from the
standard monomial theory for reductive dual pairs developed in [Jackson]; the
monomials associated with splits are among the generators of the ideal of
nonstandard monomials.) Rather than reproduce the technical definition here,
we show a useful way to visualize a split in Figure 2. (This new picture may
also help in understanding the original proofs in [EricksonHunzikerMOC2024].)

In Figure 2, note that all of the $d_{i}$’s are less than all of the
$c_{i}$’s, which are less than all of the $b_{i}$’s. Also, all of the
$d_{i}$’s are less than all of the $a_{i}$’s. For $\operatorname{GL}_{k}$, the
asterisks indicate row indices in $\mathbf{L}$ rather than column indices; for
$\operatorname{Sp}_{2k}$, we deal only with row indices. The $b_{i}$’s and
$c_{i}$’s determine a collection of _inner points_ lying in $\mathbf{L}$,
which we depict in blue or red; note that these are pairwise
northeast/southwest of each other. The $a_{i}$’s and $d_{i}$’s determine a
collection of _outer points_ , shown along the boundary $\mathbf{B}$; these
outer points should be viewed as the starting points of tentacles. (The
$a_{i}$, $b_{i}$, $c_{i}$ and $d_{i}$ sequences, with or without asterisks,
are allowed to be empty.) From now on, when we say _split_ , we mean this
collection of points (rather than the $a_{i}$’s, $b_{i}$’s, etc. themselves).
Note that we depict the $a_{i}$’s as hollow dots, while all the others are
solid. The important feature in the definition of a split is that the number
of solid dots equals $k+1$.

We now briefly re-prove the claim that no jellyfish contains a split. Let
$(\mathbf{F},T)$ be a jellyfish, and suppose, toward contradiction, that there
is a split whose inner points are contained in
${\uparrow}{\mathbf{A}}\cup\mathbf{F}$, and whose outer points are the
starting points of the tentacles in $T$. Because the split has $k+1$ solid
dots, it follows from (3.6) that its inner points lie inside
${\downarrow}{\mathbf{A}}-\mathbf{A}$. Therefore, each inner point of the
split belongs to a distinct lattice path in $\mathbf{F}$, and thus corresponds
to a distinct circled endpoint on $\mathbf{B}$, lying weakly to its southeast.
By Definition 3.4, each of the outer points of the split also attaches to a
distinct circled point on $\mathbf{B}$, lying weakly to its northwest. In
particular, the circled points at which the _solid_ outer points attach are
disjoint from the circled endpoints corresponding to the inner points. Hence
we must have $k+1$ distinct circled points on $\mathbf{B}$, which contradicts
(3.6).

The rest of the proof (showing that each standard monomial lies in exactly one
Stanley space corresponding to a jellyfish) is identical to those of Theorems
3.7 and 3.13 in [EricksonHunzikerMOC2024], via the Stanley–Reisner theory of
order complexes. ∎

$d_{w}$$\ldots$$d_{1}$$a_{v}$$\cdots$$a_{1}$$a_{s}^{*}$$\vdots$$a_{1}^{*}$$d_{t}^{*}$$\vdots$$d_{1}^{*}$$b_{u}^{*}$

…

$b_{1}^{*}$$c_{1}^{*}$$c_{r}^{*}$$\vdots$$c_{u}$$c_{1}$$\cdots$$b_{1}$$b_{r}$$\cdots$
(a) $H=\operatorname{GL}_{k}$

$c_{1}$$c_{2}$

$\cdots$

$c_{r}$$b_{1}$$b_{2}$…$b_{r}$$a_{s}$$\vdots$$a_{1}$$d_{t}$$\vdots$$d_{1}$ (b)
$H=\operatorname{Sp}_{2k}$

Figure 2. Depiction of a split, with notation following
[EricksonHunzikerMOC2024]*Defs. 7.3, 7.7. The number of solid dots equals
$k+1$.

##  4\. Proof of Theorem 2.4

Define the _dimension_ of a jellyfish $(\mathbf{F},T)$ to be
$|{\uparrow}{\mathbf{A}}\cup\mathbf{F}|$. This definition is motivated by the
dimension of a Stanley space, defined above in Section 3.2; indeed, the
dimension of a jellyfish coincides with the dimension of its corresponding
Stanley space in Theorem 3.6.

######  Lemma 4.1.

Let $H=\operatorname{GL}_{k}$ or $\operatorname{Sp}_{2k}$ in the setting of
Table 1, for fixed positive integers $k,n,p,q$. Among all jellyfish of shape
$\sigma$, a jellyfish $(\mathbf{F},T)$ has maximal dimension if and only if
$T\in\mathcal{R}(\sigma)$.

###### Proof.

Since the subset ${\uparrow}{\mathbf{A}}$ depends only on $k$ and not on the
choice of $\sigma$, it follows that $(\mathbf{F},T)$ has maximum dimension if
and only if $|\mathbf{F}|$ is the maximum among all jellyfish of shape
$\sigma$. Hence we must show that $|\mathbf{F}|$ attains this maximum if and
only if $T\in\mathcal{R}(\sigma)$.

  1. (a)

Let $H=\operatorname{GL}_{k}$. In any jellyfish of shape $\sigma$, the family
$\mathbf{F}$ consists of exactly $|\mathbf{A}|$ lattice paths. Hence
$\mathbf{F}$ has maximum size if and only if it contains the last
$|\mathbf{A}|$ many antidiagonals in $\mathbf{L}$. Observe that the $i$th-to-
last antidiagonal contains exactly $i$ points. Since lattice paths travel down
and to the right, the $i$th-to-last antidiagonal is contained in $\mathbf{F}$
if and only if it contains one point from each of $i$ many distinct lattice
paths. This condition is equivalent to each of the following:

     * •

$\mathbf{F}$ has exactly $i$ endpoints contained in the last $i$ antidiagonals
of $\mathbf{L}$.

     * •

By (3.6), there are no more than $k-i$ tentacles of $T$ that are attached
_outside_ the last $i$ antidiagonals of $\mathbf{L}$.

     * •

The following sum is no more than $k-i$: the number of vertical tentacles that
attach weakly left of the $(q-i)$th column of $\mathbf{L}$, plus the number of
horizontal tentacles that attach weakly above the $(p-i)$th row of
$\mathbf{L}$.

     * •

$\\#\\{x\in T^{+}_{1}:x\leq q-i\\}+\\#\\{x\in T^{-}_{1}:x\leq p-i\\}\leq k-i$.
(4.1)

Hence $|\mathbf{F}|$ is maximal if and only if (• ‣ (a)) holds for each $1\leq
i\leq|\mathbf{A}|$. This is precisely the condition in Definition 2.1 for
membership in $\mathcal{R}(\sigma)$. Note that
$|\mathbf{A}|=\min\\{k,\>r,\>s-k+1\\}$ if $k\leq s$, and 0 if $k>s$. In fact,
whenever $|\mathbf{A}|=s-k+1$, the $|\mathbf{A}|$th-to-last antidiagonal of
$\mathbf{L}$ is $\mathbf{A}$ itself, which is necessarily contained in
$\mathbf{F}$, so in this case we need only check for $1\leq i\leq s-k$.

  2. (b)

Let $H=\operatorname{Sp}_{2k}$. In any jellyfish of shape $\sigma$, the family
$\mathbf{F}$ consists of exactly $|\mathbf{A}|$ lattice paths. Hence
$\mathbf{F}$ has maximum size if and only if it contains the $(2i-1)$th
antidiagonal from the bottom, for all $1\leq i\leq|\mathbf{A}|$. Observe that,
counting from the bottom, the $(2i-1)$th antidiagonal contains exactly $i$
points; therefore it is contained in $\mathbf{F}$ if and only if it contains
one point from each of $i$ many distinct lattice paths. Equivalently, we
require exactly $i$ paths in $\mathbf{F}$ to end among the bottommost $2i-1$
many points in $\mathbf{B}$. Equivalently, there must be no more than $k-i$
tentacles of $T$ that are attached among the topmost $n-2i$ many points in
$\mathbf{B}$. This is precisely the condition for $T\in\mathcal{R}(\sigma)$
given in Definition 2.1. Verification of the range $1\leq
i\leq\min\\{k,\>r,\>s-k\\}$ is identical to the end of the proof in part (a)
above. ∎

######  Example 4.2.

We refer the reader back to Figure 1 (page 1) for a concrete illustration of
the proof of Lemma 4.1. In that figure, we intentionally chose
$T\notin\mathcal{R}(\sigma)$, so that the jellyfish $(\mathbf{F},T)$ are _not_
of maximal dimension.

  1. (a)

At the bottom of Figure 1(a), one can immediately see that the jellyfish is
not of maximal dimension, because $\mathbf{F}$ does not contain the last
$|\mathbf{A}|=6$ antidiagonals in $\mathbf{L}$. In particular, the point
$(5,9)$ does not lie in $\mathbf{F}$. In fact, since this point lies in the
3rd-to-last antidiagonal, we know that $T$ violates the restriction in
Definition (2.1) precisely when $i=3$: indeed, the restriction violated by $T$
is

$\\#\\{x\in T^{+}_{1}:x\leq 6\\}+\\#\\{x\in T^{-}_{1}:x\leq 4\\}\leq 7.$

Note that increasing the bottom entry of either $T^{+}_{1}$ or $T^{-}_{1}$
would rectify this violation, resulting in an element of
$\mathcal{R}(\sigma)$. We can see this from the jellyfish as well: by moving
the rightmost vertical tentacle to the right, or moving the bottommost
horizontal tentacle downward, we would shift the lattice paths enough so that
$\mathbf{F}$ would contain the one additional point.

  2. (b)

Likewise, at the bottom of Figure 1(b), we see that $\mathbf{F}$ does not
contain every point in the last $2|\mathbf{A}|-1=5$ antidiagonals of
$\mathbf{L}$. In particular, the point $(7,8)$ does not lie in $\mathbf{F}$.
Since this point lies in an antidiagonal of size $2$, we know that $T$
violates Definition (2.1) for $i=2$: indeed, the restriction violated by $T$
is

$\\#\\{x\in T_{1}:x\leq 5\\}\leq 3.$

In the picture, moving the fourth tentacle downward (i.e., increasing the
fourth entry of $T_{1}$ so that it is strictly greater than $5$, resulting in
a tableau in $\mathcal{R}(\sigma)$) would also force its attached endpoint to
move downward, which would force the lattice paths in $\mathbf{F}$ to shift
and thus contain the one additional point.

######  Lemma 4.3.

Fix a tableau $T\in\mathcal{R}(\sigma)$, in the setting of Lemma 4.1. The
number of jellyfish $(\mathbf{F},T)$ equals the number of families of
$|\mathbf{A}|$ many nonintersecting lattice paths, starting on $\mathbf{A}$
and ending on the $m$th-to-last antidiagonal of $\mathbf{L}$, where

$m=\begin{cases}|\mathbf{A}|,&H=\operatorname{GL}_{k},\\\
2|\mathbf{A}|-1,&H=\operatorname{Sp}_{2k}.\end{cases}$

###### Proof.

In each case, by Lemma 4.1 we know that $\mathbf{F}$ contains every point in
the last $m$ antidiagonals. Restricting our attention to these last $m$
antidiagonals, we thus have a family of $|\mathbf{A}|$ many lattice paths,
whose starting points and endpoints are all fixed (the endpoints are fixed by
$T$). Therefore, since these paths contain every point in these last
antidiagonals, there is no freedom in choosing these paths. Hence for fixed
$T\in\mathcal{R}(\sigma)$, every jellyfish $(\mathbf{F},T)$ has the same
restriction of $\mathbf{F}$ to the last $m$ antidiagonals of $\mathbf{L}$. ∎

###### Proof of Theorem 2.4.

By Lemmas 3.1 and 4.1, the Bernstein degree of $(\mathbb{C}[W]\otimes
U_{\sigma})^{H}$ equals the number of jellyfish $(\mathbf{F},T)$ such that
$T\in\mathcal{R}(\sigma)$. By Lemma 4.3, for each $T\in\mathcal{R}(\sigma)$,
the number of jellyfish $(\mathbf{F},T)$ equals the number of families of
$|\mathbf{A}|$ many nonintersecting lattice paths which start on $\mathbf{A}$
and end on the $m$th-to-last antidiagonal of $\mathbf{L}$. But (again by Lemma
4.3) this is precisely the number of jellyfish $(\mathbf{F},\varnothing)$ in
the special case where $\sigma=0$, and where
$\mathcal{R}(\sigma)=\\{\varnothing\\}$ contains only the empty tableau. Hence
this number is the Bernstein degree of $(\mathbb{C}[W]\otimes
U_{0})^{H}=\mathbb{C}[W]^{H}$, and the result follows. In particular, if
$k\geq r$, then there is only one family of nonintersecting lattice paths from
$\mathbf{A}$ to the $m$th-to-last antidiagonal, and so in this case
$\operatorname{Deg}\mathbb{C}[W]^{H}=1$. ∎

##  Appendix A Howe duality

As described in Section 1.2, the modules of covariants $(\mathbb{C}[W]\otimes
U_{\sigma})^{H}$ can be viewed as $(\mathfrak{g},K)$-modules of unitary
highest weight representations of a real reductive group $G_{\mathbb{R}}$. The
theory uniting these two viewpoints is called _Howe duality_ , which we
summarize below; see also [Howe89, KV].

Consider any of the three settings in Table 1. Let $\mathfrak{g}$ denote the
complexified Lie algebra of $G_{\mathbb{R}}$, and $K$ the complexification of
a maximal compact subgroup. A _$(\mathfrak{g},K)$ -module_ is a complex vector
space carrying representations of both $\mathfrak{g}$ and $K$, such that $K$
acts locally finitely, and also acts compatibly with the action of
$\mathfrak{g}$. Harish-Chandra showed [HC55, HC56] that $G_{\mathbb{R}}$
admits irreducible highest weight $(\mathfrak{g},K)$-modules if and only if
$G_{\mathbb{R}}/K_{\mathbb{R}}$ is a Hermitian symmetric space. Recall from
(1.6) the decomposition of $\mathbb{C}[W]$ as an
$H\times(\mathfrak{g},K)$-module:

$\mathbb{C}[W]\cong\bigoplus_{\sigma\in\Sigma\subseteq\widehat{H}}U_{\sigma}^{*}\otimes\underbrace{(\mathbb{C}[W]\otimes
U_{\sigma})^{H}}_{\cong L_{\lambda(\sigma)}}.$

There is an injective map from $\Sigma$ into the set of dominant integral
weights for $\mathfrak{k}$, which we write above as
$\sigma\mapsto\lambda(\sigma)$. Here $L_{\lambda(\sigma)}$ is the simple
$\mathfrak{g}$-module with highest weight $\lambda(\sigma)$. For the explicit
action of $\mathfrak{g}$ on $(\mathbb{C}[W]\otimes U_{\sigma})^{H}$ by
differential operators, see the appendix in [EricksonHunzikerMOC2024].

The $H$-spectrum $\Sigma$ of $\mathbb{C}[W]$, along with the explicit map
$\sigma\mapsto\lambda(\sigma)$, is given in Table 3. Note that if
$G_{\mathbb{R}}=\operatorname{U}(p,q)$, then $\lambda(\sigma)$ is written out
as a $(p+q)$-tuple, with the first $p$ coordinates written before the
semicolon. If $G_{\mathbb{R}}=\operatorname{O}^{*}(2n)$ or
$\operatorname{Sp}(2n,\mathbb{R})$, then $\lambda(\sigma)$ is an $n$-tuple. We
write $\overleftarrow{\sigma}$ to denote the tuple obtained from $\sigma$ by
reversing its coordinates, and padding with zeros to the left. Similarly, we
write $\overrightarrow{\sigma}$ to denote the result of padding $\sigma$ with
zeros to the right. In each case, we write $\omega_{i}$ to denote the $i$th
fundamental weight for $\mathfrak{g}$ (following Bourbaki’s conventions).

$H$ | $G_{\mathbb{R}}$ |  $(\mathfrak{g},K)$ | $\sigma\in\Sigma$ | $\lambda(\sigma)$
---|---|---|---|---
[2pt] $\operatorname{GL}_{k}$ | $\operatorname{U}(p,q)$ |  $(\mathfrak{gl}_{p+q},\operatorname{GL}_{p}\times\operatorname{GL}_{q})$ | $\ell(\sigma^{+})+\ell(\sigma^{-})\leq k,$ |
$\ell(\sigma^{+})\leq q,$ |  |  |  |
$\ell(\sigma^{-})\leq p$ | $-k\omega_{p}+(-\overleftarrow{\sigma^{-}};\overrightarrow{\sigma^{+}})$ |  |  |
$\operatorname{Sp}_{2k}$ | $\operatorname{O}^{*}(2n)$ |  $(\mathfrak{so}_{2n},\operatorname{GL}_{n})$ | $\ell(\sigma)\leq\min\\{k,n\\}$ | $-2k\omega_{n}-\overleftarrow{\sigma}$
$\operatorname{O}_{k}$ | $\operatorname{Sp}(2n,\mathbb{R})$ |  $(\mathfrak{sp}_{2n},\operatorname{GL}_{n})$ |  $\sigma^{\prime}_{1}+\sigma^{\prime}_{2}\leq k$, |
$\ell(\sigma)\leq n$ | $-\frac{k}{2}\omega_{n}-\overleftarrow{\sigma}$ |  |  |
Table 3. Howe duality data.

By inspecting the coordinates of $\lambda(\sigma)$ given in Table 3, we see
that $F_{\lambda(\sigma)}$, which denotes the $\mathfrak{k}$-module with
highest weight $\lambda(\sigma)$, has the same dimension as another
$\mathfrak{k}$-module whose highest weight is expressed in terms of $\sigma$
alone:

$\dim F_{\lambda(\sigma)}=\dim\begin{cases}F^{(p)}_{\sigma^{-}}\otimes
F^{(q)}_{\sigma^{+}},&G_{\mathbb{R}}=\operatorname{U}(p,q),\\\\[4.30554pt]
F^{(n)}_{\sigma},&G_{\mathbb{R}}=\operatorname{O}^{*}(2n)\text{ or
}\operatorname{Sp}(2n,\mathbb{R}),\end{cases}$

where, for example, $F^{(p)}_{\sigma^{-}}$ denotes the simple
$\mathfrak{gl}_{p}$-module with highest weight $\sigma^{-}$. It is a standard
fact that, for any partition $\pi$, the elements of
$\operatorname{SSYT}(\pi,n)$ can be viewed as elements of a weight basis for
the simple $\mathfrak{gl}_{n}$-module with highest weight $\pi$. From this it
follows that in all three dual pair settings, we have

(A.1) $\dim F_{\lambda(\sigma)}=|\mathcal{S}(\sigma)|,$

where $\mathcal{S}(\sigma)$ is given in Table 1. By Remark 2.2, if $k\geq s$,
then $\operatorname{Deg}L_{\lambda(\sigma)}=\dim F_{\lambda(\sigma)}$ for all
$\sigma\in\widehat{H}$.

## References