On volume functions of special flow polytopes

TL;DR

This paper investigates the volume of specific flow polytopes, demonstrating it satisfies differential equations and providing an inductive formula related to root system rank, advancing understanding of their geometric properties.

Contribution

It establishes a differential equation framework for flow polytope volumes and derives an inductive formula based on root system rank, offering new computational tools.

Findings

Volume satisfies a unique differential system

Inductive formula for volume based on root system rank

Provides a new approach to computing flow polytope volumes

Abstract

In this paper, we consider the volume of a special kind of flow polytope. We show that its volume satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a constant multiple. In addition, we give an inductive formula for the volume with respect to the rank of the root system of type A.

Equations

$$A_r^{+}=\{e_i-e_j|1\le i<j\le r+1\}$$

$$\displaystyle P_{A_{r}^{+},m}(a)=\left\{(y_{i,j,k})\in\mathbb{R}^{M}\,\middle|\,\begin{array}[]{l}1\leq i<j\leq r+1\,,\,\,1\leq k\leq m_{i,j}\,,\\ y_{i,j,k}\geq 0\,,\,\,\sum_{1\leq i<j\leq r+1}\sum_{1\leq k\leq m_{i,j}}y_{i,j,k}(e_{i}-e_{j})=a\end{array}\right\}.$$

$${\mathfrak{c}}_{\text{nice}}:=\{a=a_1{e}_1+\cdots +a_r{e}_r-(a_1+\cdots +a_r)e_{r+1}\in {\mathbb{R}}^{r+1}|a_i>0,i=1,\dots ,r\}$$

$$\{\begin{array}{l}{\partial }_r^{m_{r,r+1}}v=0\\ ({\partial }_{r-1}-{\partial }_r{)}^{m_{r-1,r}}{\partial }_{r-1}^{m_{r-1,r+1}}v=0\\ ⋮\\ ({\partial }_1-{\partial }_2{)}^{m_{1,2}}({\partial }_1-{\partial }_3{)}^{m_{1,3}}\cdots ({\partial }_1-{\partial }_r{)}^{m_{1,r}}{\partial }_1^{m_{1,r+1}}v=0,\end{array}$$

$$L{v}_{A_r^{+},m}(a)=\delta (a)$$

$$V=\left\{a=\sum _{i=1}^{r+1}{a}_i{e}_i\in {\mathbb{R}}^{r+1}|\sum _{i=1}^{r+1}{a}_i=0\right\}.$$

$$A_r^{+}=\{e_i-e_j|1\le i<j\le r+1\}.$$

$$C(A_r^{+})=\{a=a_1{e}_1+\cdots +a_r{e}_r-(a_1+\cdots +a_r)e_{r+1}|a_1,\dots ,a_r\in {\mathbb{R}}_{\ge 0}\}.$$

$$\displaystyle P_{A_{r}^{+},m}(a)=\left\{(y_{i,j,k})\in\mathbb{R}^{M}\,\middle|\,\begin{array}[]{l}1\leq i<j\leq r+1\,,\,\,1\leq k\leq m_{i,j}\,,\\ y_{i,j,k}\geq 0\,,\,\,\sum_{1\leq i<j\leq r+1}\sum_{1\leq k\leq m_{i,j}}y_{i,j,k}(e_{i}-e_{j})=a\end{array}\right\},$$

$$f_{\sigma }(x):=\frac{1}{{\prod }_{\alpha \in \sigma }\alpha (x)}$$

$$R_{A_r}=\partial (U)R_{A_r}\oplus S_{A_r}.$$

$$F:=\frac{e^{a_1{x}_1+\cdots +a_r{x}_r}}{{\prod }_{i=1}^r{x}_i^{m_{i,r+1}}{\prod }_{1\le i<j\le r}(x_i-x_j{)}^{m_{i,j}}}$$

$$J_{A_r^{+},m}(a)={\mathrm{Tres}}_{A_r}F.$$

$${\mathrm{Ires}}_{x=0}f={\mathrm{Res}}_{x_1=0}{\mathrm{Res}}_{x_2=0}\cdots {\mathrm{Res}}_{x_r=0}f(x_1,\dots ,x_r).$$

$${\mathrm{Ires}}_{x=0}{J}_{A_r^{+},m}(a)={\mathrm{Ires}}_{x=0}F.$$

$$v_{A_r^{+},m}(a)=⟨⟨\mathfrak{c},J_{A_r^{+},m}(a)⟩⟩.$$

$${\mathfrak{c}}_{\mathrm{nice}}=\{a\in C(A_r^{+})|a_i>0(i=1,\dots ,r)\}.$$

$$⟨⟨{\mathfrak{c}}_{\mathrm{nice}},f⟩⟩={\mathrm{Ires}}_{x=0}f.$$

$$v_{A_r^{+},m,{\mathfrak{c}}_{\mathrm{nice}}}(a)={\mathrm{Ires}}_{x=0}F.$$

$$P_{A_1^{+},m}(a)=\left\{(y_{i,j,k})\in {\mathbb{R}}^{m_{1,2}}|y_{i,j,k}\ge 0,y_{1,2,1}+y_{1,2,2}+\cdots +y_{1,2,m_{1,2}}=a_1\right\}.$$

$$v_{A_1^{+},m,{\mathfrak{c}}_{\mathrm{nice}}}(a)$$

$$=\frac{1}{(m_{1,2}-1)!}{a}_1^{m_{1,2}-1}.$$

$$\displaystyle P_{A_{2}^{+},m}(a)=\left\{(y_{i,j,k})\in\mathbb{R}^{n+2}\,\middle|\,\begin{array}[]{ll}y_{i,j,k}\geq 0\\ y_{1,2,1}+y_{1,2,2}+\cdots+y_{1,2,n}+y_{1,3,1}=a_{1}\\ -y_{1,2,1}-y_{1,2,2}-\cdots-y_{1,2,n}+y_{2,3,1}=a_{2}\end{array}\right\}.$$

$$v_{A_2^{+},m,{\mathfrak{c}}_{\mathrm{nice}}}(a)$$

$$={\mathrm{Res}}_{x_1=0}{\mathrm{Res}}_{x_2=0}\left(\frac{e^{a_1{x}_1+a_2{x}_2}}{x_1{x}_2(x_1-x_2{)}^n}\right)$$

$$=\frac{1}{n!}{a}_1^n.$$

$$\displaystyle P_{A_{3}^{+},m}(a)=\left\{(y_{i,j,k})\in\mathbb{R}^{9}\,\middle|\,\begin{array}[]{ll}y_{i,j,k}\geq 0\\ y_{1,2,1}+y_{1,3,1}+y_{1,4,1}+y_{1,4,2}=a_{1}\\ -y_{1,2,1}+y_{2,3,1}+y_{2,4,1}+y_{2,4,2}=a_{2}\\ -y_{1,3,1}-y_{2,3,1}+y_{3,4,1}+y_{3,4,2}=a_{3}\end{array}\right\}.$$

$$v_{A_3^{+},m,{\mathfrak{c}}_{\mathrm{nice}}}(a)$$

$$=\frac{1}{360}{a}_1^3(a_1^3+6a_1^2{a}_2+3a_1^2{a}_3+15a_1{a}_2^2+15a_1{a}_2{a}_3+10a_2^3+30a_2^2{a}_3).$$

$$\{\begin{array}{l}{\partial }_r^{m_{r,r+1}}v=0\\ ({\partial }_{r-1}-{\partial }_r{)}^{m_{r-1,r}}{\partial }_{r-1}^{m_{r-1,r+1}}v=0\\ ⋮\\ ({\partial }_1-{\partial }_2{)}^{m_{1,2}}({\partial }_1-{\partial }_3{)}^{m_{1,3}}\cdots ({\partial }_1-{\partial }_r{)}^{m_{1,r}}{\partial }_1^{m_{1,r+1}}v=0,\end{array}$$

$$P({\partial }_1,\dots ,{\partial }_r)({\mathrm{Ires}}_{x=0}F)$$

$$={\mathrm{Ires}}_{x=0}(P(x_1,\dots ,x_r)F)$$

$${\partial }_r^{m_{r,r+1}}v$$

Full text

1 Introduction

The number of lattice points and the volume of a convex polytope are important and interesting objects and have been studied from various points of view (see, e.g., [4]). For example, the number of lattice points of a convex polytope associated to a root system is called the Kostant partition function, and it plays an important role in representation theory of Lie groups (see, e.g., [7]).

In this paper, we consider a convex polytope associated to the root system of type $A$, which is called a flow polytope. As explained in [2, 3], the cone spanned by the positive roots is divided into several polyhedral cones called chambers, and the combinatorial property of a flow polytope depends on a chamber. Moreover, there is a specific chamber called the nice chamber, which plays a significant role in [9]. Also in [2, 3], a number of theoretical results related to the Kostant partition function and the volume function of a flow polytope can be found. In particular, it is shown that these functions for the nice chamber are written as iterated residues ([3, Lemma 21]). We also refer to [1] for similar formulas for other chambers in more general settings.

The purpose of this paper is to characterize the volume function of a flow polytope for the nice chamber in terms of a system of differential equations, based on a result in [3]. In order to state the main results, we give some notation. Let $e_{i}$ be the standard basis of $\mathbb{R}^{r+1}$ and let

[TABLE]

be the positive root system of type $A$ with rank $r$. We assign a positive integer $m_{i,j}$ to each $i$ and $j$ with $1\leq i<j\leq r+1$. Let us set $m=(m_{i,j})$ and $M=\sum_{1\leq i<j\leq r+1}m_{i,j}$. For $a=a_{1}e_{1}+\cdots+a_{r}e_{r}-(a_{1}+\cdots+a_{r})e_{r+1}\in\mathbb{R}^{r+1}$, where $a_{i}\in\mathbb{R}_{\geq 0}\,\,(i=1,\dots,r)$, the following polytope $P_{A_{r}^{+},m}(a)$ is called the flow polytope:

[TABLE]

Note that the flow polytopes in [3] include the case that some of $m_{i,j}$’s are zero, whereas we exclude such cases in this paper. We denote the volume of $P_{A_{r}^{+},m}(a)$ by $v_{A_{r}^{+},m}(a)$.

The open set

[TABLE]

in ${\mathbb{R}}^{r+1}$ is called the nice chamber. We are interested in the volume $v_{A_{r}^{+},m}(a)$ when $a$ is in the closure of the nice chamber, and then it is written by $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}$. It is a homogeneous polynomial of degree $M-r$. The first result of this paper is the following.

Theorem 1.1

Let $a=\sum_{i=1}^{r}a_{i}(e_{i}-e_{r+1})\in\overline{\mathfrak{c}_{{\rm nice}}}$, and let $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$ be the volume of $P_{A_{r}^{+},m}(a)$. Then $v=v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$ satisfies the system of differential equations as follows:

[TABLE]

where $\partial_{i}=\frac{\partial}{\partial a_{i}}$ for $i=1,\dots,r$. Conversely, the polynomial $v=v(a)$ of degree $M-r$ satisfying the above equations is equal to a constant multiple of $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$.

We remark that it is known the volume function $v_{A_{r}^{+},m}(a)$ of $P_{A_{r}^{+},m}(a)$, as a distribution on $\mathbb{R}^{r}$, satisfies the differential equation

[TABLE]

in general, where $L=\prod_{i<j}(\partial_{i}-\partial_{j})^{m_{i,j}}$ and $\delta(a)$ is the Dirac delta function on $\mathbb{R}^{r}$ ([6, 9]). Note that $\partial_{r+1}$ in the definition of $L$ is supposed to be zero. The above theorem characterizes the function $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$ on $\overline{\mathfrak{c}_{\rm nice}}$ more explicitly.

In addition, in Theorem 3.6, we show the volume $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$ is written by a linear combination of $v_{A_{r-1}^{+},m^{\prime},\mathfrak{c}^{\prime}_{\rm nice}}(a)$ and its partial derivatives, where $m^{\prime}=(m_{i,j})_{2\leq i<j\leq r+1}$ and $\mathfrak{c}^{\prime}_{\rm nice}$ is the nice chamber of $A_{r-1}^{+}$.

This paper is organized as follows. In Section 2, we recall the iterated residue, the Jeffrey-Kirwan residue, and the nice chamber based on [2], [3], [5] and [8]. Also, we give the some examples of $P_{A_{r}^{+},m}(a)$ and the calculations of the volume $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$. In Section 3, we prove the main theorems.

2 Preliminaries

In this section, we set up the tools to prove the main theorems based on [2], [3], [5] and [8].

2.1 Flow polytopes and its volumes

Let $e_{1},\dots,e_{r+1}$ be the standard basis of $\mathbb{R}^{r+1}$, and let

[TABLE]

We consider the positive root system of type $A$ with rank $r$ as follows:

[TABLE]

Let $C(A_{r}^{+})$ be the convex cone generated by $A_{r}^{+}$:

[TABLE]

We assign a positive integer $m_{i,j}$ to each $i$ and $j$ with $1\leq i<j\leq r+1$, and it is called a multiplicity. Let us set $m=(m_{i,j})$ and $M=\sum_{1\leq i<j\leq r+1}m_{i,j}$.

Definition 2.1

Let $a=a_{1}e_{1}+\cdots+a_{r}e_{r}-(a_{1}+\cdots+a_{r})e_{r+1}\in C(A_{r}^{+})$. We consider the following polytope:

[TABLE]

which is called the flow polytope.**

Remark 2.2

The flow polytopes in [3] include the case that $m_{i,j}=0$ for some $i$ and $j$.**

The elements of $A_{r}^{+}$ generate a lattice $V_{\mathbb{Z}}$ in $V$. The lattice $V_{\mathbb{Z}}$ determines a measure $da$ on $V$.

Let $du$ be the Lebesgue measure on $\mathbb{R}^{M}$. Let $[\alpha_{1},\dots,\alpha_{M}]$ be a sequence of elements of $A_{r}^{+}$ with multiplicity $m_{i,j}$, and let $\varphi$ be the surjective linear map from $\mathbb{R}^{M}$ to $V$ defined by $\varphi(e_{k})=\alpha_{k}$. The vector space $\ker(\varphi)=\varphi^{-1}(0)$ is of dimension $d=M-r$ and it is equipped with the quotient Lebesgue measure $du/da$. For $a\in V$, the affine space $\varphi^{-1}(a)$ is parallel to $\ker(\varphi)$, and thus also equipped with the Lebesgue measure $du/da$. Volumes of subsets of $\varphi^{-1}(a)$ are computed for this measure. In particular, we can consider the volume $v_{A_{r}^{+},m}(a)$ of the polytope $P_{A_{r}^{+},m}(a)$.

2.2 Total residue and iterated residue

Let $A_{r}=A_{r}^{+}\cup(-A_{r}^{+})$, and let $U$ be the dual vector space of $V$. We denote by $R_{A_{r}}$ the ring of rational functions $f(x_{1},\dots,x_{r})$ on the complexification $U_{\mathbb{C}}$ of $U$ with poles on the hyperplanes $x_{i}-x_{j}=0\,\,(1\leq i<j\leq r+1)$ or $x_{i}=0\,\,(1\leq i\leq r)$. A subset $\sigma$ of $A_{r}$ is called a basis of $A_{r}$ if the elements $\alpha\in\sigma$ form a basis of $V$. In this case, we set

[TABLE]

and call such a element a simple fraction. We denote by $S_{A_{r}}$ the linear subspace of $R_{A_{r}}$ spanned by simple fractions. The space $U$ acts on $R_{A_{r}}$ by differentiation: $(\partial(u)f)(x)=(\frac{d}{d\varepsilon})f(x+\varepsilon u)|_{\varepsilon=0}$. We denote by $\partial(U)R_{A_{r}}$ the space spanned by derivatives of functions in $R_{A_{r}}$. It is shown in [5, Proposition 7] that

[TABLE]

The projection map ${\rm Tres}_{A_{r}}:R_{A_{r}}\to S_{A_{r}}$ with respect to this decomposition is called the total residue map.

We extend the definition of the total residue to the space $\hat{R}_{A_{r}}$ consisting of functions $P/Q$ where $Q$ is a finite product of powers of the linear forms $\alpha\in A_{r}$ and $P=\sum_{k=0}^{\infty}P_{k}$ is a formal power series with $P_{k}$ of degree $k$. As the total residue vanishes outside the homogeneous component of degree $-r$ of $A_{r}$, we can define ${\rm Tres}_{A_{r}}(P/Q)={\rm Tres}_{A_{r}}(P_{q-r}/Q)$, where $q$ is degree of $Q$. For $a\in V$ and multiplicities $m=(m_{i,j})\in(\mathbb{Z}_{\geq 0})^{M}$ of elements of $A_{r}^{+}$, the function

[TABLE]

is in $\hat{R}_{A_{r}}$. We define $J_{A_{r}^{+},m}(a)\in S_{A_{r}}$ by

[TABLE]

Next, we describe the iterated residue.

Definition 2.3

For $f\in R_{A_{r}}$, we define the iterated residue by

[TABLE]

Since the iterated residue ${\rm Ires}_{x=0}f$ vanishes on the space $\partial(U)R_{A_{r}}$ as in [3], we have

[TABLE]

2.3 Chambers and Jeffrey–Kirwan residue

Definition 2.4

Let $C(\nu)$ be the closed cone generated by $\nu$ for any subset $\nu$ of $A_{r}^{+}$ and let $C(A_{r}^{+})_{{\rm sing}}$ be the union of the cones $C(\nu)$ where $\nu$ is any subset of $A_{r}^{+}$ of cardinal strictly less than $r=\dim V$. By definition, the set $C(A_{r}^{+})_{{\rm reg}}$ of $A_{r}^{+}$-regular elements is the complement of $C(A_{r}^{+})_{{\rm sing}}$. A connected component of $C(A_{r}^{+})_{{\rm reg}}$ is called a chamber.**

The Jeffrey–Kirwan residue [8] associated to a chamber $\mathfrak{c}$ of $C(A_{r}^{+})$ is a linear form $f\mapsto\langle\langle\mathfrak{c},f\rangle\rangle$ on the vector space $S_{A_{r}}$ of simple fractions. Any function $f$ in $S_{A_{r}}$ can be written as a linear combination of functions $f_{\sigma}$, with a basis $\sigma$ of $A_{r}$ contained in $A_{r}^{+}$. To determine the linear map $f\mapsto\langle\langle\mathfrak{c},f\rangle\rangle$, it is enough to determine it on this set of functions $f_{\sigma}$. So we assume that $\sigma$ is a basis of $A_{r}$ contained in $A_{r}^{+}$.

Definition 2.5

For a chamber $\mathfrak{c}$ and $f_{\sigma}\in S_{A_{r}}$, we define the Jeffrey–Kirwan residue $\langle\langle\mathfrak{c},f_{\sigma}\rangle\rangle$ associated to a chamber $\mathfrak{c}$ as follows:

• •

If $\mathfrak{c}\subset C(\sigma)$, then $\langle\langle\mathfrak{c},f_{\sigma}\rangle\rangle=1$.

• •

If $\mathfrak{c}\cap C(\sigma)=\emptyset$, then $\langle\langle\mathfrak{c},f_{\sigma}\rangle\rangle=0$,

where $C(\sigma)$ is the convex cone generated by $\sigma$.**

Remark 2.6

More generally, as in [3, Definition 11], the Jeffrey–Kirwan residue $\langle\langle\mathfrak{c},f_{\sigma}\rangle\rangle$ is defined to be $\frac{1}{{\rm vol}(\sigma)}$ if $\mathfrak{c}\subset C(\sigma)$, where ${\rm vol}(\sigma)$ is the volume of the parallelepiped $\oplus_{\alpha\in\sigma}[0,1]\alpha$, relative to our Lebesgue measure $da$. In our case, the volume ${\rm vol}(\sigma)$ is equal to $1$ since $A_{r}$ is unimodular.**

The volume $v_{A_{r}^{+},m}(a)$ of the flow polytope $P_{A_{r}^{+},m}(a)$ is written by the function $J_{A_{r}^{+},m}(a)$ and the Jeffrey–Kirwan residue in the following.

Theorem 2.7** ([3])**

Let $\mathfrak{c}$ be a chamber of $C(A_{r}^{+})$. Then, for $a\in\bar{\mathfrak{c}}$, the volume $v_{A_{r}^{+},m}(a)$ of $P_{A_{r}^{+},m}(a)$ is given by

[TABLE]

We denote by $v_{A_{r}^{+},m,\mathfrak{c}}(a)$ the polynomial function of $a$ coinciding with $v_{A_{r}^{+},m}(a)$ when $a\in\bar{\mathfrak{c}}$. It is a homogeneous polynomial of degree $M-r$.

2.4 Nice chamber

Definition 2.8

The open subset $\mathfrak{c}_{{\rm nice}}$ of $C(A_{r}^{+})$ is defined by

[TABLE]

The set $\mathfrak{c}_{{\rm nice}}$ is in fact a chamber for the root system $A_{r}^{+}$ ([3]). The chamber $\mathfrak{c}_{{\rm nice}}$ is called the nice chamber.**

Lemma 2.9** ([3])**

For the nice chamber $\mathfrak{c}_{{\rm nice}}$ of $A_{r}^{+}$ and $f\in S_{A_{r}}$, we have

[TABLE]

From Theorem 2.7, Lemma 2.9 and (2.1), we have the following corollary.

Corollary 2.10

Let $a\in\overline{\mathfrak{c}_{{\rm nice}}}$. Then the volume function $v_{A_{r}^{+},m,\mathfrak{c}_{{\rm nice}}}(a)$ is given by

[TABLE]

2.5 Examples

In this subsection, we give some examples of the flow polytopes for $A_{1},A_{2}$, and $A_{3}$, and calculate their volumes.

Example 2.11

When $r=1$, the nice chamber of $A_{1}^{+}$ is $\mathfrak{c}_{{\rm nice}}=\{a=a_{1}(e_{1}-e_{2})\,|\,a_{1}>0\}$. For $a=a_{1}(e_{1}-e_{2})\in\overline{\mathfrak{c}_{{\rm nice}}}$,**

[TABLE]

From Corollary 2.10, we have**

[TABLE]

Example 2.12

When $r=2$, there are two chambers $\mathfrak{c}_{1},\mathfrak{c}_{2}$ of $A_{2}^{+}$ as below, and the nice chamber $\mathfrak{c}_{{\rm nice}}$ of $A_{2}^{+}$ is $\mathfrak{c}_{1}$.**

e_{1}-e_{2}$$e_{1}-e_{3}$$e_{2}-e_{3}$$\mathfrak{c}_{2}$$\mathfrak{c}_{1}$$\mathbf{O}

Figure 1  :  The chamber of $A_{2}^{+}$.**

For example, we set $m_{1,2}=n\,\,(n\in\mathbb{Z}_{>0})$, $m_{1,3}=1$, and $m_{2,3}=1$. For $a=a_{1}e_{1}+a_{2}e_{2}-(a_{1}+a_{2})e_{3}\in\overline{\mathfrak{c}_{{\rm nice}}}$,**

[TABLE]

From Corollary 2.10, we have**

[TABLE]

Example 2.13

When $r=3$, there are seven chambers of $A_{3}^{+}$ as below ([1]), and the nice chamber $\mathfrak{c}_{{\rm nice}}$ of $A_{3}^{+}$ is $\mathfrak{c}_{1}$.**

e_{1}-e_{3}$$e_{2}-e_{4}$$e_{2}-e_{3}$$e_{1}-e_{4}$$e_{1}-e_{2}$$e_{3}-e_{4}$$\mathfrak{c}_{7}$$\mathfrak{c}_{1}$$\mathfrak{c}_{2}$$\mathfrak{c}_{3}$$\mathfrak{c}_{4}$$\mathfrak{c}_{5}$$\mathfrak{c}_{6}

Figure 2  :  The chamber of $A_{3}^{+}$.**

For example, we set $m_{1,2}=1$, $m_{1,3}=1$, $m_{1,4}=2$, $m_{2,3}=1$, $m_{2,4}=2$, and $m_{3,4}=2$. For $a=\sum_{i=1}^{3}a_{i}(e_{i}-e_{4})\in\overline{\mathfrak{c}_{{\rm nice}}}$,**

[TABLE]

From Corollary 2.10, we have**

[TABLE]

3 Main theorems

In this section, we prove the main theorems of this paper. Let $\mathfrak{c}_{{\rm nice}}$ be the nice chamber of $A_{r}^{+}$ and let $a=\sum_{i=1}^{r}a_{i}(e_{i}-e_{r+1})\in\overline{\mathfrak{c}_{{\rm nice}}}$.

Theorem 3.1

For $a\in\overline{\mathfrak{c}_{{\rm nice}}}$, let $P_{A_{r}^{+},m}(a)$ be the flow polytope as in Definition $2.1$ and let $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$ be the volume of $P_{A_{r}^{+},m}(a)$. Then $v=v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$ satisfies the system of differential equations as follows:

[TABLE]

where $\partial_{i}=\frac{\partial}{\partial a_{i}}$ for $i=1,\dots,r$.

Proof. Let $F=\frac{e^{a_{1}x_{1}+\cdots+a_{r}x_{r}}}{\prod_{i=1}^{r}x_{i}^{m_{i,r+1}}\prod_{1\leq i<j\leq r}(x_{i}-x_{j})^{m_{i,j}}}$. It is easy to see that

[TABLE]

where $P$ is a polynomial. Therefore, from Corollary 2.10, we obtain

[TABLE]

and

[TABLE]

where we used

[TABLE]

for $k=1,\dots,r$. Similarly, we can check the left expressions.    $\square$

Remark 3.2

In general, it is known that the volume function $v_{A_{r}^{+},m}(a)$ of $P_{A_{r}^{+},m}(a)$, as a distribution on $V$, satisfies the differential equation

[TABLE]

where $L=\prod_{i<j}(\partial_{i}-\partial_{j})^{m_{i,j}}$ and $\delta(a)$ is the Dirac delta function on $V$ ([6, 9]). Note that $\partial_{r+1}$ in the definition of $L$ is supposed to be zero. The above theorem, together with Proposition 3.3 and Theorem 3.4 as below, characterizes the function $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$ on $\overline{\mathfrak{c}_{\rm nice}}$ more explicitly.**

Let $M_{l}=\sum_{i=l+1}^{r+1}m_{l,i}\,\,(l=1,\dots,r)$. Then we have the following proposition.

Proposition 3.3

The coefficient of $a_{1}^{M_{1}-1}a_{2}^{M_{2}-1}\cdots a_{r-1}^{M_{r-1}-1}a_{r}^{M_{r}-1}$ in the volume function $v_{A_{r}^{+},m}(a)$ is given by

[TABLE]

Proof. From Corollary 2.10, we have

[TABLE]

where $|i|=i_{1}+\cdots+i_{r}$. When $i_{l}=M_{l}-1$ for $l=1,\dots,r$,

[TABLE]

Thus we obtain the proposition.    $\square$

Theorem 3.4

Let $\phi_{r}=\phi(a_{1},\dots,a_{r})$ be a homogeneous polynomial of $a_{1},\dots,a_{r}$ with degree $d$ and let $M=\sum_{1\leq i<j\leq r+1}m_{i,j}$. Suppose $\phi_{r}$ satisfies the system of differential equations as follows:

[TABLE]

• $(1)$

If $M-r<d$, then $\phi_{r}=0$.

• $(2)$

If $0\leq d\leq M-r$, then there is a non trivial homogeneous polynomial $\phi_{r}$ satisfying $(3.2)$.

• $(3)$

If $d=M-r$ in particular, $\phi_{r}$ is equal to a constant multiple of $v=v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$.

Proof. We argue by induction on $r$. In the case that $r=1$, we write

[TABLE]

where $p$ is a constant. If $m_{1,2}-1<d$ and $\phi_{1}$ satisfies the differential equation $\partial_{1}^{m_{1,2}}\phi_{1}=0$, then $p=0$ and hence $\phi_{1}=0$. If $0\leq d\leq m_{1,2}-1$, then for any $p\neq 0$, $\partial_{1}^{m_{1,2}}\phi_{1}=0$. Also, if $d=m_{1,2}-1$, in particular, then $\phi_{1}=pa_{1}^{m_{1,2}-1}$, while $v=\frac{1}{(m_{1,2}-1)!}a_{1}^{m_{1,2}-1}$ as in Example 2.11. Hence $\phi_{1}$ is equal to a constant multiple of $v$.

We assume that the statement of this theorem holds for $r-1$. We write $\phi_{r}$ as

[TABLE]

where $g_{k}$ is a homogeneous polynomial of $a_{2},\dots,a_{r}$ with degree $k$ for $k=0,1,\dots,d$. Then for $k=0,1,\dots,d$, $g_{k}$ satisfies the differential equations as follows:

[TABLE]

We set $h=(\sum_{2\leq i<j\leq r+1}m_{i,j})-(r-1)$. From the inductive assumption, if $0\leq k\leq h$, then $g_{k}$ is a homogeneous polynomial. On the other hand, if $h+1\leq k\leq d$, then $g_{k}=0$, namely,

[TABLE]

(1) We consider the case of $M-r<d$. Let $M_{1}=\sum_{i=2}^{r+1}m_{1,i}$. Now we compare the coefficients of $a_{1}^{d-h-M_{1}+n}$ in $(\partial_{1}-\partial_{2})^{m_{1,2}}(\partial_{1}-\partial_{3})^{m_{1,3}}\cdots(\partial_{1}-\partial_{r})^{m_{1,r}}\partial_{1}^{m_{1,r+1}}\phi_{r}$ for $n=0,\dots,h$. For $q=1,\dots,M_{1}-m_{1,r+1}$, we define

[TABLE]

Then we have the following equation:

[TABLE]

When $n=0$, from (3.4) and (3.6), we have

[TABLE]

When $n=1$, we have

[TABLE]

Thus we have

[TABLE]

Similarly, we have

[TABLE]

and hence $\phi_{r}=0$.

(2) We consider the case of $0\leq d\leq M-r$. By the inductive assumption, there is a non trivial homogeneous polynomial $g_{h-n_{1}+i}$ satisfying (3.3) for $i=1,\dots,n_{1}$, where $n_{1}=M-r-d+1$. We can take

[TABLE]

When $n=n_{1}$, from (3.4) and (3.6),

[TABLE]

When $n=n_{1}+1$,

[TABLE]

Similarly, for $n=n_{1}+2,\dots,h$, we can express $g_{h-j}(a_{2},\dots,a_{r})\,\,(j=n_{1},n_{1}+1,\dots,h)$ in terms of $g_{h-j+i}(a_{2},\dots,a_{r})\,\,(i=1,\dots,j)$ and their partial derivatives. Namely, we can express $\phi_{r}$ in terms of $g_{h-n_{1}+i}(a_{2},\dots,a_{r})$ and their partial derivatives. It follows that $\phi_{r}\neq 0$ when $0\leq d\leq M-r$.

(3) If $d=M-r$ in particular, then $n_{1}=1$, and $g_{h-j}\,\,(j=1,\dots,h)$ becomes linear combination of $g_{h}$ and their partial derivatives. Therefore $\phi_{r}$ is uniquely determined by $g_{h}$. Moreover, from the inductive assumption, $g_{h}=C\cdot v_{A_{r-1}^{+},m^{\prime},\mathfrak{c}^{\prime}_{\rm nice}}$, where $C$ is a constant, $m^{\prime}=(m_{i,j})_{2\leq i<j\leq r+1}$, and $\mathfrak{c}^{\prime}_{\rm nice}$ is a nice chamber of $A_{r-1}^{+}$. Hence the solution of (3.2) is unique up to a constant multiple. On the other hand, by Theorem 3.1, $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}$ satisfies the system of differential equations (3.2). Hence $\phi_{r}$ is equal to a constant multiple of $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}$.    $\square$

Remark 3.5

Let $M_{1}=\sum_{i=2}^{r+1}m_{1,i}$ and let $D_{q}\,\,(q=1,\dots,h)$ be as in $(3.5)$. When $d=M-r$, from the proof of Theorem $3.4\,(3)$, $g_{h-j}\,\,(j=1,\dots,h)$ is uniquely determined as follows:

[TABLE]

Let $m^{\prime}=(m_{i,j})_{2\leq i<j\leq r+1}$, $\mathfrak{c}^{\prime}_{{\rm nice}}$ a nice chamber of $A_{r-1}^{+}$ and $a^{\prime}=\sum_{i=2}^{r}a_{i}(e_{i}-e_{r+1})\in\overline{\mathfrak{c}^{\prime}_{{\rm nice}}}$. From Proposition 3.3 and Remark 3.5, we obtain the following theorem.

Theorem 3.6

Let $h=(\sum_{2\leq i<j\leq r+1}m_{i,j})-(r-1)$ and let $D_{q}\,\,(q=1,\dots,h)$ be as in $(3.5)$. Then $v_{A_{r}^{+},m,\mathfrak{c}_{\rm nice}}(a)$ is written by linear combination of $v_{A_{r-1}^{+},m^{\prime},\mathfrak{c}^{\prime}_{\rm nice}}(a^{\prime})$ and its partial derivatives as follows:

[TABLE]

Example 3.7

Let $r=3$, let $a=\sum_{i=1}^{3}a_{i}(e_{i}-e_{4})\in\overline{\mathfrak{c}_{{\rm nice}}}$ and let $a^{\prime}=\sum_{i=2}^{3}a_{i}(e_{i}-e_{4})\in\overline{\mathfrak{c}^{\prime}_{{\rm nice}}}$. We set $m_{1,2}=1,m_{1,3}=1,m_{1,4}=2,m_{2,3}=1,m_{2,4}=2$ and $m_{3,4}=2$ as in Example 2.13. Then we have**

[TABLE]

We can check that $v=v_{A_{3}^{+},m,\mathfrak{c}_{\rm nice}}(a)$ satisfies the system of differential equations as follows:**

[TABLE]

Also, from Proposition 3.2, the coefficient of the term $a_{1}^{3}a_{2}^{2}a_{3}$ is $\frac{1}{3!2!1!}=\frac{1}{12}$. When $r=2$,**

[TABLE]

Therefore, we have**

[TABLE]

Hence when $r=3$, we can check the equation (3.7) in Theorem 3.6.**

Acknowledgements

The third author was supported by JSPS KAKENHI Grant Number JP16K05137.