# MULTIMODAL OBJECT DETECTION VIA PROBABILISTIC A PRIORI INFORMATION
INTEGRATION

1st Hafsa El Hafyani  IMT Atlantique
Lab-STICC, UMR CNRS 6285
Brest F-29238, France
<EMAIL_ADDRESS>2nd Bastien Pasdeloup  3rd Camille Yver
IMT Atlantique
Lab-STICC, UMR CNRS 6285
Brest F-29238, France
<EMAIL_ADDRESS>InPixal
Rennes, France
<EMAIL_ADDRESS>4th Pierre Romenteau  InPixal
Rennes, France
<EMAIL_ADDRESS>
###### Abstract

Multimodal object detection has shown promise in remote sensing. However,
multimodal data frequently encounter the problem of low-quality, wherein the
modalities lack strict cell-to-cell alignment, leading to mismatch between
different modalities. In this paper, we investigate multimodal object
detection where only one modality contains the target object and the others
provide crucial contextual information. We propose to resolve the alignment
problem by converting the contextual binary information into probability maps.
We then propose an early fusion architecture that we validate with extensive
experiments on the DOTA dataset.

###### Index Terms:

Object detection, Multimodal fusion, deep learning, alignment, remote sensing

##  I Introduction

Object detection has made remarkable progress thanks to the advances of deep
neural detectors. While this thematic is generally developed based on ground
level sensors images, applying classical object detection techniques ([1] [2])
directly to aerial or space images (e.g. satellite and/or drone images)
results in reduced performance. Remote sensing images present more complicated
background information, larger scale variance and densely small targets.
Therefore, practical applications, such as military surveillance of areal
images leverage multimodal data to enhance object detection capabilities, by
incorporating complementary information modelling the structure and context of
the objects in the scene.

However, when complementary information is collected by multiple sensors or
sources (e.g., land cover/use, SAM [3]), it may present problems of low
quality or misalignment with the acquired images, due to, for instance,
angular acquisition, leading to potential errors in detection. Figure 1
presents an example of low-quality problem of the contextual information (i.e.
masks) acquired by three different sensors. Neither of the contextual
information matches perfectly the aerial image. The masks of roundabouts (i.e.
left) and vehicles (i.e. right) collected by two different sensors show a
shift w.r.t. the aerial image (i.e. center). Therefore, it is crucial to
present the combination of different modalities in a cohesive manner. Failure
to align one modality with the others could lead to increased errors in
detection.

While showing considerable promise in multimodal tasks ([4, 5]), the
performance of object detectors in mis-aligned multimodal geospatial settings
is to be verified. Therefore, this paper first investigates the performance of
multimodal data compared to unimodal data for geospatial tasks, since
multimodal data do not necessarily improve performance [6]. Second, we suggest
to resolve the mis-alignment issue by transforming contextual information into
probability maps. Third, we examine the proposed alignment approach
performance compared to no-alignment settings using the DOTA dataset[7].

The rest of this paper is organized as follows. We introduce the related work
in Section II. Section III gives a thorough problem description, describes the
proposed approach for resolving the alignment issue, and discuss the proposed
fusion models. Section IV presents the experimental results and evaluation of
the multimodal fusion detectors. Last but not least, in section V, we
summarize our conclusions and provide directions for future work.

Figure 1: Example of misalignment of contextual data with an acquired RGB
image. On the left, the mask of roundabout does not match the RGB image
(center). On the right, the mask of vehicles shows a shift w.r.t. the RGB
image.

##  II Related work

###  II-A Generic Object Detection Methods

Deep learning-based algorithms are widely used for object detection with two
main categories: two-stage and single-stage approaches. Two-stage methods,
such as Faster-RCNN [8], have better detection accuracy as they initially
generate candidate boxes, and then identify the object category and refine the
object location. One-stage methods, such as SSD [1] and YOLO series [2], offer
faster inference speeds since the proposals generation is not needed. Although
these methods have shown a good performance on images taken from the ground
level, the same performance is not achieved on remote sensing images due to
the complex nature of areal images.

On top of that, modern detectors adopt an encoder-decoder architecture based
on transformers [9, 10, 11]. Specifically, Carion et al. [12] and Zhu et al.
propose new models called DETR and Deformable DETR, respectively, which have
shown competitive results to Faster-RCNN model on large objects. DETR performs
less effectively for small objects [12]. Additionaly, Kirillov et al. unveiled
SAM [3], a highly accurate pre-trained model for diverse image types,
including remote sensing images. Researchers are exploring its potential in
handling multimodal data scenarios.

###  II-B Deep Multimodal Fusion

Researchers have explored various approaches for implementing multimodal
fusion to enhance the performance of object detectors in remote sensing. Zhang
et al. [4] propose a contextual bidirectional enhancement method to enhance
objects’ features. Chen et al. [5] propose to combine scene-contextual
information in a convolutional neural network to improve the relationship
between scene and target. Besides aerial images, several works on the
multimodal fusion exist in the literature. Gao et al. [13] review different
approaches to represent multimodal data, and Chu et al. [14] present MT-DETR,
a new fusion model through early and late fusion. Even though these methods
have shown good performances when combined with contextual information or
other modalities, the alignment problem was not addressed.

###  II-C Image Alignment

Image alignment is not a new task for the computer vision research community.
Brown [15] provides a comprehensive review of methods involving geometric
transformations to align images based on criteria like similarity. Building
upon these approaches, Wang et al. [16] successfully aligned and merged RGB
images with thermal images. Girard et al. [17] proposed a fully convolutional
architecture to correct the alignment of building registries (_e.g._ , from
OpenStreetMap) on the image. To handle the weakly aligned multispectral data,
which makes one object has different positions in different modalities, Zhang
et al. [18] proposed a novel Aligned Region CNN (AR-CNN), which includes a
region feature alignment module to align the region features. following the
work of Zhang et al. [18], Yuan et al. [19] propose an end-to-end model called
CAGT to solve weak misalignment and fusion imprecision problem in regional
level in multispectral data. Therefore, image alignment has proven successful
in these studies, particularly when dealing with two images that depict the
same aspects (_e.g._ , RGB images and thermal images) and both exhibit the
target objects. However, when attempting to align RGB images with binary
masks, the task becomes more and more challenging.

##  III Contributions

###  III-A Problem Statement

We consider a problem that involves two domains, a multimodal object detection
problem [13] of RGB images and contextual information represented by binary
masks retrieved by sensors or segmentation models (e.g. SAM), showing the land
cover/use data, and image alignment problem ([17, 20, 21]) between different
modalities.

Existing work on modality alignment mainly emphasizes datasets where objects
are visible across all modalities. However, in our case, objects are
exclusively visible within one modality. The alignment of these unique
characteristic modalities requires the development of specialized techniques
tailored to address this specific circumstance. Traditional approaches may not
be directly applicable here, since they are based on the assumption of object
presence across multiple modalities. Therefore, innovative solutions are
necessitated to effectively align modalities and fill the gap of modality
alignment in the case of the absence of object visibility across all
modalities in the dataset.

Our objective is to train a detector that can combine the multimodal data
while ensuring a sound cross-modality representation. Specifically, we desire
to obtain an agnostic shift alignment representation that works equally
whether the multi modalities are shifted or not. Therefore, we propose a
multimodal fusion model for object detection based on probability maps.
Indeed, instead of having binary values (_i.e._ , 0 and 1), the probability
map assigns a continuous value between 0 and 1 to each pixel, indicating the
probability of that pixel hosting the object. Hence, even if the modalities
are not perfectly aligned, the probability maps can provide guidance to the
model and inform it about the likelihood of the object’s presence within the
cell.

###  III-B Probability Maps

One straightforward way to generate probabilistic maps is the Euclidean
distance transformation. Inspired by common sens, it consists of calculating
the Euclidean distance of each pixel to the nearest boundary (pixel with a
value of 1) in the binary image. As the distance between a pixel and the
object’s boundary increases, its probability of hosting the object decreases.
Formally, the task involves processing a single channel image represented as a
tensor $I_{c}$ of size $H*W*1$, where each element $I_{c}(i,j)$ is either 0 or
1. $H$ and $W$ denote image’s height and the width, respectively. We want to
compute the probability map $P$ of the same size, where each element $P(i,j)$
represents the probability of the pixel at position $(i,j)$ belonging to the
target class:

$\begin{cases}P(i,j)=1-d_{norm}(i,j)\\\ d_{norm}(i,j)=\frac{d(i,j)}{max(d)}\\\
d(i,j)=min_{i^{\prime},j^{\prime}s.t.(i^{\prime},j^{\prime})\in
I_{c}}||(i,j)-(i^{\prime},j^{\prime})||_{2}\\\ \end{cases}\\\ $ (1)

where $d_{norm}(i,j)$ is normalized distance from each cell to the object, and
$d(i,j)$ is the Euclidean distance of pixel at position $(i,j)$ to the nearest
boundary.

(a)

(b)

(c)

Figure 2: Example of the construction of probability maps from a binary mask
(a) based on Eq. 1 (b) and Eq. 2 (c).

Figure 2 shows an example of the generation of these maps (Figure 2b) from a
binary mask (Figure 2a). The probability is higher for pixels closer to the
object boundary and decreases as the distance from the object boundary
increases. As a matter of fact, the above representation of probability maps
based on the Euclidean distance shows a smooth propagation of the mask through
the whole image. However, the pixel with the coordinates $(1,1)$ in Figure 2b
should have a higher probability since it is a neighbour of two cells hosting
the object (_i.e._ , cells with coordinates $(0,1)$ and $(1,2)$), as it would
be resilient to misalignments along two axes rather than one. Therefore, we
propose an alternative distance for constructing the probability maps from the
binary masks. It takes into consideration both the distance to the boundaries,
and the neighbouring cells as follows:

$\forall p\in I_{c},\;\;\;d(p)=\frac{\sum_{\begin{subarray}{c}r\in
R(p)\exp(-\alpha d(r,p))\end{subarray}}}{\sum_{\begin{subarray}{c}m\in
M\exp(-\alpha d(m,p))\end{subarray}}}$ (2)

where $R(p)$ is the set of neighbouring cells in a specific radius (to be
defined by the user), $\alpha\in\mathbf{R}$ (to be set by the user as well),
and $M$ is the set of all the cells containing the mask. Therefore, with this
new representation, the example shown in Figure 2a becomes as indicated in
Figure 2c with $\alpha=1$ and the radius set to 1 cell in every direction
(_i.e._ , left, right, up, down, and diagonal).

###  III-C Proposed Method

The misalignment problem is resolved with the probability maps, we introduce
thereafter the flow diagram of our framework. As presented in Figure 3, the
framework provides an illustration of the fusion’s logic. First, we start by
generating the probability maps from the binary masks using either equation 1
or equation 2. Then, the generated probabilities maps are concatenated to the
RGB image, which is then fed to an object detector (e.g. Faster-RCNN).
Finally, The detector produces the candidate classes and their bounding boxes
with a feature pyramid network (FPN) based network.

Figure 3: The pipeline of our framework.

##  IV Experiments

###  IV-A Dataset

Because of the sensibility of military data, we recreate the multi modality
miss-alignment problem with a subset of the DOTA dataset [7] with its 15
classes as shown in Table I. The binary masks are extracted from the iSAID
dataset [22].

###  IV-B Experimental Setup

We kept the training set as established by the DOTA dataset. However, we split
the validation set between the validation set and testing set, since the
binary masks are only provided for the training and validation sets.
Therefore, 2/3 of the DOTA validation set goes to the validation set and 1/3
goes to the test set. For the evaluation metrics, we employ the widely used
average precision (AP) to evaluate the performance of our model over every
class, and the mean average precision (mAP), which is computed over all
classes. Additionally, a candidate bounding box is considered as a true
positive if the value of Intersection-Over-Union (IoU) metric is greater than
0.5; otherwise, it is a false positive. All the experiments were made by
training a Faster-RCNN [8] model, initialized with COCO pre-trained weights.
We inject a random shift between $5\%$ and $10\%$ of the image width in the
binary mask to simulate the misalignment problems. Our implementation is based
on the PyTorch framework under Ubuntu 20.04. The experiments were executed on
NVIDIA GeForce RTX 3090. The models were trained on 50 epochs, with a batch
size of 8, SGD optimizer, weight decay of 5e-4, momentum of 0.9, and the
learning rate set to $0.001$. The source code will be publicly available at:
https://github.com/helhafyani/Multimodal_fusion.

###  IV-C Experimental Results

We perform two types of fusion experiments to evaluate the model’s response to
the integration of contextual information. The first type involves adding
direct contextual information such as the inclusion of shifted, low-quality
masks of objects already present in the RGB images (e.g., adding a shifted
mask of boats to enhance boat’s detection). The second type explores fusion
with indirect contextual information, such as shifted masks of elements like
harbors to observe their impact on the model’s ability to detect specific
objects (e.g. boats).

####  IV-C1 Does fusion improve detection performance?

We perform the direct fusion model with 15-class object detection and 15
binary masks (a binary mask for every class). For the indirect fusion model,
we keep the 15 classes in the detection target classes, and use 3 masks: the
mask of `harbors` is leveraged for the detection of `ships`, and the masks of
`bridges` and `roundabouts` are adopted to to enhance the detection of `small`
and `large vehicles`. We report the results in Table I. Basic model refers to
the model without fusion. “Distance in Eq1” points to the fusion with the
probability maps based on Equation 1, and “Distance in Eq2” relates to the
fusion with probability maps based on Equation 2. Basic fusion model is the
baseline which refers to the standard fusion model without adopting the
probability maps solution.

TABLE I: Performance of the different models compared to the model without
fusion (_i.e._ , Basic). Bold entries denote the best detection per target
class. Green/orange/red entries show the models that are better/equal/worse
compared to the Basic model.

Model | Plane | Ship | ST | BD | TC | BC | GTF | Harbor | Bridge | LV | SV | Heli. | Round. | SBF | SP | mAP
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Basic | 0.76 | 0.47 | 0.63 | 0.62 | 0.77 | 0.40 | 0.39 | 0.65 | 0.26 | 0.59 | 0.34 | 0.30 | 0.46 | 0.50 | 0.14 | 0.48
Basic fusion | Direct | 0.78 | 0.47 | 0.62 | 0.66 | 0.77 | 0.54 | 0.50 | 0.62 | 0.28 | 0.60 | 0.33 | 0.24 | 0.39 | 0.53 | 0.13 | 0.50
Indirect | 0.76 | 0.48 | 0.64 | 0.66 | 0.80 | 0.49 | 0.50 | 0.64 | 0.30 | 0.59 | 0.34 | 0.30 | 0.42 | 0.50 | 0.15 | 0.50
Distance in Eq.1 | Direct | 0.79 | 0.47 | 0.66 | 0.60 | 0.80 |  0.44 | 0.62 | 0.64 | 0.41 | 0.58 | 0.34 | 0.46 | 0.53 | 0.48 | 0.18 | 0.53
Indirect | 0.74 |  0.49 | 0.63 | 0.72 | 0.79 | 0.37 | 0.30 | 0.62 | 0.38 | 0.58 | 0.33 | 0.38 | 0.44 | 0.57 | 0.17 | 0.50
Distance in Eq.2 | Direct | 0.78 | 0.48 | 0.62 | 0.67 | 0.78 | 0.54 | 0.53 | 0.68 | 0.35 | 0.61 | 0.34 | 0.31 | 0.49 | 0.55 | 0.21 | 0.53
Indirect | 0.77 | 0.50 | 0.58 | 0.67 | 0.81 | 0.45 | 0.45 | 0.62 | 0.33 | 0.57 | 0.37 | 0.22 | 0.50 | 0.54 | 0.16 | 0.50

Overall, all of the fusion models perform better than the basic model (_i.e._
, without fusion) when looking at the mAP, which indicates that adding either
direct or indirect contextual information brings along a better distinction
between classes. Also, although the direct fusion models show undoubtedly the
highest performances, the indirect fusion models show a remarkable performance
with the few added contextual information. The indirect model does not only
demonstrate a better performance for the related classes, especially `ship`
and `large vehicle`, but it facilitates the discrimination between the other
classes. Compared to the direct fusion model (i.e. without the probability
maps), both direct models perform better, which means adding probability maps
improves the detection performance of the models. However, the indirect models
have a similar performance in terms of mAP. Still, the models with probability
maps perform better in detecting the classes with the contextual information,
especially `ship` and `small vehicle`.

TABLE II: Performance of the fusion model with one class at a time. Bold
entries denote the best detection per target class. Green/orange/red entries
show the classes that are better/equal/worse compared to the Basic model
reported in Table I.

Model Class | Plane | Ship | ST | BD | TC | BC | GTF | Harbor | Bridge | LV | SV | Heli. | Round. | SBF | SP | mAP
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Plane | 0.75 | 0.47 | 0.61 | 0.67 | 0.78 | 0.52 | 0.51 | 0.62 | 0.27 | 0.61 | 0.35 | 0.26 | 0.44 | 0.47 | 0.15 | 0.50
Ship | 0.75 | 0.49 | 0.62 | 0.62 | 0.79 | 0.43 | 0.42 | 0.62 | 0.24 | 0.57 | 0.33 | 0.33 | 0.47 | 0.51 | 0.19 | 0.49
Storage tank (ST) | 0.78 | 0.48 | 0.62 | 0.64 | 0.79 | 0.50 | 0.47 | 0.63 | 0.36 | 0.60 | 0.34 | 0.35 | 0.48 | 0.53 | 0.19 | 0.52
Baseball diamond (BD) | 0.77 | 0.49 | 0.62 | 0.60 | 0.81 | 0.40 | 0.48 | 0.65 | 0.29 | 0.60 | 0.33 | 0.31 | 0.44 | 0.43 | 0.19 | 0.49
Tennis court (TC) | 0.78 | 0.48 | 0.61 | 0.64 | 0.81 | 0.42 | 0.44 | 0.65 | 0.29 | 0.60 | 0.34 | 0.33 | 0.39 | 0.43 | 0.15 | 0.49
Basketball court (BC) | 0.78 | 0.47 | 0.64 | 0.66 | 0.77 | 0.50 | 0.48 | 0.66 | 0.31 | 0.59 | 0.34 | 0.39 | 0.44 | 0.57 | 0.18 | 0.52
Ground track field (GTF) | 0.79 | 0.48 | 0.63 | 0.53 | 0.75 | 0.47 | 0.50 | 0.60 | 0.35 |  0.60 | 0.33 | 0.34 | 0.44 | 0.45 | 0.15 | 0.49
Harbor | 0.78 | 0.48 | 0.63 | 0.57 | 0.79 | 0.46 | 0.32 | 0.64 | 0.28 | 0.59 | 0.35 | 0.35 | 0.44 | 0.40 | 0.19 | 0.49
Bridge | 0.77 | 0.48 | 0.65 | 0.58 | 0.80 | 0.50 | 0.43 | 0.65 | 0.34 | 0.59 | 0.34 | 0.33 | 0.41 | 0.47 | 0.20 | 0.50
Large vehicle (LV) | 0.77 | 0.47 | 0.65 | 0.65 | 0.79 | 0.43 | 0.37 | 0.63 | 0.35 | 0.59 | 0.35 | 0.34 | 0.42 | 0.53 | 0.16 | 0.50
Small vehicle (SV) | 0.78 | 0.48 | 0.64 | 0.72 | 0.79 | 0.48 | 0.40 | 0.63 | 0.30 | 0.62 | 0.32 | 0.34 | 0.54 | 0.51 | 0.17 | 0.51
Helicopter | 0.78 | 0.49 | 0.60 | 0.65 | 0.79 | 0.46 | 0.43 | 0.65 | 0.31 | 0.60 | 0.34 | 0.36 | 0.35 | 0.48 | 0.18 | 0.50
Roundabout | 0.72 | 0.48 | 0.60 | 0.58 | 0.79 | 0.46 | 0.47 | 0.61 | 0.29 | 0.61 | 0.34 | 0.29 | 0.57 | 0.45 | 0.15 | 0.50
Soccer ball field (SBF) | 0.78 | 0.49 | 0.63 | 0.62 | 0.79 | 0.42 | 0.52 | 0.61 | 0.34 | 0.61 | 0.34 | 0.37 | 0.45 | 0.52 | 0.19 | 0.51
Swimming pool (SP) | 0.76 | 0.47 | 0.64 | 0.57 | 0.81 | 0.40 | 0.47 | 0.65 | 0.30 | 0.61 | 0.35 | 0.38 | 0.45 | 0.50 | 0.23 | 0.51

####  IV-C2 Which contextual information improves class discrimination?

To answer this question, we only add a single mask as additional input, and
report the obtained performance for each class individually in Table II. The
matrix diagonal can be understood as a direct fusion process with only the
target class used as contextual information, while off-diagonal entries show
the indirect fusion, _i.e._ , the impact of the contextual class for detecting
other classes.

While it appears that adding an input layer for a probability map helps in the
detection process in many cases, straightforward conclusions are not so easy.
Indeed, the original reasoning behind this work was that presence of a
`bridge` or a `roundabout` would probably lead to improved detection
performance for `small` and `large vehicles`. However, here, we observe
marginal or no improvement at all. The same observation holds for `harbor`,
that only brings a marginal gain for detection of `ships`. However, we observe
that adding a layer for `planes` brings a 12pt gain for `basketball courts`,
or that adding a layer for `small vehicles` improves detection of baseball
diamonds by 10pt, which is hard to understand semantically speaking. Regarding
the direct fusion though, we observe that the best performance was achieved
for 4 classes out of the 15, and that performance was either improved or
stayed the same for 10 of these classes. This suggests again that a priori
knowledge of an element in the image improves its detection, even when we have
an approximate idea of its location due to misalignments.

##  V Conclusion & perspectives

We explore the effectiveness of multimodal data compared to unimodal for
geospatial detection. We introduce a solution to mitigate the misalignment
issue, involving the transformation of contextual information into probability
maps. We validate the proposed alignment approach with two fusion type
settings using the DOTA dataset. For future work, we intend to explore other
fusion methods such as mid-fusion at different layers of the architecture.
Additionally, we plan to evaluate the performance of the proposed approach on
other datasets.

## Acknowledgment

This work has been supported by the French National Research Agency (ANR),
under the grant agreement ANR-21-ASIA-0004.

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Because of Roy’s Theorem 3.1, the proof of the $a$-theorem for $k>1$ is
reduced to the statement: $\Delta a=a(\lambda)-a(\mu)>0$ for any pair
$\lambda<\mu$ of dominant coweights of either of the following two forms:

  * •

Type _i_): $\mu=\lambda+\widehat{\alpha}_{I}^{\vee}$ where $I$ is an
irreducible component of the subdiagram of zeros of $\lambda$.

  * •

Type _ii_): $\mu=\lambda+\alpha_{i}^{\vee}$ where $n_{j}>0$ for all $j$ which
are connected to $i$ by an edge.

The proof comes down to a case by case check for each of the 53 proper
connected subdiagrams of the affine Dynkin diagram in Type _i_), and each of
the nine simple coroots in Type _ii_). Although one could in principle deal
with these cases by hand, many of the calculations become rather complicated;
we therefore checked the details with a computer. (We used the computational
algebra platform GAP.) We will however sketch an argument that covers most of
the cases in Type _i_). Note that this type is often more straightforward to
compute, because various coefficients $n_{i}$ must be zero.

Let $\lambda<\mu$ be a pair as in Type _i_). For simplicity, assume until
further notice that $I$ does not contain the affine node $\alpha_{1}$. Then
$\lambda=(k,\overline{\lambda},n)$ and
$\mu=(k,\overline{\lambda}+\widehat{\alpha}_{I}^{\vee},n)$ (i.e. the value of
$n$ does not change – there are no instanton transitions in this RG flow).
Hence the second and third lines in the formula for the $a$ anomaly (5.2) are
unchanged in passing from $\lambda$ to $\mu$. It follows from our earlier
discussion that $\overline{\mu}(\rho)=\overline{\lambda}(\rho)+h_{I}-1$, where
$h_{I}$ is the Coxeter number of the root subsystem spanned by $I$. Moreover,
by assumption we have
$\langle\overline{\lambda},\widehat{\alpha}_{I}^{\vee}\rangle=0$ and hence

$\langle\overline{\mu},\overline{\mu}\rangle=\langle\overline{\lambda}+\widehat{\alpha}_{I}^{\vee},\overline{\lambda}+\widehat{\alpha}_{I}^{\vee}\rangle=\langle\overline{\lambda},\overline{\lambda}\rangle+\langle\widehat{\alpha}_{I}^{\vee},\widehat{\alpha}_{I}^{\vee}\rangle=\langle\overline{\lambda},\overline{\lambda}\rangle+2\
.$ (5.3)

These observations lead to a simple formula for the change in the first line
of (5.2). For the final line, we need to analyze the change in values of
$\sum_{\alpha\in\Phi^{+}}\overline{\lambda}(\alpha)^{3}$ and
$\sum_{\alpha\in\Phi^{+}}\overline{\lambda}(\alpha)^{5}$.

####  5.2.1 Considerations on sums of powers of roots

Fix an integer $m\geq 1$. Let

$F_{m}(\overline{\lambda})=\sum_{\alpha>0}\overline{\lambda}(\alpha)^{m}\ ,$
(5.4)

which we think of as a homogeneous form (i.e. a multivariate polynomial in the
coefficients $n_{i}$) of degree $m$. Introduce bihomogeneous forms of degree
$(m-i,i)$, defined as follows:

$F_{m}^{(i)}(\overline{\lambda},\theta)=\frac{m!}{i!(m-i)!}\sum_{\alpha\in\Phi^{+}}\overline{\lambda}(\alpha)^{m-i}\theta(\alpha)^{i}\
.$ (5.5)

Then
$F_{m}(\overline{\lambda}+\theta)=\sum_{i=0}^{m}F_{m}^{(i)}(\overline{\lambda},\theta)$
for all $\overline{\lambda}$, $\theta$. In particular we have
$F_{m}^{(0)}(\overline{\lambda},\theta)=F_{m}(\overline{\lambda})$ and
$F_{m}^{(m)}(\overline{\lambda},\theta)=F_{m}(\theta)$. We will carry out a
general analysis of the linear and quadratic functions
$F_{m}^{(1)}(\overline{\lambda},-)$ and $F_{m}^{(2)}(\overline{\lambda},-)$.
We assume $\overline{\lambda}=\sum_{i}n_{i}\varpi_{i}^{\vee}$ where $n_{i}\geq
0$ for all $i$. (In this section, all such sums are over $i\neq 1$, i.e.
$i\in\Delta$.) We start with the following observations.

######  Lemma 5.1.

If $\langle\overline{\lambda},\theta\rangle=0$ then
$F_{m}^{(1)}(\overline{\lambda},\theta)=0$ for $m>1$.

###### Proof.

By definition,
$F_{m}^{(1)}(\overline{\lambda},\theta)=m\sum_{\alpha>0}\overline{\lambda}(\alpha)^{m-1}\theta(\alpha)$.
Since $\theta$ is a linear combination of coroots $\alpha_{i}^{\vee}$ with
$\overline{\lambda}(\alpha_{i})=0$, it will suffice to prove the Lemma for
$\theta=\alpha_{i}^{\vee}$. The positive roots
$\beta\in\Phi^{+}\setminus\\{\alpha_{i}\\}$ are either orthogonal to
$\alpha_{i}$ (in which case $\overline{\lambda}(\beta)\theta(\beta)=0$) or
belong to a chain $\beta,\beta+\alpha_{i}$ of positive roots, with
$\theta(\beta)=\alpha_{i}^{\vee}(\beta)=-1$. Then
$\overline{\lambda}(\beta)^{m-1}\theta(\beta)+\overline{\lambda}(\beta+\alpha_{i})^{m-1}\theta(\beta+\alpha)=-\overline{\lambda}(\beta)^{m-1}+\overline{\lambda}(\beta)^{m-1}=0$.
Summing up over all such chains, we obtain
$F_{m}^{(1)}(\overline{\lambda},\theta)=0$. ∎

To understand the quadratic form $F_{m}^{(2)}(\overline{\lambda},-)$, it will
be useful to express $\overline{\lambda}$ simultaneously in terms of
fundamental coweights and simple coroots:

$\overline{\lambda}=\sum_{i}n_{i}\varpi_{i}^{\vee}=\sum_{i}b_{i}\alpha_{i}^{\vee}\
.$ (5.6)

By assumption, $n_{i}\geq 0$ for all $i$. Writing $\theta$ similarly as
$\sum_{i}c_{i}\alpha_{i}^{\vee}$, we have:

$F_{m}^{(1)}(\overline{\lambda},\theta)=\sum_{i}\frac{\partial F_{m}}{\partial
b_{i}}c_{i}\ .$ (5.7)

We therefore obtain:

######  Corollary 5.2.

For any $m\geq 2$, $\frac{\partial F_{m}}{\partial b_{i}}$ is divisible by
$n_{i}$.

We note the special case $\frac{\partial F_{2}}{\partial b_{i}}=60n_{i}$. (In
fact, we have
$F_{2}(\overline{\lambda})=30\langle\overline{\lambda},\overline{\lambda}\rangle$,
which can be deduced from inspection of (5.1).) For $m>2$, we write
$\frac{\partial F_{m}}{\partial b_{i}}=p_{i}n_{i}$, where $p_{i}$ has degree
$m-2$. Similarly to the above, we have:

$F_{m}^{(2)}(\overline{\lambda},\theta)=\frac{1}{2}\sum_{i}\frac{\partial^{2}F_{m}}{\partial
b_{i}^{2}}c_{i}^{2}+\sum_{i\neq j}\frac{\partial^{2}F_{m}}{\partial
b_{i}\partial b_{j}}c_{i}c_{j}\ ,$ (5.8)

where the second sum is over the unordered pairs $i\neq j$. We are especially
interested in the case $\langle\overline{\lambda},\theta\rangle=0$, which is
equivalent to $c_{i}n_{i}=0$ for all $i$. Hence we only need to know the
coefficients $\frac{\partial^{2}F_{m}}{\partial b_{i}^{2}}$ modulo multiples
of $n_{i}$ and $\frac{\partial^{F}_{m}}{\partial b_{i}\partial b_{j}}$ modulo
linear combinations of $n_{i}$ and $n_{j}$ (written: $\bmod(n_{i},n_{j})$).

Recall that the support of $\theta=\sum_{i}c_{i}\alpha_{i}^{\vee}$ is the set
$\\{i\in\Delta:c_{i}\neq 0\\}$. The support of a coroot is a connected subset
of the Dynkin diagram.

######  Lemma 5.3.

a) If $i-j$ is an edge then $p_{i}(\overline{\lambda})\equiv
p_{j}(\overline{\lambda})\bmod(n_{i},n_{j})$.

b) Assuming $\langle\overline{\lambda},\theta\rangle=0$, we have
$F_{m}^{(2)}(\overline{\lambda},\theta)=\sum_{i}p_{i}(\overline{\lambda})c_{i}^{2}-\sum_{i-j}p_{i}(\overline{\lambda})c_{i}c_{j}$,
where the second sum is over all edges $i-j$, taken in any order. (This is
well-defined, by (a).)

c) If in addition the support $J$ of $\theta$ is connected then
$F_{m}^{(2)}(\overline{\lambda},\theta)=\frac{1}{2}p_{j}(\overline{\lambda})\langle\theta,\theta\rangle$
for any choice of $j\in J$. In particular, if $\theta$ is a coroot then
$F_{m}^{(2)}(\overline{\lambda},\theta)=p_{j}(\overline{\lambda})$.

###### Proof.

Clearly, $\frac{\partial}{\partial b_{i}}(n_{i})=2$ and, for $j\neq i$,

$\frac{\partial}{\partial b_{j}}(n_{i})=\begin{cases}-1&\mbox{if $i-j$ an
edge,}\\\ 0&\mbox{otherwise.}\end{cases}.$ (5.9)

From the above, we have $\frac{\partial^{2}F_{m}}{\partial
b_{i}^{2}}=2p_{i}+n_{i}\frac{\partial p_{i}}{\partial b_{i}}$ and

$\frac{\partial^{2}F_{m}}{\partial b_{i}\partial
b_{j}}=\begin{cases}n_{i}\frac{\partial p_{i}}{\partial b_{j}}-p_{i}&\mbox{if
$i-j$ an edge,}\\\ n_{i}\frac{\partial p_{i}}{\partial
b_{j}}&\mbox{otherwise.}\end{cases}.$ (5.10)

We see immediately that if $i$ is not connected to $j$ by an edge then
$\frac{\partial^{2}F_{m}}{\partial b_{i}\partial b_{j}}$ is divisible by
$n_{i}$, so the $c_{i}c_{j}$ term can be omitted from the sum when
$\langle\overline{\lambda},\theta\rangle=0$. Furthermore, if $i-j$ is an edge
then we have $\frac{\partial^{2}F_{m}}{\partial b_{i}\partial b_{j}}\equiv-
p_{i}\equiv-p_{j}\bmod(n_{i},n_{j})$, giving (a) and (b). For (c), we observe
that if $i,j\in J$ and $i-j$ is an edge then (since
$\langle\overline{\lambda},\theta\rangle=0$) we have $n_{i}=n_{j}=0$ and
$\frac{1}{2}\frac{\partial^{2}F_{n}}{\partial
b_{i}^{2}}=p_{i}=p_{j}=\frac{1}{2}\frac{\partial^{2}F_{m}}{\partial
b_{j}^{2}}=-\frac{\partial^{2}F_{m}}{\partial b_{i}\partial b_{j}}$. By
assumption, the indices $i,j,k,$ etc. in $J$ can be connected by edges $i-j$,
$j-k$, etc. and we therefore obtain
$F_{m}^{(2)}(\overline{\lambda},\theta)=p_{i}(\overline{\lambda})(c_{i}^{2}-c_{i}c_{j}+c_{j}^{2}-c_{j}c_{k}+c_{k}^{2}-\ldots)=p_{i}(\overline{\lambda})\langle\theta,\theta\rangle$.
The final assertion follows from the fact that
$\langle\alpha^{\vee},\alpha^{\vee}\rangle=2$ for any coroot $\alpha^{\vee}$.
∎

The above observations apply to $F_{m}$ for any $m>1$. Note that if $m=2$ then
equality of the second order mixed derivatives and the connectedness of the
Dynkin diagram imply that $p_{i}=p_{j}$ for all $i,j$. (It is straightforward
to calculate that $p_{i}=60$.) The observations in the proof of Lemma 5.3 are
also quite useful when $m=3$.

######  Lemma 5.4.

Assume $m=3$. If $i$ is not connected to $j$ via an edge then $\frac{\partial
p_{i}}{\partial b_{j}}=\frac{\partial p_{j}}{\partial b_{i}}=0$.

###### Proof.

By the remark in the proof of the above Lemma, we have $n_{i}\frac{\partial
p_{i}}{\partial b_{j}}=n_{j}\frac{\partial p_{j}}{\partial b_{i}}$. Since
$p_{i}$ is linear, it follows that $\frac{\partial p_{i}}{\partial b_{j}}$ and
$\frac{\partial p_{j}}{\partial b_{i}}$ are scalars, hence they must both be
zero. ∎

The previous two lemmas suffice to determine (up to a common scalar multiple)
the polynomials $p_{i}$ modulo $n_{i}$. Although computer verification of the
following Lemma is very straightforward, we have retained the conceptual proof
because the same argument can be applied to an arbitrary simply-laced Lie
algebra.

######  Lemma 5.5.

In the case $m=3$, we have $p_{i}\equiv 12\,b_{6}$ for
$i=6,3^{\prime},4^{\prime},5$ and:

$\displaystyle p_{2^{\prime}}\equiv 12(b_{6}+n_{4^{\prime}})\bmod
n_{2^{\prime}},\;\;p_{4}\equiv 12(b_{6}+n_{5})\bmod n_{4}\ ,$ (5.11)
$\displaystyle p_{3}\equiv 12(b_{6}+n_{5}+2n_{4})\bmod n_{3},\;\;p_{2}\equiv
12(b_{6}+n_{5}+2n_{4}+3n_{3})\bmod n_{2}\ .$ (5.12)

###### Proof.

By Lemma 5.4, $p_{i}$ depends only on $b_{i}$ and the $b_{j}$ such that $i-j$
is an edge. It follows in particular that $p_{3^{\prime}}$ is a linear
combination of $b_{3^{\prime}}$ and $b_{6}$, and $p_{6}$ is a linear
combination of $b_{6},b_{3^{\prime}},b_{4^{\prime}},b_{5}$. Since $p_{6}\equiv
p_{3^{\prime}}\bmod(n_{3^{\prime}},n_{6})$, then $p_{6}$ must be of the form
$\xi b_{5}+\xi b_{4^{\prime}}+\eta b_{6}+\nu b_{3^{\prime}}$. By exactly the
same argument using $p_{4^{\prime}}$, we must have $\nu=\xi$. Hence
$p_{6}=\xi(b_{5}+b_{4^{\prime}}+b_{3^{\prime}})+\eta b_{6}=-\xi
n_{6}+(\eta+2\xi)b_{6}$. Writing $p_{3^{\prime}}=ya_{3^{\prime}}+zb_{6}$
similarly, we obtain (by an equality above)
$zn_{3^{\prime}}-yb_{3^{\prime}}-zb_{6}=\xi
n_{6}-\xi(b_{5}+b_{4^{\prime}}+b_{3^{\prime}})-\eta b_{6}$, so $\xi=0$. Thus
$p_{6}=\eta b_{6}$. Subject to calculation of $\eta$, our previous
observations now allow us to determine all the coefficients $p_{i}$ modulo
$n_{i}$. In particular, $p_{3^{\prime}}$ is congruent to $\eta b_{6}$ modulo
$(n_{3^{\prime}},n_{6})$, but is also a linear combination of
$b_{3^{\prime}},b_{6}$, hence we must have $p_{3^{\prime}}\equiv\eta
b_{6}\bmod n_{3^{\prime}}$. Similarly, $p_{4^{\prime}}\equiv\eta
b_{6}\bmod(n_{4^{\prime}})$ and $p_{5}\equiv\eta b_{5}\bmod(n_{5})$. Now,
$p_{2^{\prime}}$ is congruent to $\eta b_{6}$ modulo
$(n_{2^{\prime}},n_{4^{\prime}})$; the only linear combination of
$b_{2^{\prime}},b_{4^{\prime}}$ which has this property is
$\eta(b_{6}+n_{4^{\prime}})=\eta(2b_{4^{\prime}}-b_{2^{\prime}})$. Similarly,
we must have $p_{4}\equiv\eta(b_{6}+n_{5})\equiv\eta(2b_{5}-b_{4})\bmod
n_{4}$. With the same argument, we obtain
$p_{3}\equiv\eta(b_{6}+n_{5}+2n_{4})\equiv\eta(3b_{4}-2b_{3})\bmod n_{3}$ and
$p_{2}\equiv\eta(b_{6}+n_{5}+2n_{4}+3n_{3})\equiv\eta(4b_{3}-3b_{2})\bmod
n_{2}$.

It only remains to determine $\eta$. It follows from our earlier observations
that $\frac{\partial^{3}F_{3}}{\partial b_{6}^{3}}=4\frac{\partial
p_{6}}{\partial b_{6}}=4\eta$. Hence the coefficient of $c_{6}^{3}$ in
$F_{3}(\theta)$ is $\frac{2}{3}\eta$. But $F_{3}(\alpha_{6}^{\vee})=8$, hence
$\eta=12$. ∎

######  Corollary 5.6.

If $\overline{\lambda}(\beta)=0$, then
$F_{3}^{(2)}(\overline{\lambda},\beta^{\vee})\geq 12b_{6}$, with equality if
the support of $\beta$ contains at least one of $6,5,4^{\prime},3^{\prime}$.

Note that the condition in the Corollary fails precisely when
$\beta=\alpha_{2^{\prime}}$ or $\beta$ is in the root subsystem generated by
$\alpha_{2},\alpha_{3},\alpha_{4}$.

###### Proof.

This follows immediately from Lemma 5.3 and Cor. 5.6. ∎

Cor. 5.6 completes our analysis of $F_{3}$. For $F_{5}$, we will require a
certain lower bound on $F_{5}^{(2)}(\overline{\lambda},\beta^{\vee})$. In
theory, a similar analysis to Lemma 5.5 could be applied to determine
$F_{5}^{(2)}(\overline{\lambda},\theta)$ as a quadratic polynomial in the
$c_{i}$. By Lemma 5.3, we only need to find $p_{i}(\overline{\lambda})$ modulo
$n_{i}$ for each $i$. The calculations involved are rather intricate, hence we
turn to computer verification for the following result. Recall that

$k=\sum_{i=1}^{6}in_{i}+2n_{2^{\prime}}+3n_{3^{\prime}}+4n_{4^{\prime}}=n_{1}+\langle\overline{\lambda},\varpi_{2}^{\vee}\rangle=n_{1}+b_{2}\
.$ (5.13)

We define various integers $\tau_{i,j}$, as follows. For fixed $i$, the number
$r(i)$ of such integers is $i-2$ if $i$ is an unprimed integer, $5$ if
$i=3^{\prime},4^{\prime}$ and $6$ if $i=2^{\prime}$. For $2\leq i\leq 6$ we
have

$\tau_{i,1}=\sum_{2\leq j<i}n_{j}\ ,\quad\tau_{i,2}=\sum_{3\leq j<i}n_{j}\
,\quad\ldots\quad,\tau_{i,i-2}=n_{i-1}\ .$ (5.14)

Similarly,

$\tau_{3^{\prime},1}=\sum_{j=2}^{6}n_{j}\
,\quad\tau_{3^{\prime},2}=\sum_{j=3}^{6}n_{j}\
,\quad\ldots,\tau_{3^{\prime},5}=n_{6}$ (5.15)

and $\tau_{4^{\prime},j}=\tau_{3^{\prime},j}$ for all $j$. Finally,
$\tau_{2^{\prime},j}=\tau_{3^{\prime},j}+n_{3^{\prime}}+n_{4^{\prime}}$ for
$1\leq j\leq 5$ and $\tau_{2^{\prime},6}=n_{4^{\prime}}$. Then we have:

######  Lemma 5.7.

Assume $m=5$. Then we have:

$\frac{1}{20}p_{i}(\overline{\lambda})\equiv-2(b_{2}-\tau_{i,1})^{3}+6(b_{2}-\tau_{i,1})\langle\overline{\lambda},\overline{\lambda}\rangle+2\sum_{j=1}^{r(i)}\tau_{i,j}^{3}\mod(n_{i})\
.$ (5.16)

###### Proof.

This can be verified by a computational proof, differentiating $F_{5}$ and
dividing by $n_{i}$. ∎

######  Corollary 5.8.

For any root $\beta$ with $\overline{\lambda}(\beta)=0$, we have
$F_{5}^{(2)}(\overline{\lambda},\beta^{\vee})+20(2b_{2}^{3}-6b_{2}\langle\overline{\lambda},\overline{\lambda}\rangle)\geq
0$.

###### Proof.

By Lemma 5.3(c), we only need to show that
$\frac{1}{20}p_{i}(\overline{\lambda})+2b_{2}^{3}-6b_{2}\langle\overline{\lambda},\overline{\lambda}\rangle\geq
0$ for each $i$. By Lemma 5.7, it is enough to prove that
$G_{i}+2b_{2}^{3}-6b_{2}\langle\overline{\lambda},\overline{\lambda}\rangle\geq
0$, where

$G_{i}=-2(b_{2}-\tau_{i,1})^{3}+6(b_{2}-\tau_{i,1})\langle\overline{\lambda},\overline{\lambda}\rangle+2\sum_{j=1}^{r}\tau_{i,j}^{3}\
.$ (5.17)

This is a straightforward computer verification. In fact the following
stronger property holds: $G_{i}$ is non-decreasing as $i$ moves along the
Dynkin diagram from $i=2$ to $i=2^{\prime}$. ∎

Note that
$p_{2}(\overline{\lambda})+2b_{2}^{3}-6b_{2}\langle\overline{\lambda},\overline{\lambda}\rangle=0$,
so the inequality in the Corollary is sharp. We need one final observation
about $F_{5}$.

######  Lemma 5.9.

For any root $\beta$ with $\overline{\lambda}(\beta)=0$, we have
$F_{5}^{(3)}(\overline{\lambda},\beta^{\vee})=0$ and
$F_{5}^{(4)}(\overline{\lambda},\beta^{\vee})=\frac{5}{3}F_{3}^{(2)}(\overline{\lambda},\beta^{\vee})$.

###### Proof.

We observe that the positive roots $\alpha$ come in three classes:

  * •

we have $\beta^{\vee}(\alpha)=2$ if and only if $\alpha=\beta$, in which case
$\overline{\lambda}(\beta)=0$;

  * •

the remaining roots with $\beta^{\vee}(\alpha)\neq 0$ come in pairs
$\\{\alpha,\alpha+\beta\\}$, where $\beta^{\vee}(\alpha)=-1$ and
$\beta^{\vee}(\alpha+\beta)=+1$ (note that
$\overline{\lambda}(\alpha+\beta)=\overline{\lambda}(\alpha)$);

  * •

otherwise, $\beta^{\vee}(\alpha)=0$.

Let $S_{\beta}$ be the set of positive roots satisfying
$\beta^{\vee}(\alpha)=-1$. Then

$\sum_{\alpha\in\Phi^{+}}\overline{\lambda}(\alpha)^{m-i}\beta^{\vee}(\alpha)^{i}=2\,\sum_{\alpha\in
S_{\beta}}\overline{\lambda}(\alpha)^{m-i}(1+(-1)^{i})\ .$ (5.18)

In particular, we have $F_{5}^{(3)}(\overline{\lambda},\beta^{\vee})=0$ and

$F_{3}^{(2)}(\overline{\lambda},\beta^{\vee})=6\sum_{\alpha\in
S_{\beta}}\overline{\lambda}(\alpha),\quad
F_{5}^{(4)}(\overline{\lambda},\beta^{\vee})=10\sum_{\alpha\in
S_{\beta}}\overline{\lambda}(\alpha)\ ,$ (5.19)

which completes our proof. ∎

######  Corollary 5.10.

Suppose further that the support of $\beta^{\vee}$ contains at least one of
$6,5,4^{\prime},3^{\prime}$. Then we have
$F_{3}(\overline{\lambda}+\beta^{\vee})=F_{3}(\overline{\lambda})+12\,b_{6}+8$
and

$F_{5}(\overline{\lambda}+\beta^{\vee})\geq
F_{5}(\overline{\lambda})+20(6b_{2}\langle\overline{\lambda},\overline{\lambda}\rangle-2b_{2}^{3})+20b_{6}+32\
.$ (5.20)

###### Proof.

It is straightforward to check
$F_{3}^{(3)}(\overline{\lambda},\beta^{\vee})=F_{3}(\beta^{\vee})=8$, since
(using the argument in the proof of Lemma 5.9) the only class of positive
roots which has a non-zero contribution to the sum is $\alpha=\beta$.
Similarly, $F_{5}(\beta^{\vee})=32$. ∎

We use the notation $b_{i}$ for brevity in the above formulas. In the sketch
of the proof of the $a$-theorem which follows, it will be important to bear in
mind the equalities:

$\displaystyle
b_{2}=2n_{2^{\prime}}+3n_{3^{\prime}}+4n_{4^{\prime}}+\sum_{i=2}^{6}in_{i}=k-n_{1}\
,$ (5.21) $\displaystyle
b_{6}=\langle\overline{\lambda},\alpha_{6}^{\vee}\rangle=10n_{2^{\prime}}+15n_{3^{\prime}}+20n_{4^{\prime}}+\sum_{2}^{6}6in_{i}=5k-\sum_{i=1}^{5}(6-i)n_{i}\
.$ (5.22)

####  5.2.2 Finalizing the proof

Let $(\lambda,\mu=\lambda+\widehat{\alpha}_{I}^{\vee})$ where $I$ is an
irreducible component of the subdiagram of zeros of $\lambda$. Assume the
support of $I$ contains at least one of $6,5,4^{\prime},3^{\prime}$, so that
Cor. 5.10 applies. We recall the requirement that $n+k-N_{\rho}\leq 0$; in
particular, we have $n\leq 0$. We here outline the uniform argument which
establishes $\Delta a=a(\lambda)-a(\mu)>0$ in these cases; the remaining (and
indeed all) cases have been checked by computer.

Inspecting the $a$ anomaly (5.2) and using Cor. 5.10, we can rewrite it as

$\displaystyle\Delta a=$ $\displaystyle\
\frac{3}{2}(h_{I}-1)+\frac{32}{7}(6kn^{2}-6k^{2}n-15n+2k^{3}+13k)-\frac{192}{7}\left(\langle\overline{\lambda},\overline{\lambda}\rangle+1\right)(k-n)\
+$
$\displaystyle-\frac{80}{7}b_{6}-\frac{40}{21}(h_{I}+2)+\frac{8}{35}F_{5}^{(2)}(\overline{\lambda},\widehat{\alpha}_{I}^{\vee})+\frac{32}{7}b_{6}+\frac{16}{35}(h_{I}+14)\
.$ (5.23)

Subtracting the strictly positive term $\frac{11}{210}h_{I}+\frac{229}{210}$,
it will suffice to show that the sum of the remaining terms is non-negative.
Multiplying by $\frac{7}{32}$, we obtain a quadratic polynomial in $n$ of the
following form:

  * •

the coefficient of $n^{2}$ is $6k$, hence is positive;

  * •

the coefficient of $n$ is
$-6(k^{2}-\langle\overline{\lambda},\overline{\lambda}\rangle)+9$.

Note that the linear coefficient here is always negative (but $n$ is non-
positive). In fact, we observe that
$\langle\overline{\lambda},\overline{\lambda}\rangle\leq k^{2}$. Indeed, we
have $k\geq\sum_{i\neq 1}in_{i}$ and

$\langle\overline{\lambda},\overline{\lambda}\rangle=\sum_{i}\langle\varpi_{i}^{\vee},\varpi_{i}^{\vee}\rangle
n_{i}^{2}+2\sum_{i\neq j}\langle\varpi_{i}^{\vee},\varpi_{j}^{\vee}\rangle
n_{i}n_{j}\ ,$ (5.24)

and one sees easily that
$\langle\varpi_{i}^{\vee},\varpi_{j}^{\vee}\rangle\leq ij$. Adding the non-
negative term $6kn^{2}-9(n+k-N_{\rho})$, it will suffice to show that the
following expression is non-negative:

$2k^{3}-6k\langle\overline{\lambda},\overline{\lambda}\rangle-\frac{3}{2}b_{6}+16k+\frac{1}{20}F_{5}^{(2)}(\overline{\lambda},\widehat{\alpha}_{I}^{\vee})-9N_{\rho}\
.$ (5.25)

Recalling that $k=b_{2}+n_{1}$, we note that this expression is increasing
with $n_{1}$, hence we may assume $k=b_{2}$. The sum of the cubic terms is
$2b_{2}^{3}-6b_{2}\langle\overline{\lambda},\overline{\lambda}\rangle+\frac{1}{20}F_{5}^{(2)}(\overline{\lambda},\widehat{\alpha}_{I}^{\vee})$,
which is non-negative by Cor. 5.8.

It only remains to check that $\frac{3}{2}b_{6}+9N_{\rho}\leq 16b_{2}$. But it
can be easily verified that:

$32b_{2}-3b_{6}=46n_{2}+60n_{3}+74n_{4}+88n_{5}+102n_{6}+51n_{3^{\prime}}+68n_{4^{\prime}}+34n_{2^{\prime}}\
,$ (5.26)

from which the required bound follows immediately.

This proves the desired inequality in the cases where
$\overline{\mu}=\overline{\lambda}+\widehat{\alpha}_{I}^{\vee}$ and $I$
contains at least one of the nodes $3^{\prime},4^{\prime},6,5$. It is worth
remarking again that the sharp equality in Cor. 5.8 is exactly what was needed
to deal with the cubic terms in $\Delta a$. We used the computational algebra
platform GAP to check in all cases (including these, and the Type _ii_) cases
$\overline{\mu}=\overline{\lambda}+\alpha_{i}^{\vee}$) that the bound $\Delta
a>0$ holds.

##  6 Conclusions

In this paper we have realized the hierarchy of Higgs branch RG flows between
the A-type orbi-instantons as the Hasse diagram of the stratification of the
double affine Grassmannian of $E_{8}$. The strata that we have considered are
given by the orbi-instantons (with no decoupled E-strings) at fixed $k$ but
different holonomy at infinity $\rho_{\infty}$ and (potentially) number $N$ of
full instantons (M5-branes); the transverse slices among them are the flavor
Higgsings reducing the $E_{8}$ flavor symmetry of the UV SCFT on its Higgs
branch. Exploiting the natural partial order defined on this Hasse diagram, we
have proven analytically the $a$-theorem for all such RG flows.

Several extensions of this work can be considered. In this work we only
considered the case where $\Gamma_{\text{ADE}}$ is cyclic, i.e. of type A. For
$\Gamma_{\text{ADE}}$ of type D or E, the problem of classifying homomorphisms
$\Gamma_{\text{ADE}}\to G$ is (in general) open. The homomorphisms from
$\Gamma_{E_{8}}$ to exceptional type $G$ have been classified in a series of
papers by Frey [145, 146, 147]. Some partial results on homomorphisms from
$\Gamma_{D_{5}}$ and $\Gamma_{D_{7}}$ to $E_{8}$ were also obtained in [145].
In subsequent work with Rudelius [127], a conjectural classification of
homomorphisms $\Gamma_{\textrm{D}}$ to $E_{8}$ was given in terms of orbi-
instantons (and their F-theory electric quivers), leading to a partial order
on such homomorphisms via RG flows. However, it is unclear what is the correct
mathematical language to understand the flows.

For binary dihedral groups, i.e. for $\Gamma$ of type D, a “Kac-style”
classification of homomorphisms $\Gamma\rightarrow{\rm Aut}\,G$ is possible,
using the theory of involutions of simple Lie algebras. The idea is as
follows: let $\lambda_{\rm Kac}$ be a Kac diagram with an associated
homomorphism $\Gamma:{\mathbb{Z}}_{k}\rightarrow T\subset G$ into the standard
maximal torus of $E_{8}$. There is an element $w_{0}$ of the Weyl group which
acts as $-1$ on the Cartan subalgebra (i.e. as $t\mapsto t^{-1}$ on $T$).
Furthermore, $w_{0}$ has a representative $n_{0}$ in the normalizer of $T$,
and $n_{0}^{2}=1$. It follows that $\Gamma$ extends to a homomorphism from the
dihedral group

$\langle a,b:a^{k}=b^{2}=1\,,\quad bab^{-1}=a^{-1}\rangle$ (6.1)

to $E_{8}$, sending $a$ to $\Gamma(1)$ and $b$ to $n_{0}$. One can modify this
construction by sending $b$ to $n_{0}g$ for any $g$ which commutes with
$\Gamma(a)$, i.e. which lies in the zero subalgebra of $\lambda_{\rm Kac}$.
The challenge now is to determine which $g$ lead to homomorphisms from the
binary dihedral group $\Gamma_{D_{k-2}}$ to $E_{8}$, and when two choices of
$g$ lead to conjugate homomorphisms. We will return to this in future work.

We remark here an obvious connection with unipotent orbits in $G$. (This
connection was hinted to at various places in the physics literature, perhaps
starting with [75, Sec. 5].) Recall that the conjugacy classes of such
homomorphisms are in one-to-one correspondence with the homomorphisms
$\operatorname{SU}(2)\to G$. Specifically, for each unipotent element $x\in G$
there exists a homomorphism $\phi:\operatorname{SU}(2)\to G$ such that
$\phi\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right)=u$, and any
two such homomorphisms are conjugate by an element of the centralizer of $u$.
If $u\neq 1$ then $\phi$ is either injective or has kernel $\\{-I\\}$. Thus,
any finite subgroup $\Gamma_{\text{ADE}}\subset\operatorname{SU}(2)$ has image
$\phi(\Gamma_{\text{ADE}})\subset G$ which is isomorphic to
$\Gamma_{\text{ADE}}$ or $\Gamma_{\text{ADE}}/\\{\pm I\\}$. For a fixed
subgroup $\Gamma_{\text{ADE}}$, this construction (ranging over various
unipotent orbits) produces various homomorphisms $\Gamma_{\text{ADE}}\to G$.
It is clear from Kac’s classification that not all homomorphisms from a cyclic
group $\Gamma$ to $G$ can be extended to a homomorphism from
$\operatorname{SU}(2)$ to $G$. By considering an involution which acts as $-1$
on a fixed Cartan subalgebra, we observe that this also holds for binary
dihedral groups. On the other hand, in Frey’s classification it appears that
all homomorphisms from the binary icosahedral group $I$ to $E_{8}$ arise as
the restriction of a homomorphism $\operatorname{SU}(2)\to G$. However, it is
quite possible for two distinct homomorphisms
$\phi_{1},\phi_{2}:\operatorname{SU}(2)\to G$ to have the same restriction to
a finite subgroup $\Gamma_{\text{ADE}}\subset\operatorname{SU}(2)$. We mention
for example that in Frey’s classification [145, 147] there is exactly one
homomorphism $I\to G$ whose image has trivial connected centralizer. By a
straightforward calculation involving standard results on centralizers [148],
one observes that this holds for the restriction of the $\operatorname{SU}(2)$
for each of the following seven (distinguished) unipotent orbits: those with
Bala–Carter labels $E_{8}$ (regular), $E_{8}(a_{1})$ (subregular),
$E_{8}(a_{2})$, $E_{8}(a_{3})$, $E_{8}(a_{4})$, $E_{8}(b_{5})$ and
$E_{8}(a_{7})$. (These are the distinguished orbits for which the adjoint
action of the corresponding image of $\operatorname{SU}(2)$ has no summands
$S^{i}V$ for $i=12$, $20$, $24$, $30$, $32$, $36$, $40$ etc., where $V$ is the
natural representation for $\operatorname{SU}(2)$; such a summand would give
rise to a subspace fixed by $I$.) There is no apparent pattern to this
repetition. The last observation complicates the relationship between
unipotent orbits and homomorphisms $I\rightarrow G$: while one might expect
such homomorphisms to be encapsulated in the set of unipotent elements (the
unipotent cone), this is _not_ consistent with the repetition observed above.

Another direction worth exploring is the study of the Coulomb branch of the
magnetic quiver of 6d orbi-instantons as a quiver variety. The problem, as we
mentioned in the introduction, is that the former is neither of finite nor
affine type, so it is beyond reach using results already available in the
mathematics literature. However, given its shape (see (A.7)), a possible trick
would be to “embed” in it a long quiver of type D to study (at least
partially) the root space, which is needed to apply the results of [77].

One final comment regards the study of the “full” Higgs branch of the orbi-
instantons, by which we mean also including flavor Higgsings of the _right_
$[\operatorname{SU}(k)]$ factor in (2.5).393939Some examples have been studied
in [76, 80, 108]. As is by now well known, these are classified by nilpotent
orbits of its ${\mathfrak{su}({k})}$ algebra, so the study of the full Higgs
branch would involve mixing left homomorphisms
$\text{Hom}(\mathbb{Z}_{k},E_{8})$ and right orbits
$\mathcal{O}_{\mathfrak{su}({k})}$. However we also remark that the
$a$-theorem for such a combined flow would still boil down to $a$-theorems for
its constituents, which are either already proven [103, 70] (right orbit) or
have been proven here (left homomorphism).

## Acknowledgments

We would like to thank A. Bourget, S. Giacomelli, J. Grimminger, D. Juteau, Y.
Tachikawa, A. Tomasiello and especially H. Nakajima for helpful discussions
and correspondence. MF would like to thank the University of Milano–Bicocca,
the Pollica Physics Center, the University of Lancaster, Kavli IPMU at the
University of Tokyo for the warm hospitality at various stages of this work.
MF and SG acknowledge support from the Simons Center for Geometry and Physics,
Stony Brook University (2022 workshops “5d $\mathcal{N}=1$ SCFTs and Gauge
Theories on Brane Webs” and “Supersymmetric Black Holes, Holography and
Microstate Counting”, MF; 2023 Simons Physics Summer Workshop, MF and SG) at
which some of the research for this paper was performed. MF would like to
thank the organizers of the “Symplectic Singularities and Supersymmetric QFT”
conference held at UPJV in Amiens for providing a stimulating environment. SG
would like to thank the University of Milano–Bicocca for hospitality during
early stages of this work. The work of MF is supported in part by the Knut and
Alice Wallenberg Foundation under grant KAW 2021.0170, the Swedish Research
Council grant VR 2018-04438, the Olle Engkvists Stiftelse grant No. 2180108,
the European Union’s Horizon 2020 research and innovation programme under the
European Research Council (ERC) grant agreement No. 851931 - MEMO. The work of
SG was conducted with funding awarded by the Swedish Research Council grant VR
2022-06157.

##  Appendix A Magnetic quivers

In this appendix we summarize the results of [132], which constructed 3d
magnetic quivers for 6d orbi-instantons. The former are 3d $\mathcal{N}=4$
quiver gauge theories which, in the case of orbi-instantons, flow in the IR to
SCFTs. (They are “good” quivers in the sense of [125].)

On the tensor branch of the 6d orbi-instanton (i.e. at finite coupling for all
gauge algebras in (2.6) or (2.8)), the associated magnetic quiver is star-
shaped with many arms:

${\displaystyle
1-2-\cdots-(k-1)-\overset{\overset{\displaystyle\overbrace{\displaystyle
1\cdots 1}^{\widetilde{M}}}{\rotatebox[origin={c}]{190.0}{$\setminus$}\
\rotatebox[origin={c}]{-190.0}{$/$}}}{k}-r_{1}-r_{2}-r_{3}-r_{4}-r_{5}-\overset{\overset{\displaystyle
r_{3^{\prime}}}{|}}{r_{6}}-r_{4^{\prime}}-r_{2^{\prime}}}$ (A.1)

with $\widetilde{M}\coloneqq N+\sum_{i=1}^{6}n_{i}$, or, equivalently,

${\scriptstyle
1-2-\cdots-(k-1)-\overset{\overset{\scriptstyle\overbrace{\scriptstyle 1\
\cdots\ 1}^{M}}{\rotatebox[origin={c}]{190.0}{$\setminus$}\
\rotatebox[origin={c}]{-190.0}{$/$}}}{k}-(r_{1}-p)-(r_{2}-2p)-(r_{3}-3p)-(r_{4}-4p)-(r_{5}-5p)-\overset{\overset{\scriptstyle(r_{3^{\prime}}-3p)}{|}}{(r_{6}-6p)}-(r_{4^{\prime}}-4p)-(r_{2^{\prime}}-2p)}$
(A.2)

with $M\coloneqq\widetilde{M}+p=N+N_{\rho}$, remembering the definition in
(2.7). The ranks $r_{i},r_{i^{\prime}}$ are given by:

$\displaystyle
r_{j}=(1-\delta_{j6})\sum_{i=1}^{6-j}in_{i+j}+2n_{2^{\prime}}+3n_{3^{\prime}}+4n_{4^{\prime}}=k-\sum_{i=1}^{6}in_{i}+(1-\delta_{j6})\sum_{i=1}^{6-j}in_{i+j}$
(A.3)

for $j=1,\ldots,6$ and

$\displaystyle r_{2^{\prime}}=n_{3^{\prime}}+n_{4^{\prime}}\ ,\quad
r_{3^{\prime}}=n_{2^{\prime}}+n_{3^{\prime}}+2n_{4^{\prime}}\ ,\quad
r_{4^{\prime}}=n_{2^{\prime}}+2n_{3^{\prime}}+2n_{4^{\prime}}\ .$ (A.4)

Notice that $r_{2^{\prime}}$ also equals $2(p+r)$ in the notation of [132];
that is, $r$ is defined as

$r:=\frac{r_{2^{\prime}}}{2}-p=\frac{r_{2^{\prime}}}{2}-\min\left(\lfloor\left(n_{3^{\prime}}+n_{4^{\prime}}\right)/2\rfloor,\lfloor\left(n_{2^{\prime}}+n_{3^{\prime}}+2n_{4^{\prime}}\right)/3\rfloor\right)\
.$ (A.5)

The number $r_{2^{\prime}}=2(p+r)$ is the total number of half-NS5’s stuck on
the O8- in a Type IIA engineering of the orbi-instanton. This number is
important, as starting at level $k=7$ the following happens: there are some
Kac diagrams for which the resulting ranks $r_{i},r_{i^{\prime}}$ are _all_
greater or equal to the Coxeter label of affine $E_{8}$ in position
$i,i^{\prime}$. That is, $p\neq 0$. Let us call these the “saturating
diagrams”.404040Since for this to be possible we must have
$r_{2^{\prime}}=n_{3^{\prime}}+n_{4^{\prime}}\geq 2\ ,\
r_{3^{\prime}}=n_{2^{\prime}}+n_{3^{\prime}}+2n_{4^{\prime}}\geq 3\ ,\
r_{4^{\prime}}=n_{2^{\prime}}+2n_{3^{\prime}}+2n_{4^{\prime}}\geq 4$ (A.6)
(which automatically imply that
$r_{6}=2n_{2^{\prime}}+3n_{3^{\prime}}+4n_{4^{\prime}}\geq 6$), we see that
the first case is the $[3^{\prime},4^{\prime}]$ diagram at $k=7$ (which
saturates the above bounds). A simple case-by-case analysis shows that the
saturating diagrams must have
$(n_{3^{\prime}},n_{4^{\prime}})\geq(1,1),(0,2),(2,0)$ where in the last case
we should also impose that $n_{2^{\prime}}\geq 1$.  For these diagrams the
bouquet in the finite-coupling magnetic quiver (A.1) has $p$ extra 1’s. Or,
the total number of full instantons (NS5’s in the orientifold of Type IIA) is
increased by $p$. Notice that this is already taken care of by our definition
of $M"=N+N_{\rho}$, since $N_{\rho}=\sum_{i=1}^{6}n_{i}+p$: for the saturating
diagrams, this $p\neq 0$ precisely accounts for the extra full instantons that
we get in the Type IIA picture. Then, we simply remove the $p$ “extra” affine
$E_{8}$ Dynkins from the right tail, obtaining (A.2).

The SCFT at the origin of the tensor branch has a magnetic quiver obtained by
performing $\widetilde{M}$ small $E_{8}$ instanton transitions. Performing the
transitions in (A.1), the proposal of [122] is that we obtain the following
quiver:

${\scriptstyle
1-2-\cdots-k-(r_{1}+\widetilde{M})-(r_{2}+2\widetilde{M})-(r_{3}+3\widetilde{M})-(r_{4}+4\widetilde{M})-(r_{5}+5\widetilde{M})-\overset{\overset{\scriptstyle
r_{3^{\prime}}+3\widetilde{M}}{|}}{(r_{6}+6\widetilde{M})}-(r_{4^{\prime}}+4\widetilde{M})-(r_{2^{\prime}}+2\widetilde{M})}\
.$ (A.7)

The Coulomb branch dimension of the above quiver was computed in [76], and
reads:

$\dim_{\mathbb{H}}\widehat{\mathcal{M}}_{\text{C}}=30(N+k)+\frac{k}{2}(k+1)-\overline{\lambda}(\rho)-1\
,$ (A.8)

for all $k>1$, where $\overline{\lambda}(\rho)$ is the height function
associated with the diagram $\lambda_{\text{Kac}}=(k,\overline{\lambda})$
defined in section 5.1:

$\overline{\lambda}(\rho)=29n_{2}+57n_{3}+84n_{4}+110n_{5}+135n_{6}+46n_{2^{\prime}}+68n_{3^{\prime}}+91n_{4^{\prime}}\
,$ (A.9)

which obviously vanishes only for the diagram $[1^{k}]$.

###  A.1 $\operatorname{SU}(k)$ hyperkähler quotient of orbi-instanton
magnetic quiver

We can now prove414141This proof is due to H. Nakajima. that the hyperkähler
quotient
$(\widehat{\mathcal{M}}_{\text{C}}\times\mathcal{O}_{\bm{\xi}})/\\!/\\!/\operatorname{SU}(k)$,
with $\widehat{\mathcal{M}}_{\text{C}}$ the Coulomb branch of (A.7), equals
the moduli space of $E_{8}$-instantons on the deformation/resolution of
$\mathbb{C}^{2}/\mathbb{Z}_{k}$, i.e. the Coulomb branch
$\mathcal{M}_{\text{C}}^{\text{inst}}$ of

${\scriptstyle\boxed{\scriptstyle
k}-(r_{1}+\widetilde{M})-(r_{2}+2\widetilde{M})-(r_{3}+3\widetilde{M})-(r_{4}+4\widetilde{M})-(r_{5}+5\widetilde{M})-\overset{\overset{\scriptstyle
r_{3^{\prime}}+3\widetilde{M}}{|}}{(r_{6}+6\widetilde{M})}-(r_{4^{\prime}}+4\widetilde{M})-(r_{2^{\prime}}+2\widetilde{M})}\
.$ (A.10)

(As is well known, that the Coulomb branch of the above quiver gives the
instanton moduli space was originally proposed by [115] for $k=1$ and [149]
for any $k$.) Because of the hyperkähler quotient, we have:

$\dim_{\mathbb{H}}\mathcal{M}_{\text{C}}^{\text{inst}}=30(N+k)-\overline{\lambda}(\rho)\
.$ (A.11)

(There is no overall $U(1)$ gauge group that decouples because of the presence
of the flavor node $\boxed{k}$.)

We simply need to apply [133, Rem. 5.21], which in our present context
directly gives the wanted statement:

$(\mathcal{M}_{\text{C}}(T(\operatorname{SU}(k))\times\widehat{\mathcal{M}}_{\text{C}})/\\!/\\!/\operatorname{SU}(k)\cong\mathcal{M}_{\text{C}}^{\text{inst}}$
(A.12)

where all Coulomb branches are understood as Nakajima quiver varieties [87,
77] associated with the two quivers above and
$T(\operatorname{SU}(k))=1-\ldots-(k-1)-\boxed{k}$, and $\cong$ means that the
coordinate rings of the two algebraic varieties are isomorphic.

##  Appendix B RG flows for the decoupled system

We will now show that the $a$ central charge decreases along an RG flow from
an orbi-instanton defined by $(N+M,k,\rho_{\infty})$ to a decoupled system
given by $(N,k,\rho_{\infty})$ and $n_{i}$ decoupled rank $m_{i}$ E-strings
such that $\sum_{i}n_{i}m_{i}=M$, i.e. all integer partitions of $M$. We will
do this in three steps.

  1. _i_)

We will first show that $\Delta a>0$ for the top partition
$i=1,(m_{1},n_{1})=(M,1)$ i.e. a single rank-$M$ E-string.

  2. _ii_)

We will then show that $a$ decreases for all flows from a single rank-$M$
E-string to two E-strings of rank $p$ and $q$ such that $p+q=M$ i.e.
$i=2,(m_{1},n_{1})=(p,1)$, and $(m_{2},n_{2})=(q,1)$.

  3. _iii_)

Applying step _(i)_ and then step _(ii)_ recursively implies $\Delta a>0$ for
all partitions $(m_{i},n_{i})$ of $M$ and completes the proof.

######  Lemma B.1.

The $a$ anomaly decreases along the RG flow from the orbi-instanton
$(N+M,k,\rho_{\infty})$ to $(N,k,\rho_{\infty})$ and the top partition of $M$:
$i=1,(m_{1},n_{1})=(M,1)$, i.e. a single rank-$M$ E-string:

$\Delta a^{(i)}\coloneqq
a\left(N+M,k,\rho_{\infty}\right)-a\left(N,k,\rho_{\infty}\right)-a\left(M,1,\left[1^{1}\right]\right)>0\
.$ (B.1)

###### Proof.

$\displaystyle\Delta a^{(i)}$ $\displaystyle\coloneqq
a\left(N+M,k,\rho_{\infty}\right)-a\left(N,k,\rho_{\infty}\right)-a\left(M,1,\left[1^{1}\right]\right)$
(B.2)
$\displaystyle=\frac{48M}{7}\left[\frac{k^{2}}{6}\left(2M^{2}-3\right)+\left(\frac{5}{2}+k(M+2N)+\underbrace{k^{2}-\langle\overline{\lambda},\overline{\lambda}\rangle}_{\geq
0}\right)^{2}-\left(\frac{7}{4}+3M+\frac{4M^{2}}{3}\right)\right]$
$\displaystyle\geq\frac{48M}{7}\left[\frac{k^{2}}{6}\left(2M^{2}-3\right)+\left(\frac{5}{2}+kM\right)^{2}-\left(\frac{7}{4}+3M+\frac{4M^{2}}{3}\right)\right]$
$\displaystyle\geq 0\quad\forall\,k\geq 1\textrm{ and }M\geq 1\ .$

∎

######  Lemma B.2.

The $a$ anomaly decreases when a single rank-$M$ E-string breaks into two
E-strings of rank $p$ and $q$ respectively such that $p+q=M$.

$\Delta a^{(ii)}\coloneqq
a\left(M,1,\left[1^{1}\right]\right)-a\left(p,1,\left[1^{1}\right]\right)-a\left(q,1,\left[1^{1}\right]\right)>0\
.$ (B.3)

###### Proof.

$\Delta a^{(ii)}=\frac{96}{7}pq\left(3+2p+2q\right)>0\ .$ (B.4)

∎

In particular, the above lemma proves the case of degeneration _iv)_ in
section 3.2.5.

Now, every nontrivial integer partition of $M$ can be written as a sum of at
least two integers. Therefore, applying lemma B.2 repeatedly it is easy to see
that

$a(M,1,[1^{1}])>\sum_{i}n_{i}a(m_{i},1,[1^{1}])\ .$ (B.5)

Finally, consider the quantity

$\Delta a^{(iii)}\coloneqq
a\left(N+M,k,\rho_{\infty}\right)-a\left(N,k,\rho_{\infty}\right)-\sum_{i}n_{i}a(m_{i},1,[1^{1}])\
.$ (B.6)

Subtracting $\Delta a^{(i)}$ defined in B.1, we have

$\Delta a^{(iii)}-\Delta
a^{(i)}=a\left(M,1,\left[1^{1}\right]\right)-\sum_{i}n_{i}a(m_{i},1,[1^{1}])>0\
,$ (B.7)

where the second inequality is due to (B.5). This implies

$\Delta a^{(iii)}-\Delta a^{(i)}>0\Rightarrow\Delta a^{(iii)}>\Delta
a^{(i)}>0\Rightarrow\Delta a^{(iii)}>0\ ,$ (B.8)

where the last inequality follows from lemma B.1. This completes the proof.

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