# Auxiliary Variables for Multi-Dirichlet Priors

Christoph Carl Kling
GESIS, Cologne
<EMAIL_ADDRESS>
(August 2017)

###### Abstract

Bayesian models that mix multiple Dirichlet prior parameters, called Multi-
Dirichlet priors (MD) in this paper, are gaining popularity [3, 2]. Inferring
mixing weights and parameters of mixed prior distributions seems tricky, as
sums over Dirichlet parameters complicate the joint distribution of model
parameters.

This paper shows a novel auxiliary variable scheme which helps to simplify the
inference for models involving hierarchical MDs and MDPs. Using this scheme,
it is easy to derive fully collapsed inference schemes which allow for an
efficient inference.

##  1 Introduction

In order to develop a collapsed variational inference for hierarchical
Dirichlet processes, Teh et al. [5] introduced an auxiliary variable scheme
based on truncated Dirichlet processes. The same auxiliary variables can be
used for efficiently inferring hierarchical models of multinomial
distributions with Dirichlet-distributed parameters.

Dirichlet priors can mix multiple parameters to realise predictions of
multinomial parameters based on multiple influence factors [3, 2]. In this
paper, we call these priors Multi-Dirichlet priors (MD). The sampled
parameters can serve as parts of Dirichlet priors in a lower level of
hierarchical Dirichlet distributions or Dirichlet processes, paving the way
for arbitrarily complex hierarchical MD models.

So far, no collapsed variational inference scheme has been proposed for
hierarchical MD distributions. In this paper, an auxiliary variable scheme is
proposed which allows to directly employ the collapsed variational inference
scheme for hierarchical Dirichlet processes given in [5].

##  2 Multinomial distributions and Dirichlet priors

If we sample $n$ single values from a multinomial distribution with parameters
$\theta_{1},\dots,\theta_{K}$, the likelihood of observations is is given by

$\displaystyle\operatorname{Mult}(n_{1},\dots,n_{n}\mid\bm{\theta})=\prod_{k=1}^{K}\theta_{k}^{n_{k}}$
(1)

where $n_{k}$ are counts which give the frequency of observing the $k$th
category in the observations $\bm{x}$. The Dirichlet distribution is the
conjugate prior distribution of the multinomial:

$\displaystyle\operatorname{Dir}\left(\bm{\theta}\mid\bm{\alpha}\right)=\frac{\Gamma\left(\sum_{k=1}^{K}\alpha_{k}\right)}{\prod_{k=1}^{K}\Gamma\left(\alpha_{k}\right)}\prod_{k=1}^{K}\theta_{k}^{\alpha_{k}-1}$
(2)

The storyline behind a multinomially distributed variable $x$ with a Dirichlet
prior is

$\displaystyle\bm{\theta}\mid\bm{\alpha}\sim$
$\displaystyle\operatorname{Dir}\left(\alpha_{1},\dots,\alpha_{K}\right)$
$\displaystyle x_{1},\dots,x_{n}\mid\bm{\theta}\sim$
$\displaystyle\operatorname{Mult}\left(\theta_{1},\dots,\theta_{K}\right).$
(3)

The graphical model is shown in Fig. 1(a).

The joint distribution over observations and multinomial parameters of the
model is:

$\displaystyle\operatorname{Mult}(n_{1},\dots,n_{K}\mid\bm{\theta})\cdot\operatorname{Dir}(\bm{\theta}\mid\bm{\alpha})=$
$\displaystyle\frac{\Gamma(\sum_{k=1}^{K}\alpha_{k})}{\prod_{k=1}^{K}\Gamma(\alpha_{k})}\prod_{k=1}^{K}\theta_{k}^{n_{k}+\alpha_{k}-1}.$
(4)

The posterior of a Dirichlet-multinomial distribution is a Dirichlet
distribution with parameters $\alpha_{1}+n_{1},\dots,\alpha_{K}+n_{K}$ where
the Dirichlet parameters of the prior act as pseudo-counts:

$\displaystyle\bm{\theta}\sim\operatorname{Dir}(\alpha_{1}+n_{1},\dots,\alpha_{K}+n_{K})$
(5)

Integrating out $\bm{\theta}$ yields

$\displaystyle
p(\bm{n}\mid\bm{\alpha})=\frac{\Gamma(\sum_{k=1}^{K}\alpha_{k})}{\Gamma(\sum_{k=1}^{K}\alpha_{k}+n_{k})}\prod_{k=1}^{K}\frac{\Gamma(\alpha_{k}+n_{k})}{\Gamma(\alpha_{k})}$
(6)

For efficient inference schemes, auxiliary variables $m_{1},\dots,m_{K}$ can
be introduced [1, 5]:

$\displaystyle
p(\bm{n},\bm{m}\mid\bm{\alpha})=\frac{\Gamma(\sum_{k=1}^{K}\alpha_{k})}{\Gamma(\sum_{k=1}^{K}\alpha_{k}+n_{k})}\prod_{k=1}^{K}\operatorname{s}(m_{k},n_{k})\cdot\left(\alpha_{k}\right)^{m_{k}}$
(7)

where the following equality is used:

$\displaystyle\sum_{m_{k}=0}^{n_{k}}\operatorname{s}(m_{k},n_{k})\cdot\left(\alpha_{k}\right)^{m_{k}}=\frac{\Gamma(\alpha_{k}+n_{k})}{\Gamma(\alpha_{k})}.$
(8)

The auxiliary variables $\bm{m}$ behave like balls in the Polya urn scheme (or
like tables in the Chinese restaurant process for truncated Dirichlet
processes) [5]. The expected values of $\bm{m}$ are

$\displaystyle\operatorname{E}\left[m_{k}\right]=\alpha_{k}\cdot\left(\Psi(\alpha_{k}+n_{k})-\Psi(\alpha_{k})\right).$
(9)

If $\alpha_{k}$ contains multinomial parameters, their inference is simple, as
$m_{1},\dots,m_{K}$ behave like observed counts of a multinomial. For non-
multinomial parameters, further auxiliary variables allow for an elegant
inference using gamma distributions [5].

$\bm{\alpha}$$\bm{\theta}$$x$$n$ (a) Dirichlet prior

$\bm{\alpha}_{j}$$\dots$$\bm{\alpha}_{2}$$\bm{\alpha}_{1}$$\dots$$\bm{\alpha}_{J}$$\bm{\theta}$$x$$n$
(b) Multi-Dirichlet prior

Figure 1: Plate notation for a multinomial distribution with 1(a) a Dirichlet
and 1(b) a multi-Dirichlet prior distribution. The multi-Dirichlet prior is
created by summing over parameters $\alpha_{k1},\dots,\alpha_{jk}$ for each
category $k$ of the multinomial.

##  3 The multi-Dirichlet prior

Given that we have a multinomial distribution with parameters
$\theta_{1},\dots,\theta_{K}$ from which we sample $n$ single values, stored
in $x_{1},\dots,x_{n}$. We place a Dirichlet prior over the multinomial
parameters $\bm{\theta}$ which has $K$ parameters. The Dirichlet parameters
are calculated by summing over $J$ different $K$-dimensional vectors called
parent prior parameters $\bm{\alpha_{1}},\dots,\bm{\alpha_{J}}$. Then the
$k$th parameter of our Dirichlet prior distribution is a sum over parent
parameters $\alpha_{1k},\dots,\alpha_{Jk}$. In the following, this prior
distribution will be called a Multi-Dirichlet (MD) prior. Formally:

$\displaystyle\bm{\theta}\mid\bm{\alpha_{1}},\dots,\bm{\alpha_{J}}\sim$
$\displaystyle\operatorname{Dir}\left(\sum_{j=1}^{J}\alpha_{j1},\dots,\sum_{j=1}^{J}\alpha_{jK}\right);\quad\alpha_{jk}\in\mathbb{R}_{>0}\,\forall
j,k$ $\displaystyle x_{1},\dots,x_{n}\mid\bm{\theta}\sim$
$\displaystyle\operatorname{Mult}\left(\theta_{1},\dots,\theta_{K}\right)$
(10)

where we show vectors in bold. The graphical model for a multi-Dirichlet
distribution is shown in Fig. 1(b).

If we observed counts $n_{1},\dots,n_{K}$ for the $K$ categories from n draws,
the joint distribution of observations and parameters becomes:

$\displaystyle\frac{\Gamma(\sum_{k=1}^{K}\sum_{j=1}^{J}\alpha_{jk})}{\prod_{k=1}^{K}\Gamma(\sum_{j=1}^{J}\alpha_{jk})}\prod_{k=1}^{K}\theta_{k}^{n_{k}+\left(\sum_{j=1}^{J}\alpha_{jk}\right)-1}$
(11)

The same holds for a Multi-Dirichlet Process (MDP) [2] with truncation level
$K$.

After integrating over the multinomial parameters $\theta$, the joint
distribution is:

$\displaystyle p(\bm{n}\mid\bm{\alpha_{1}},\dots,\bm{\alpha_{J}})=$
$\displaystyle\frac{\Gamma(\sum_{k=1}^{K}\sum_{j=1}^{J}\alpha_{jk})}{\Gamma(\sum_{k=1}^{K}\sum_{j=1}^{J}\alpha_{jk}+n_{k})}\prod_{k=1}^{K}\frac{\Gamma(\sum_{j=1}^{J}\alpha_{jk}+n_{k})}{\Gamma(\sum_{j=1}^{J}\alpha_{jk})}$
(12)

###  3.1 Aggregation property

An important feature of the Dirichlet distribution is the aggregation
property: Every sum of Dirichlet-distributed parameters follows a Dirichlet
distribution where the corresponding parameters were summed up. For our
Dirichlet-multinomial distribution we have a Dirichlet-distributed posterior
where the parameters are counts plus pseudo counts (the Dirichlet parameters).

$\displaystyle(\theta_{1},\dots,\theta_{j-1},\theta_{j},\dots,\theta_{J})\sim$
$\displaystyle\operatorname{Dir}(\alpha_{1},\dots,\alpha_{j-1},\alpha_{j},\dots,\alpha_{J})$
$\displaystyle\Rightarrow(\theta_{1},\dots,\theta_{j-1}+\theta_{j},\dots,\theta_{J})\sim$
$\displaystyle\operatorname{Dir}(\alpha_{1},\dots,\alpha_{j-1}+\alpha_{j},\dots,\alpha_{J})$
(13)

Since we have to learn about the contribution of each parent to the Dirichlet
prior for learning their parameters, we would like to have individual counts:
Instead of storing counts $n_{k}$ telling us how often we saw category $k$, we
would like to store counts $n^{\prime}_{jk}$ telling how often we saw category
$k$ caused by parent $j$. Now if those counts are part of the parameters of a
Dirichlet-multinomial posterior, we know that summing over those counts with
$\sum_{j=1}^{J}n^{\prime}_{jk}=n_{k}$ yields the original Dirichlet posterior
of our model. To do so, we introduce probabilities $\theta^{\prime}_{jk}$
corresponding to the probability of seeing category $k$ explained by the $j$th
parent parameter. Then

$\displaystyle\sum_{j=1}^{J}n^{\prime}_{jk}=n_{k};\qquad\sum_{j=1}^{J}\theta^{\prime}_{jk}=\theta_{k}$
(14)

and the posterior of a multinomial distribution over $\bm{\theta^{\prime}}$
with a Dirichlet prior distribution is

$\displaystyle\bm{\theta^{\prime}}\sim$
$\displaystyle\operatorname{Dir}(\alpha_{11}+n^{\prime}_{11},\dots,\alpha_{1K}+n^{\prime}_{1K},\dots,\alpha_{J1}+n^{\prime}_{J1},\dots,\alpha_{JK}+n^{\prime}_{JK})$
$\displaystyle\Rightarrow\bm{\theta}\sim$
$\displaystyle\operatorname{Dir}(\sum_{j=1}^{J}\alpha_{j1}+n_{1},\dots,\sum_{j=1}^{J}\alpha_{jK}+n_{K})$
(15)

which directly follows from the aggregation property.

###  3.2 Auxiliary variables for category counts

Based on the aggregation property of the Dirichlet distribution we introduce
the auxiliary variables $n^{\prime}_{jk}$ from Eq. 14 corresponding to the
share of counts assigned to factor $k$ explained by the $j$th parent
parameter. We have to account for the possible orderings of parent counts
$\bm{n^{\prime}}$ and get:

$\displaystyle
p(\bm{n},\bm{n^{\prime}}\mid\bm{\alpha_{1}},\dots,\bm{\alpha_{J}})=$
$\displaystyle\frac{\Gamma(\sum_{k=1}^{K}\sum_{j=1}^{J}\alpha_{jk})}{\Gamma(\sum_{k=1}^{K}\sum_{j=1}^{J}\alpha_{jk}+n_{k})}\prod_{k=1}^{K}{n_{k}\choose
n^{\prime}_{1k},\dots,n^{\prime}_{Jk}}\prod_{j=1}^{J}\frac{\Gamma(\alpha_{jk}+n^{\prime}_{jk})}{\Gamma(\alpha_{jk})}$
(16)

which is a product of $K$ Dirichlet-multinomial distributions with prior
parameters $\alpha_{1k},\dots,\alpha_{Jk}$ and therefore

$\operatorname{E}[n^{\prime}_{jk}]=\frac{\alpha_{jk}}{\sum_{j^{\prime}=1}^{J}\alpha_{jk^{\prime}}}\cdot
n_{k}.$ (17)

Summing over all possible values of $n^{\prime}_{jk}$ yields the original Eq.
12.

In order to get rid of the gamma functions in Eq. 12, Teh et al. [5]
introduced auxiliary variables $m_{k}$ and $m_{jk}$, corresponding to tables
per topic and tables per topic and per parent prior, respectively (see Eq. 8):

$\displaystyle
p(\bm{n},\bm{m}\mid\bm{\alpha_{1}},\dots,\bm{\alpha_{J}})=\frac{\Gamma(\sum_{k=1}^{K}\sum_{j=1}^{J}\alpha_{jk})}{\Gamma(\sum_{k=1}^{K}\sum_{j=1}^{J}\alpha_{jk}+n_{k})}\prod_{k=1}^{K}\operatorname{s}(m_{k},n_{k})\cdot\left(\sum_{j=1}^{J}\alpha_{jk}\right)^{m_{k}}$
(18)

where $s(m,n)$ are the unsigned Stirling numbers of the first kind.

###  3.3 Auxiliary parent-level counts

For MD prior distributions, Eq. 18 complicates the inference for
$\alpha_{jk}$, as the single parts of the parent prior parameters are hidden
in exponentiated sums. However, introducing the parent counts
$\bm{n^{\prime}}$ from Eq. 16 and using the auxiliary variables from Eq. 7 we
can readily see that

$\displaystyle
p(\bm{n},\bm{n^{\prime}},\bm{m^{\prime}}\mid\bm{\alpha_{1}},\dots,\bm{\alpha_{J}})=$
$\displaystyle\frac{\Gamma(\sum_{k=1}^{K}\sum_{j=1}^{J}\alpha_{jk})}{\Gamma(\sum_{k=1}^{K}\sum_{j=1}^{J}\alpha_{jk}+n_{k})}\prod_{k=1}^{K}{n_{k}\choose
n^{\prime}_{1k},\dots,n^{\prime}_{Jk}}\prod_{j=1}^{J}\operatorname{s}(m^{\prime}_{jk},n^{\prime}_{jk})\cdot\alpha_{jk}^{m^{\prime}_{jk}}$
(19)

after which the exponentiated sums disappear. The auxiliary variables
$\bm{m^{\prime}}$ behave like balls in a Polya urn scheme (or like tables in a
Chinese restaurant process for truncated Dirichlet processes) where we
distinguish not only between $K$ categories, but also distinguish between the
$J$ parent prior parameters.

As the choice of parent priors parameters in the Polya urn scheme directly
depends on their relative sizes, the expectations for $\bm{m^{\prime}}$ are:

$\operatorname{E}[m^{\prime}_{jk}]=\frac{\alpha_{jk}}{\sum_{j^{\prime}=1}^{J}\alpha_{j^{\prime}k}}\cdot
m_{k}=\alpha_{jk}\cdot\left(\Psi(\alpha_{k}+n_{k})-\Psi(\alpha_{k})\right).$
(20)

If $\alpha_{jk}$ contains multinomial parameters, it now is simple to infer
them, as $\bm{m^{\prime}}$ behave like observed counts of a multinomial. For
non-multinomial parameters, further auxiliary variables allow for an elegant
inference using gamma distributions [5].

##  4 Inference

For inference on categories, one can directly work with the summed parameters
and counts from Eq. 18. These sums can be calculated in advance and can be
updated after a fixed number of inference steps. The resulting inference
scheme is identical as for standard Dirichlet-multinomial models [5].

For learning about the auxiliary variables for the parent prior parameters, we
use the expectations from Eq. 17 and Eq. 20 which now can be directly
calculated given category counts. We do not have to explicitly calculate
variational distributions over the new auxiliary variables $\bm{n^{\prime}}$
and $\bm{m^{\prime}}$, which allows for an efficient inference. If we separate
our parent prior parameters into a mean and a precision part as in [4], the
multinomial mean part of the parent parameters follow a Dirichlet
distribution, while the precision part follows a Gamma function, as shown in
[5]. Using the auxiliary variable scheme, it is also possible to integrate
over Dirichlet distributed means of parent priors, which allows for collapsed
inference schemes for hierarchical multi-Dirichlet and multi-Dirichlet process
models.

## References

  * [1] Charles E. Antoniak.  Mixtures of dirichlet processes with applications to bayesian nonparametric problems.  Annals of Statistics, 2(6), November 1974.
  * [2] Christoph Carl Kling, Jérôme Kunegis, Sergej Sizov, and Steffen Staab.  Detecting non-Gaussian geographical topics in tagged photo collections.  In Proceedings of the 7th ACM international conference on Web search and data mining, pages 603–612. ACM, 2014.
  * [3] Dahua Lin and John W Fisher.  Coupling nonparametric mixtures via latent dirichlet processes.  In Advances in Neural Information Processing Systems, pages 55–63, 2012.
  * [4] Thomas Minka.  Estimating a Dirichlet distribution.  Technical report, MIT, 2000.
  * [5] Yee Whye Teh, Kenichi Kurihara, and Max Welling.  Collapsed variational inference for HDP.  In Advances in Neural Information Processing Systems (NIPS), 2008\.


# Degrees of the stretched Kostka quasi-polynomials

Shiliang Gao  Department of Mathematics, University of Illinois at Urbana-
Champaign, Urbana, IL 61801<EMAIL_ADDRESS>and  Yibo Gao  Department of
Mathematics, University of Michigan, Ann Arbor, MI 48109<EMAIL_ADDRESS>
###### Abstract.

We provide a type-uniform formula for the degree of the stretched Kostka
quasi-polynomial $K_{\lambda,\mu}(N)$ in all classical types, improving a
previous result by McAllister in $\mathfrak{sl}_{r}(\mathbb{C})$. Our proof
relies on a combinatorial model for the weight multiplicity by Berenstein and
Zelevinsky.

##  1\. Introduction

The Kostka numbers $K_{\lambda,\mu}$ have great importance in many branches of
mathematics, including representation theory, algebraic geometry, symmetric
function theory, polytope theory and so on. They can be defined as the
dimension of the weight subspace of weight $\mu$ in the irreducible
representation $V_{\lambda}$ with highest weight $\lambda$. In type $A$ (i.e.
$\mathfrak{sl}_{r}(\mathbb{C})$), they are also well-known as the coefficient
of the expansion of Schur functions $s_{\lambda}$ into monomial symmetric
functions $m_{\mu}$, the number of semistandard Young tableaux of shape
$\lambda$ and weight $\mu$, the number of lattice points in the Gelfand-
Tsetlin polytope $\mathrm{GT}_{\lambda}$ of weight $\mu$, and many others [7,
14].

One method to study a family of numbers is to organize them together. Define
the _stretched Kostka quasi-polynomial_ via
$K_{\lambda,\mu}(N)=K_{N\lambda,N\mu}$ (Definition 1.1), which is known to be
a quasi-polynomial. In type $A$, it is a polynomial and can be investigated
via Ehrhart theory on the Gelfand-Tsetlin polytope, which enjoys rich
combinatorial and polyhedral geometric properties including its face
structures [1, 5, 10], its connection to order polytopes [2, 13] , its
combinatorial diameter and automorphism group [8]. In particular, McAllister
utilized the combinatorics of the Gelfand-Tsetlin polytope to compute the
degree of the stretched Kostka polynomial in type $A$ [11], resolving a
conjecture by King, Tollu, and Toumazet [9]. However, although an analogue of
Gelfand-Tsetlin polytopes is defined in other classical types [3], which we
call _Berenstein-Zelevinsky_ polytope, its various properties are not so well-
understood.

In this paper, we provide a type-uniform formula for the degree of the
stretched Kostka quasi-polynomial $K_{\lambda,\mu}(N)$ in all classical types
(Theorem 1.2), improving a result by McAllister [11]. Along the way, we study
relevant properties of the corresponding Berenstein-Zelevinsky polytope,
especially its interior points. In the remainder of the section, we state our
main result (Theorem 1.2). We provide further background in Section 2 and
prove the main result in Section 3.

We adopt the following notations throughout the paper. More background will be
provided in Section 2. Let $\mathfrak{g}$ be a complex semisimple Lie algebra
with Cartan subalgebra $\mathfrak{h}$ and root system
$\Phi\subset\mathfrak{h}^{*}$. Let $r=\dim\mathfrak{g}$, which is also the
rank of $\Phi$, denoted $\mathrm{rk}(\Phi)$. Let $\Phi_{+}$ be the positive
roots, $\Pi=\\{\alpha_{1},\ldots,\alpha_{r}\\}$ be the simple roots and
$\\{\omega_{1},\ldots,\omega_{r}\\}$ be the corresponding fundamental weights.
For dominant weights $\lambda$ and $\mu$, let $V_{\lambda}$ be the irreducible
finite-dimensional representation of $\mathfrak{g}$ with highest weight
$\lambda$, and let $V_{\lambda}(\mu)\subset V_{\lambda}$ be its subspace of
weight $\mu$.

######  Definition 1.1.

The _stretched Kostka quasi-polynomial_ (in $N$) is defined as

$K_{\lambda,\mu}(N):=\dim V_{N\lambda}(N\mu).$

It is known that $K_{\lambda,\mu}(N)$ is a quasi-polynomial in general, which
is a polynomial in type $A$. It is also a classical result that $\dim
V_{\lambda}(\mu)\neq 0$ if and only if $\lambda-\mu$ is a nonnegative integral
linear combination of the simple roots [3].

######  Theorem 1.2.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra of classical type and
dimension $r$. For dominant weights $\lambda,\mu\in\mathfrak{h}^{*}$ such that
$\lambda-\mu=\sum_{i=1}^{r}c_{i}\alpha_{i}$ with $c_{i}\in\mathbb{Z}_{\geq 0}$
for all $i$, write $\lambda=\sum_{i=1}^{r}d_{i}\omega_{i}$ where
$d_{i}\in\mathbb{Z}_{\geq 0}$ for all $i$. Let

$\begin{cases}\Phi^{(1)}\subset\Phi\text{ be the root subsystem spanned by
}\\{\alpha_{i}\>|\>c_{i}\neq 0\\},\\\ \Phi^{(2)}\subset\Phi^{(1)}\text{ be the
root subsystem spanned by }\\{\alpha_{i}\>|\>c_{i}\neq
0,d_{i}=0.\\}\end{cases}$

Then the degree of the stretched Kostka quasi-polynomial equals

$\deg
K_{\lambda,\mu}(N)=|\Phi^{(1)}_{+}|-\mathrm{rk}(\Phi^{(1)})-|\Phi^{(2)}_{+}|.$

######  Conjecture 1.3.

Theorem 1.2 holds for all complex semisimple Lie algebras (of arbitrary
types).

##  2\. Background

We discuss more background on root systems, and then talk about a model in [3]
that allows us to compute $K_{\lambda,\mu}(N)$ in classical types.

###  2.1. Root system and Weyl groups

Following Section 1, we continue with more background on root systems. Recall
that the positive roots $\Pi=\\{\alpha_{1},\ldots,\alpha_{r}\\}$ form a basis
of our ambient vector space $\mathfrak{h}^{*}$, which is also called the
_weight space_. It is sometimes more convenient to identify this space with
its dual, by equipping it with an inner product $\langle-,-\rangle$. The
_simple coroots_ are defined as

$\alpha_{i}^{\vee}=\frac{2\alpha_{i}}{\langle\alpha_{i},\alpha_{i}\rangle}$

for $i=1,\ldots,r$. And the fundamental weights
$\\{\omega_{1},\ldots,\omega_{r}\\}$ are defined as the dual basis to the
coroots, i.e. $\langle\alpha_{i}^{\vee},\omega_{j}\rangle=\delta_{i,j}$ where
$\delta$ is the Kronecker delta. A weight $\lambda\in\mathfrak{h}^{*}$ is an
_integral weight_ if $\lambda$ is written as $\sum_{i=1}^{r}d_{i}\omega_{i}$
where $d_{i}\in\mathbb{Z}$ for all $i$. All integral weights form the _weight
lattice_ $Q$. We also say that a weight $\lambda=\sum_{i=1}d_{i}\omega_{i}$ is
an _dominant integral weight_ if $d_{i}\in\mathbb{Z}_{\geq 0}$ for all $i$.
Let $\Lambda^{+}$ be the set of all dominant integral weights.

We define a partial order, called the _root poset_ , on the positive roots
$\Phi_{+}$, where $\alpha\leq\beta$ if $\beta-\alpha$ is a nonnegative
(integral) linear combination of the simple roots $\Pi$. Thus, the minimal
elements of the root poset $\Phi_{+}$ are precisely the simple roots. Define
the _support_ of a positive root $\alpha$ to be

$\operatorname{Supp}(\alpha):=\\{\alpha_{i}\in\Pi\>|\>\alpha_{i}\leq\alpha\\}\subset\Pi.$

We also say that a positive root $\alpha$ is _supported on_ $\alpha_{i}\in\Pi$
if $\alpha_{i}\in\operatorname{Supp}(\alpha)$.

Similarly, for $\lambda,\mu\in\Lambda^{+}$, we say that $\lambda$ _dominates_
$\mu$, if

(1) $\lambda-\mu=\sum_{i=1}^{r}c_{i}\alpha_{i},$

for some $c_{i}\in\mathbb{Z}_{\geq 0}$. We say that a pair $(\lambda,\mu)$ is
_primitive_ if $c_{i}\in\mathbb{Z}_{>0}$ for all $i$ in (1). We call a
primitive pair _simple_ if the underlying Lie algebra is simple.

We adopt the following conventions on the root systems for classical types.

  * •

Type $A_{r-1}$ ($\mathfrak{g}=\mathfrak{sl}_{r}$):
$\Phi=\\{e_{i}-e_{j}\>|\>1\leq i<j\leq r\\}$,
$\Phi_{+}=\\{e_{i}-e_{j}\>|\>1\leq i<j\leq r\\}$, $\alpha_{i}=e_{i}-e_{i+1}$
and $\omega_{i}=e_{1}+\cdots+e_{i}$ for $i=1,\ldots,r-1$.

  * •

Type $B_{r}$ ($\mathfrak{g}=\mathfrak{so}_{2r+1}$): $\Phi=\\{\pm e_{i}\pm
e_{j}\>|\>1\leq i<j\leq r\\}\cup\\{\pm e_{i}\>|\>1\leq i\leq r\\}$,
$\Phi_{+}=\\{e_{i}\pm e_{j}\>|\>1\leq i<j\leq r\\}\cup\\{e_{i}\>|\>1\leq i\leq
r\\}$, $\alpha_{i}=e_{i}-e_{i+1}$ and $\omega_{i}=e_{1}+\cdots+e_{i}$ for
$i=1,\ldots,r-1$, and $\alpha_{r}=e_{r}$ and
$\omega_{r}=\frac{1}{2}(e_{1}+\cdots+e_{r})$.

  * •

Type $C_{r}$ ($\mathfrak{g}=\mathfrak{sp}_{2r}$): $\Phi=\\{\pm e_{i}\pm
e_{j}\>|\>1\leq i<j\leq r\\}\cup\\{\pm 2e_{i}\>|\>1\leq i\leq r\\}$,
$\Phi_{+}=\\{e_{i}\pm e_{j}\>|\>1\leq i<j\leq r\\}\cup\\{2e_{i}\>|\>1\leq
i\leq r\\}$, $\alpha_{i}=e_{i}-e_{i+1}$ for $i=1,\ldots,r-1$ and
$\alpha_{r}=2e_{r}$, and $\omega_{i}=e_{1}+\cdots+e_{i}$ for $i=1,\ldots,r$.

  * •

Type $D_{r}$ ($\mathfrak{g}=\mathfrak{so}_{2r}$): $\Phi=\\{\pm e_{i}\pm
e_{j}\>|\>1\leq i<j\leq r\\}$, $\Phi_{+}=\\{e_{i}\pm e_{j}\>|\>1\leq i<j\leq
r\\}$, $\alpha_{i}=e_{i}-e_{i+1}$ for $i=1,\ldots,r-1$,
$\alpha_{r}=e_{r-1}+e_{r}$, $\omega_{i}=e_{1}+\cdots+e_{i}$ for
$i=1,\ldots,r-2$, $\omega_{r-1}=\frac{1}{2}(e_{1}+\cdots+e_{r-1}-e_{r})$ and
$\omega_{r}=\frac{1}{2}(e_{1}+\cdots+e_{r})$.

###  2.2. Kostka coefficients

Here we review some basic notions and results related to Kostka coefficients.
The Kostka coefficients can be computed via the _Kostant multiplicity formula_
, and can be expressed as tensor product multiplicities.

######  Definition 2.1.

The _Kostant partition function_ is defined by

(2)
$P(\lambda-\mu):=\\#\\{(m_{\alpha})_{\alpha\in\Phi^{+}}:m_{\alpha}\in\mathbb{Z}_{\geq
0},\sum_{\alpha\in\Phi^{+}}m_{\alpha}\alpha=\lambda-\mu\\}.$

For a pair $\lambda,\mu\in\Lambda^{+}$ such that $\lambda$ dominates $\mu$,
the _Kostant multiplicity formula_ express $K_{\lambda,\mu}$ as alternate sum
of Kostant partition functions:

######  Theorem 2.2.

$K_{\lambda,\mu}=\sum_{w\in W}\det(w)P(w(\lambda+\rho)-\mu-\rho)$, where $W$
is the Weyl group and $\rho=\frac{1}{2}\sum_{\alpha\in\Phi^{+}}\alpha$.

Consider the root space decomposition of $\mathfrak{g}$:

(3)
$\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha}.$

For $\alpha\in\Pi$, let $e_{\pm\alpha}\in\mathfrak{g}_{\pm\alpha}$ and
$h_{\alpha}=[e_{\alpha},e_{-\alpha}]\in\mathfrak{h}$ be the standard
generator. For each weight space $V_{\lambda}(\beta)\subset V_{\lambda}$ and
highest weight $\nu=\sum_{\alpha\in\Pi}n_{\alpha}\alpha$, let
$V_{\lambda}(\beta,\nu)\subseteq V_{\lambda}(\beta)$ be the subspace defined
by

(4) $V_{\lambda}(\beta,\nu):=\\{v\in
V_{\lambda}(\beta):e_{\alpha}^{n_{\alpha}+1}v=0\text{ for }\alpha\in\Pi\\}.$

Let $c_{\lambda,\nu}^{\mu}$ be the multiplicity of $V_{\mu}$ in the tensor
product $V_{\lambda}\otimes V_{\nu}$. It is known (see e.g.[15]) that
$c^{\mu}_{\lambda,\nu}$ can be realized as the dimension of certain subspace
of $V_{\lambda}$:

(5) $c^{\mu}_{\lambda,\nu}=\dim(V_{\lambda}(\mu-\nu,\nu)).$

######  Lemma 2.3.

For $\lambda,\mu\in\Lambda^{+}$ such that $\lambda$ dominates $\mu$, let
$c_{i}$’s be as in (1) and set

$\nu=\sum_{i=1}^{r}c_{i}\omega_{i}\text{ and }\mu^{\prime}=\nu+\mu.$

Then $K_{\lambda,\mu}=c^{\mu^{\prime}}_{\lambda,\nu}$.

###### Proof.

For any $v\in V_{\lambda}(\mu)$ and any $i\in[r]$,

$e_{\alpha_{i}}^{c_{i}+1}v\in V_{\lambda}(\mu+(c_{i}+1)\alpha_{i}).$

Since $\lambda-\mu-(c_{i}+1)\alpha_{i}=(\sum_{j\neq
i}c_{j}\alpha_{j})-\alpha_{i}$, $\lambda$ does not dominate
$\mu+(c_{i}+1)\alpha_{i}$ for any $i\in[r]$. Therefore

$\dim(V_{\lambda}(\mu+(c_{i}+1)\alpha_{i}))=0\text{ and
}e_{\alpha_{i}}^{c_{i}+1}v=0\text{ for all }i\in[r].$

Now by (4),

$V_{\lambda}(\mu^{\prime}-\nu,\nu)=\\{v\in
V_{\lambda}(\mu):e_{\alpha_{i}}^{c_{i}+1}v=0\text{ for all
}i\in[r]\\}=V_{\lambda}(\mu).$

We are then done by (5) as

$K_{\lambda,\mu}=\dim(V_{\lambda}(\mu))=\dim(V_{\lambda}(\mu^{\prime}-\nu,\nu))=c^{\mu^{\prime}}_{\lambda,\nu}.\qed$

###  2.3. BZ-patterns

######  Definition 2.4.

A real _BZ-pattern_ in type $B_{r}/C_{r}$ (where BZ stands for Berenstein-
Zelevinsky) of highest weight
$\lambda=\lambda_{11}e_{1}+\cdots+\lambda_{1r}e_{r}$ is an array of real
numbers $\\{\lambda_{i,j},\eta_{i,j}\\}$ with $1\leq i\leq j\leq r$,
satisfying the inequalities

(6)
$\min(\lambda_{i,j-1},\lambda_{i+1,j-1})\geq\eta_{i,j-1}\geq\max(\lambda_{i,j},\lambda_{i+1,j})$

whenever the indices are defined, and $\eta_{i,r}\geq 0$ for $1\leq i\leq r$.
Denote $|\lambda_{i}|:=\sum_{j}\lambda_{i,j}$ and
$|\eta_{i}|:=\sum_{j}\eta_{i,j}$. We say that such a BZ-pattern has _weight_
$\mu=\mu_{1}e_{1}+\cdots+\mu_{r}e_{r}$ where
$\mu_{i}=|\lambda_{i}|+|\lambda_{i+1}|-2|\eta_{i}|$ with the convention that
$|\lambda_{r+1}|=0$.

A real BZ-pattern in type $D_{r}$ of highest weight
$\lambda=\lambda_{11}e_{1}+\cdots+\lambda_{1r}e_{r}$ is an array of real
numbers $\\{\lambda_{i,j}\\}_{1\leq i\leq j\leq r}\cup\\{\eta_{i,j}\\}_{1\leq
i\leq j<r}$, satisfying (6) and

(7)
$\lambda_{i,r}+\lambda_{i+1,r}+\min(\lambda_{i,r-1},\lambda_{i+1,r-1})\geq\eta_{i,r-1}$

for all $i$ with the convention that $\lambda_{i+1,i}=+\infty$. The weight of
a real type $D_{r}$ BZ-pattern is defined in the same way as in type
$B_{r}/C_{r}$.

It is visually more convenient to view BZ-patterns in the following way:

$\begin{pmatrix}\lambda_{11}&&\lambda_{12}&\cdots&\cdots&\lambda_{1r}&\\\
&\eta_{11}&&\eta_{12}&\cdots&\cdots&\eta_{1r}\\\
&&\lambda_{22}&\cdots&\cdots&\lambda_{2r}&\vdots\\\
&&&\ddots&&\vdots&\vdots\\\ &&&&&\lambda_{rr}&\vdots\\\
&&&&&&\eta_{rr}\end{pmatrix}$

where the inequalities in (6) dictate that adjacent entries (in the
$45^{\circ}$ manner) increase from left to right.

######  Definition 2.5.

For a dominant weight $\lambda=\lambda_{11}e_{1}+\cdots+\lambda_{1r}e_{r}$,
define $\operatorname{BZ}_{\lambda}$, the _BZ-polytope_ of highest weight
$\lambda$, to be the polytope cut off by the inequalities specified in
Definition 2.4. For another dominant weight $\mu$, let the BZ-polytope of
highest weight $\lambda$ and weight $\mu$, $\operatorname{BZ}_{\lambda,\mu}$,
be a section of $\operatorname{BZ}_{\lambda}$ by imposing the equality
$\mu_{i}=|\lambda_{i}|+|\lambda_{i+1}|-2|\eta_{i}|$ for $i=1,\ldots,r$, as in
Definition 2.4.

The notion of $\operatorname{BZ}$-patterns allows us to work with classical
types almost uniformly. Note that a $\operatorname{BZ}$-pattern of type $D$
has different coordinates and additional inequalities comparing to type $B$
and $C$.

######  Definition 2.6.

A real BZ-pattern $\\{\lambda_{i,j},\eta_{i,j}\\}$ is _integral_ of

  * •

type $C_{r}$, if $\lambda_{i,j},\eta_{i,j}\in\mathbb{Z}$;

  * •

type $B_{r}$, if all of $\lambda_{i,j}$, $1\leq i\leq j\leq r$ and
$\eta_{i,j}$, $1\leq i\leq j<r$ lie simultaneously in either $\mathbb{Z}$ or
$\frac{1}{2}+\mathbb{Z}$, and $\eta_{i,r}\in\frac{1}{2}\mathbb{Z}$;

  * •

type $D_{r}$, if all of $\lambda_{i,j}$ and $\eta_{i,j}$ lie simultaneously in
either $\mathbb{Z}$ or $\frac{1}{2}+\mathbb{Z}$.

######  Theorem 2.7 ([3]).

In types $B_{r},C_{r},D_{r}$, $K_{\lambda,\mu}$ equals the number of integral
BZ-patterns of highest weight $\lambda$ and weight $\mu$.

##  3\. Proof of the main theorem, Theorem 1.2

We first provide the general framework to out proof, before going into each
subsection for the corresponding step:

  1. (1)

Reduce to the case where the pair $(\lambda,\mu)$ is primitive.

  2. (2)

Compute the dimension of the polytope $\operatorname{BZ}_{\lambda}$.

  3. (3)

Identify a point
$P\in\operatorname{BZ}_{\lambda}^{\circ}\cap\operatorname{BZ}_{\lambda,\mu}$
to compute $\dim\operatorname{BZ}_{\lambda,\mu}$.

  4. (4)

Finish by establishing the equality $\deg
K_{\lambda,\mu}=\dim\operatorname{BZ}_{\lambda,\mu}$.

Throughout, consider a root system $\Phi$ of rank $r$ of classical type.

###  3.1. Reduction to simple primitive pairs

For any $\Pi^{\prime}\subset\Pi$, let $\Phi^{\prime}\subset\Phi$ be the root
subsystem generated by $\Pi^{\prime}$. Let
$\mathfrak{g}^{\prime}\subset\mathfrak{g}$ be the subalgebra defined by

(8)
$\mathfrak{g}^{\prime}=\mathfrak{h}^{\prime}\oplus\bigoplus_{\alpha\in\Phi^{\prime}}\mathfrak{g}_{\alpha},$

where $\mathfrak{h}^{\prime}$ is spanned by
$\\{h_{\alpha}\in\mathfrak{h}:\alpha\in\Phi^{\prime}\\}$. Let
$p:\mathfrak{h}^{*}\rightarrow(\mathfrak{h}^{\prime})^{*}$ be the natural
projection.

######  Lemma 3.1.

$p(\omega_{i})=0$ if $\alpha_{i}\notin\Pi^{\prime}$ and $p(\omega_{i})$ is a
fundamental weight of $\mathfrak{g}^{\prime}$ if $\alpha_{i}\in\Pi^{\prime}$.
In fact, all fundamental weights of $\mathfrak{g}^{\prime}$ can be written as
$p(\omega_{i})$ for some $i$.

###### Proof.

For $\alpha_{i}\notin\Pi^{\prime}$, $\langle\omega_{i},\alpha\rangle=0$ for
all $\alpha\in\Pi^{\prime}$. Since $\\{\alpha\in\Pi^{\prime}\\}$ span
$(\mathfrak{h}^{\prime})^{*}$, $\omega_{i}$ is orthogonal to
$(\mathfrak{h}^{\prime})^{*}$ and $p(\omega_{i})=0$. Now for
$\alpha_{i}\in\Pi^{\prime}$, we have
$\alpha_{i}\in(\mathfrak{h}^{\prime})^{*}$. Therefore
$\langle\omega_{i},\alpha_{i}\rangle=\langle
p(\omega_{i}),\alpha_{i}\rangle=1$. For any $\alpha_{j}\in\Pi^{\prime}$ such
that $i\neq j$, since $\langle\omega_{i},\alpha_{j}\rangle=0$ and
$\alpha_{j}\in(\mathfrak{h}^{\prime})^{*}$, we get $\langle
p(\omega_{i}),\alpha_{j}\rangle=0$. Therefore $\langle
p(\omega_{i}),\alpha_{j}\rangle=\delta_{i,j}$ and $p(\omega_{i})$ is a
fundamental weight of $\mathfrak{g}^{\prime}$. Since the number of fundamental
weights of $\mathfrak{g}^{\prime}$ equals number of simple roots,
$\\{p(\omega_{i}):\alpha_{i}\in\Pi^{\prime}\\}$ are the fundamental weights of
$\mathfrak{g}^{\prime}$. ∎

######  Lemma 3.2 ([3], Proposition 1.3).

Let $\lambda,\nu,\mu$ be dominant integral weights of $\mathfrak{g}$ such that
$\lambda+\nu-\mu\in\Phi^{\prime}$. Denote
$c_{p(\lambda),p(\nu)}^{p(\mu)}(\mathfrak{g}^{\prime})$ the multiplicity of
$V_{p(\mu)}\subset V_{p(\lambda)}\otimes V_{p(\nu)}$ as
$\mathfrak{g}^{\prime}$-module. Then

$c^{\mu}_{\lambda,\nu}=c^{p(\mu)}_{p(\lambda),p(\nu)}(\mathfrak{g}^{\prime}).$

######  Lemma 3.3.

For $\lambda,\mu\in\Lambda^{+}$ such that $\lambda$ dominates $\mu$ and
$\lambda-\mu\in\Phi^{\prime}$,

$K_{\lambda,\mu}=K_{p(\lambda),p(\mu)}(\mathfrak{g}^{\prime}),$

where $K_{p(\lambda),p(\mu)}(\mathfrak{g}^{\prime})$ is the weight
multiplicity of $p(\mu)$ in the $\mathfrak{g}^{\prime}$-module
$V_{p(\lambda)}$.

###### Proof.

Since $\lambda$ dominates $\mu$ and $\lambda-\mu\in\Phi^{\prime}$, we can
write

$\lambda-\mu=\sum_{\\{i:\alpha_{i}\in\Pi^{\prime}\\}}c_{i}\alpha_{i}.$

By Lemma 2.3, $K_{\lambda,\mu}=c^{\mu^{\prime}}_{\lambda,\nu}$, where
$\nu=\sum_{\\{i:\alpha_{i}\in\Pi^{\prime}\\}}c_{i}\omega_{i}$ and
$\mu^{\prime}=\mu+\nu$. Since
$\lambda+\nu-\mu^{\prime}=\lambda-\mu\in\Phi^{\prime}$, by Lemma 3.2,
$c^{\mu^{\prime}}_{\lambda,\nu}=c^{p(\mu^{\prime})}_{p(\lambda),p(\nu)}(\mathfrak{g}^{\prime})$.
Since $\lambda-\mu\in(\mathfrak{h}^{\prime})^{*}$,
$p(\lambda)-p(\mu)=p(\lambda-\mu)=\lambda-\mu=\sum_{\\{i:\alpha_{i}\in\Pi^{\prime}\\}}c_{i}\alpha_{i}$.
By Lemma 3.1,
$p(\nu)=\sum_{\\{i:\alpha_{i}\in\Pi^{\prime}\\}}c_{i}p(\omega_{i})$ and each
$p(\omega_{i})$ is a fundamental weight of $\mathfrak{g}^{\prime}$ that
corresponds to $\alpha_{i}\in(\mathfrak{h}^{\prime})^{*}$. Since
$p(\mu^{\prime})=p(\mu+\nu)=p(\mu)+p(\nu)$, we can apply Lemma 2.3 again and
get
$K_{p(\lambda),p(\mu)}(\mathfrak{g}^{\prime})=c^{p(\mu^{\prime})}_{p(\lambda),p(\nu)}(\mathfrak{g}^{\prime})$.
Therefore

$K_{\lambda,\mu}=c^{\mu^{\prime}}_{\lambda,\nu}=c^{p(\mu^{\prime})}_{p(\lambda),p(\nu)}(\mathfrak{g}^{\prime})=K_{p(\lambda),p(\mu)}(\mathfrak{g}^{\prime}).\qed$

We now return to the context of non-primitive pairs. For a pair
$(\lambda,\mu)$ that is not primitive, let $\Pi_{0}=\\{\alpha_{i}:c_{i}\neq
0\\}$ be the subset of $\Pi$ consisting of simple roots that appears in (1).
Let $\Phi_{0}\subset\Phi,W_{0}\subset W$ be the corresponding root subsystem
and Weyl subgroup and $\mathfrak{g}_{0}\subset\mathfrak{g}$ the Lie subalgebra
as in (8). Write $r_{0}=rk(\Phi_{0})$ and
$\\{\omega_{\alpha}:\alpha\in\Pi_{0}\\}$ the fundamental weights of
$\mathfrak{g}_{0}$. Since $\mathfrak{g}_{0}$ is also semisimple, we can write
$\mathfrak{g}_{0}=\bigoplus_{k=1}^{m}\mathfrak{g}_{k}$ where each
$\mathfrak{g}_{k}$ is a simple Lie algebra. Let
$\mathfrak{h}_{k},\Pi_{k},\Phi_{k},W_{k}$ be the Cartan subalgebra, simple
roots, root system and Weyl group of $\mathfrak{g}_{k}$. In particular,
$\Pi_{0}=\bigsqcup_{k=1}^{m}\Pi_{k}$ and $W_{0}=W_{1}\times\ldots\times
W_{m}$. Let $\rho_{k}=\frac{1}{2}\sum_{\alpha\in\Phi^{+}_{k}}\alpha$ be the
half sum of positive roots in $\Phi_{k}$. Define
$p_{k}:\mathfrak{h}^{*}\rightarrow(\mathfrak{h_{k}})^{*}$ to be the natural
projection.

######  Proposition 3.4.

There exists pairs $(\lambda^{(k)},\mu^{(k)})_{k\in[m]}$ such that each
$(\lambda^{(k)},\mu^{(k)})$ is a primitive pair of $\mathfrak{g}_{k}$ and

$K_{\lambda,\mu}=\prod_{k=1}^{m}K_{\lambda^{(k)},\mu^{(k)}}(\mathfrak{g}_{k}).$

###### Proof.

Let $\lambda^{(i)}=p_{i}(\lambda)$ and $\mu^{(i)}=p_{i}(\mu)$ for all
$i\in[0,m]$. By Lemma 3.3,

(9) $K_{\lambda,\mu}=K_{\lambda^{(0)},\mu^{(0)}}(\mathfrak{g}_{0}).$

For any $k\in[m]$, set

$\tilde{\lambda}^{(k)}=\sum_{\alpha\in\Pi_{k}}\langle\lambda^{(0)},\alpha\rangle\omega_{\alpha}\text{
and
}\tilde{\mu}^{(k)}=\sum_{\alpha\in\Pi_{k}}\langle\mu^{(0)},\alpha\rangle\omega_{\alpha}.$

Notice that
$\tilde{\lambda}^{(k)},\tilde{\mu}^{(k)}\in(\mathfrak{h}_{k})^{*}\subset\mathfrak{h}^{*}$
and $\lambda^{(0)}=\sum_{k=1}^{m}\tilde{\lambda}^{(k)}$ and
$\mu^{(0)}=\sum_{k=1}^{m}\tilde{\mu}^{(k)}$. By (2.2),

$K_{\lambda^{0},\mu^{(0)}}(\mathfrak{g}_{0})=\sum_{w=(w_{1},\ldots,w_{m})\in
W_{0}}(\prod_{k=1}^{m}\det(w_{k}))P(w(\sum_{k=1}^{m}(\tilde{\lambda}^{(k)}+\rho_{k}))-\sum_{k=1}^{m}(\tilde{\mu}^{(k)}-\rho_{k}))$

Since $\tilde{\lambda}^{(k)},\tilde{\mu}^{(k)}$ are invariant under $W_{j}$
for all $j\neq k$, for $w=(w_{1},\ldots,w_{m})\in W$,

$P(w(\sum_{k=1}^{m}(\tilde{\lambda}^{(k)}+\rho_{k}))-\sum_{k=1}^{m}(\tilde{\mu}^{(k)}-\rho_{k}))=P(\sum_{k=1}^{m}(w_{k}(\tilde{\lambda}^{(k)}+\rho_{k})-\tilde{\mu}^{(k)}-\rho_{k})).$

Since
$w_{k}(\tilde{\lambda}^{(k)}+\rho_{k})-\tilde{\mu}^{(k)}-\rho_{k})\in(\mathfrak{h}_{k})^{*}$
for all $k\in[m]$,

$P(\sum_{k=1}^{m}(w_{k}(\tilde{\lambda}^{(k)}+\rho_{k})-\tilde{\mu}^{(k)}-\rho_{k}))=\prod_{k=1}^{m}P(w_{k}(\tilde{\lambda}^{(k)}+\rho_{k})-\tilde{\mu}^{(k)}-\rho_{k}).$

Therefore

(10) $\displaystyle K_{\lambda^{0},\mu^{(0)}}(\mathfrak{g}_{0})$
$\displaystyle=\sum_{(w_{1},\ldots,w_{m})\in
W}\prod_{k=1}^{m}(\det(w_{k})P(w_{k}(\tilde{\lambda}^{(k)}+\rho_{k})-\tilde{\mu}^{(k)}-\rho_{k}))$
$\displaystyle=\prod_{k=1}^{m}\sum_{w_{k}\in
W_{k}}(\det(w_{k})P(w_{k}(\tilde{\lambda}^{(k)}+\rho_{k})-\tilde{\mu}^{(k)}-\rho_{k}))$
$\displaystyle=\prod_{k=1}^{m}K_{\tilde{\lambda}^{(k)},\tilde{\mu}^{(k)}}(\mathfrak{g}_{0}).$

Define $\lambda^{(k)}=p_{k}(\tilde{\lambda}^{(k)})$ and
$\mu^{(k)}=p_{k}(\tilde{\mu}^{(k)})$. Since
$\lambda^{(k)}-\mu^{(k)}=p_{k}(\tilde{\lambda}^{(k)}-\tilde{\mu}^{(k)})=p_{k}(\lambda^{(0)}-\mu^{(0)})$
and every simple root of $\mathfrak{g}_{k}$ appear in the expansion of
$\lambda^{(0)}-\mu^{(0)}$, $(\lambda^{(k)},\mu^{(k)})$ is a primitive pair of
$\mathfrak{g}_{k}$ for all $k\in[m]$. By Lemma 3.3,
$K_{\tilde{\lambda}^{(k)},\tilde{\mu}^{(k)}}(\mathfrak{g}_{0})=K_{\lambda^{(k)},\mu^{(k)}}(\mathfrak{g}_{k})$.
We are then done by (9) and (10). ∎

An immediate consequence of Proposition 3.4 is the following:

######  Corollary 3.5.

$\deg K_{\lambda,\mu}(\mathfrak{g})(N)=\sum_{k=1}^{m}\deg
K_{\lambda^{(k)},\mu^{(k)}}(\mathfrak{g}_{k})(N)$.

By induction on $k$, it is now enough to prove Theorem 1.2 when
$(\lambda,\mu)$ is a simple primitive pair.

###  3.2. Dimension of the BZ-polytope $\operatorname{BZ}_{\lambda}$

Recall that the BZ-polytopes have coordinates indxed by

$\begin{cases}\\{\lambda_{i,j}\\}_{1<i\leq j\leq r}\text{ and
}\\{\eta_{i,j}\\}_{1\leq i\leq j\leq r}\text{ for type }B_{r}\text{ and
}C_{r},\\\ \\{\lambda_{i,j}\\}_{1<i\leq j\leq r}\text{ and
}\\{\eta_{i,j}\\}_{1\leq i\leq j<r}\text{ for type }D_{r}.\end{cases}$

The number of coordinates equals $|\Phi_{+}|$, where $\Phi$ is the root system
that we are working with. Note that since $\lambda=\sum\lambda_{1i}e_{i}$ is
fixed, we choose not to consider $\lambda_{11},\ldots\lambda_{1r}$ as
coordinates for $\operatorname{BZ}_{\lambda}$ and
$\operatorname{BZ}_{\lambda,\mu}$.

######  Lemma 3.6.

Let $\lambda=\sum_{i=1}^{r}d_{i}\omega_{i}$ be a dominant weight in a root
system $\Phi$ where $d_{i}\in\mathbb{Z}_{\geq}0$ for all $i$. Let
$\Phi^{\prime}\subset\Phi$ be the root subsystem of $\Phi$ spanned by
$\\{\alpha_{i}\>|\>d_{i}=0\\}$. Then we have that

$\dim\operatorname{BZ}_{\lambda}=|\Phi_{+}|-|\Phi^{\prime}_{+}|.$

We prove Lemma 3.6 using two steps: upper bounding the dimension by figuring
out conditions on the coordinates; and lower bounding the dimension by
providing a certain neighborhood around an interior point.

###### Proof of Lemma 3.6 in type $B/C$.

We adopt the conventions in Section 2. Let the simple roots of $\Phi^{\prime}$
be $S^{\prime}$ and let $S^{\prime}=Z_{1}\sqcup\cdots\sqcup Z_{m}$ be its
partition into the connected components of the Dynkin diagram of
$\Phi^{\prime}$ where the indices are in order. Write $z_{i}=|Z_{i}|$ for all
$i$. Since $\omega_{j}=e_{1}+\cdots+e_{j}$ for all $j$, if $j\in S^{\prime}$,
$\lambda_{1,j}=\lambda_{1,j+1}$, with the convention that $\lambda_{1,r+1}=0$.

Case 1: $r\in S^{\prime}$. Here $Z_{1},\ldots,Z_{m-1}$ are of type $A$ and
$Z_{m}$ is of type $B/C$ with

$|\Phi^{\prime}_{+}|=\sum_{i=1}^{m-1}{z_{i}+1\choose 2}+z_{m}^{2}.$

For each $Z_{i}=\\{t+1,\ldots,t+z_{i}\\}$ where $i<m$, we have that
$\lambda_{1,t+1}=\cdots=\lambda_{1,t+z_{i}+1}$, and thus all the
${z_{i}+1\choose 2}$ coordinates in the triangle below must take on this
value. For $Z_{m}=\\{r-z_{m}+1,\ldots,r\\}\subset\Phi^{\prime}$, we know that
$\lambda_{1,r-z_{m}}=\cdots=\lambda_{1,r}=0$. Since $\eta_{i,r}\geq 0$ for all
$i$, we again deduce that $\lambda_{i,j}=0$, $\eta_{i,j}=0$ for $j-i\geq
r-z_{m}-1$. There are $z_{m}^{2}$ such coordinates, which is the number of
positive roots in a type $B_{z_{m}}$ root system. As a result, we have
$|\Phi^{\prime}_{+}|$ fixed coordinates so
$\dim\operatorname{BZ}_{\lambda}\leq|\Phi_{+}|-|\Phi^{\prime}_{+}|$.

To obtain an upper bound, we construct an interior point
$T\in\operatorname{BZ}_{\lambda}^{\circ}$, as follows. The coordinates
$\lambda_{1,1},\ldots,\lambda_{1,r}$ have been given. For all the other
coordinates, take $\lambda_{i,j}=(\eta_{i-1,j-1}+\eta_{i-1,j})/2$ and
$\eta_{i,j}=(\lambda_{i,j}+\lambda_{i,j+1})/2$ to be the average of the two
values above it, where $\eta_{i,r}=\lambda_{i,r}/2$. See Figure 1 for an
example.

$\begin{pmatrix}3&&1&&1&&0&&0&\\\
&2&&\underline{1}&&1/2&&\underline{0}&&\underline{0}\\\
&&3/2&&3/4&&1/4&&\underline{0}&\\\ &&&9/8&&1/2&&1/8&&\underline{0}\\\
&&&&13/16&&5/16&&1/16&\\\ &&&&&\cdots&&&&\end{pmatrix}$

Figure 1. An interior point in $\operatorname{BZ}_{\lambda}$ of type $B/C$,
where $\lambda=(3,1,1,0,0)$ with $\S^{\prime}=\\{2,4,5\\}$. The fixed values
are underlined.

In the previous paragraph, we analyzed the coordinates that are fixed. On the
other hand, from this construction, for each coordinate $\lambda_{i,j}$ or
$\eta_{i,j}$ that is not fixed, all of the inequalities involved (see
Definition 2.4 for the inequalities) are not tight, and we can thus find a
small cube of dimension $|\Phi_{+}|-|\Phi^{\prime}_{+}|$ around $T$ with side
length $\epsilon>0$ that is small enough. As a result,
$\dim\operatorname{BZ}_{\lambda}\geq|\Phi_{+}|-|\Phi^{\prime}_{+}|$ as well.

Case 2: $r\notin S^{\prime}$. Here, $Z_{1},\ldots,Z_{m}$ are all of type $A$.
It is a simpler case than case 1. As above, we have
$\sum_{i=1}^{n}{z_{i}+1\choose 2}$ fixed coordinates coming from the triangles
below each consecutive chunk of same values in the first row. We can
analogously construct an interior point
$T\in\operatorname{BZ}_{\lambda}^{\circ}$ by assigning the average value of
the two values above each coordinate with the convention that
$\lambda_{i},r+1=0$ for all $i$. Both the upper and lower bounds of
$\dim\operatorname{BZ}_{\lambda}$ are established completely analogously. So
$\dim\operatorname{BZ}_{\lambda}=|\Phi_{+}|-|\Phi^{\prime}_{+}|$ as desired. ∎

###### Proof of Lemma 3.6 in type $D$.

Again, let $S^{\prime}=Z_{1}\sqcup\cdots\sqcup Z_{m}$ be the partition of
simple roots in $\Phi^{\prime}$ into connected components. The main extra
difficulty is the additional inequalities and the fact that $\lambda_{1,r}$
(and some other coordinates) can be negative since
$\omega_{r-1}=\frac{1}{2}(1,1,\ldots,1,-1)$.

Case 1: $r-1,r\notin S^{\prime}$. All of the $Z_{1},\ldots,Z_{m}$ are of type
$A$. As above, we have $\sum_{i=1}^{n}{z_{i}+1\choose 2}=|\Phi^{\prime}_{+}|$
fixed coordinates so
$\dim\operatorname{BZ}_{\lambda}\leq|\Phi_{+}|-|\Phi^{\prime}_{+}|$.

We need some different treatment in this case to obtain an interior point
$T\in\operatorname{BZ}_{\lambda}^{\circ}$. Notice that
$\lambda_{1,r-1}=\frac{1}{2}(d_{r-1}+d_{r})$ and
$\lambda_{1,r}=\frac{1}{2}(d_{r}-d_{r-1})$. Fix a positive number
$\delta<<d_{r-1},d_{r}$. Set

$\begin{cases}\eta_{1,r-1}=\frac{1}{2}(d_{r-1}+d_{r})-2\delta,\\\
\lambda_{2,r-1}=\frac{1}{2}(d_{r-1}+d_{r})-\delta,\\\
\lambda_{2,r}=\frac{1}{2}(d_{r-1}+d_{r})-3\delta,\\\
\lambda_{i,j}=\frac{1}{2}(\eta_{i-1,j-1}+\eta_{i-1,j})\text{ for other
}i,j\text{'s with }\lambda_{i,r}=\frac{1}{2}\eta_{i-1,r-1},\\\
\eta_{i,j}=\frac{1}{2}(\lambda_{i,j}+\lambda_{i,j+1})\text{ for other
}i,j\text{'s}.\end{cases}$

We directly see that
$\eta_{1,r-2}>\lambda_{2,r-1}>\eta_{1,r-1}>\lambda_{2,r}$, where the first
inequality holds as
$\eta_{1,r-2}\geq\lambda_{1,r-1}=\frac{1}{2}(d_{r-1}+d_{r})$, and that
$\lambda_{1,r-1}>\eta_{1,r-1}>\lambda_{1,r}$. We now check the more obscure-
looking inequality (7)

$\lambda_{i,r}+\lambda_{i+1,r}+\min(\lambda_{i,r-1},\lambda_{i+1,r-1})\geq\eta_{i,r-1}.$

When $i=1$, we have
$\min(\lambda_{1,r-1},\lambda_{2,r-1})=\lambda_{2,r-1}=\frac{1}{2}(d_{r-1}+d_{r})-\delta$.
The left hand side equals $\frac{1}{2}d_{r-1}+\frac{3}{2}d_{r}-4\delta$ while
the right hand side equals $\frac{1}{2}d_{r-1}+\frac{1}{2}d_{r}-2\delta$, so
we indeed have a strict inequality. Moreover, since $\lambda_{2,r}>0$, by the
averaging construction, $\lambda_{i,j}>0$ for $i>1$ and $\eta_{i,j}>0$ for all
$i,j$. The only potentially non-positive entry is $\lambda_{1,r}$. As
$\min(\lambda_{i,r-1},\lambda_{i+1,r-1})\geq\eta_{i,r-1}$, this inequality (7)
is always strict when $i\geq 2$ because $\lambda_{i,r},\lambda_{i+1,r}>0$.

As a result, we now constructed an interior point
$T\in\operatorname{BZ}_{\lambda}^{\circ}$, and all inequalities involved with
non-fixed coordinates are strict. As before, there is a hypercube of dimension
$|\Phi_{+}|-|\Phi^{\prime}_{+}|$ around $T$ of side length $\epsilon<<\delta$.
See Figure 2 for an example of this construction.

$\begin{pmatrix}4&&3&&3&&3&&-2\\\
&7/2&&\underline{3}&&\underline{3}&&3{-}2\delta&\\\
&&13/4&&\underline{3}&&3{-}\delta&&3{-}3\delta\\\
&&&25/8&&3{-}\frac{1}{2}\delta&&3{-}2\delta&\\\ &&&&\cdots&&&&\end{pmatrix}$

Figure 2. An interior point in $\operatorname{BZ}_{\lambda}$ of type $D$,
where $\lambda=(4,3,3,3,-2)$ with $S^{\prime}=\\{2,3\\}$. The fixed
coordinates are underlined.

Case 2: $r-1\in S^{\prime}$ and $r\notin S^{\prime}$ Now $\lambda$ has no
contribution from $\omega_{r-1}=\frac{1}{2}(1,1,\ldots,1,-1)$ and positive
contribution from $\omega_{r}=\frac{1}{2}(1,1,\ldots,1)$ so all coordinates
$\lambda_{1,1},\ldots,\lambda_{1,r}$ are strictly positive. Therefore, as
$\min(\lambda_{i,r-1},\lambda_{i+1,r-1})\geq\eta_{i,r-1}$, inequalities (7)
always hold and are always strict, for all $i$. This is the simplest case,
which follows the same argument as in case 2 of type $B/C$ (and in fact type
$A$). There are $|\Phi^{\prime}_{+}|$ fixed coordinates coming from the
triangles below the same values in $\lambda_{1,1},\ldots,\lambda_{1,r}$ so
$\dim\operatorname{BZ}_{\lambda}\leq|\Phi_{+}|-|\Phi^{\prime}_{+}|$. We again
use the averaging technique to create an interior point
$T\in\operatorname{BZ}_{\lambda}^{\circ}$ and find a neighborhood of dimension
$|\Phi_{+}|-|\Phi^{\prime}_{+}|$ around $T$, as all inequalities involving the
coordinates that are not fixed are always strict. We conclude that
$\dim\operatorname{BZ}_{\lambda}=|\Phi_{+}|-|\Phi^{\prime}_{+}|$.

Case 3: $r-1\notin S^{\prime}$ and $r\in S^{\prime}$. All the $Z_{i}$’s are of
type $A$. We have $\lambda_{1,r-1}+\lambda_{1,r}=0$ where $\lambda_{1,r-1}>0$
and $\lambda_{1,r}<0$. More specifically,
$\lambda_{1,r-z_{m}}=\cdots=\lambda_{1,r-1}=-\lambda_{1,r}>0$. Let this value
be $t=\frac{1}{2}d_{r-1}=\lambda_{1,r-1}$. The inequality (7) says that
$\lambda_{1,r-1}+\lambda_{1,r}+\lambda_{2,r}\geq\eta_{1,r-1}$, which
simplifies to $t-t+\lambda_{2,r}\geq\eta_{1,r-1}$,
$\lambda_{2,r}\geq\eta_{1,r-1}$. At the same time,
$\lambda_{2,r}\leq\eta_{1,r-1}$ so $\lambda_{2,r}=\eta_{1,r-1}$. If $z_{m}=1$,
then this equality already reduces the dimension by $1$, which is the number
of positive roots in a type $A_{1}$ root system. If $z_{m}>1$, we further have
$\lambda_{2,r-1}+\lambda_{2,r}+\lambda_{1,r}\geq\eta_{1,r-1}$, which
simplifies to $\lambda_{2,r-1}\geq t$. When $z_{m}>1$,
$\lambda_{2,r-1}\leq\eta_{1,r-2}\leq\lambda_{1,r-2}=t$ so $\lambda_{2,r-1}=t$.
As a result, in the triangle below $\lambda_{1,r-z_{m}},\ldots,\lambda_{1,r}$
with ${z_{m}+1\choose 2}$ entries, all coordinates except $\eta_{1,r-1}$ are
fixed to be $t$. Since the equality $\eta_{1,r-1}=\lambda_{2,r}\in[-t,t]$
reduces the dimension by $1$, there is a decrease of ${z_{m}+1\choose 2}$ in
the dimension. The other $Z_{1},\ldots,Z_{m-1}$ are dealt with exactly as
before. Therefore,
$\dim\operatorname{BZ}_{\lambda}\leq|\Phi_{+}|-|\Phi_{+}^{\prime}|$.

To obtain a point $T\in\operatorname{BZ}_{\lambda,\mu}^{\circ}$, we use the
averaging construction
$\lambda_{i,j}=\frac{1}{2}(\eta_{i-1,j-1}+\eta_{i-1,j})$ where
$\lambda_{i,r}=\frac{1}{2}\eta_{i,r}$ and
$\eta_{i,j}=\frac{1}{2}(\lambda_{i,j}+\lambda_{i,j+1})$ for $z_{m}\geq 2$. In
the edges case where $z_{m}=1$, we slightly modify $\lambda_{2,r-1}=t+\delta$
for $0<\delta<<t$, and use the averaging construction for other coordinates.
Again, all inequalities involving non-fixed entries are all strict by
considering $\eta_{1,r-1}=\lambda_{2,r}$ as the same variable, and we can thus
find a neighborhood of dimension $|\Phi_{+}|-|\Phi_{+}^{\prime}|$ around $T$
to conclude
$\dim\operatorname{BZ}_{\lambda}\geq|\Phi_{+}|-|\Phi_{+}^{\prime}|$. See
Figure 4 for this construction.

$\setcounter{MaxMatrixCols}{12}\begin{pmatrix}3&&2&&1&&1&&1&&-1\\\
&5/2&&3/2&&\underline{1}&&\underline{1}&&\underline{0}&\\\
&&2&&5/4&&\underline{1}&&\underline{1}&&0\\\
&&&13/8&&9/8&&\underline{1}&&1/2&\\\ &&&&11/8&&17/16&&3/4&&1/4\\\
&&&&&\cdots&&&&&&\end{pmatrix}$

Figure 3. An interior point in $\operatorname{BZ}_{\lambda}$ of type $D$,
where $\lambda=(3,2,1,1,1,-1)$ with $S^{\prime}=\\{3,4,6\\}$. The fixed
coordinates are underlined.

Case 4: both $r-1$ and $r$ are in $S^{\prime}$. This means
$\lambda_{1,r-1}=\lambda_{1,r}=0$. If $r-2\in S^{\prime}$, then $Z_{m}$ is of
type $D_{z_{m}}$ where $z_{m}\geq 3$. If $r-2\notin S^{\prime}$, then $r-1$
and $r$ are in their own connected components, in which case we let
$Z_{m}=\\{r-1,r\\}$ regardless. Note that a type $D_{z}$ root system has
$z^{2}-z$ positive roots, and this is also true if we adopt the convention
that a type $D_{2}$ root system is isomorphic to $A_{1}\oplus A_{1}$. In this
case here, we let $r-1,r\in Z_{m}$, which is a root system of type
$D_{z_{m}}$. As before, $Z_{1},\ldots,Z_{m-1}$ are of type $A$ and can be
dealt similarly as before.

Now $\lambda_{1,r-z_{m}+1},\ldots,\lambda_{1,r}$ all equal to $0$. We then use
induction on the row number $i$ to show that $\lambda_{i,j}=0$, $\eta_{i,j}=0$
for $j-i\geq r-z_{m}$. The base case $i=1$ is given by the assumption. If
$\lambda_{i,j}=0$ for a fixed $i$ and $j\geq r-z_{m}+i$, then we have
$\eta_{i,j}=0$ since it is between $\lambda_{i,j}$ and $\lambda_{i,j+1}$,
which are both $0$. If $\eta_{i,j}=0$ for a fixed $i$ and $j\geq r-z_{m}+i$,
then we similarly have $\eta_{i,j}\leq\lambda_{i+1,j+1}\leq\eta_{i,j+1}$ so
$\lambda_{i+1,j+1}=0$ for all $j+1<r$. It suffices to argue that
$\lambda_{i+1,r}=0$. By the inequality (7),
$\lambda_{i+1,r}+\lambda_{i+1,r-1}+\lambda_{i,r}\geq\eta_{i,r-1}$, which
simplifies to $\lambda_{i+1,r}\geq 0$. At the same time,
$\lambda_{i+1,r}\leq\eta_{i,r-1}=0$. Therefore, $\lambda_{i+1,r}=0$ as
desired. The induction step goes through. The number of resulting fixed
coordinates is $z_{m}+z_{m}+(z_{m}-1)+(z_{m}-1)+\cdots+1+1=z_{m}^{2}-z_{m}$,
which is the number of positive roots in a type $D_{z_{m}}$ root system.

We have now $|\Phi^{\prime}_{+}|$ fixed coordinates. To obtain
$T\in\operatorname{BZ}_{\lambda}^{\circ}$, we apply the averaging construction
$\lambda_{i,j}=\frac{1}{2}(\eta_{i-1,j-1}+\eta_{i-1,j})$,
$\eta_{i,j}=\frac{1}{2}(\lambda_{i,j}+\lambda_{i,j+1})$ and notice that all
inequalities involving unfixed coordinates are strict to conclude
$\dim\operatorname{BZ}_{\lambda}=|\Phi_{+}|-|\Phi^{\prime}_{+}|$. See Figure
4.

$\begin{pmatrix}2&&1&&0&&0&&0\\\ &3/2&&1/2&&\underline{0}&&\underline{0}&\\\
&&1&&1/4&&\underline{0}&&\underline{0}\\\ &&&5/8&&1/8&&\underline{0}&\\\
&&&&3/8&&1/16&&\underline{0}\\\ &&&&&\cdots&&&&\end{pmatrix}$

Figure 4. An interior point in $\operatorname{BZ}_{\lambda}$ of type $D$,
where $\lambda=(2,1,0,0,0)$ with $S^{\prime}=\\{3,4,5\\}$. The fixed
coordinates are underlined.

∎

###  3.3. Dimension of the BZ-polytope $\operatorname{BZ}_{\lambda,\mu}$

Let $\lambda$ be a dominant weight in a root system $\Phi$ of rank $r$ as
above. Consider the linear projection
$\operatorname{wt}:\operatorname{BZ}_{\lambda}\rightarrow\mathbb{R}^{r}$ which
sends a BZ-pattern to its weight (See Definition 2.4). Denote the image of
$\operatorname{wt}$ by $P_{\lambda}$, called the _weight polytope_ in the
literature. Recall that we have defined the $\operatorname{BZ}$-polytope
$\operatorname{BZ}_{\lambda,\mu}$ to be $\operatorname{wt}^{-1}(\mu)$. The
following theorem is well-known, see for example [4].

######  Theorem 3.7.

The image of $P_{\lambda}=\operatorname{wt}(\operatorname{BZ}_{\lambda})$, is
a polytope defined by

$P_{\lambda}=\\{\mu\in\mathbb{R}^{r}\>|\>\langle\lambda,\omega_{i}\rangle\geq\langle\mu,u\omega_{i}\rangle,\text{
for all }i=1,\ldots,r\text{ and }u\in W\\},$

or equivalently, by

$P_{\lambda}=\mathrm{Conv}(\\{u\lambda\>|\>u\in W\\}),$

where $W$ is the Weyl group. Moreover, $V_{\lambda}(\mu)\neq 0$ if and only if
$\mu\in P_{\lambda}$ and that $\lambda-\mu$ is a nonnegative integral linear
combination of the simple roots.

We need another lemma that is purely polytopal.

######  Lemma 3.8 ([11], Lemma 3.1).

Given a polytope $P\in\subset\mathbb{R}^{m}$ and a linear map
$\pi:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$, we have
$(\pi(P))^{\circ}\subset\pi(P^{\circ})$, where $P^{\circ}$ is the interior of
$P$.

We are now ready to deduce the dimension of $\operatorname{BZ}_{\lambda,\mu}$.

######  Lemma 3.9.

Let $(\lambda,\mu)$ be a primitive pair of dominant weights in a root system
$\Phi$ of rank $r$. Then
$\dim\operatorname{BZ}_{\lambda,\mu}=\dim\operatorname{BZ}_{\lambda}-r$.

###### Proof.

Since $\mu$ is dominant,
$\langle\mu,u\omega_{i}\rangle\leq\langle\mu,\omega_{i}\rangle$ for all $u\in
W$ and $i=1,\ldots,r$. Since $(\lambda,\mu)$ is a primitive pair,
$\lambda-\mu$ is a positive linear combination of simple roots, meaning that
$\langle\lambda-\mu,\omega_{i}\rangle>0$ for all $i$. As a result,
$\langle\lambda,\omega_{i}\rangle>\langle\mu,u\omega_{i}\rangle$ for all $u\in
W$ and $i=1,\ldots,r$. All the defining inequalities of $P_{\lambda}$ in
Theorem 3.7 are strictly for $\mu$, so $\mu\in P_{\lambda}^{\circ}$. Recall
$P_{\lambda}=\operatorname{wt}(\operatorname{BZ}_{\lambda})$. By Lemma 3.8,
$\mu\in(\operatorname{wt}(\operatorname{BZ}_{\lambda}))^{\circ}\subset\operatorname{wt}(\operatorname{BZ}_{\lambda}^{\circ})$.
Thus, there exists a BZ-pattern $T\in\operatorname{BZ}_{\lambda}^{\circ}$ such
that $\operatorname{wt}(T)=\mu$. Let the coordinates of $T$ be
$\\{\lambda_{i,j}\\}\cup\\{\eta_{i,j}\\}$ as before.

As a quick summary, we have a point $T$ in the interior of
$\operatorname{BZ}_{\lambda}$ with weight $\mu$. Now the polytope
$\operatorname{BZ}_{\lambda,\mu}$ can be viewed by intersecting
$\operatorname{BZ}_{\lambda}$ with $r$ hyperplanes $H_{1},\ldots,H_{r}$ which
are the $r$ defining equations (see Definition 2.4) of the weight $\mu$ that
all pass through $T$. One directly sees that these defining equations are
upper triangular with respect to the coordinates
$\\{\lambda_{i,j}\\}\cup\\{\eta_{i,j}\\}$ so the normals of
$H_{1},\ldots,H_{r}$ are linearly independent and $K:=H_{1}\cap\cdots\cap
H_{r}$ has codimension $r$. As the linear subspace $K$ passes through an
interior point of $\operatorname{BZ}_{\lambda}$,
$\operatorname{BZ}_{\lambda,\mu}=K\cap\operatorname{BZ}_{\lambda}$ must have
codimension $r$ in $\operatorname{BZ}_{\lambda}$. Therefore,
$\dim\operatorname{BZ}_{\lambda,\mu}=\dim\operatorname{BZ}_{\lambda}-r$ as
desired. ∎

###  3.4. Finishing the proof

We use the following classical result in rational Ehrhart theory.

######  Theorem 3.10 ([6, 12]).

Let $Q$ be a rational polytope, i.e. the convex hull of finitely many points
in $\mathbb{Q}^{n}$. Then $|NQ\cap\mathbb{Z}^{n}|$ is a quasi-polynomial in
$N$ of degree $\dim Q$, for a positive integer $N$.

We are now ready for the final step of our main proof.

######  Lemma 3.11.

Let $(\lambda,\mu)$ be a primitive pair of dominant weight in a root system of
classical type. Then $\deg
K_{\lambda,\mu}(N)=\dim\operatorname{BZ}_{\lambda,\mu}$.

###### Proof.

From Definition 2.5, we know that $\operatorname{BZ}_{\lambda,\mu}$ is bounded
by equalities and inequalities with integral coefficients, and is thus a
rational polytope. Since all the equalities and inequalities involved do not
have constant terms,
$\operatorname{BZ}_{N\lambda,N\mu}=N\operatorname{BZ}_{\lambda,\mu}$ for a
positive integer $N$.

By Definition 2.6 and Theorem 2.7, in type $C_{r}$,
$K_{\lambda,\mu}(N)=K_{N\lambda,N\mu}=|\mathbb{Z}^{|\Phi_{+}|}\cap
N\operatorname{BZ}_{\lambda,\mu}|$ which is a quasi-polynomial in $N$ of
degree $\dim\operatorname{BZ}_{\lambda,\mu}$. In type $D_{r}$,

$K_{\lambda,\mu}(N)=K_{N\lambda,N\mu}=\left|\mathbb{Z}^{|\Phi_{+}|}\cap
N\operatorname{BZ}_{\lambda,\mu}\right|+\left|(\mathbb{Z}+\frac{1}{2})^{|\Phi_{+}|}\cap
N\operatorname{BZ}_{\lambda,\mu}\right|$

which is the sum of two quasi-polynomials in $N$ of degree
$\dim\operatorname{BZ}_{\lambda,\mu}$ (by properly shifting the coordinate
axis). As for type $B_{r}$, we use the same argument in type $D_{r}$ after
scaling the coordinates $\eta_{i,r}$ by a factor of $2$, for $i=1,\ldots,r$. ∎

Finally, the proof of Theorem 1.2 follows from Corollary 3.5, Lemma 3.6, Lemma
3.9 and Lemma 3.11.

## Acknowledgements

This project is initiated during the 2022 Graduate Research Workshop in
Combinatorics (GRWC) hosted by University of Denver and CU Denver. We are
really grateful for the opportunities created by this workshop. We would like
to thank Tyrrell McAllister and Igor Pak for helpful conversations. SG was
partially supported by NSF RTG grant DMS 1937241 and NSF Graduate Research
Fellowship under grant No. DGE-1746047.

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