The primitive derivation and discrete integrals

The modules of logarithmic derivations for the (extended) Catalan and Shi arrangements associated with root systems are known to be free. However, except for a few cases, explicit bases for such modules are not known. In this paper, we construct explicit bases for type $A$ root systems. Our construction is based on Bandlow-Musiker's integral formula for a basis of the space of quasiinvariants. The integral formula can be considered as an expression for the inverse of the primitive derivation introduced by K. Saito. We prove that the discrete analogues of the integral formulas provide bases for Catalan and Shi arrangements.

for the purpose of computing Gauss-Manin connections, and D(A, m) was introduced by Ziegler [25] for studying restrictions of free arrangements. The algebraic structures of these modules are thought to reflect the combinatorial nature of A (see [11,24]). Now we assume that A is a Coxeter arrangement, that is, the set of reflecting hyperplanes of a finite irreducible real reflection group W ⊂ GL(V ). The filtration (1.2) is closely related to several important structures. First, taking W -invariant parts, we have This filtration is known to be equal to the semi-infinite Hodge filtration studied by K. Saito [15,16,21,22], which is a crucial structure in his theory of primitive forms [14]. In particular, the inverse operator of the so-called primitive derivation ∇ D describes the filtration as As indicated by Misha Feigin (see forthcoming paper [1] for details), these spaces are also isomorphic to the isotypic component of the spaces of mquasiinvariants, which were introduced in the study of the Calogero-Moser system [4,5,8].
Around 2000, Terao proved that the module D(A, m) is an S-free module using these structures [20]. Terao's results on the freeness of D(A, m) opened new perspectives between the primitive derivation and enumerative combinatorics of Catalan/Shi arrangements.
Catalan and Shi arrangements are classes of finite truncations of affine Weyl arrangements for root systems. The terminology "Catalan arrangement" is explained by the fact that the number of chambers in the fundamental region of a type A root system is equal to the Catalan number [12]. The Shi arrangement was introduced by J. -Y. Shi in [17] in the study of affine Weyl groups. In 1996, Edelman-Reiner [6] posed a conjecture concerning the freeness of cones of Catalan and Shi arrangements for root systems.
This conjecture was proved for type A root system by Edelman-Reiner [6] and Athanasiadis [3], and later for all root systems by [23]. The freeness of D(A, m) played crucial role in the proof of [23] because D(A, m) can be regarded as the "leading terms" of the logarithmic vector fields for Catalan and Shi arrangements. In the present paper: we focus on the construction of explicit bases for these modules.

1.2.
Constructions of explicit bases. In this section, we introduce Catalan and Shi arrangements (of type A ℓ−1 ). We define m-Catalan arrangement Cat ℓ (m) as We also denote Cat ℓ (0) by B ℓ . The arrangement B ℓ is defined as the polynomial 1≤i<j≤ℓ (x i − x j ) and called the braid arrangement. The cones c Cat ℓ (m) and c Shi ℓ (m) are defined by the homogeneous polynomials respectively. As we have already noted, the modules D(c Cat ℓ (m)) and D(c Shi ℓ (m)) are free. The exponents, that is, the degrees of the homogeneous bases of the modules, are as follows.
We note that explicit bases were not constructed in the known proofs. Indeed, the proofs by Edelman-Reiner [6] and Athanasiadis [3] used Terao's addition-deletion theorem of freeness [11,Theorem 4.51], and that in [23] used cohomological arguments to guarantee the existence of global sections of certain coherent sheaves associated with the graded module D(A). Since then, a number of efforts have been made to construct explicit bases for D(c Cat ℓ (m)) and D(c Shi ℓ (m)). First, in [18], a basis for D(c Shi ℓ (1)) was constructed using the Bernoulli polynomial. Subsequently, in [10] and [19], similar bases were constructed for root systems of type B, C, and D. Note that these works are for Shi arrangements with m = 1. Catalan arrangements and Shi arrangements with m > 1 have not been covered. For larger m, the type A 2 was the only known case. Namely, explicit bases were constructed for c Cat 3 (m) and c Shi 3 (m), m ≥ 1, in [2].
Our purpose in the present paper is to construct an explicit basis for D(c Cat ℓ (m)) and D(c Shi ℓ (m)), for all ℓ ≥ 2 and m ≥ 1. The paper is organized as follows. The starting point of our study is Bandlow-Musiker's integral expression [4] (which goes back to Felder-Veselov's integral expression [9]) for a basis of the space of quasiinvariants introduced in [5,8]. Misha Feigin [7] communicated to us that Bandlow-Musiker's formula provides a basis for the multiarrangement D(B ℓ , 2m + 1). More precisely, for appropriate choices of k, the integral expressions provide a basis for D(B ℓ , 2m + 1). We discuss these facts in addition to a basis for D(B ℓ , 2m) in §2.
After introducing the notion of "discrete integrals" in §3, we present the main results in §4: that is, we prove the following: a basis for D(c Cat ℓ (m)) is obtained from (1.10) by simply replacing the integration " b a dt" with the discrete integration " b a ∆t" as follows.
(1.11) ℓ i,j=1 x j (To be precise, we need to homogenize the above polynomial vector field, see §3 and §4 for details.) We also provide a basis for D(c Shi ℓ (m)).

BANDLOW-MUSIKER'S EXPRESSION
Recall that B ℓ denotes the braid arrangement defined by 1≤i<j≤ℓ (x i − x j ). The symmetric group W = S ℓ naturally acts on B ℓ by the permutation of coordinates. In this section, we construct a basis for D(B ℓ , m), m ≥ 1. Note that because the vector field . Following Bandlow-Musiker [4] and Feigin [7], for m, k ≥ 0, we introduce the following vector field.  Here we prove Corollary 2.2 in order to see the relationship between integral expression (2.2) and the primitive derivation. First, we prove η m k ∈ D(B ℓ , 2m + 1). Since η m k is W -symmetric, it is sufficient to show that . This can be checked by a straightforward calculation using the change of variables t ′ = t − x 1 .
x 2 Next we describe the action of the primitive derivation on the vector field η m k . Let ∇ denote the integrable connection with flat sections ∂ 1 , . . . , ∂ ℓ .

DISCRETE INTEGRALS
In this section we only consider polynomial functions. For a function f (t), we define the difference operator ∆ as ∆f (t) = f (t+1)−f (t). When ∆F (t) = f (t), F (t) is called an indefinite summation (or antidifference) of f (t), and denoted by Let F (t) be an indefinite summation of f (t). Then we define the definite summation as Obviously we have the following.
Note that if b−a = n is a positive integer, the definite summation is nothing but the finite sum Example 3.1. Recall that the Bernoulli polynomial B n (t) is a monic of rational coefficients defined by The Bernoulli polynomial B n (t) satisfies ∆B n (t) = nt n−1 . Therefore, the monomial t n has an indefinite summation B n+1 (t) n+1 . Furthermore, arbitrary polynomial f (t) has an indefinite summation.
The leading part of a definite summation is equal to a definite integral. More precisely, we have the following. Let f (x 0 , x 1 , . . . , x n ) ∈ C[x 0 , . . . , x n ] be a homogeneous polynomial of degree d. Let   F (y 1 , y 2 , x 1 , . . . , x n ) := Then F (y 1 , y 2 , x 1 , . . . , x n ) is a (not necessarily homogeneous) polynomial of degree d + 1 in y 1 , y 2 , x 1 , . . . , x n whose highest degree part is Proof. This is straightforward from the fact that the leading term of B n+1 (t) n+1 is t n+1 n+1 , which is an indefinite integral of t n .
The following is a discrete analogue of the power. Let n > 0 be a positive integer. We define the falling power f (t) n as

A basis for the Catalan arrangement. Let
. , x ℓ )∂ i be a polynomial vector field (f 1 , . . . , f ℓ are not necessarily homogeneous). Let d := max{deg f 1 , . . . , deg f ℓ }. We define the homogenization of δ by Using the notion of discrete integrals, we define ζ m k as follows.
x j Note that the definition of ζ m k is a discrete analogue of (2.2). The next result shows that the homogenizations of these vector fields form a basis for the Catalan arrangement.
Proof. In view of Ziegler's characterization of freeness [25] (see also [24,Corollary 1.35]), it is sufficient to prove the following.
First we prove (b). By Proposition 3.2, we have , we need to show that (4.5) x 2 +p It is easy to check using (3.3) that the left-hand side is equal to 0 for 0 < p ≤ m. The case −m ≤ p < 0 is also proved using a similar argument.

4.2.
A basis for the Shi arrangement. To construct a basis for the Shi arrangement, we need the following. For 0 ≤ k ≤ ℓ and m ≥ 0, let We define the following vector field which is a discrete analogue of of (2.6).
(4.7) τ m k = ℓ i,j=1 x j Using these vector fields, we can construct an explicit basis for the Shi arrangement.
Proof. The strategy is similar to the proof of Theorem 4.1. It is sufficient to prove the following. To prove (a), we need to show that for 1 ≤ u < v ≤ ℓ, is divisible by (x u − x v − p) for any 1 − m ≤ p ≤ m. The case p = 0 is obvious. We assume 0 < p. In this case, we need to show that k (x v + s).
Suppose 0 < p < m. In this case, we have m > 1. Then g(t) (m−1) vanishes when t = x v + s, 0 ≤ s ≤ m − 2. The remaining case is p = m. Then x u = x v + m and t = x v + m − 1. In this case, the product (4.11) is equal to zero. Indeed, if v > k, the factor (t − x v − m + 1) vanishes. If v ≤ k, then u < k. Then the factor (t − x u + 1) vanishes.