# An Error-Guided Correction Model for Chinese Spelling Error Correction

Rui Sun1,2, Xiuyu Wu1,2, Yunfang Wu1,3
1MOE Key Laboratory of Computational Linguistics, Peking University Beijing,
China
2School of Software and Microelectronics, Peking University, Beijing, China
3School of Computer Science, Peking University, Beijing, China
{sunrui0720<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding author.

###### Abstract

Although existing neural network approaches have achieved great success on
Chinese spelling correction, there is still room to improve. The model is
required to avoid over-correction and to distinguish a correct token from its
phonological and visually similar ones. In this paper, we propose an error-
guided correction model (EGCM) to improve Chinese spelling correction. By
borrowing the powerful ability of BERT, we propose a novel zero-shot error
detection method to do a preliminary detection, which guides our model to
attend more on the probably wrong tokens in encoding and to avoid modifying
the correct tokens in generating. Furthermore, we introduce a new loss
function to integrate the error confusion set, which enables our model to
distinguish easily misused tokens. Moreover, our model supports highly
parallel decoding to meet real application requirements. Experiments are
conducted on widely used benchmarks. Our model achieves superior performance
against state-of-the-art approaches by a remarkable margin, on both the
correction quality and computation speed. Our codes are publicly available at
https://github.com/ruisun1/Mask-Predict-main

##  1 Introduction

Chinese spelling correction (CSC) attracts wide attention in recent years,
which is significant for many real applications, such as search engine Martins
and Silva (2004), optical character recognition(OCR) Afli et al. (2016) and
automatic speech recognition(ASR) Hinton et al. (2012).

Given an input sentence with spelling errors, the model is trained to detect
and correct these errors and output a correct sentence. According to Liu et
al. (2010), phonologically and visually similar characters are major
contributing factors for errors in Chinese text. As shown in Figure 1, in the
first example, the error is caused by the misuse of ”派”(send) and ”拍”(take)
which have similar Chinese pronunciation. In the second example, the error is
caused by the misuse of ”门”(door) and ”们”(they) which have similar shapes.

Recently, the advanced neural network models and pre-trained models have
achieved great success in CSC, such as PLOME Liu et al. (2021), REALISE Xu et
al. (2021), PHMOSpell Huang et al. (2021), SpellBert Ji et al. (2021) , GAD
Guo et al. (2021), MLM-phonetics Cheng et al. (2020), RoBERTa-DCN Wang et al.
(2021) and ECSpell Lv et al. (2022). Although much progress has been made,
there are still limitations in previous methods.

First, given an input sequence, only a small fragment might be misspelled.
However, for most of the previous models, they are totally blind to the errors
at start, and so they attend on all tokens equally in encoding and generate
every token from left to right for inference. As a result, previous models are
inefficient and might create over-correction. As these models obtain a
stronger ability to correct the errors, they also tend to modify the correct
tokens by mistake.

Figure 1: Examples of Chinese spelling errors. Misspelled characters and their
corresponding corrections are marked in red.

Second, the confusion set, where a set of phonological and visual similar
tokens are defined for each Chinese token, provides valuable knowledge for
spelling correction, as shown in Figure 2. But the methodology to use it
should be further improved. For example, Liu et al. (2021) propose a Confusion
Set based Masking Strategy, in which they remove a token and replace the token
with a random character in the confusion set. As the model randomly chooses a
token from the confusion set each time, some tokens might be ignored. Besides,
this method can’t pay more attention to the token that is more easily to be
misused. Wang et al. (2019) propose to generate a character from the confusion
set rather than the entire vocabulary. In this hard restriction, the model
cannot generate tokens that are not in the confusion set.

Third, when a CSC model is deployed in real applications, the time cost of
inference is a critical problem to be considered. However, most previous
models try to improve the generation quality but ignore the computation speed.

Figure 2: An example of the confusion set.

To address these issues mentioned above, we propose an Error-Guided Correction
Model (EGCM) for CSC. Firstly, taking advantage of the strong ability of BERT
Devlin et al. (2018), we propose a novel zero-shot error detection method to
do a preliminary detection, which provides precise guidance signals to the
correction model. Following the guidance, our model attends more on the
probably wrong tokens in encoding, and fixes the probably correct tokens
during generation to avoid over-correction. Furthermore, we introduce a new
loss function that effectively integrates the confusion set. By applying this
loss function, every similar token in the confusion set is learned to be
distinguished from the target token, and the most similar token with a high
possibility of being misused is given more attention. To speed up the
inference, we apply a mask-predict strategy Ghazvininejad et al. (2019) to
support parallel decoding, where the tokens with low generation probability
are masked and predicted iteratively.

We conduct extensive experiments on the widely used benchmark dataset SIGHAN
Wu et al. (2013); Yu and Li (2014); Tseng et al. (2015). Experimental results
show that our model significantly outperforms all previous approaches,
achieving a new state-of-the-art performance for Chinese spelling correction.
Moreover, our model has a distinct speed advantage over other models, which is
6.3 times faster than the standard Transformer and 1.5 times faster than the
recent non-autoregressive model TtT Li and Shi (2021).

We summarize our contributions as follows:

  * •

We propose a novel zero-shot error detection method, which guides the
correction model to attend more on the probably wrong tokens in encoding and
fix the probably correct tokens in inference to avoid over-correction.

  * •

We propose a new loss function to take advantage of the confusion set, which
enables our model to distinguish similar tokens and attach more importance to
the easily misused tokens.

  * •

We apply an error-guided mask-predict decoding strategy for spelling
correction, which supports highly parallel decoding and greatly accelerates
the computation speed.

  * •

We integrate all modules into a unified model, which achieves a new state-of-
the-art performance for both correction quality and inference speed.

Figure 3: The architecture of our proposed model. ”M” denotes the [MASK]
token. Guidance for Inference and Guidance Attention Mask are generated from
Zero-shot error detection as shown in Figure 4.

##  2 Related work

CSC is a task that detect and correct wrong tokens in Chinese Sentences. It’s
an active topic that varieties of approaches have been proposed to tackle the
task Wang et al. (2019); Cheng et al. (2020); Li and Shi (2021); Xu et al.
(2021); Liu et al. (2021); Huang et al. (2021).

Figure 4: An example of how our model obtains guidance signals using a zero-
shot method. The original wrong token is marked red.

Earlier work in CSC focuses mainly on unsupervised methods, which typically
adopts a confusion set to find correct candidates and employs a language model
to select the correct one Chen et al. (2013); Yu and Li (2014). Recently,
sequence translation and sequence tagging are the two most widely used methods
in CSC. Wang et al. (2018) treats the CSC task as a sequence labeling problem,
and use a bidirectional LSTM to predict the correct characters. Liu et al.
(2021); Ji et al. (2021); Xu et al. (2021); Lv et al. (2022) try to enrich the
representation generated by the encoder by introducing visual and phonetic
features. Softmax operation is utilized to find a substitution for each token
in the sentence. As the rapid development of neural machine translation
Vaswani et al. (2017), seq2seq encoder-decoder frameworks have been introduced
to the CSC task in Ji et al. (2017); Chollampatt et al. (2016); Wang et al.
(2019).

Recent work tends to utilize character similarity as an external knowledge.
The confusion set where similar characters are stored is widely used Liu et
al. (2021); Zhang et al. (2020); Wang et al. (2019); Yu and Li (2014); Cheng
et al. (2020); Lv et al. (2022). There are several ways of using the confusion
set. The first is to augment the training data by replacing the original token
with it’s similar tokens Liu et al. (2021); Zhang et al. (2020). Wang et al.
(2019) proposes to generate a character from the confusion set rather than the
entire vocabulary. Yu and Li (2014) proposes to produce candidates by
retrieving the confusion set and then filter them via language models. Cheng
et al. (2020) uses similarity graphs derived from the confusion set and use
graph convolution operation to absorb the information from neighboring
characters in the graph.

##  3 Methodology

The proposed Error-Guided Correction Model (EGCM) is illustrated in Figure 3.
We apply the conditional masked language model (CMLM) Ghazvininejad et al.
(2019) as a backbone, which is an encoder-decoder architecture trained with a
masked language model objective Devlin et al. (2018); Conneau and Lample
(2019). In the CMLM architecture, the source wrong sentence with n tokens is
denoted as $X=(x_{1},x_{2},x_{3},...,x_{n})$, the target sentence is denoted
as $Y=(y_{1},y_{2},y_{3},...,y_{n})$. Several tokens in $Y$ are replaced with
[MASK]. These masked tokens construct the set $Y_{mask}$. And the rest of the
tokens in $Y$ that are unmasked construct the set of $Y_{obs}$. For Chinese
spelling correction, given a source sentence $X$ and the set of unmasked
target tokens $Y_{obs}$, the objec is to predict the probability
$P(y|X,Y_{obs})$ and generate token y for each $y\in Y_{mask}$.

We first propose a zero-shot spelling error detection method to provide two
guidance signals to the correction model, as shown in Figure 4. The first
guidance signal is the Guidance Attention Mask that is used in the error-
focused encoder, in which the probably correct tokens are masked to push our
model to attend more on the wrong tokens. The second guidance signal is the
Guidance for Inference that serves as the start of decoding to avoid modifying
correct tokens by mistake. Moreover, we introduce a new loss function to take
advantage of the confusion set. During inference, we apply an error-guided
mask-predict strategy in which the correct tokens are fixed and the probably
wrong tokens are masked and repredicted iteratively.

###  3.1 Zero-shot Error Detection

Given a sentence $X=(x_{1},x_{2},x_{3},...,x_{n})$ that contains $n$ tokens,
we want to make a preliminary decision on which tokens are probably wrong and
which are correct.

As shown in Figure 4, firstly, we construct a $n\times n$ matrix by repeating
the original sentence $n$ times, where the $k^{th}$ token is masked in the
$k^{th}$ row in the matrix ($k$ is from 1 to $n$). Then, we employ BERT Devlin
et al. (2018) to predict each masked position condition on the unmasked tokens
in the same row. Thus, for each position from $x_{1}$ to $x_{n}$ in the
sentence $X$, we obtain the predicted tokens along with their probabilities.
The tokens with the top-$k$ probabilities are selected as candidates of
modification. We assume that if the original token $x_{i}$ occurs in the
candidates list, the token is considered correct. Otherwise, the token is
probably wrong and needs to be corrected.

Based on the output of error detection, we construct two guidance signals
namely Guidance Attention Mask and Guidance for Inference, as shown in Figure
4. The Guidance Attention Mask (GAM) is a matrix constructed by:

$GAM_{ij}=\begin{cases}0,&\texttt{$x_{ij}$ is probably wrong}\\\
1,&\texttt{$otherwise$}\end{cases}$ (1)

where $x_{ij}$ denotes the $j^{th}$ token in the $i^{th}$ sentence. $GAM_{ij}$
denotes the element of the $i^{th}$ row and the $j^{th}$ colomn in GAM. The
Guidance for Inference (GFI) is constructed by masking all the probably wrong
tokens in the original sentence. Further, GAM will be projected into the
error-focused encoder, and GFI will be utilized to initialize the decoder.

###  3.2 Error-aware Encoder

We adopt the Transformer Vaswani et al. (2017) encoder-decoder framework for
Chinese spelling correction. We deviate from the standard Transformer encoder
by fusing an error-focused encoder, as shown in the left part of Figure 3.
$Encoder_{s}$ is a standard Transformer encoder, and on top of that we
introduce an error-focused encoder $Encoder_{ef}$, which utilizes Guidance
Attention Mask to expose the probably wrong tokens and divert the attention of
our model from the correct tokens. The output of $Encoder_{s}$ is input into
the error-focused encoder $Encoder_{ef}$. The Guidance Attention Mask is used
as an extra attention mask in calculating self-attention in $Encoder_{ef}$,
which informs the model which error part of the sentence should be focused on.
Concretely, the output of the $Encoder_{s}$ and $Encoder_{ef}$ is calculated
respectively as:

$H^{s}={\rm Encoder_{s}}({\rm Embedding}(X))$ (2) $H^{ef}={\rm
Encoder_{ef}}(H^{s},atten\\_mask=GAM)$ (3)

###  3.3 Integrating Error Confusion Set for Training

During training, the tokens in $Y_{mask}$ are randomly selected among the
target correct sentence as shown in Figure 3. To better fit the requirements
of correcting both single-character errors and multi-character errors in
Chinese spelling correction, we adopt two masking strategies, namely mask-
separate and mask-range. In mask-separate, we first sample the number of
masked tokens from a uniform distribution between [1, len($X$)], and then
randomly choose that number of tokens. For mask-range, we select $l\in[2,3]$,
and randomly select a span with length $l$. We replace the tokens in
$Y_{mask}$ with a special [MASK] token, which is the generation object of the
model.

There are three attention blocks in the Transformer decoder layer. After the
self-attention block, the decoder will first attend to $H^{s}$, the
representation of the source wrong sentence. Then, the decoder will attend to
$H^{ef}$, the representation of the sentence with correct tokens being masked.
The output of the previous decoder layer is then input into the next decoder
layer.

$H^{l}_{d_{1}}={\rm selfAttention}(H^{l-1})$ (4) $H^{l}_{d_{2}}={\rm
Attention}(Q=H^{l}_{d_{1}},K=H^{s},V=H^{s})$ (5)

$H^{l}={\rm Attention}(Q=H^{l}_{d_{2}},K=H^{ef},V=H^{ef})$ (6)

where $H^{0}=Embedding(Y_{obs})$. Q, K, V represents the Query, Key, Value
matrix. $Y_{obs}$ is the set of unmasked tokens in the target sentence. The
output probability distribution $P$ is generated from the decoder over the
vocabulary $V$:

$P={\rm softmax}(H^{l}W+b)$ (7)

where $H^{l}\in\mathbb{R}^{t\times d}$, $W\in\mathbb{R}^{d\times\lvert
v\rvert}$, $b\in\mathbb{R}^{t\times\lvert v\rvert}$. $t$ denotes the sequence
length.

We optimize the model over every token in $Y_{mask}$. Besides the traditional
loss function, we introduce a new loss to integrate the confusion set
knowledge.

We employ Maximum Likelihood Estimation (MLE) to conduct parameter learning
and utilize negative log-likelihood (NLL) as the loss function, which is
computed as:

$L_{nll}=-\sum_{y_{i}\in{Y_{mask}}}\log{P(y_{i}|X,Y_{obs})}$ (8)

To make full use of the confusion set knowledge, we introduce a new loss
function $L_{cs}$. We adopt the confusion set constructed by Lv et al. (2022).
For each token $y_{i}$ in $Y_{mask}$, we find out the set of the similar
tokens of $y_{i}$ based on the confusion set, namely $Y_{conf}$. The tokens in
$Y_{conf}$ are regarded as negative samples of $y_{i}$. We use these negative
samples to help our model better learn the difference between the target token
and its similar ones. The optimization objective for the confusion loss
$L_{cs}$ is defined as:

$L_{cs}=-\sum_{y_{i}\in{Y_{mask}}}\frac{\log{P(y_{i}|X,Y_{obs})}}{\sum_{y_{c}\in{Y_{conf}}}\log{P(y_{c}|X,Y_{obs})}}$
(9)

where $y_{c}$ denotes the similar token of $y_{i}$ in the confusion set.

Overall, the final optimization objective of our model is:

$L_{f}=L_{nll}+\gamma\times L_{cs}$ (10)

where $\gamma$ is a hyperparameter to balance two loss functions.

###  3.4 Error-Guided Generation

In the inference stage, we apply a mask-predict approach Ghazvininejad et al.
(2019), where the tokens with low probability are masked and predicted within
a constant number of iterations.

To provide the model a good start point for generation, we exploit the
Guidance for Inference (GFI) as an initialization for decoding. GFI produces a
draft sentence, where the probably wrong tokens are masked and the probably
correct ones are remained unmasked. During generation, the unmasked tokens
will be fixed, and only the masked tokens are taken into consideration for
modification in each iteration. Fixing these correct tokens will effectively
teach our model to avoid over-correction. Figure 5 shows how does our model
correct a wrong sentence in 3 iterations.

The model runs for a pre-determined number of iterations $T$. The number of
[MASK] in the draft sentence is denoted as $N_{ori}$. Accordingly, the number
of tokens that are masked in the $t_{th}$ iteration is defined as
$N_{t}={N_{ori}}\times\frac{T-t}{T}$.

Figure 5: An example of Error-Guided Generation. In Guidance for Inference,
the masked tokens are highlighted. In later iterations, the highlighted tokens
are of lowest probabilities and are masked and repredicted. The wrong tokens
are marked in red.

Formally, $Y_{mask}^{(0)}$ is the set of masked tokens in the Guidance for
Inference. At a later iteration $t$, we choose $N_{t}$ tokens among the masked
tokens in the previous iteration $t-1$ that has the lowest probability scores:

$Y_{mask}^{(t)}=\mathop{\arg\min}\limits_{y_{i}\in
Y_{mask}^{(t-1)}}(p_{i},N_{t})$ (11)

$Y_{obs}^{(t)}=Y\backslash Y_{mask}^{(t)}$ (12)

Where $p_{i}$ is the probability score of $y_{i}$ calculated in Equation 13,
14. $Y_{mask}^{(t)}$ is the set of masked tokens that are probably wrong at
the $t_{th}$ iteration, and $Y_{obs}^{(t)}$ is the set of unmasked tokens that
are considered correct and fixed in later iterations. At each iteration, the
model predicts the probably wrong tokens in $Y_{mask}^{(t)}$ conditioned on
the source text $X$ and $Y_{obs}^{(t)}$. We select the prediction with the
highest probability for each masked token $y_{i}\in Y_{mask}^{(t)}$, and
update its probability score accordingly:

$y_{i}^{(t)}=\mathop{\arg\max}\limits_{w\in V}P(y_{i}=w|X,Y_{obs}^{(t)})$ (13)
$p_{i}^{(t)}=\mathop{\max}\limits_{w\in V}P(y_{i}=w|X,Y_{obs}^{(t)})$ (14)

where $P(y_{i}=w|X,Y_{obs}^{(t)})$ is the conditional probability of $y_{i}$
being predicted as the token $w$ in the vocabulary set $V$.

##  4 Experimental Setup

###  4.1 Dataset and Metrics

Training dataset Following Liu et al. (2021), the training data is composed of
10K manually annotated samples from SIGHAN Wu et al. (2013) and 271K
automatically generated samples from Wang et al. (2018).

Evaluation dataset Following previous works, the SIGHAN15 test dataset Tseng
et al. (2015) is used to evaluate the proposed model. Statistics of the used
datasets please refer to Appendix A.

Evaluation Metrics We evaluate model performance of detection and correction
at sentence-level, with accuracy, precision, recall and F1 scores. We evaluate
these metrics using the script from Cheng et al. (2020)
111https://github.com/ACL2020SpellGCN/SpellGCN. Moreover, following Liu et al.
(2021), we also report the sentence-level results evaluated by SIGHAN official
tool 222http://nlp.ee.ncu.edu.tw/resource/csc. html.

Method |  | Detection Level |  |  | Correction Level |
---|---|---|---|---|---|---
Acc. | Pre. | Rec. | F1. | Acc. | Pre. | Rec. | F1
Confusionset (2019) | - | 66.8 | 73.1 | 69.8 | - | 71.5 | 59.5 | 64.9
FASPell (2019) | 74.2 | 67.6 | 60.6 | 63.5 | 73.7 | 66.6 | 59.1 | 62.6
SpellGCN (2020) | - | 74.8 | 80.7 | 77.7 | - | 72.1 | 77.7 | 75.9
Chunk2020 (2020) | 76.8 | 88.1 | 62.0 | 72.8 | 74.6 | 87.3 | 57.6 | 69.4
SM BERT (2020) | 80.9 | 73.7 | 73.2 | 73.5 | 77.4 | 66.7 | 66.2 | 66.4
RoBERTa-DCN (2021) | - | 76.6 | 79.8 | 78.2 | - | 74.2 | 77.3 | 75.7
ECSpell (2022) | 83.4 | 76.4 | 79.9 | 78.1 | 82.4 | 74.4 | 77.9 | 76.1
PLOME* (2021) | - | 77.4 | 81.5 | 79.4 | - | 75.3 | 79.3 | 77.2
REALISE* (2021) | 84.7 | 77.3 | 81.3 | 79.3 | 84.0 | 75.9 | 79.9 | 77.8
PHMOSpell* (2021) | - | 90.1 | 72.7 | 80.5 | - | 89.6 | 69.2 | 78.1
MLM-phonetics* (2021) | - | 77.5 | 83.1 | 80.2 | - | 74.9 | 80.2 | 77.5
GAD* (2021) | - | 75.6 | 80.4 | 77.9 | - | 73.2 | 77.8 | 75.4
SpellBert* (2021) | - | 87.5 | 73.6 | 80.0 | - | 87.1 | 71.5 | 78.5
BERT-finetune | 82.4 | 74.2 | 78.0 | 76.1 | 81.0 | 71.6 | 75.3 | 73.4
Our EGCM | 86.4 | 82.7 | 77.6 | 80.0 | 85.8 | 80.6 | 74.7 | 77.5
Pre-Tn EGCM* | 87.2 | 83.4 | 79.8 | 81.6 | 86.3 | 81.4 | 78.4 | 79.9
Table 1:  Performance on the SIGHAN15 test set. Best results are in bold. The first group lists the models that are not pretrained, and the second group lists the methods that are pretrained (denoted with ”*” ).  Method | Detection level | Correction level
---|---|---
Pre | Rec | F1 | Pre | Rec | F1
SpellGCN | 85.9 | 80.6 | 83.1 | 85.4 | 77.6 | 81.3
ECSpell | 85.7 | 78.4 | 81.9 | 85.4 | 76.6 | 80.7
TtT | 85.4 | 78.1 | 81.6 | 85.0 | 75.6 | 80.0
GAD | 86.0 | 80.4 | 83.1 | 85.6 | 77.8 | 81.5
Pre-Tn EGCM | 93.5 | 76.7 | 84.3 | 91.4 | 74.5 | 82.1
Table 2:  Performance on the SIGHAN15 test evaluated by the official tools.
Best results are in bold.

###  4.2 Comparing Methods

We compare the performance of our model with several strong baseline methods
as follows:

Confusionset introduces a copy mechanism into seq2seq and generates characters
from the confusionset Wang et al. (2019).

FASPell utilizes a denoising autoencoder to generate candidates Hong et al.
(2019).

SpellGCN incorporates phonological and visual knowledge via a graph
convolutional network Cheng et al. (2020).

Chunk proposes a chunk-based decoding method with global optimization Bao et
al. (2020).

SM BERT uses soft-masking technique to connect the network of detection and
correction Zhang et al. (2020).

TtT employs a Transformer Encoder with a Conditional Random Fields layer
stacked Li and Shi (2021).

PLOME proposes a confusion set based masking strategy Liu et al. (2021).

REALISE leverages the multimodal information and mixes them electively Xu et
al. (2021).

PHMOSpell integrates pinyin and glyph with a multi-modal method Huang et al.
(2021).

ECSpell adopts the Error Consistent masking strategy for pretraining Lv et al.
(2022).

MLM-phonetics integrates phonetic features by leveraging pre-training and
fine-tuning Zhang et al. (2021).

RoBERTa-DCN generates the candidates via a Pinyin Enhanced Generator Wang et
al. (2021).

SpellBert employs a graph neural network to introduce visual and phonetic
features Ji et al. (2021).

GAD learns the global relationships of the potential correct input characters
and the candidates of potential error characters Guo et al. (2021).

BERT We also implement classical methods for comparison. We fine-tune the
Chinese BERT model Devlin et al. (2018) on the CGEC corpus directly.

###  4.3 Hyperparameter Setting

We follow most of the standard hyperparameters for transformers in the base
configuration Vaswani et al. (2017) and follow the weight initialization
scheme from BERT Devlin et al. (2018). For regularization, we use 0.3 dropout,
0.01 L2 weight decay. The hyperparameter $\gamma$ which is used to weight the
confusion loss is set to 2 after tuning. Adam optimizer Kingma and Ba (2014)
with $\beta=(0.9,0.999),\varepsilon=1e^{-6}$ is used to conduct the parameter
learning. The learning rate is set to $5e^{-5}$, and the model is trained with
learning rate warming up and linear decay.

##  5 Results and Analysis

Method | Detection Level | Correction Level
---|---|---
Pre. | Rec. | F1. | Pre. | Rec. | F1.
EGCM | 82.7 | 77.6 | 80.0 | 80.6 | 74.7 | 77.5
\- EFEnc | 80.5 | 75.2 | 77.8 | 78.7 | 72.3 | 75.4
\- CFL | 79.2 | 72.7 | 75.8 | 77.3 | 69.9 | 73.4
\- GFI | 77.5 | 76.9 | 77.2 | 77.1 | 73.9 | 75.7
Table 3:  Ablation study on SIGHAN15. ”-EFEnc” means removing the error-focused encoder. ”-CFL” means removing the confusion set loss. ”-GFI” means not using Guidance for Inference in the inference stage. top-k | $P_{error\&mask/error}$ | $P_{correct/unmask}$
---|---|---
k=1 | 94% | 99.8%
k=2 | 90% | 99.7%
k=3 | 88% | 99.7%
Table 4: An evaluation on Zero-shot error detection.

###  5.1 Overall Performance

Table 1 reports the performance of our proposed EGCM model and baseline models
on the SIGHAN15 test set. For a fair comparison, we also employ the pre-
trained model cBERT Liu et al. (2021) which has the same architecture with
BERT and pre-trained via the confusion set based masking strategy. Our model
with pretrained cBERT (Pre-Tn EGCM) outperforms all existing approaches,
achieving a 81.6 F1 at detection and 79.9 F1 at correction. Compared with the
BERT baseline, Pre-Tn EGCM achieves 5.5% performance gain on detection F1 and
6.5% gain on correction F1. Among un-pretrained methods, EGCM also outperforms
all competitor models by a wide margin.

We also evaluate the model performance using the official tool, and report the
results in Table 2. Our model Pre-Tn EGCM obtains the best results for both
detection and correction. Especially, it greatly outperforms previous methods
in precision.

It should be emphasized that, our model EGCM is trained on 270k HybirdSet and
outperforms several models that are pre-trained on a big size of synthetic
data, such as PLOME Liu et al. (2021) which is pre-trained using 162 million
sentences. This demonstrates that our model effectively learns to correct
spelling errors without relying on heavyweight data. An example output of our
EGCM comparing with BERT is listed in Appendix B.

###  5.2 Ablation Study

We explore the contribution of each component in our EGCM model by conducting
ablation studies with the following settings: (1) Removing the error-focused
encoder mentioned in 3.2. (2) Removing the confusion set loss $L_{cs}$ in
equation 9. (3) Initialize the start sequence of inference with all [MASK]
instead of using the Guidance for Inference. The results are shown in Table 3.

Specifically, the confusion set loss leads to the biggest improvement to our
model with 4.2 points for detection and 4.1 points for correction. By removing
the error-focused encoder, the drop of performance indicates that this encoder
does learn to pay attention to the probably wrong tokens of the sentence and
impel our model to correct the wrong tokens actively. Also, without the use of
Guidance for Inference as the start of decoding for inference, the performance
drops especially on precision, which indicates that by fixing the tokens that
are correct can effectively avoid over-correction and improve precision.

###  5.3 Evaluation on Zero-shot Error Detection

We employ a zero-shot detection approach to do a preliminary detection, in
which all the tokens are divided into two groups, the probably wrong tokens
and the probably correct one. In the inference stage, the probably correct
ones are unmasked and will not be modified to avoid over-correction, while the
probably wrong ones are masked and repredicted . We want to ensure that
unmasked tokens are truly correct that don’t need to be modified, and at the
same time, the errors in the sentences are masked as many as possible.

As shown in Table 4, $P_{error\&mask/error}$ denotes the percentage of errors
that are masked, $P_{correct/unmask}$ denotes the percentage of truly correct
tokens in the unmasked tokens. In our zero-shot error detection, the BERT
predicted tokens with top-$k$ probabilities are selected as candidates, if the
original token is not in the candidates list, it is considered as wrong. We
try different $k$ and conduct experiments. Our method achieves promising
results with high accuracy, which guarantees correct signals for further
processing. Obviously, the smaller $k$ is, the more tokens are masked and less
tokens are fixed, this might lead to over-correction. We want the errors are
masked as many as possible, and at the same time, fewer tokens are masked.
Therefore, in our model, we set $k=2$.

###  5.4 Analysis on Different Confusion Sets

Confusion set | Method | Detection level | Correction level
---|---|---|---
Pre. | Rec. | F1. | Pre. | Rec. | F1.
[1] (2022) | ECSpell | 76.4 | 79.9 | 78.1 | 74.4 | 77.9 | 76.1
EGCM | 82.7 | 77.6 | 80.0 | 80.6 | 74.7 | 77.5
[2] (2013) | SpellGCN | 74.8 | 80.7 | 77.7 | 72.1 | 77.7 | 75.9
EGCM | 80.6 | 78.2 | 79.4 | 78.2 | 76.3 | 77.2
PLOME* | 77.4 | 81.5 | 79.4 | 75.3 | 79.3 | 77.2
Pre-Tn EGCM* | 81.6 | 79.6 | 80.6 | 79.8 | 76.4 | 78.1
[3] (2018) | Confusionset | 66.8 | 73.1 | 69.8 | 71.5 | 59.5 | 64.9
EGCM | 79.5 | 74.7 | 77.0 | 77.4 | 71.7 | 74.4
Table 5:  Effects of different confusion sets.

To further prove the effectiveness of the confusion loss we proposed, and to
show that this loss function can be generalized, we conduct experiments on
three different confusion sets, including the confusion set proposed by Lv et
al. (2022) 333https://github.com/Aopolin-Lv/ECSpell, Wu et al. (2013)
444http://nlp.ee.ncu.edu.tw/resource/csc.html, and Wang et al. (2018)
555https://github.com/wdimmy/Automatic-Corpus-Generation. For each confusion
set, we compare our model with the models that use the same confusion set but
in different way.

As shown in table 5. For all three confusion sets, our model outperforms the
model that utilizes the same confusion set. Compared with previous methods,
our model takes every token in the confusion set into consideration by
computing it’s possibility of being misused. Moreover, the results indicate
that our model has strong generalization ability and is not limited to any
specific confusion set.

###  5.5 Analysis on Decoding Iterations

With a predefined decoding iteration $T=10$, we show the F1 score of previous
iterations $t(t<T)$ to illustrate how the mask-predict strategy detects and
corrects the wrong tokens step by step. As shown in figure 6, the F1 score of
detection and correction improves as the decoding iteration goes up. This
indicates that, by masking and repredicting the tokens of low probability in
each iteration, our model corrects the tokens that are wrongly predicted
during previous iterations. And as the number of unmasked tokens increases,
more information is given to help predict the hard masked tokens. With 8
iterations, our model achieves state-of-the-art performance.

Model | Time(ms) | Speedup
---|---|---
Transfomer (2017) | 63ms | 1x
PLOME (2021) | 45ms | 1.4x
REALISE (2021) | 17ms | 3.7x
TtT (2021) | 15ms | 4.2x
EGCM | 10ms | 6.3x
Table 6:  Comparisons of the computing efficiency.

###  5.6 Analysis on Computing Efficiency

Chinese spelling correction can be applied in many real-life applications,
such as writing assistant and search engine. Therefore, the time cost
efficiency of models is a key point to be considered. We implement both the
baseline models and our model on the single NVIDIA RTX 2080 GPU. Table 6
depicts the time cost per sample of our model comparing with some previous
approaches. Our model runs faster than all previous approaches.

Figure 6: Results of different decoding iterations

##  6 Conclusion

We propose an error-guided correction model for the CSC task. A zero-shot
error detection method is proposed to provide guidance signals for training
and inference. We apply a conditional masked language model as a backbone,
where we improve the encoder-decoder architecture by adding an error-focused
encoder which pushes our model to focus on the wrong tokens. During training,
we introduce a new confusion loss to help the model distinguish similar
tokens. During inference, the error-guided mask-predict decoding strategy is
adopted to mask and repredict the tokens that are probably wrong. Experimental
results show that our model not only achieves superior performance against
state-of-the-art approaches but also is cost-saving and green.

## Limitation

In this paper, we use the results from zero-shot spelling error detection as a
guidance signal. The sentence with probably wrong tokens masked and the other
tokens fixed are used as a start of decoding. This means that if a wrong token
is not assigned with a [MASK] token, it will never be corrected in later
iterations. Even though we conduct experiments and the result shows that up to
94% of the wrong tokens are masked in the guidance signal, there are still
some wrong tokens missed by our model. To limit the number of tokens that are
free to be modified is one of our ways to improve precision, but we are also
looking forward to a way to further improve recall.

What’s more, even though we make full use of the confusion set, we still think
that’s not enough. Now we are using the confusion set in which every token has
a set of predefined similar tokens. And these sets of similar tokens are
isolated with each other. However, Chinese has various kinds of spelling
errors, the target token might not be in the predetermined similar tokens set
of the original token. And this kind of mistakes can never be learned to
correct by the model. We think a better design for the data structure of the
confusion set needs to be proposed, in which the sets are not isolated and we
are able to calculate the similarity distance between each pair of tokens
using particular algorithms, for example, UnionFind on a dynamic Graph. This
kind of dynamic confusion knowledge can help avoid ignoring the probably
misused tokens.

## Acknowledgement

This work is supported by the National Hi-Tech RD Program of China
(No.2020AAA0106600), the National Natural Science Foundation of China
(62076008) and the Key Project of Natural Science Foundation of China
(61936012).

## References

  * Afli et al. (2016) Haithem Afli, Zhengwei Qiu, Andy Way, and Páraic Sheridan. 2016.  Using smt for ocr error correction of historical texts. In _LREC_. European Language Resources Association (ELRA).
  * Bao et al. (2020) Z. Bao, C. Li, and R. Wang. 2020.  Chunk-based chinese spelling check with global optimization.  In _Findings of the Association for Computational Linguistics: EMNLP 2020_.
  * Chen et al. (2013) Kuan-Yu Chen, Hung-Shin Lee, Chung-Han Lee, Hsin-Min Wang, and Hsin-Hsi Chen. 2013\.  A study of language modeling for chinese spelling check. In _SIGHAN@IJCNLP_ , pages 79–83. Asian Federation of Natural Language Processing.
  * Cheng et al. (2020) Xingyi Cheng, Weidi Xu, Kunlong Chen, Shaohua Jiang, Feng Wang, Taifeng Wang, Wei Chu, and Yuan Qi. 2020.  Spellgcn: Incorporating phonological and visual similarities into language models for chinese spelling check. _CoRR_ , abs/2004.14166.
  * Chollampatt et al. (2016) Shamil Chollampatt, Kaveh Taghipour, and Hwee Tou Ng. 2016.  Neural network translation models for grammatical error correction. _CoRR_ , abs/1606.00189.
  * Conneau and Lample (2019) Alexis Conneau and Guillaume Lample. 2019.  Cross-lingual language model pretraining. In _NeurIPS_ , pages 7057–7067.
  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  Bert: Pre-training of deep bidirectional transformers for language understanding.  Cite arxiv:1810.04805.
  * Ghazvininejad et al. (2019) Marjan Ghazvininejad, Omer Levy, Yinhan Liu, and Luke Zettlemoyer. 2019.  Mask-predict: Parallel decoding of conditional masked language models. In _EMNLP/IJCNLP (1)_ , pages 6111–6120. Association for Computational Linguistics.
  * Guo et al. (2021) Zhao Guo, Yuan Ni, Keqiang Wang, Wei Zhu, and Guotong Xie. 2021.  Global attention decoder for chinese spelling error correction.  In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , pages 1419–1428.
  * Hinton et al. (2012) Geoffrey Hinton, Li Deng, Dong Yu, George E Dahl, Abdel-rahman Mohamed, Navdeep Jaitly, Andrew Senior, Vincent Vanhoucke, Patrick Nguyen, Tara N Sainath, et al. 2012.  Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups.  _IEEE Signal processing magazine_ , 29(6):82–97.
  * Hong et al. (2019) Y. Hong, X. Yu, N. He, N. Liu, and J. Liu. 2019.  Faspell: A fast, adaptable, simple, powerful chinese spell checker based on dae-decoder paradigm.  In _Proceedings of the 5th Workshop on Noisy User-generated Text (W-NUT 2019)_.
  * Huang et al. (2021) L. Huang, J. Li, W. Jiang, Z. Zhang, and J. Xiao. 2021.  Phmospell: Phonological and morphological knowledge guided chinese spelling check.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_.
  * Ji et al. (2017) Jianshu Ji, Qinlong Wang, Kristina Toutanova, Yongen Gong, Steven Truong, and Jianfeng Gao. 2017.  A nested attention neural hybrid model for grammatical error correction. _CoRR_ , abs/1707.02026.
  * Ji et al. (2021) Tuo Ji, Hang Yan, and Xipeng Qiu. 2021.  Spellbert: A lightweight pretrained model for chinese spelling check.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 3544–3551.
  * Kingma and Ba (2014) D. Kingma and J. Ba. 2014.  Adam: A method for stochastic optimization.  _Computer Science_.
  * Li and Shi (2021) P. Li and S. Shi. 2021.  Tail-to-tail non-autoregressive sequence prediction for chinese grammatical error correction.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_.
  * Liu et al. (2010) C. L. Liu, M. H. Lai, Y. H. Chuang, and C. Y. Lee. 2010.  Visually and phonologically similar characters in incorrect simplified chinese words.  In _COLING 2010, 23rd International Conference on Computational Linguistics, Posters Volume, 23-27 August 2010, Beijing, China_.
  * Liu et al. (2021) Shulin Liu, Tao Yang, Tianchi Yue, Feng Zhang, and Di Wang. 2021.  PLOME: Pre-training with misspelled knowledge for Chinese spelling correction.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 2991–3000, Online. Association for Computational Linguistics.
  * Lv et al. (2022) Q. Lv, Z. Cao, L. Geng, C. Ai, X. Yan, and G. Fu. 2022.  General and domain adaptive chinese spelling check with error consistent pretraining.
  * Martins and Silva (2004) Bruno Martins and Mário J. Silva. 2004.  Spelling correction for search engine queries. In _EsTAL_ , volume 3230 of _Lecture Notes in Computer Science_ , pages 372–383. Springer.
  * Tseng et al. (2015) Yuen-Hsien Tseng, Lung-Hao Lee, Li-Ping Chang, and Hsin-Hsi Chen. 2015.  Introduction to sighan 2015 bake-off for chinese spelling check. In _SIGHAN@IJCNLP_ , pages 32–37. Association for Computational Linguistics.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, _Advances in Neural Information Processing Systems 30_ , page 5998–6008. Curran Associates, Inc.
  * Wang et al. (2021) Baoxin Wang, Wanxiang Che, Dayong Wu, Shijin Wang, Guoping Hu, and Ting Liu. 2021\.  Dynamic connected networks for chinese spelling check.  In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , pages 2437–2446.
  * Wang et al. (2018) Dingmin Wang, Yan Song, Jing Li, Jialong Han, and Haisong Zhang. 2018.  A hybrid approach to automatic corpus generation for chinese spelling check.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 2517–2527.
  * Wang et al. (2019) Dingmin Wang, Yi Tay, and Li Zhong. 2019.  Confusionset-guided pointer networks for chinese spelling check. In _ACL (1)_ , pages 5780–5785. Association for Computational Linguistics.
  * Wu et al. (2013) Shih-Hung Wu, Chao-Lin Liu, and Lung-Hao Lee. 2013.  Chinese spelling check evaluation at sighan bake-off 2013. In _SIGHAN@IJCNLP_ , pages 35–42. Asian Federation of Natural Language Processing.
  * Xu et al. (2021) H. D. Xu, Z. Li, Q. Zhou, C. Li, and X. L. Mao. 2021.  Read, listen, and see: Leveraging multimodal information helps chinese spell checking.
  * Yu and Li (2014) Junjie Yu and Zhenghua Li. 2014.  Chinese spelling error detection and correction based on language model, pronunciation, and shape. In _CIPS-SIGHAN_ , pages 220–223. Association for Computational Linguistics.
  * Zhang et al. (2021) Ruiqing Zhang, Chao Pang, Chuanqiang Zhang, Shuohuan Wang, Zhongjun He, Yu Sun, Hua Wu, and Haifeng Wang. 2021.  Correcting chinese spelling errors with phonetic pre-training.  In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , pages 2250–2261.
  * Zhang et al. (2020) Shaohua Zhang, Haoran Huang, Jicong Liu, and Hang Li. 2020.  Spelling error correction with soft-masked bert. In _ACL_ , pages 882–890. Association for Computational Linguistics.

##  Appendix A Statistics of Datasets

Training Set | #Sent | Avg.Length | #Errors
---|---|---|---
SIGHAN13 | 700 | 41.8 | 343
SIGHAN14 | 3437 | 49.6 | 5122
SIGHAN15 | 2338 | 31.3 | 3037
Wang271K | 271329 | 42.6 | 381962
Test Set | #Sent | Avg.Length | #Errors
SIGHAN15 | 1100 | 30.6 | 704
Table 7: Statistics of used datasets.

##  Appendix B Case Study

Figure 7: Case Study. The wrong tokens are marked red.

We list several cases of Chinese Spelling Correction. We present the source
wrong sentence and the target correct sentence. We also present the
corrections made by BERT and our EGCM.


# Explicit formulas for Eigenvalues of Capelli operators for the Lie
superalgebra $\mathfrak{osp}(1|2n)$

Dene Lepine Department of Mathematics and Statistics, University of Ottawa
<EMAIL_ADDRESS>
###### Abstract.

We define a natural basis for the algebra of $\mathfrak{gosp}(1|2n)$-invariant
differential operators on the affine superspace $\mathbb{C}^{1|2n}$. We prove
that these operators lie in the image of the centre of the enveloping algebra
of $\mathfrak{gosp}(1|2n)$. Using this result, we compute explicit formulas
for the eigenvalues of these operators on irreducible summands of
$\mathcal{P}(\mathbb{C}^{1|2n})$. This settles the Capelli eigenvalue problem
for orthosymplectic Lie superalgebras in the cases that were not addressed in
[SSS20] and [SSS21]. Our main technique relies on an explicit calculation of a
certain determinant with polynomial entries.

###### Contents

  1. 1 Introduction
  2. 2 Main Results
  3. 3 The matrices $M_{d}$ and $M_{d}^{\prime}$.
  4. 4 Factorization of $\det(M_{d})$
  5. 5 Solution of the Capelli Eigenvalue Problem

##  1\. Introduction

11footnotetext: This paper grew out of the results of my Masters Thesis at the
University of Ottawa under the supervision of Professor Hadi Salmasian. The
author would like thank Professor Siddhartha Sahi for his suggestions on an
earlier draft of this article, which improved the presentation of Theorem 2.6.
He also thanks Jérémy Champagne for the useful discussions that eventually
lead to the idea for Lemma 3.16.

The Capelli identity, discovered by Alfredo Capelli [Cap87], is an equality
between a differential operator on the space of $n\times n$ matrices, and a
non-commutative determinant whose entries are natural generators of the
universal enveloping algebra of $\mathfrak{gl}(n)$. This identity played an
integral role in Hermann Weyl’s proof of the FFT in invariant theory [Wey39].
Howe and Umeda’s seminal work [HU91] transcends this identity from the
viewpoint of multiplicity-free actions and formulates the _abstract Capelli
problem_ , see Problem 2.1. In [Sah94], Sahi introduced a natural basis (the
_Capelli operators_) for the algebra of invariant polynomial coefficient
differential operators on a multiplicity-free module of a reductive Lie
algebra, and a multi-parameter family of polynomials whose special values
yield the eigenvalues of these operators. The Capelli operators of [Sah94]
include the Capelli element as a special case. The _Capelli eigenvalue
problem_ is the problem of describing the eigenvalues of this natural basis,
see Problem 2.5. Other examples of Capelli operators were explored by
Konstant, Sahi, and Wallach, see [KS91] and [Wal92].

The solution to the Capelli eigenvalue problem was followed by a further
investigation of _interpolation Jack polynomials_ by Sahi, Knop-Sahi, and
around the same time by Okounkov-Olshanski [Sah94], [KS96],[OO97]. In the past
few years, the Capelli eigenvalue problem has emerged again in the setting of
Lie superalgebras, and nearly completely solved for multiplicity-free actions
that arise from the Tits-Kantor-Koecher construction for Jordan superalgebras
[ASS18], [SS16], [SSS20], [SSS21]. However, for a small number of cases the
uniform methods used in the aforementioned works did not apply. In particular,
for the Lie superalgebra
$\mathfrak{gosp}(m,2n):=\mathbb{C}\oplus\mathfrak{osp}(m,2n)$ acting on its
natural representation $\mathbb{C}^{m|2n}$, which corresponds to the Jordan
superalgebra $(m-1,2n)_{+}$ in Kac’s notation [Kac77], the problem was solved
for all cases in [SSS20] and [SSS21], except for when $m=1$ and $n>0$. The
reason why the case $m=1$ is different is that the highest weights of the
irreducible components in the decomposition of
$\mathcal{S}(\mathbb{C}^{m|2n})$ into $\mathfrak{gosp}(m|2n)$-modules look
different from the generic case: they are of the form
$\delta_{1}+\dotsb+\delta_{i}$, unlike the generic case which has highest
weights of the form $k\delta_{1}$. Furthermore when $m=1$, the number of
irreducible components in $\mathcal{S}^{d}(\mathbb{C}^{1|2n})$ does not tend
to infinity. Therefore, a priori it is not clear how to parametrize
irreducible components suitably by partitions to obtain a polynomial formula
for the eigenvalues of the Capelli operators. Another issue is that the
general argument of [SSS20] for surjectivity of the map from the centre of the
enveloping algebra $\mathfrak{U}(\mathfrak{gosp}(1|2n))$ onto the algebra of
invariant differential operators fails. This surjectivity is a crucial step in
the proofs of [SSS20].

In this paper, we circumvent the above issues and give solutions to both the
Capelli eigenvalue problem and the abstract Capelli problem for the Lie
superalgebra $\mathfrak{osp}(1|2n)$, see Theorem 2.6 and Theorem 2.7
respectively. The two new ideas that arise are a proper indexing of the
irreducible components by partitions and an explicit formula for the
determinant of a matrix whose entries are the eigenvalues of the operators
$C^{i}E^{j}$, where $C$ is the Casimir operator and $E$ is the Euler operator.

##  2\. Main Results

Let $V:=\mathbb{C}^{1|2n}$ be the super vector space over $\mathbb{C}$ of
dimension $1|2n$. Equip $V$ with the non-degenerate even supersymmetric
bilinear form $B\colon V\times V\to\mathbb{C}$ given by the $(1|n+n)$-block
matrix

$\displaystyle\mathfrak{J}_{1|2n}$ $\displaystyle=\begin{bmatrix}1&0&0\\\
0&0&I_{n}\\\ 0&-I_{n}&0\end{bmatrix}.$

Denote by $\mathfrak{osp}(1|2n)$ the orthosymplectic Lie superalgebra that
leaves $B$ invariant. Let $\mathfrak{h}$ be the standard Cartan subalgebra of
$\mathfrak{osp}(1|2n)$ with basis
$\mathscr{B}=\\{E_{i+1,i+1}-E_{i+n+1,i+n+1}\\}_{i=1}^{n}$, where the $E_{i,j}$
denote the usual matrix units. Let $\\{\delta_{1},\dotsc,\delta_{n}\\}$ be the
basis of $\mathfrak{h}^{*}$ dual to $\mathscr{B}$. Let $\mathfrak{b}$ be the
Borel subalgebra of $\mathfrak{osp}(1|2n)$ corresponding to the fundamental
system

$\displaystyle\Sigma:=\\{\delta_{i}-\delta_{i+1}\\}_{i=1}^{n-1}\cup\\{\delta_{n}\\}.$

Set

$\displaystyle\mathfrak{g}:=\mathfrak{gosp}(1|2n):=\mathbb{C}Z\oplus\mathfrak{osp}(1|2n),$

where $Z$ is a central element of $\mathfrak{g}$. We can extend the natural
$\mathfrak{osp}(1|2n)$-module structure on $V$ to $\mathfrak{g}$ by defining
the action of $Z$ to be $-I_{V}$.

Let $\mathcal{P}(V)$ be the superalgebra of (super)polynomials on $V$. We
denote the generators of $V$ dual to the standard basis of $V$ by
$\\{y,x_{1},\dotsc,x_{2n}\\}$, where $y$ is even and $x_{1},\dotsc,x_{2n}$ are
odd. Define the (super)derivations
$\\{\partial_{y},\partial_{1},\dotsc,\partial_{2n}\\}$ for all $1\leq i,j\leq
2n$ and homogeneous $u,v\in\mathcal{P}(V)$ by the relations

$\displaystyle\partial_{y}(y)$ $\displaystyle:=1,$
$\displaystyle\partial_{y}(x_{j}):=0,$ $\displaystyle\partial_{y}(uv)$
$\displaystyle:=\partial_{y}(u)v+u\partial_{y}(v),$
$\displaystyle\partial_{i}(y)$ $\displaystyle:=0,$
$\displaystyle\partial_{i}(x_{j}):=\delta_{i,j},$
$\displaystyle\partial_{i}(uv)$
$\displaystyle:=\partial_{i}(u)v+(-1)^{|u|}u\partial_{i}(v).$

Let $\mathcal{D}(V)$ be the superalgebra of constant-coefficient
(super)differential operators generated by
$\\{\partial_{y},\partial_{1},\dotsc,\partial_{2n}\\}$ and let
$\mathcal{PD}(V)$ be the superalgebra of (super)polynomial coefficient
(super)differential operators. We can consider $\mathcal{PD}(V)$ as the
subalgebra of $\operatorname{End}(\mathcal{P}(V))$ generated by
$y,x_{1},\dotsc,x_{2n},\partial_{y},\partial_{1},\dotsc,\partial_{2n}$. As a
vector space we have $\mathcal{PD}(V)\cong\mathcal{P}(V)\otimes\mathcal{D}(V)$
and we use $\mathcal{PD}^{k}(V)$ to denote the image of
$\mathcal{P}^{k}(V)\otimes\mathcal{D}^{k}(V)$ in $\mathcal{PD}(V)$, where
$\mathcal{P}^{k}(V)$ and $\mathcal{D}^{k}(V)$ are the homogenous degree $k$
components. For more details on our notation see [SS16, Sec. 2].

The $\mathfrak{g}$-module structure on $V$ induces a $\mathfrak{g}$-module
structure on $\mathcal{P}(V)$. Explicitly, the action of
$\mathfrak{osp}(1|2n)$ is given by the restricted action from
$\mathfrak{gl}(1|2n)$ to $\mathfrak{osp}(1|2n)$, where the action of
$\mathfrak{gl}(1|2n)$ on $\mathcal{P}(V)$ is defined for all $1\leq i,j\leq
2n$ by

$\displaystyle E_{1,1}$ $\displaystyle\mapsto y\partial_{y},$ $\displaystyle
E_{1,j+1}$ $\displaystyle\mapsto y\partial_{j},$ $\displaystyle E_{j+1,1}$
$\displaystyle\mapsto x_{j}\partial_{y},$ $\displaystyle E_{i+1,j+1}$
$\displaystyle\mapsto x_{i}\partial_{j}.$

Furthermore, $Z$ acts on $\mathcal{P}(V)$ by the _degree operator_. That is,

$\displaystyle Z\mapsto E:=y\partial_{y}+\sum_{i=1}^{2n}x_{i}\partial_{i}.$

Since $V$ is self-dual, then
$\mathcal{D}(V)\cong\mathcal{P}(V^{*})\cong\mathcal{P}(V)$. Thus, the action
of $\mathfrak{g}$ on $\mathcal{P}(V)$ induces an action on $\mathcal{D}(V)$
and hence on $\mathcal{PD}(V)\cong\mathcal{P}(V)\otimes\mathcal{D}(V)$.
Observe that the action of $\mathfrak{g}$ on $\mathcal{P}(V)$ induces a map

$\displaystyle\mathfrak{U}(\mathfrak{g})\to\mathcal{PD}(V),$

where $\mathfrak{U}(\mathfrak{g})$ denotes the universal enveloping algebra of
$\mathfrak{g}$. An element $D\in\mathcal{PD}(V)$ is $\mathfrak{g}$-invariant
if and only if $Dx=(-1)^{|D||x|}xD$, for all $x\in\mathfrak{g}$. Denote by
$\mathcal{PD}(V)^{\mathfrak{g}}$ the subalgebra of $\mathfrak{g}$-invariant
polynomial coefficient differential operators. Notice that the image of the
centre of $\mathfrak{U}(\mathfrak{g})$, denoted by
$\mathcal{Z}(\mathfrak{g})$, lies in $\mathcal{PD}(V)^{\mathfrak{g}}$. That
is, we have a map

(2.1)
$\displaystyle\mathcal{Z}(\mathfrak{g})\to\mathcal{PD}(V)^{\mathfrak{g}}.$

######  Problem 2.1 (Abstract Capelli Problem for $\mathfrak{osp}(1|2n)$).

Determine whether or not the map (2.1) is surjective.

To put Problem 2.1 in perspective, recall that in the non-super setting, this
problem was investigated in the work of Howe and Umeda [HU91]. In [SSS20] this
problem was studied in the general setting of multiplicity-free modules of
basic classical Lie superalgebras that arise from the super Tits-Kantor-
Koecher construction. More precisely, in [SSS20, Corollary 1.18] the authors
addressed Problem 2.1 for pairs $(\mathfrak{g},V)$ associated to a simple
unital Jordan superalgebra $J$ such that $J_{1}\neq\\{0\\}$ and $J$ is not of
type $(0,2n)_{+}$ or $JP(0,n)$, in the Kac notation. The answer turns out to
be affirmative for all such pairs $(\mathfrak{g},V)$, except when $V$ is a
$\mathfrak{gosp}(2|4)$-module constructed from the exceptional Jordan
superalgebra of type $F$. The proof technique of [SSS20] uses the connection
between the image of the Harish-Chandra homomorphism and Sergeev-Veselov
interpolation polynomials. This technique is not available when
$\mathfrak{g}=\mathfrak{osp}(1|2n)$ and $V=\mathbb{C}^{1|2n}$, corresponding
to $J=(0,2n)_{+}$.

Our first main result of this paper is an affirmative answer to Problem 2.1
(see Theorem 2.7). This resolves the abstract Capelli problem for pairs
$(\mathfrak{g},V)$ associated to the Jordarn superalgebra $(0,2n)_{+}$ which
were left out in [SSS20].

To explain our second main result, we begin by constructing the Capelli basis
of $\mathcal{PD}(V)^{\mathfrak{g}}$. Let us define two operators $R^{2}$ and
$\nabla^{2}$ on $\mathcal{P}(V)$ as follows: $R^{2}$ is the operator of left
multiplication by $y^{2}-2\sum_{1\leq i\leq n}x_{i+n}x_{i}$ and $\nabla^{2}$
is the constant coefficient differential operator $\partial_{y}-2\sum_{1\leq
i\leq n}\partial_{i+n}\partial_{i}.$ We call $\nabla^{2}$ the _super Laplace
operator_. These operators admit the following relations

$\displaystyle\left[(1/2)R^{2},-(1/2)\nabla^{2}\right]$
$\displaystyle=E+\frac{1-2n}{2},$
$\displaystyle\left[E+\frac{1-2n}{2},(1/2)R^{2}\right]$ $\displaystyle=R^{2},$
$\displaystyle\left[E+\frac{1-2n}{2},-(1/2)\nabla^{2}\right]$
$\displaystyle=\nabla^{2}.$

That is, the elements
$\left\\{E+\frac{1-2n}{2},(1/2)R^{2},-(1/2)\nabla^{2}\right\\}$ form a
standard $\mathfrak{sl}_{2}(\mathbb{C})$-triple. Furthermore, one can check
directly that the actions of $\mathfrak{sl}_{2}(\mathbb{C})$ and
$\mathfrak{osp}(1|2n)$ commute, so that $\mathcal{P}(V)$ is an
$\mathfrak{osp}(1|2n)\times\mathfrak{sl}_{2}(\mathbb{C})$-module. There is a
natural grading of $\mathcal{P}(V)$ by degree and we have $Ep=kp$ for
$p\in\mathcal{P}^{k}(V)$. Furthermore, define $\mathcal{H}=\ker(\nabla^{2})$
and $\mathcal{H}_{k}=\mathcal{P}^{k}(V)\cap\mathcal{H}$. The elements of
$\mathcal{H}$ are called _harmonic superpolynomials_. The next Lemma is a
special case of [Cou13, Th. 5.2].

######  Lemma 2.2.

For $0\leq k\leq 2n+1$, the space $\mathcal{H}_{k}$ is an irreducible
$\mathfrak{osp}(1|2n)$-module of highest weight

$\displaystyle\lambda_{k}$
$\displaystyle=\sum_{j=1}^{\min(2n-k+1,k)}\delta_{j}.$

Using Lemma 2.2 and the fact that multiplication by $R^{2}$ commutes with the
$\mathfrak{osp}(1|2n)$ action it follows that $R^{2\ell}\mathcal{H}_{k}$ is an
irreducible $\mathfrak{osp}(1|2n)$-module of highest weight $\lambda_{k}$, for
all $\ell\in\mathbb{Z}_{\geq 0}$. For the following Lemma see [Cou13, Lemma
2.1].

######  Lemma 2.3 (Fischer Decomposition).

For $k\in\mathbb{Z}_{\geq 0}$ and $\ell=\lfloor\frac{k}{2}\rfloor$, the
decomposition of $\mathcal{P}(V)$ as an $\mathfrak{osp}(1|2n)$-module is given
by

$\displaystyle\mathcal{P}^{k}(V)$
$\displaystyle\simeq\begin{cases}\mathcal{H}_{k}\oplus
R^{2}\mathcal{H}_{k-2}\oplus\dotsb\oplus
R^{2\ell}\mathcal{H}_{k-2\ell}&\textrm{ for }k\leq 2n+1\textrm{ and }\\\
R^{2}\mathcal{P}^{k-2}(V)&\textrm{ for }k>2n+1.\end{cases}$

Thus as a $\mathfrak{g}$-module, the space $\mathcal{P}(V)$ admits a
multiplicity-free decomposition. Define

$\mathbb{P}:=\\{(\nu_{1},\nu_{2})\colon\nu_{1},\nu_{2}\in\mathbb{Z}_{\geq
0},~{}\nu_{1}\geq\nu_{2}\\},$

and for any $\nu\in\mathbb{P}$ set $|\nu|:=\nu_{1}+\nu_{2}$. We will
parametrize the irreducible submodules of $\mathcal{P}^{k}(V)$ by the set

(2.2) $\displaystyle\mathbb{P}_{k,n}^{*}$
$\displaystyle:=\left\\{\nu\in\mathbb{P}\colon|\nu|=k\textrm{ and
}\nu_{2}\geq\left\lfloor\frac{k}{2}\right\rfloor-n\right\\},$

via the assignment $\nu\mapsto
V_{\nu}:=R^{2\nu_{2}}\mathcal{H}_{\nu_{1}-\nu_{2}},$ for
$\nu\in\mathbb{P}_{k,n}^{*}$. Then we have

(2.3)
$\displaystyle\mathcal{P}^{k}(V)=\bigoplus_{\nu\in\mathbb{P}_{k,n}^{*}}V_{\nu}.$

Let $\mathcal{D}^{k}(V)\otimes\mathcal{P}^{k}(V)\to\mathbb{C}$ be the
canonical non-degenerate pairing between $\mathcal{D}^{k}(V)$ and
$\mathcal{P}^{k}(V)$. This pairing induces a $\mathfrak{g}$-module isomorphism
$\mathcal{D}^{k}(V)\simeq\mathcal{P}^{k}(V)^{*}.$ That is,

(2.4)
$\displaystyle\mathcal{D}^{k}(V)\simeq\bigoplus_{\nu\in\mathbb{P}_{k,n}^{*}}V_{\nu}^{*}.$

We remark that the latter isomorphism can be chosen such that for every
$\nu\in\mathbb{P}_{k,n}^{*}$, the restriction of the natural map
$\mathcal{D}^{k}(V)\otimes\mathcal{P}^{k}(V)\to\mathbb{C}$ to
$V_{\nu}^{*}\otimes V_{\nu}$ is the standard duality pairing. Define

(2.5)
$\displaystyle\Lambda_{d,n}^{*}:=\bigcup_{k=0}^{d}\mathbb{P}_{k,n}^{*}\quad\textrm{and}\quad\Lambda_{n}^{*}:=\bigcup_{k\geq
0}\mathbb{P}_{k,n}^{*}.$

Using (2.3) and (2.4) we have

(2.6)
$\displaystyle\mathcal{PD}(V)^{\mathfrak{g}}\simeq\bigoplus_{\nu,\nu^{\prime}\in\Lambda_{n}^{*}}(V_{\nu}\otimes
V_{\nu^{\prime}}^{*})^{\mathfrak{g}}\simeq\bigoplus_{\nu,\nu^{\prime}\in\Lambda_{n}^{*}}\operatorname{Hom}_{\mathfrak{g}}(V_{\nu^{\prime}},V_{\nu})\simeq\bigoplus_{\nu\in\Lambda_{n}^{*}}\operatorname{Hom}_{\mathfrak{g}}(V_{\nu},V_{\nu}).$

######  Definition 2.4 (Capelli Operator).

For $\nu\in\Lambda_{n}^{*}$, define $D_{\nu}\in\mathcal{PD}(V)^{\mathfrak{g}}$
to be the element mapped to
$I_{\nu}\in\operatorname{Hom}_{\mathfrak{g}}(V_{\nu},V_{\nu})$ by (2.6). The
operator $D_{\nu}$ is called a _Capelli Operator_.

Indeed, Schur’s Lemma implies
$\dim\operatorname{Hom}_{\mathfrak{g}}(V_{\nu},V_{\nu})=1$ and therefore the
family $\\{D_{\nu}\\}_{\nu\in\Lambda_{n}^{*}}$ forms a basis of
$\mathcal{PD}(V)^{\mathfrak{g}}$. Schur’s Lemma also implies that for any
$\mu,\nu\in\Lambda_{n}^{*}$ the Capelli operator $D_{\mu}$ acts on $V_{\nu}$
by a scalar. This gives rise to the following problem called the _Capelli
Eigenvalue problem_ for $\mathfrak{osp}(1|2n)$:

######  Problem 2.5 (Capelli Eigenvalue Problem for $\mathfrak{osp}(1|2n)$).

Let $\mu,\nu\in\Lambda_{n}^{*}$. Determine the scalar $c_{\nu}(\mu)$ by which
$D_{\nu}$ acts on $V_{\mu}$.

Our answer to Problem 2.5 makes use of a special case of _Knop-Sahi
polynomials_. These polynomials are uniquely defined by certain vanishing
conditions. In particular, we let $\rho:=(-n-\frac{1}{2},0)$ and let
$P_{\nu}^{\rho}$, for $\nu\in\mathbb{P}$, be the polynomial defined in [KS96]
that satisfies the following conditions:

  1. (1)

$P_{\nu}^{\rho}$ is symmetric in two variables $x$ and $y$,

  2. (2)

$\deg(P^{\rho}_{\nu})\leq|\nu|$,

  3. (3)

$P_{\nu}^{\rho}(\mu+\rho)=0$ for all $\mu\in\mathbb{P}$ with $|\mu|\leq|\nu|$
and $\mu\neq\nu$,

  4. (4)

$P_{\nu}^{\rho}(\nu+\rho)=(\nu_{1}-\nu_{2})!(\nu_{2}!)(\nu_{1}-(n+\frac{1}{2}))^{\underline{\nu_{2}}}$,

where $x^{\underline{i}}=x(x-1)\dotsb(x-i+1)$.

######  Theorem 2.6.

Let $\mu,\nu\in\Lambda^{*}_{n}$. Then
$c_{\nu}(\mu)=\frac{P_{\nu}^{\rho}(\mu_{1}-n-\frac{1}{2},\mu_{2})}{(\nu_{1}-\nu_{2})!(\nu_{2}!)(\nu_{1}-(n+\frac{1}{2}))^{\underline{\nu_{2}}}}$.
In particular, the explicit formula is given by $c_{\nu}(\mu)=ab$, where

$\displaystyle a$
$\displaystyle=\frac{\mu_{2}^{\underline{\nu_{2}}}(\mu_{1}-(n+\frac{1}{2}))^{\underline{\nu_{2}}}}{(\nu_{2}!)(\nu_{1}-(n+\frac{1}{2}))^{\underline{\nu_{2}}}(n+\frac{1}{2})^{\underline{\nu_{1}-\nu_{2}}}}$

and

$\displaystyle
b=\sum_{i=0}^{\nu_{1}-\nu_{2}}{n+\frac{1}{2}\choose\nu_{1}-\nu_{2}-i}{n+\frac{1}{2}\choose
i}(\mu_{1}-\nu_{2}-i)^{\underline{\nu_{1}-\nu_{2}-i}}(\mu_{2}-\nu_{2})^{\underline{i}}.$

Even though the formula obtained for eigenvalues is typical of Capelli
operators, i.e. Knop-Sahi polynomials for a specific $\rho$, we note that the
standard technique of verifying the vanishing conditions (used in [SSS20])
does not apply. The reason is that for $\nu$ such that $|\nu|>2n+1$, there are
not enough vanishing conditions to allow us to use the uniqueness of the Knop-
Sahi polynomials. However, we circumvent this issue by reducing the formula
for $D_{\nu}$ to one for another operator, for which there are enough
vanishing conditions.

The proof of Theorem 2.6 needs Theorem 2.7 below, which answers the abstract
Capelli problem (Problem 2.1) for the action of $\mathfrak{gosp}(1|2n)$ on
$\mathbb{C}^{1|2n}$ affirmatively. Define

(2.7) $\displaystyle\mathbb{P}_{k,n}$
$\displaystyle:=\\{\mu\in\mathbb{P}\colon|\mu|=k\textrm{ and }\mu_{2}\leq
n\\}\quad\textrm{and}\quad\Lambda_{d,n}:=\bigcup_{k=0}^{d}\mathbb{P}_{k,n}.$

Let $C$ denote the Casimir operator of $\mathfrak{osp}(1|2n)$.

######  Theorem 2.7.

Let $d\in\mathbb{Z}_{\geq 0}$ and $\nu\in\Lambda_{d,n}^{*}$. The Capelli
operator $D_{\nu}$ can be expressed as a linear combination of elements from
the image of $B_{d,n}$ under the map (2.1), where

(2.8) $\displaystyle
B_{d,n}:=\\{C^{\mu_{2}}Z^{\mu_{1}-\mu_{2}}\colon\mu\in\Lambda_{d,n}\\}.$

We remark that Theorem 2.7 cannot be proven by the general technique of
[SSS20].

##  3\. The matrices $M_{d}$ and $M_{d}^{\prime}$.

The strategy to prove Theorem 2.7 is as follows. We define a family of square
matrices denoted by $M_{d}$, for $d\in\mathbb{Z}_{\geq 0}$, with coefficients
in the polynomial ring $\mathbb{Z}[x]$. For all $d\in\mathbb{Z}_{\geq 0}$ the
matrix $M_{d}$ will have the following property: if $\det(M_{d})(n)\neq 0$
then $D_{\nu}$, for all $\nu\in\Lambda_{d,n}^{*}$, can be written as a linear
combination of images of elements of $\emph{B}_{d,n}$.

We equip $\Lambda_{d,n}$, from (2.7), and $\Lambda_{d,n}^{*}$, from (2.5),
with the following total orderings.

######  Definition 3.1.

For $\mu,\mu^{\prime}\in\Lambda_{d,n}$, define

$\displaystyle\mu\preceq\mu^{\prime}\textrm{ if and only if
}\mu_{2}<\mu_{2}^{\prime}\textrm{ or }(\mu_{2}=\mu_{2}^{\prime}\textrm{ and
}\mu_{1}\leq\mu_{1}^{\prime})$

and for $\nu,\nu^{\prime}\in\Lambda_{d,n}^{*}$ define

$\displaystyle\nu\leqslant\nu^{\prime}\textrm{ if and only if
}\left\lfloor\frac{\nu_{1}-\nu_{2}}{2}\right\rfloor<\left\lfloor\frac{\nu_{1}^{\prime}-\nu_{2}^{\prime}}{2}\right\rfloor\textrm{
or
}\left(\left\lfloor\frac{\nu_{1}-\nu_{2}}{2}\right\rfloor=\left\lfloor\frac{\nu_{1}^{\prime}-\nu_{2}^{\prime}}{2}\right\rfloor\textrm{
and }|\nu|\leq|\nu^{\prime}|\right).$

######  Lemma 3.2.

For any $d\in\mathbb{Z}_{\geq 0}$, we have
$|\Lambda_{d,n}|=|\Lambda_{d,n}^{*}|.$

###### Proof.

This follows from the definition of $\Lambda_{d,n}^{*}$ and $\Lambda_{d,n}$
and the observation that for all $k\in\mathbb{Z}_{\geq 0}$,
$|\mathbb{P}_{k,n}|=\min\left(\left\lfloor\frac{k}{2}\right\rfloor,n\right)+1=|\mathbb{P}_{k,n}^{*}|.$
∎

######  Remark 3.3.

In particular, Lemma 3.2 implies
$|B_{d,n}|=|\\{D_{\nu}\colon\nu\in\Lambda_{d,n}^{*}\\}|$.

For $0\leq j\leq\min\left(\lfloor\frac{d}{2}\rfloor,n\right)$ define a family
of subsets of $\Lambda_{d,n}$ and $\Lambda_{d,n}^{*}$ by

(3.1) $\displaystyle\Lambda_{d,n,j}$
$\displaystyle:=\\{\mu\in\Lambda_{d,n}\colon\mu_{2}\geq j\\},$
$\displaystyle\Lambda_{d,n,j}^{*}$
$\displaystyle:=\left\\{\nu\in\Lambda_{d,n}^{*}\colon\nu_{2}\leq\left\lfloor\frac{|\nu|}{2}\right\rfloor-j\right\\},$

and

(3.2) $\displaystyle\partial\Lambda_{d,n,j}$
$\displaystyle:=\\{(\mu_{1},j)\colon\mu_{1}=j,j+1,\dotsc,d-j\\},$
$\displaystyle\partial\Lambda_{d,n,j}^{*}$
$\displaystyle:=\left\\{\left(k-\left\lfloor\frac{k}{2}\right\rfloor+j,\left\lfloor\frac{k}{2}\right\rfloor-j\right)\colon
k=2j,2j+1,\dotsc,d\right\\}.$

Observe that the following unions are disjoint:

(3.3) $\displaystyle\Lambda_{d,n,j}$
$\displaystyle=\Lambda_{d,n,j+1}\cup\partial\Lambda_{d,n,j}$
$\displaystyle\Lambda_{d,n,j}^{*}$
$\displaystyle=\Lambda_{d,n,j+1}^{*}\cup\partial\Lambda_{d,n,j}^{*}.$

The next corollary follows inductively from Equations (3.3), Lemma 3.2 and the
equalities $\Lambda_{d,n}=\Lambda_{d,n,0}$ and
$\Lambda_{d,n}^{*}=\Lambda_{d,n,0}^{*}$.

######  Corollary 3.4.

We have $|\Lambda_{d,n,j}|=|\Lambda_{d,n,j}^{*}|$, for $0\leq
j\leq\min\left(\lfloor\frac{d}{2}\rfloor,n\right)$.

######  Definition 3.5.

For $\mu\in\Lambda_{d,n}$ and $\nu\in\Lambda_{d,n}^{*}$, set

(3.4) $\displaystyle\ell_{\nu}:=\min(\nu_{1}-\nu_{2},2n+1-(\nu_{1}-\nu_{2}))$

and set

(3.5) $\displaystyle\lambda_{\mu,\nu}(x)$
$\displaystyle:=((2x+1)\ell_{\nu}-\ell_{\nu}^{2})^{\mu_{2}}(\nu_{1}+\nu_{2})^{\mu_{1}-\mu_{2}}\in\mathbb{Z}[x].$

######  Definition 3.6.

Let $M_{d}$ and $M_{d}^{\prime}$ be square matrices defined by

(3.6) $\displaystyle
M_{d}:=[\lambda_{\mu,\nu}]_{\begin{subarray}{c}\mu\in\Lambda_{d,n}\\\
\nu\in\Lambda_{d,n}^{*}\end{subarray}}$

and

(3.7) $\displaystyle
M_{d}^{\prime}:=[(2\ell_{\nu})^{\mu_{2}}(\nu_{1}+\nu_{2})^{\mu_{1}-\mu_{2}}]_{\begin{subarray}{c}\mu\in\Lambda_{d,n}\\\
\nu\in\Lambda_{d,n}^{*}\end{subarray}}.$

The rows and columns of $M_{d}$ and $M_{d}^{\prime}$ are ordered according to
the total orderings given in Definition 3.1.

Notice that the entries of $M_{d}^{\prime}$ are the leading coefficients of
the polynomials $\lambda_{\mu,\nu}$.

######  Remark 3.7.

For $\nu\in\Lambda_{n}^{*}$, the Casimir operator $C$ acts on $V_{\nu}$ by the
scalar $(2n+1)\ell_{\nu}-\ell_{\nu}^{2}.$ Furthermore, for any
$\mu\in\mathbb{P}$ the central element $C^{\mu_{2}}Z^{\mu_{1}-\mu_{2}}$ of
$\mathfrak{U}(\mathfrak{g})$ acts on $V_{\nu}$ by $\lambda_{\mu,\nu}(n).$

######  Lemma 3.8.

If $\det(M_{d}^{\prime})\neq 0$, then $\det(M_{d})=\det(M_{d}^{\prime})p(x)$
where $p(x)\in\mathbb{Q}[x]$ is monic.

###### Proof.

Notice that $\deg(\lambda_{\mu,\nu}(x))=\mu_{2}$, for all
$\mu\in\Lambda_{d,n}$ and all $\nu\in\Lambda_{d,n}^{*}$. Furthermore, any
monomial in the Leibniz determinant formula of $M_{d}$ is a product of
distinct row entries from $M_{d}$. That is, all the non-zero monomials in this
formula have the same degree, which is equal to
$\sum_{\mu\in\Lambda_{d,n}}\mu_{2}.$ Thus if $\det(M_{d}^{\prime})\neq 0$ then
$\det(M_{d}^{\prime})$ is the leading coefficient of $\det(M_{d})$. ∎

######  Remark 3.9.

By the proof of Lemma 3.8, if $\det(M_{d}^{\prime})\neq 0$ then
$\deg(\det(M_{d}))=\sum_{\mu\in\Lambda_{d,n}}\mu_{2}.$

The remainder of this section is dedicated to showing that
$\det(M_{d}^{\prime})\neq 0$. The final results are given in Theorem 3.17.

######  Definition 3.10.

For $\nu\in\Lambda_{d,n}^{*}$ and $0\leq j\leq\lfloor\frac{|\nu|}{2}\rfloor$,
define

$\displaystyle\nu_{(j)}:=\left(|\nu|-\left\lfloor\frac{|\nu|}{2}\right\rfloor+j,\left\lfloor\frac{|\nu|}{2}\right\rfloor-j\right)\in\partial\Lambda_{d,n,j}^{*}.$

######  Corollary 3.11.

Let $\nu\in\Lambda_{d,n}^{*}$, $0\leq j\leq\lfloor\frac{|\nu|}{2}\rfloor$, and
set $\nu^{\prime}=\nu_{(j)}$. Then $\nu_{(k)}^{\prime}=\nu_{(k)}$ for $0\leq
k\leq\lfloor\frac{|\nu|}{2}\rfloor$.

######  Lemma 3.12.

Let $\nu,\nu^{\prime}\in\Lambda_{d,n}^{*}$, such that $\nu\neq\nu^{\prime}$
and $|\nu|=|\nu^{\prime}|$. Then $\ell_{\nu}\neq\ell_{\nu^{\prime}}.$

###### Proof.

Straightforward by contradiction and considering all the possibilities of
$\ell_{\nu}$ and $\ell_{\nu^{\prime}}$ according to Equation (3.4). ∎

######  Corollary 3.13.

Let $1\leq j\leq\min\left(\lfloor\frac{d}{2}\rfloor,n\right)-1$, let
$\nu\in\Lambda_{d,n,j+1}^{*}$ and set $\nu^{\prime}=\nu_{(j)}$. Then
$\ell_{\nu}\neq\ell_{\nu^{\prime}}.$

###### Proof.

Notice that
$\nu_{2}\leq\left\lfloor\frac{|\nu|}{2}\right\rfloor-(j+1)=\nu_{2}^{\prime}-1.$
That is, $\nu\neq\nu^{\prime}$. Furthermore, $|\nu|=|\nu^{\prime}|$ and
therefore Lemma 3.12 implies $\ell_{\nu}\neq\ell_{\nu^{\prime}}$. ∎

######  Definition 3.14.

Let $0\leq j\leq\min\left(\lfloor\frac{d}{2}\rfloor,n\right)$.

  1. (i)

Let $\mu\in\Lambda_{d,n,j}$ and $\nu\in\Lambda_{d,n,j}^{*}$. Define

$\displaystyle S_{\mu,\nu,j}$
$\displaystyle:=\begin{cases}\ell_{\nu}^{\mu_{2}}&\textrm{ if }j=0,\\\
1&\textrm{ if }\mu_{2}=j,\\\
\sum_{t_{0}=j}^{\mu_{2}-1}\sum_{t_{1}=j-1}^{t_{0}-1}\dotsb\sum_{t_{j}=0}^{t_{j-1}-1}\ell_{\nu_{(0)}}^{\mu_{2}-1-t_{0}}\ell_{\nu_{(1)}}^{t_{0}-1-t_{1}}\dotsb\ell_{\nu_{(j-1)}}^{t_{j-1}-1-t_{j}}\ell_{\nu}^{t_{j}}&\textrm{
otherwise.}\end{cases}$

  2. (ii)

Define the square matrix

$\displaystyle N_{j}$
$\displaystyle:=[2^{\mu_{2}-j}(\nu_{1}+\nu_{2})^{\mu_{1}-\mu_{2}}S_{\mu,\nu,j}]_{\begin{subarray}{c}\mu\in\Lambda_{d,n,j}\\\
\nu\in\Lambda_{d,n,j}^{*}\end{subarray}}.$

######  Lemma 3.15.

Let $0\leq j\leq\min\left(\lfloor\frac{d}{2}\rfloor,n\right)-1$,
$\nu\in\Lambda_{d,n,j}$, and $\nu\in\Lambda_{d,n,j}^{*}$. Then

$\displaystyle
S_{\mu,\nu,j}-S_{\mu,\nu_{(j)},j}=(\ell_{\nu}-\ell_{\nu_{(j)}})S_{\mu,\nu,j+1}.$

###### Proof.

Using Corollary 3.11 we have

$\displaystyle S_{\mu,\nu,j}-S_{\mu,\nu_{(j)},j}$
$\displaystyle=\sum_{t_{0}=j+1}^{\mu_{2}-1}\sum_{t_{1}=j}^{t_{0}-1}\dotsb\sum_{t_{j}=1}^{t_{j-1}-1}\ell_{\nu_{(0)}}^{\mu_{2}-1-t_{0}}\ell_{\nu_{(1)}}^{t_{0}-1-t_{1}}\dotsb\ell_{\nu_{(j-1)}}^{t_{j-1}-1-t_{j}}(\ell_{\nu}^{t_{j}}-\ell_{\nu_{(j)}}^{t_{j}}).$

Now apply the identity

$\displaystyle(\ell_{\nu}^{t_{j}}-\ell_{\nu_{(j)}}^{t_{j}})=(\ell_{\nu}-\ell_{\nu_{(j)}})\left(\sum_{t_{j+1}=0}^{t_{j}-1}\ell_{\nu_{(j)}}^{t_{j}-1-t_{j+1}}\ell_{\nu}^{t_{j+1}}\right)$

to obtain

$\displaystyle S_{\mu,\nu,j}-S_{\mu,\nu_{(j)},j}$
$\displaystyle=(\ell_{\nu}-\ell_{\nu_{(j)}})\sum_{t_{0}=j+1}^{\mu_{2}-1}\sum_{t_{1}=j}^{t_{0}-1}\dotsb\sum_{t_{j}=1}^{t_{j-1}-1}\sum_{t_{j+1}=0}^{t_{j}-1}\ell_{\nu_{(0)}}^{\mu_{2}-1-t_{0}}\ell_{\nu_{(1)}}^{t_{0}-1-t_{1}}\dotsb\ell_{\nu_{(j-1)}}^{t_{j-1}-1-t_{j}}\ell_{\nu_{(j)}}^{t_{j}-1-t_{j+1}}\ell_{\nu}^{t_{j+1}}$
$\displaystyle=(\ell_{\nu}-\ell_{\nu_{(j)}})S_{\mu,\nu,j+1}.\qed$

For the next lemma, recall the Vandermonde matrix defined for
$\alpha_{1},\dotsc,\alpha_{s}\in\mathbb{C}$ by

$\displaystyle V(\alpha_{1},\dotsc,\alpha_{s})$
$\displaystyle:=[\alpha_{j}^{i-1}]_{1\leq i,j\leq s},$

where $0^{0}:=1$. Indeed, $\det(V(\alpha_{1},\dotsc,\alpha_{s}))=\prod_{1\leq
j<i\leq s}(\alpha_{i}-\alpha_{j})$.

######  Lemma 3.16.

For $0\leq j\leq\min\left(\lfloor\frac{d}{2}\rfloor,n\right)-1$,

$\displaystyle\det(N_{j})$
$\displaystyle=\det(V(2j,2j+1,\dotsc,d))\det(N_{j+1})\left(\prod_{\nu\in\Lambda_{d,n,j+1}^{*}}2\left(\ell_{\nu}-\ell_{\nu_{(j)}}\right)\right).$

###### Proof.

Fix $0\leq j\leq\min\left(\lfloor\frac{d}{2}\rfloor,n\right)-1$. Observe that
$N_{j}$ has a submatrix

$[(\nu_{1}+\nu_{2})^{\mu_{1}-j}]_{\begin{subarray}{c}\mu\in\partial\Lambda_{d,n,j}\\\
\nu\in\partial\Lambda_{d,n,j}^{*}\end{subarray}}=V(2j,2j+1,\dotsc,d).$

Therefore for each $\nu\in\Lambda_{d,n,j+1}^{*}$ we have
$\nu_{(j)}\in\partial\Lambda_{d,n,j}^{*}$ and when we subtract the column
indexed by $\nu_{(j)}$ from the column indexed by $\nu$, the entries at the
$(\mu,\nu)$ position, where $\mu\in\partial\Lambda_{d,n,j}$, will vanish. The
matrix obtained from $N_{j}$ after the above column operations is block lower
triangular of the form $\begin{bmatrix}*&0\\\ *&*\end{bmatrix}$. The
determinant of the latter matrix is equal to $\det(N_{j})$ and hence

$\displaystyle\det(N_{j})$
$\displaystyle=\det(V(2j,2j+1,\dotsc,d))\det\left([2^{\mu_{2}-j}(\nu_{1}+\nu_{2})^{\mu_{1}-\mu_{2}}(S_{\mu,\nu,j}-S_{\mu,\nu_{(j)},j})]_{\begin{subarray}{c}\mu\in\Lambda_{d,n,j+1}\\\
\nu\in\Lambda_{d,n,j+1}^{*}\end{subarray}}\right).$

Applying Lemma 3.15 to the columns of the second factor on the right hand side

$\displaystyle\det(N_{j})$
$\displaystyle=\det(V(2j,2j+1,\dotsc,d))\det\left([2^{\mu_{2}-j}(\nu_{1}+\nu_{2})^{\mu_{1}-\mu_{2}}(\ell_{\nu}-\ell_{\nu_{(j)}})S_{\mu,\nu,j+1}]_{\begin{subarray}{c}\mu\in\Lambda_{d,n,j+1}\\\
\nu\in\Lambda_{d,n,j+1}^{*}\end{subarray}}\right)$
$\displaystyle=\det(V(2j,2j+1,\dotsc,d))\det(N_{j+1})\left(\prod_{\nu\in\Lambda_{d,n,j+1}^{*}}2\left(\ell_{\nu}-\ell_{\nu_{(j)}}\right)\right).\qed$

######  Theorem 3.17.

Let $M_{d}^{\prime}$ be as defined in Equation (3.7). Then
$\det(M_{d}^{\prime})=ab$ where

$\displaystyle
a=\prod_{j=1}^{\min\left(\lfloor\frac{d}{2}\rfloor,n\right)-1}\det(V(2j,2j+1,\dotsc,d))\left(\prod_{\nu\in\Lambda_{d,n,j+1}^{*}}2\left(\ell_{\nu}-\ell_{\nu_{(j)}}\right)\right)$

and

$\displaystyle
b=\det\left(V\left(2\min\left(\left\lfloor\frac{d}{2}\right\rfloor,n\right),2\min\left(\left\lfloor\frac{d}{2}\right\rfloor,n\right)+1,\dotsc,d\right)\right).$

###### Proof.

This follows from the fact that $N_{0}=M_{d}^{\prime}$, applying Lemma 3.16
$\min(\lfloor\frac{d}{2}\rfloor,n)-1$ times and observing that

$\displaystyle\det\left(N_{\min\left(\lfloor\frac{d}{2}\rfloor,n\right)}\right)$
$\displaystyle=\det\left(V\left(2\min\left(\left\lfloor\frac{d}{2}\right\rfloor,n\right),2\min\left(\left\lfloor\frac{d}{2}\right\rfloor,n\right)+1,\dotsc,d\right)\right).\qed$

##  4\. Factorization of $\det(M_{d})$

Theorem 3.17 implies that $\det(M_{d}^{\prime})\neq 0$. Hence by Lemma 3.8
there exists a monic polynomial $p(x)\in\mathbb{Q}[x]$ such that

$\displaystyle\det(M_{d})$ $\displaystyle=\det(M_{d}^{\prime})p(x).$

This section is dedicated to factoring $p(x)$ into linear terms. The final
results are given in Theorem 4.7.

######  Lemma 4.1.

Let $\mu\in\Lambda_{d,n}$ and let $\nu,\nu^{\prime}\in\Lambda_{d,n}^{*}$, such
that $\nu\neq\nu^{\prime}$ and $|\nu|=|\nu^{\prime}|$. Then the polynomial
$\lambda_{\mu,\nu}-\lambda_{\mu,\nu^{\prime}}$ is divisible by
$\left(x-\frac{\ell_{\nu}+\ell_{\nu^{\prime}}-1}{2}\right)$, where
$\lambda_{\mu,\nu}$ is defined in Equation (3.5).

###### Proof.

Let $\mu\in\Lambda_{d,n}$ and $\nu,\nu^{\prime}\in\Lambda_{d,n}^{*}$, such
that $|\nu|=|\nu^{\prime}|$. Then
$\nu_{1}+\nu_{2}=\nu_{1}^{\prime}+\nu_{2}^{\prime}$ and

$\displaystyle\lambda_{\mu,\nu}-\lambda_{\mu,\nu^{\prime}}$
$\displaystyle=(\nu_{1}+\nu_{2})^{\mu_{1}-\mu_{2}}\left(((2x+1)\ell_{\nu}-\ell_{\nu}^{2})^{\mu_{2}}-((2x+1)\ell_{\nu^{\prime}}-\ell_{\nu^{\prime}}^{2})^{\mu_{2}}\right)$
$\displaystyle=(\nu_{1}+\nu_{2})^{\mu_{1}-\mu_{2}}(2(\ell_{\nu}-\ell_{\nu^{\prime}}))\left(x-\frac{\ell_{\nu}+\ell_{\nu^{\prime}}-1}{2}\right)$
$\displaystyle\hskip
72.26999pt\left(\sum_{t=0}^{\mu_{2}-1}((2x+1)\ell_{\nu}-\ell_{\nu}^{2})^{\mu_{2}-1-t}((2x+1)\ell_{\nu^{\prime}}-\ell_{\nu^{\prime}}^{2})^{t}\right).\qed$

The next lemma will justify that Definition 4.3 is valid.

######  Lemma 4.2.

Let $\nu,\nu^{\prime}\in\Lambda_{d,n}^{*}$, such that $\nu\neq\nu^{\prime}$
and $|\nu|=|\nu^{\prime}|$. Then

$\ell_{\nu}+\ell_{\nu^{\prime}}-1\in\\{0,1,\dotsc,2n-2\\}.$

###### Proof.

Let $0\leq k\leq d$ and $\nu,\nu^{\prime}\in\Lambda_{d,n}^{*}$, such that
$k=|\nu|=|\nu^{\prime}|$. This proof is divided into three cases:

  * •

If $k\leq n$, then $\ell_{\nu},\ell_{\nu^{\prime}}\in\\{k-2j\colon
j=0,1,\dotsc,\lfloor\frac{k}{2}\rfloor\\}$. Lemma 3.12 implies that
$\ell_{\nu}\neq\ell_{\nu^{\prime}}$ and therefore

$\displaystyle\ell_{\nu}+\ell_{\nu^{\prime}}-1\in\left\\{2k-2j-1\colon
j=1,2,\dotsc,2\left\lfloor\frac{k}{2}\right\rfloor-1\right\\}.$

  * •

If $n+1\leq k\leq 2n+1$, then $\ell_{\nu}$ and $\ell_{\nu^{\prime}}$ take
distinct values in the sets

(4.1) $\displaystyle\left\\{k-2j\colon
j=k-\left\lfloor\frac{n+k}{2}\right\rfloor,\dotsc,\left\lfloor\frac{k}{2}\right\rfloor\right\\}$
(4.2) $\displaystyle\left\\{2n+1-k+2j\colon
j=0,\dotsc,k-\left\lfloor\frac{n+k}{2}\right\rfloor-1\right\\}.$

    * –

If $\ell_{\nu},\ell_{\nu^{\prime}}$ are both in (4.1) then

$\displaystyle\ell_{\nu}+\ell_{\nu^{\prime}}-1\in\left\\{2k-2j-1\colon
j=2k-2\left\lfloor\frac{k}{2}\right\rfloor+1,\dotsc,2\left\lfloor\frac{k}{2}\right\rfloor-1\right\\}.$

    * –

For there to be atleast two distinct $\ell_{\nu},\ell_{\nu^{\prime}}$ in (4.2)
then $k\geq n+3$. In which case we have

$\displaystyle\ell_{\nu}+\ell_{\nu^{\prime}}-1\in\left\\{4n-2k+2j+1\colon
j=1,2,\dotsc,2k+2\left\lfloor\frac{n+1-k}{2}\right\rfloor-2n-3\right\\}.$

    * –

If $\ell_{\nu}$ is in (4.1) and $\ell_{\nu^{\prime}}$ in (4.2) then

$\displaystyle\ell_{\nu}+\ell_{\nu^{\prime}}-1\in\left\\{2n-2j\colon
j=1,2,\dotsc,\left\lfloor\frac{k}{2}\right\rfloor\right\\}.$

  * •

If $k\geq 2n+2$ then $\ell_{\nu}$ and $\ell_{\nu^{\prime}}$ take values from
$\\{0,1,\dotsc,n\\}$. It then follows for distinct $\ell_{\nu}$ and
$\ell_{\nu^{\prime}}$ that

$\displaystyle\ell_{\nu}+\ell_{\nu^{\prime}}$
$\displaystyle-1\in\\{0,1,\dotsc,2n-2\\}.\qed$

For $s\in\\{0,1,\dotsc,2n-2\\}$ and $\nu\in\Lambda_{d,n}^{*}$ there is at most
one $\nu^{\prime}\in\Lambda_{d,n}^{*}$ such that
$\ell_{\nu^{\prime}}<\ell_{\nu}$, $|\nu|=|\nu^{\prime}|$, and
$\ell_{\nu}+\ell_{\nu^{\prime}}-1=s$. Thus we can define the following matrix.

######  Definition 4.3.

For $s=0,1,\dotsc,2n-2$, define the square matrix

$M_{d,s}:=[\lambda_{\mu,\nu,s}]_{\begin{subarray}{c}\mu\in\Lambda_{d,n}\\\
\nu\in\Lambda_{d,n}^{*}\end{subarray}}$

where $\lambda_{\mu,\nu,s}\in\mathbb{Z}[x]$ is defined by

$\displaystyle\lambda_{\mu,\nu,s}(x)$
$\displaystyle:=\begin{cases}\lambda_{\mu,\nu}-\lambda_{\mu,\nu^{\prime}}&\textrm{if
there
is}~{}\nu^{\prime}\in\Lambda_{d,n}^{*},~{}|\nu|=|\nu^{\prime}|,~{}\ell_{\nu^{\prime}}<\ell_{\nu},~{}s=\ell_{\nu}+\ell_{\nu^{\prime}}-1,\\\
\lambda_{\mu,\nu}&\textrm{otherwise.}\end{cases}$

######  Lemma 4.4.

Let $s\in\\{0,1,\dotsc,2n-2\\}$. Then $\det(M_{d})=\det(M_{d,s}).$
Furthermore, $\det(M_{d})$ is divisible, in $\mathbb{Q}[x]$, by
$\left(x-\frac{s}{2}\right)^{f(d,s)},$ where

(4.3) $\displaystyle
f(d,s)=\big{|}\\{(\nu,\nu^{\prime})\in\Lambda_{d,n}^{*}\times\Lambda_{d,n}^{*}\colon|\nu|=|\nu^{\prime}|,\ell_{\nu^{\prime}}<\ell_{\nu},\ell_{\nu}+\ell_{\nu^{\prime}}-1=s\\}\big{|}$

###### Proof.

Notice that through column operations $M_{d,s}$ is obtained from $M_{d}$. In
particular, for $\nu\in\Lambda_{d,n}^{*}$, column $\nu$ of $M_{d,s}$ is either
column $\nu$ of $M_{d}$ or is column $\nu$ minus column $\nu^{\prime}$ of
$M_{d}$, where $|\nu|=|\nu^{\prime}|$, $\ell_{\nu}>\ell_{\nu^{\prime}}$, and
$\ell_{\nu}+\ell_{\nu^{\prime}}-1=s$. These operations can be done
simultaneously by starting with the largest $\ell_{\nu}$ and performing the
necessary column operations. This leaves all other columns untouched. Thus we
continue to the second largest $\ell_{\nu}$ and likewise for all $\ell_{\nu}$.
It follows that $\det(M_{d})=\det(M_{d,s}).$ Next take any

$(\nu,\nu^{\prime})\in\\{(\nu,\nu^{\prime})\in\Lambda_{d,n}^{*}\times\Lambda_{d,n}^{*}\colon|\nu|=|\nu^{\prime}|,\ell_{\nu^{\prime}}<\ell_{\nu},\ell_{\nu}+\ell_{\nu^{\prime}}-1=s\\}.$

Lemma 4.1 implies that the column $\nu$ of $M_{d,s}$ is divisible by
$(x-\frac{s}{2})$. Thus $(x-\frac{s}{2})$ divides $\det(M_{d,s})=\det(M_{d})$
with multiplicity

$\displaystyle f(d,s)$
$\displaystyle=\big{|}\\{(\nu,\nu^{\prime})\in\Lambda_{d,n}^{*}\times\Lambda_{d,n}^{*}\colon|\nu|=|\nu^{\prime}|,\ell_{\nu^{\prime}}<\ell_{\nu},\ell_{\nu}+\ell_{\nu^{\prime}}-1=s\\}\big{|}.\qed$

######  Remark 4.5.

Notice that $f(d,s)=\sum_{k=0}^{d}g_{k}(s)$ where

$\displaystyle
g_{k}(s):=\big{|}\\{(\nu,\nu^{\prime})\in\mathbb{P}_{k,n}^{*}\times\mathbb{P}_{k,n}^{*}\colon\ell_{\nu^{\prime}}<\ell_{\nu},\ell_{\nu}+\ell_{\nu^{\prime}}-1=s\\}\big{|}.$

######  Lemma 4.6.

Let $d\in\mathbb{Z}_{\geq 0}$. Then
$\sum_{s=0}^{2n-2}f(d,s)=\sum_{\mu\in\Lambda_{d,n}}\mu_{2}.$

###### Proof.

First, we have the identity

$\displaystyle\sum_{s=0}^{2n-2}g_{k}(s)$
$\displaystyle=\frac{1}{2}\min\left(\left\lfloor\frac{k}{2}\right\rfloor,n\right)\left(\min\left(\left\lfloor\frac{k}{2}\right\rfloor,n\right)+1\right),$

for $g_{k}(s)$ defined in Remark 4.5. This identity follows from the fact that
the left hand side is counting the cardinality of

$\displaystyle\bigcup_{s=0}^{2n-2}\\{(\nu,\nu^{\prime})\in\mathbb{P}_{k,n}^{*}\times\mathbb{P}_{k,n}^{*}\colon\ell_{\nu^{\prime}}<\ell_{\nu},~{}\ell_{\nu}+\ell_{\nu^{\prime}}-1=s\\}=\\{(\nu,\nu^{\prime})\in\mathbb{P}_{k,n}^{*}\times\mathbb{P}_{k,n}^{*}\colon\ell_{\nu^{\prime}}<\ell_{\nu}\\}$

and the fact that there are
$\min\left(\left\lfloor\frac{k}{2}\right\rfloor,n\right)+1$ possible
$\ell_{\nu}$. Therefore we have

$\displaystyle\sum_{s=0}^{2n-2}f(d,s)$
$\displaystyle=\sum_{s=0}^{2n-2}\sum_{k=0}^{d}g_{k}(s)=\sum_{k=0}^{d}\sum_{s=0}^{2n-2}g_{k}(s)$
$\displaystyle=\sum_{k=0}^{d}\frac{1}{2}\min\left(\left\lfloor\frac{k}{2}\right\rfloor,n\right)\left(\min\left(\left\lfloor\frac{k}{2}\right\rfloor,n\right)+1\right).$

Next $\sum_{\mu\in\Lambda_{d,n}}\mu_{2}$ is calculated for the following two
cases.

  * •

For $0\leq d\leq 2n+1$,

$\displaystyle\sum_{\mu\in\Lambda_{d,n}}\mu_{2}$
$\displaystyle=\sum_{\mu_{1}=0}^{\lfloor\frac{d}{2}\rfloor}\sum_{\mu_{2}=0}^{\mu_{1}}\mu_{2}+\sum_{\mu_{1}=\lfloor\frac{d}{2}\rfloor+1}^{d}\sum_{\mu_{2}=0}^{d-\mu_{1}}\mu_{2}$
$\displaystyle=\sum_{\mu_{1}=0}^{\lfloor\frac{d}{2}\rfloor}\frac{\mu_{1}(\mu_{1}+1)}{2}+\sum_{\mu_{1}=\lfloor\frac{d}{2}\rfloor+1}^{d}\frac{(d-\mu_{1})(d-\mu_{1}+1)}{2}=\sum_{\mu_{1}=0}^{d}\frac{\left\lfloor\frac{\mu_{1}}{2}\right\rfloor(\left\lfloor\frac{\mu_{1}}{2}\right\rfloor+1)}{2}.$

  * •

For $d\geq 2n+2$ we have the following analogous calculation

$\displaystyle\sum_{\mu\in\Lambda_{d,n}}\mu_{2}$
$\displaystyle=\sum_{\mu_{1}=0}^{n}\sum_{\mu_{2}=0}^{\mu_{1}}\mu_{2}+\sum_{\mu_{1}=n+1}^{d-n-1}\sum_{\mu_{2}=0}^{n}\mu_{2}+\sum_{\mu_{1}=d-n}^{d}\sum_{\mu_{2}=0}^{d-\mu_{1}}\mu_{2}$
$\displaystyle=\sum_{\mu_{1}=0}^{2n+1}\frac{\left\lfloor\frac{\mu_{1}}{2}\right\rfloor\left(\left\lfloor\frac{\mu_{1}}{2}\right\rfloor+1\right)}{2}+(d-2n-1)\frac{n(n+1)}{2}.$

For both of these cases, it is easy to verify that
$\sum_{s=0}^{2n-2}f(d,s)=\sum_{\mu\in\Lambda_{d,n}}\mu_{2}.$ ∎

Finally, we have all the tools needed to prove the main result of this
section.

######  Theorem 4.7.

Let $d\in\mathbb{Z}_{\geq 0}$. Then

$\displaystyle\det(M_{d})$
$\displaystyle=\det(M_{d}^{\prime})\prod_{s=0}^{2n-2}\left(x-\frac{s}{2}\right)^{f(d,s)},$

where $f(d,s)$ is given in Lemma 4.4.

###### Proof.

Recall that Lemma 3.8 and Theorem 3.17 imply that
$\det(M_{d})=\det(M_{d}^{\prime})p(x),$ for some monic $p(x)\in\mathbb{Q}[x]$.
Furthermore, Lemma 4.4 implies that $\left(x-\frac{s}{2}\right)^{f(d,s)}$
divides $p(x)$, for all $s=0,\dotsc,2n-2$. Therefore there is some monic
$q(x)\in\mathbb{Q}[x]$ such that

$\displaystyle p(x)$
$\displaystyle=q(x)\prod_{s=0}^{2n-2}\left(x-\frac{s}{2}\right)^{f(d,s)}.$

In particular, $\deg(p(x))\geq\sum_{s=0}^{2n-2}f(d,s).$ But Remark 3.9 tells
us that $\deg(p(x))=\sum_{\mu\in\Lambda_{d,n}}\mu_{2}$. It then follows from
Lemma 4.6 that

$\sum_{\mu\in\Lambda_{d,n}}\mu_{2}=\sum_{s=0}^{2n-2}f(d,s)\leq\deg(p(x))=\sum_{\mu\in\Lambda_{d,n}}\mu_{2}.$

That is, $q(x)=1$ and

$\displaystyle\det(M_{d})$
$\displaystyle=\det(M_{d}^{\prime})p(x)=\det(M_{d}^{\prime})\prod_{s=0}^{2n-2}\left(x-\frac{s}{2}\right)^{f(d,s)}.\qed$

Thus we can prove Theorem 2.7 and thereby answer Problem 2.1 in the
affirmative.

_Proof of Theorem 2.7._ Let $d\in\mathbb{Z}_{\geq 0}$. Recall that the image
of $C^{\mu_{2}}Z^{\mu_{1}-\mu_{2}}$ is a differential operator of order
$|\mu|$ and therefore is an element of
$\bigoplus_{k=0}^{|\mu|}\mathcal{PD}^{k}(\mathbb{C}^{1|2n})^{\mathfrak{g}}$.
Lemma 3.2 implies $|B_{d,n}|=|\\{D_{\nu}\colon|\nu|\leq d\\}|$. Therefore if
$B_{d,n}$ is a linearly independent set then it is a basis of
$\bigoplus_{k=0}^{d}\mathcal{PD}^{k}(\mathbb{C}^{1|2n})^{\mathfrak{g}}$.

Identify $C^{\mu_{2}}Z^{\mu_{1}-\mu_{2}}$ with its image and fix non-zero
$v_{\nu}\in V_{\nu}$, for each $\nu\in\Lambda_{d,n}^{*}$. Let
$a_{\mu}\in\mathbb{C}$ be such that

(4.4)
$\displaystyle\sum_{\mu\in\Lambda_{d,n}}a_{\mu}C^{\mu_{2}}Z^{\mu_{1}-\mu_{2}}=0.$

We evaluate (4.4) on each $v_{\nu}$. Since $C^{\mu_{2}}Z^{\mu_{1}-\mu_{2}}$
acts on $v_{\nu}$ by $\lambda_{\mu,\nu}(n)$ we have the following matrix
relation

$\displaystyle[a_{\mu}]_{\mu\in\Lambda_{d,n}}[\lambda_{\mu,\nu}(n)]_{\begin{subarray}{c}\mu\in\Lambda_{d,n}\\\
\nu\in\Lambda_{d,n}^{*}\end{subarray}}=0,$

where $[a_{\mu}]_{\mu\in\Lambda_{d,n}}$ denotes a row vector. The matrix
$[\lambda_{\mu,\nu}(n)]_{\begin{subarray}{c}\mu\in\Lambda_{d,n}\\\
\nu\in\Lambda_{d,n}^{*}\end{subarray}}$ is exactly $M_{d}$ with entries
evaluated at $x=n$. Theorem 4.7 implies that $\det(M_{d})(n)\neq 0$ and
therefore $[\lambda_{\mu,\nu}(n)]_{\begin{subarray}{c}\mu\in\Lambda_{d,n}\\\
\nu\in\Lambda_{d,n}^{*}\end{subarray}}$ is invertible. In particular,
$a_{\mu}=0$, for all $\mu\in\Lambda_{d,n}$ and hence $B_{d,n}$ is linearly
independent.∎

##  5\. Solution of the Capelli Eigenvalue Problem

In the previous section we proved Theorem 2.7 and thereby answered Problem
2.1. For the remainder of this article we will answer Problem 2.5 by proving
Theorem 2.6. We will need the following lemma, which can be found in [DBES09,
Lemma 1].

######  Lemma 5.1.

Let $w\in\mathcal{P}^{k}(V)$. Then

$\displaystyle\nabla^{2}(R^{2t}w)$
$\displaystyle=2t(2k+1-2n+2(t-1))R^{2(t-1)}w+R^{2t}\nabla^{2}w.$

###### Proof.

By induction on $t$, using $[\nabla^{2},R^{2}]=4E+2(1-2n)$ we have

$\displaystyle\nabla^{2}(R^{2}w)$
$\displaystyle=(4k+2(1-2n))w+R^{2}\nabla^{2}w.\qed$

######  Lemma 5.2.

Let $\lambda\in\Lambda_{n}^{*}$ such that $\lambda_{2}\geq 1$ and set
$\nu=(\lambda_{1}-1,\lambda_{2}-1)$. The Capelli operators $D_{\lambda}$ and
$D_{\nu}$ are related by

$\displaystyle 2\lambda_{2}(2\lambda_{1}-2n-1)D_{\lambda}$
$\displaystyle=R^{2}D_{\nu}\nabla^{2}.$

###### Proof.

Let $\\{R^{2\nu_{2}}v_{1},\dotsc,R^{2\nu_{2}}v_{\ell}\\}$, with $v_{i}$
harmonic, be a basis to $V_{\nu}$ and let
$\\{v_{1}^{*},\dotsc,v_{\ell}^{*}\\}$ be its dual basis. It follows that
$D_{\nu}=\sum_{i=1}^{\ell}\left(R^{2\nu_{2}}v_{i}\right)v_{i}^{*}.$
Furthermore, $\\{R^{2\lambda_{2}}v_{1},\dotsc,R^{2\lambda_{2}}v_{\ell}\\}$ is
a basis to $V_{\lambda}$. Notice that Lemma 5.1 implies that for all $1\leq
i,j\leq\ell$ we have

$\displaystyle v_{i}^{*}\nabla^{2}(R^{2\lambda_{2}}v_{j})$
$\displaystyle=v_{i}^{*}\left(2\lambda_{2}(2\lambda_{1}-2n-1)R^{2\nu_{2}}v_{j}+R^{2\lambda_{2}}\nabla^{2}v_{j}\right)$
$\displaystyle=2\lambda_{2}(2\lambda_{1}-2n-1)\delta_{i,j}.$

Thus $\left\\{\frac{v_{i}^{*}\nabla^{2}}{2\lambda_{2}(2\lambda_{2}-2n-1)}\mid
1\leq i\leq\ell\right\\}$ is dual to
$\\{R^{2\lambda_{2}}v_{1},\dotsc,R^{2\lambda_{2}}v_{\ell}\\}$ and

$\displaystyle D_{\lambda}$
$\displaystyle=\sum_{i=1}^{\ell}\frac{(R^{2\lambda_{2}}v_{i})(v_{i}^{*}\nabla^{2})}{2\lambda_{2}(2\lambda_{1}-2n-1)}=\frac{R^{2}D_{\nu}\nabla^{2}}{2\lambda_{2}(2\lambda_{1}-2n-1)}.\qed$

######  Proposition 5.3.

Let $\lambda,\mu\in\Lambda_{n}^{*}$ be such that $\lambda_{2},\mu_{2}\geq 1$.
Set $\nu=(\lambda_{1}-1,\lambda_{2}-1)$ and $\eta=(\mu_{1}-1,\mu_{2}-1)$. Then

$\displaystyle c_{\lambda}(\mu)$
$\displaystyle=\frac{\mu_{2}(\mu_{1}-(n+\frac{1}{2}))}{\lambda_{1}(\lambda_{1}-(n+\frac{1}{2}))}c_{\nu}(\eta).$

###### Proof.

Take any $R^{2\mu_{2}}w\in V_{\mu}$, with $w$ harmonic, then using Lemma 5.1
and Lemma 5.2 we have

$\displaystyle c_{\lambda}(\mu)R^{2\mu_{2}}w$
$\displaystyle=D_{\lambda}(R^{2\mu_{2}}w)=\frac{R^{2}D_{\nu}\nabla^{2}}{2\lambda_{2}(2\lambda_{1}-2n-1)}\left(R^{2\mu_{2}}w\right)$
$\displaystyle=\frac{\mu_{2}(2\mu_{1}-2n-1)}{\lambda_{2}(2\lambda_{1}-2n-1)}R^{2}D_{\nu}R^{2\eta_{2}}w=\frac{\mu_{2}(\mu_{1}-(n+\frac{1}{2}))}{\lambda_{2}(\lambda_{1}-(n+\frac{1}{2}))}c_{\nu}(\eta)R^{2\mu_{2}}w.\qed$

Indeed, applying Proposition 5.3 recursively relates the eigenvalues of
$D_{\lambda}$ to those of $D_{\nu}$, where $V_{\nu}$ is harmonic. That is,
this reduces Problem 2.5 to calculating the eigenvalues of $D_{\nu}$, for
$\nu$ such that $V_{\nu}$ harmonic.

######  Corollary 5.4.

Let $\lambda,\mu\in\Lambda^{*}_{n}$. Let $\ell=\min(\lambda_{2},\mu_{2})$ and
set $\nu=(\lambda_{1}-\ell,\lambda_{2}-\ell)$ and
$\eta=(\mu_{1}-\ell,\mu_{2}-\ell)$. Then

(5.1) $\displaystyle c_{\lambda}(\mu)$
$\displaystyle=\frac{\mu_{2}^{\underline{\lambda_{2}}}(\mu_{1}-(n+\frac{1}{2}))^{\underline{\lambda_{2}}}}{\lambda_{2}!(\lambda_{1}-(n+\frac{1}{2}))^{\underline{\lambda_{2}}}}c_{\nu}(\eta).$

###### Proof.

If $\mu_{2}<\lambda_{2}$, then the right hand side of Equation (5.1) is $0$.
Furthermore, Lemma 5.2 can be applied $\lambda_{2}$ times to write
$d_{\lambda,\mu}D_{\lambda}=R^{2\lambda_{2}}D_{\nu}\nabla^{2\lambda_{2}},$ for
some non-zero $d_{\lambda,\mu}\in\mathbb{Z}$. Thus for all $R^{2\mu_{2}}w\in
V_{\mu}$, with $w$ harmonic, we have the following

$\displaystyle c_{\lambda}(\mu)R^{2\mu_{2}}w$
$\displaystyle=D_{\lambda}R^{2\mu_{2}}w$
$\displaystyle=\frac{1}{d_{\lambda,\mu}}R^{2\lambda_{2}}D_{\nu}\nabla^{2\lambda_{2}}R^{2\mu_{2}}w=0,$

where the last equality follows by applying Lemma 5.1 $\mu_{2}+1$ times. Thus
for $\mu_{2}<\lambda_{2}$ we have Equation (5.1).

If $\lambda_{2}\leq\mu_{2}$, then Lemma 5.3 can be applied $\ell$-times to get

$\displaystyle c_{\lambda}(\mu)$
$\displaystyle=\left(\prod_{i=1}^{\lambda_{2}}\frac{(\mu_{2}-i+1)(2(\mu_{1}-i)+1-2n)}{(\lambda_{2}-i+1)(2(\lambda_{1}-i)+1-2n)}\right)c_{\nu}(\eta).\qed$

Thus we focus on computing the eigenvalues of $D_{\nu}$, for $V_{\nu}$
harmonic. The set of irreducible harmonic submodules of $\mathcal{P}(V)$ are
indexed by

$\Lambda_{n,\mathcal{H}}^{*}:=\\{(i,0)\mid 0\leq i\leq 2n+1\\}.$

The following lemma will allow us to relate the eigenvalues of $D_{\nu}$, for
$\nu\in\Lambda_{n,\mathcal{H}}^{*}$, to Knop-Sahi polynomials
$P_{\nu}^{\rho}$, see [KS96].

######  Lemma 5.5.

Let $\nu\in\Lambda_{n,\mathcal{H}}^{*}$ and $\mu\in\Lambda_{n}^{*}$. Then
$c_{\nu}(\mu)$ is a polynomial in $\mu_{1},\mu_{2}$ such that

  1. (1)

$c_{\nu}(\mu)$ is symmetric in $x=\mu_{1}-\left(n+\frac{1}{2}\right)$ and
$y=\mu_{2}$,

  2. (2)

$\deg(c_{\nu}(\mu))\leq|\nu|,$ and

  3. (3)

$c_{\nu}(\mu)=\delta_{\nu,\mu}$ for all $|\mu|\leq|\nu|$.

###### Proof.

Identify $C$ and $Z$ with their images under the map in Equation (2.1).
Theorem 2.7 lets us write
$D_{\nu}=\sum_{\lambda\in\Lambda_{|\nu|,n}}a_{\lambda}C^{\lambda_{2}}Z^{\lambda_{1}-\lambda_{2}}$
for some $a_{\lambda}\in\mathbb{C}$. Observe that the action of $C$ and $Z$ on
$V_{\mu}$ are given by scalar multiplication by
$(\mu_{1}-\mu_{2})(2n+1+\mu_{2}-\mu_{1})$ and $\mu_{1}+\mu_{2}$, respectively.
Indeed, the action of $D_{\nu}$ on $V_{\mu}$ is given by

$\displaystyle c_{\nu}(\mu)$
$\displaystyle=\sum_{\lambda\in\Lambda_{|\nu|,n}}a_{\lambda}(\mu_{1}-\mu_{2})^{\lambda_{2}}(2n+1+\mu_{2}-\mu_{1})^{\lambda_{2}}(\mu_{1}+\mu_{2})^{\lambda_{1}-\lambda_{2}}$
(5.2)
$\displaystyle=\sum_{\lambda\in\Lambda_{|\nu|,n}}a_{\lambda}\left(x-y+n+\frac{1}{2}\right)^{\lambda_{2}}\left(y-x+n+\frac{1}{2}\right)^{\lambda_{2}}\left(x+y-n-\frac{1}{n}\right)^{\lambda_{1}-\lambda_{2}},$

where $x=\mu_{1}-n-\frac{1}{2}$ and $y=\mu_{2}$. Condition (1) follows from
Equation (5). Furthermore, every summand of (5) has degree $|\lambda|$. This
yields Condition (2). For Condition (3), the argument is similar to [SS16,
Lemma 5.4] the point being
$\bigcup_{k=0}^{2n+1}\mathbb{P}_{k,n}^{*}=\\{\mu\in\mathbb{P}\colon|\mu|\leq
2n+1\\}$. Indeed, if $|\mu|<|\nu|$ then the order of $D_{\nu}$ is $|\nu|$ and
the degree of $V_{\mu}$ is $|\mu|$. That is, $c_{\nu}(\mu)=0$. If
$|\nu|=|\mu|$, then $D_{\nu}$ results in a $\mathfrak{g}$-module homomorphism
$V_{\mu}\to V_{\nu}$ which can be non-zero only if $\mu=\nu$. ∎

The next proposition which follows from [KS96, Prop. 3.4] gives an explicit
expression for $P_{\nu}^{\rho}$ for $\nu\in\Lambda_{n,\mathcal{H}}^{*}$.

######  Proposition 5.6.

Let $\nu\in\Lambda_{n,\mathcal{H}}^{*}$. Then

$\displaystyle P_{\nu}^{\rho}$
$\displaystyle={n+\frac{1}{2}\choose\nu_{1}}^{-1}\sum_{i=0}^{\nu_{1}}{n+\frac{1}{2}\choose\nu_{1}-i}{n+\frac{1}{2}\choose
i}\left(x+n+\frac{1}{2}-i\right)^{\underline{\nu_{1}-i}}\left(y\right)^{\underline{i}}.$

Lemma 5.5 and the uniqueness the Knop-Sahi polynomials yield the following.

######  Proposition 5.7.

Let $\nu\in\Lambda_{n,\mathcal{H}}^{*}$ and $\mu\in\Lambda_{n}^{*}$. Then

$\displaystyle c_{\nu}(\mu)$
$\displaystyle=\frac{1}{(n+\frac{1}{2})^{\underline{\nu_{1}}}}\sum_{i=0}^{\nu_{1}}{n+\frac{1}{2}\choose\nu_{1}-i}{n+\frac{1}{2}\choose
i}\left(\mu_{1}-i\right)^{\underline{\nu_{1}-i}}\left(\mu_{2}\right)^{\underline{i}}.$

The proof of Theorem 2.6 follows immediately.

_Proof of Theorem 2.6._ Notice that $D_{\nu}$ corresponds to a harmonic
irreducible module if and only if $\nu_{2}=0$. Thus we apply Proposition 5.7
and Corollary 5.3. When $\nu_{2}>0$ we use the fact that
$P_{\nu}^{\rho}(x,y)=xyP_{\nu^{\prime}}^{\rho}(x-1,y-1)$ for
$\nu^{\prime}=(\nu_{1}-1,\nu_{2}-1)$, see [KS96, Prop. 2.3]. ∎

## References

  * [ASS18] Alexander Alldridge, Siddhartha Sahi, and Hadi Salmasian.  Schur $Q$-functions and the Capelli eigenvalue problem for the Lie superalgebra $\mathfrak{q}(n)$.  In Representation theory and harmonic analysis on symmetric spaces, volume 714 of Contemp. Math., pages 1–21. Amer. Math. Soc., Providence, RI, 2018.
  * [Cap87] Alfredo Capelli.  Ueber die Zurückführung der Cayley’schen Operation $\Omega$ auf gewöhnliche Polar-Operationen.  Math. Ann., 29(3):331–338, 1887.
  * [Cou13] Kevin Coulembier.  The orthosymplectic superalgebra in harmonic analysis.  J. Lie Theory, 23(1):55–83, 2013.
  * [DBES09] H. De Bie, D. Eelbode, and F. Sommen.  Spherical harmonics and integration in superspace. II.  J. Phys. A, 42(24):245204, 18, 2009.
  * [HU91] Roger Howe and Toru Umeda.  The Capelli identity, the double commutant theorem, and multiplicity-free actions.  Math. Ann., 290(3):565–619, 1991.
  * [Kac77] V. G. Kac.  Lie superalgebras.  Advances in Math., 26(1):8–96, 1977.
  * [KS91] Bertram Kostant and Siddhartha Sahi.  The Capelli identity, tube domains, and the generalized Laplace transform.  Adv. Math., 87(1):71–92, 1991.
  * [KS96] Friedrich Knop and Siddhartha Sahi.  Difference equations and symmetric polynomials defined by their zeros.  Internat. Math. Res. Notices, (10):473–486, 1996.
  * [OO97] A. Okounkov and G. Olshanski.  Shifted Jack polynomials, binomial formula, and applications.  Math. Res. Lett., 4(1):69–78, 1997.
  * [Sah94] Siddhartha Sahi.  The spectrum of certain invariant differential operators associated to a Hermitian symmetric space.  In Lie theory and geometry, volume 123 of Progr. Math., pages 569–576. Birkhäuser Boston, Boston, MA, 1994.
  * [SS16] Siddhartha Sahi and Hadi Salmasian.  The Capelli problem for $\mathfrak{gl}(m|n)$ and the spectrum of invariant differential operators.  Adv. Math., 303:1–38, 2016.
  * [SSS20] Siddhartha Sahi, Hadi Salmasian, and Vera Serganova.  The Capelli eigenvalue problem for Lie superalgebras.  Math. Z., 294(1-2):359–395, 2020.
  * [SSS21] Siddhartha Sahi, Hadi Salmasian, and Vera Serganova.  Capelli operators for spherical superharmonics and the Dougall–Ramanujan identity.  submitted, 2021.
  * [Wal92] Nolan R. Wallach.  Polynomial differential operators associated with Hermitian symmetric spaces.  In Representation theory of Lie groups and Lie algebras (Fuji-Kawaguchiko, 1990), pages 76–94. World Sci. Publ., River Edge, NJ, 1992.
  * [Wey39] Hermann Weyl.  The Classical Groups. Their Invariants and Representations.  Princeton University Press, Princeton, N.J., 1939.