On volume functions of special flow polytopes associated to the root system of type A

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. Mathematics Subject Classifications: 52B20, 05A16


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., [9]).
We consider a flow polytope associated to the root system of type A. 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 [11]. In this paper, we call a flow polytope for the nice chamber a special flow polytope. 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. Moreover, we mention that a generalization of the Lidskii formula is shown in [3,Theorem 38], there is a geometric proof of the Lidskii formula in [12], and combinatorial applications of this formula are given in [5,7].
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 1 , . . . , e r+1 be the standard basis of R r+1 and let A + r = {e i − e j | 1 ⩽ i < j ⩽ r + 1} 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 ⩽ i < j ⩽ r + 1. Let us set m = (m i,j ) and M = ∑ 1⩽i<j⩽r+1 m i,j . For a = a 1 e 1 + · · · + a r e r − (a 1 + · · · + a r )e r+1 ∈ R r+1 , where a i ∈ R ⩾0 (i = 1, . . . , r), the following polytope P A + r ,m (a) is called the flow polytope associated to the root system of type A: 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 . The open set in 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,c nice . It is a homogeneous polynomial of degree M − r. The first result of this paper is the following.
) ∈ c nice , and let v A + r ,m,c nice (a) be the volume of P A + r ,m (a). Then v = v A + r ,m,c nice (a) satisfies the system of differential equations as follows: . We remark that it is known that the volume function v A + r ,m (a) of P A + r ,m (a), as a distribution on R r , satisfies the differential equation is the Dirac delta function on R r ( [8,11]). Note that ∂ r+1 in the definition of L is supposed to be zero. The above theorem characterizes the function v A + r ,m,c nice (a) on c nice more explicitly. It might be interesting to see what kind of properties of the volume can be derived from Theorem 1.
In addition, in Theorem 20, we show the volume v A + r ,m,c nice (a) is written by a linear combination of v A + r−1 ,m ′ ,c ′ nice (a ′ ) and its partial derivatives, where m ′ = (m i,j ) 2⩽i<j⩽r+1 , c ′ nice is the nice chamber of A + r−1 , and a ′ = ∑ r i=2 a i (e i − e r+1 ) ∈ c ′ nice . It might be interesting to ask whether there is a relaton between this theorem and the inductive formulas of Schmidt-Bincer [13, (4.1), (4.24)].
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], [6] and [10]. Also, we give some examples of P A + r ,m (a) and the calculations of the volume v A + r ,m,c nice (a). In Section 3, we prove the main theorems.

Preliminaries
In this section, we set up the tools to prove the main theorems based on [2], [3], [6] and [10].

Flow polytopes and its volumes
Let e 1 , . . . , e r+1 be the standard basis of R r+1 , and let We consider the positive root system of type A with rank r as follows: Let C(A + r ) be the convex cone generated by A + r : C(A + r ) = {a = a 1 e 1 + · · · + a r e r − (a 1 + · · · + a r )e r+1 | a 1 , . . . , a r ∈ R ⩾0 }.
We assign a positive integer m i,j to each i and j with 1 ⩽ i < j ⩽ r + 1, and it is called a multiplicity. Let us set m = (m i,j ) and M = ∑ 1⩽i<j⩽r+1 m i,j .

Definition 2.
Let a = a 1 e 1 + · · · + a r e r − (a 1 + · · · + a r )e r+1 ∈ C(A + r ). We consider the following polytope: which is called the flow polytope associated to the root system of type A.
Remark 3. The flow polytopes in [3] include the case that m i,j = 0 for some i and j.

The elements of
Let du be the Lebesgue measure on R M . Let [α 1 , . . . , α M ] be a sequence of elements of A + r with multiplicity m i,j , and let φ be the surjective linear map from R M to V defined by φ(e k ) = α k . The vector space ker(φ) = φ −1 (0) is of dimension d = M − r and it is equipped with the quotient Lebesgue measure du/da. For a ∈ V , the affine space φ −1 (a) is parallel to ker(φ), and thus also equipped with the Lebesgue measure du/da. Volumes of subsets of φ −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).

Total residue and iterated residue
, and let U be the dual vector space of V . We denote by R Ar the ring of rational functions f (x 1 , . . . , x r ) on the complexification U C of U with poles on the hyperplanes and call such a element a simple fraction. We denote by S Ar the linear subspace of R Ar spanned by simple fractions. The space U acts on R Ar by differentiation: The projection map Tres Ar : R Ar → S Ar with respect to this decomposition is called the total residue map.
We extend the definition of the total residue to the spaceR Ar consisting of functions P/Q where Q is a finite product of powers of the linear forms α ∈ A r and P = ∑ ∞ k=0 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 Tres Ar (P/Q) = Tres Ar Definition 4. For f ∈ R Ar , we define the iterated residue by Since the iterated residue Ires x=0 f vanishes on the space ∂(U )R Ar as in [3], we have the electronic journal of combinatorics 27(4) (2020), #P4.56

Chambers and Jeffrey-Kirwan residue
Definition 5. Let C(ν) be the closed cone generated by ν for any subset ν of A + r and let C(A + r ) sing be the union of the cones C(ν) where ν is any subset of A + r of cardinal strictly less than r = dim V . By definition, the set C(A + r ) reg of A + r -regular elements is the complement of C(A + r ) sing . A connected component of C(A + r ) reg is called a chamber. The Jeffrey-Kirwan residue [10] associated to a chamber c of C(A + r ) is a linear form f → ⟨⟨c, f ⟩⟩ on the vector space S Ar of simple fractions. Any function f in S Ar can be written as a linear combination of functions f σ , with a basis σ of A r contained in A + r . To determine the linear map f → ⟨⟨c, f ⟩⟩, it is enough to determine it on this set of functions f σ . So we assume that σ is a basis of A r contained in A + r . Definition 6. For a chamber c and f σ ∈ S Ar , we define the Jeffrey-Kirwan residue ⟨⟨c, f σ ⟩⟩ associated to a chamber c as follows: where C(σ) is the convex cone generated by σ.
The set c nice is in fact a chamber for the root system A + r ( [3]). The chamber c nice is called the nice chamber.  (1), we have the following corollary.

Corollary 11 (Lidskii formula [3]).
Let a ∈ c nice . Then the volume function v A + r ,m,c nice (a) is given by v A + r ,m,c nice (a) = Ires x=0 F.

Examples
In this subsection, we give some examples of the flow polytopes for A 1 , A 2 , and A 3 , and calculate their volumes.

Main theorems
In this section, we prove the main theorems of this paper. Let c nice be the nice chamber of A + r and let a =
Remark 16. 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 j and δ(a) is the Dirac delta function on V ( [8,11]). Note that ∂ r+1 in the definition of L is supposed to be zero. Theorem 15 above, together with Proposition 17 and Theorem 18 as below, characterizes the function v A + r ,m,c nice (a) on c nice more explicitly.
Theorem 18. Let ϕ r = ϕ(a 1 , . . . , a r ) be a homogeneous polynomial of a 1 , . . . , a r with degree d and let M = ∑ 1⩽i<j⩽r+1 m i,j . Suppose ϕ r satisfies the system of differential equations as follows: then there is a non trivial homogeneous polynomial ϕ r satisfying (4).
(i) We consider the case of M − r < d.
Similarly, we have g h . Moreover, from the inductive assumption, g h = C · v A + r−1 ,m ′ ,c ′ nice , where C is a constant, m ′ = (m i,j ) 2⩽i<j⩽r+1 , and c ′ nice is a nice chamber of A + r−1 . Hence the solution of (5) is unique up to a constant multiple. On the other hand, by Theorem 15, v A + r ,m,c nice satisfies the system of differential equations (5). Hence ϕ r is equal to a constant multiple of v A + r ,m,c nice . Recall that in the proof of Theorem 18, we have defined the operator We can check that v = v A + 3 ,m,c nice (a) satisfies the system of differential equations as follows: Also, from Proposition 17, the coefficient of the term a 3 1 a 2 2 a 3 is 1 3!2!1! = 1 12 . When r = 2,  (10).