A central arrangement A was termed free if the module of A-derivations was a free module. The combinatorial structure of arrangements was heavily influenced by the freeness. Yet, there has been scarce exploration into the construction of their bases. In this paper, we constructed the explicit bases for a class of free arrangements that positioned between the cone of Linial arrangements and Shi arrangements.
Citation: Meihui Jiang, Ruimei Gao. A basis construction for free arrangements between Linial arrangements and Shi arrangements[J]. AIMS Mathematics, 2024, 9(12): 34827-34837. doi: 10.3934/math.20241658
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A central arrangement A was termed free if the module of A-derivations was a free module. The combinatorial structure of arrangements was heavily influenced by the freeness. Yet, there has been scarce exploration into the construction of their bases. In this paper, we constructed the explicit bases for a class of free arrangements that positioned between the cone of Linial arrangements and Shi arrangements.
Let V be an ℓ-dimensional vector space over a field K of characteristic 0. An arrangement of hyperplanes A is a finite collection of codimension one affine subspaces in V. An arrangement A is called central if every hyperplane H∈A goes through the origin.
Let V∗ be the dual space of V, and S=S(V∗) be the symmetric algebra over V∗. A K-linear map θ:S→S is called a derivation if for f,g∈S,
θ(fg)=fθ(g)+gθ(f). |
Let DerK(S) be the S-module of derivations. When A is central, for each H∈A, choose αH∈V∗ with ker(αH)=H. Define an S-submodule of DerK(S), called the module of A-derivations by
D(A):={θ∈DerK(S)|θ(αH)∈αHSforallH∈A}. |
The arrangement A is called free if D(A) is a free S-module. Then, D(A) has a basis comprising of ℓ homogeneous elements. For an affine arrangement A in V, cA denotes the cone over A [7], which is a central arrangement in an (ℓ+1)-dimensional vector space by adding the new coordinate z.
Let E=Rℓ be an ℓ-dimensional Euclidean space with a coordinate system x1,…,xℓ, and let Φ be a crystallographic irreducible root system in the dual space E∗. Let Φ+ be a positive system of Φ. For α∈Φ+ and k∈Z, define an affine hyperplane Hα,k by
Hα,k:={v∈E∣(α,v)=k}. |
The Shi arrangement Shi(ℓ) was introduced by Shi in the study of the Kazhdan-Lusztig representation theory of the affine Weyl groups in [9] by
Shi(ℓ):={Hα,k∣α∈Φ+,0≤k≤1}, |
when the root system is of type Aℓ−1.
For m∈Z≥0, the extended Shi arrangement Shik of the type Φ is an affine arrangement defined by
Shik:={Hα,k∣α∈Φ+,−m+1≤k≤m}. |
There are a lot of researches on the freeness of the cones over the extended Shi arrangements [1,3,4]. The first breakthrough was the proof of the freeness of multi-Coxeter arrangements with constant multiplicities by Terao in [13]. Combining it with algebro-geometric method, Yoshinaga proved the freeness of the extended Shi arrangements in [15]. Nevertheless, there has been limited progress in constructing their bases, and a universal method for determining these bases remains elusive. For types Aℓ−1, Bℓ, Cℓ, and Dℓ, explicit bases for the cones over the Shi arrangements were constructed in [6,10,11]. Notably, a basis for the extended Shi arrangements of type A2 was established in [2]. Recently, Suyama and Yoshinaga constructed explicit bases for the extended Shi arrangements of type Aℓ−1 using discrete integrals in [12]. Feigin et al. presented integral expressions for specific bases of certain multiarrangements in [5]. In these studies, Suyama and Terao first constructed a basis for the derivation module of the cone over the Shi arrangement, as detailed in [11], with Bernoulli polynomials playing a central role in their approach. The following definitions are pertinent to this result.
For (k1,k2)∈(Z≥0)2, the homogenization polynomial of degree k1+k2+1 is defined by
¯Bk1,k2(x,z):=zk1+k2+1k1∑i=01k2+i+1(k1i){Bk2+i+1(xz)−Bk2+i+1}, |
where Bk(x) denotes the k-th Bernoulli polynomial and Bk(0)=Bk denotes the k-th Bernoulli number. Using this polynomial, the basis for D(cShi(ℓ)) was constructed as follows.
Theorem 1.1. [11, Theorem 3.5] The arrangement cShi(ℓ) is free with the exponents (0,1,ℓℓ−1). The homogeneous derivations
η1:=∂1+∂2+⋯+∂ℓ,η2:=x1∂1+x2∂2+⋯+xℓ∂ℓ+z∂z,ψ(ℓ)j:=(xj−xj+1−z)ℓ∑i=1∑0≤k1≤j−10≤k2≤ℓ−j−1(−1)k1+k2Ij−k1−1[1,j−1]Iℓ−j−k2−1[j+2,ℓ]¯Bk1,k2(xi,z)∂i, |
form a basis for D(cShi(ℓ)), where 1≤j≤ℓ−1 and ∂i(1≤i≤ℓ), ∂z denote ∂∂xi, ∂∂z respectively. Ik[u,v] represents the elementary symmetric function in the variables {xu,xu+1,…,xv} of degree k for 1≤u≤v≤ℓ.
The above conclusion was reached by using Saito's criterion, which is a crucial theorem for determining the basis of a free arrangement.
Theorem 1.2. [8, Saito's criterion] Let A be a central arrangement, and Q(A) be the defining polynomial of A. Given θ1,…,θℓ∈D(A), the following two conditions are equivalent:
(1)detM(θ1,…,θℓ)≐Q(A),
(2)θ1,…,θℓ form a basis for D(A) over S,
where M(θ1,…,θℓ)=(θj(xi))ℓ×ℓ denotes the coefficient matrix, and A≐B means that A=cB, c∈K∖{0}.
This theorem provides a useful tool for determining when a set of derivations forms a basis for the module of derivations associated with a central arrangement.
Let αℓ=(1,…,1)T and βℓ=(x1,…,xℓ)T be the ℓ×1 column vectors, and define ψ(ℓ)i,j:=ψ(ℓ)j(xi) for 1≤i≤ℓ, 1≤j≤ℓ−1. Suyuma and Terao in [11] proved the equality
detM(η1,η2,ψ(ℓ)1,…,ψ(ℓ)ℓ−1)=det(αℓβℓ(ψ(ℓ)i,j)ℓ×(ℓ−1)0z01×(ℓ−1))(ℓ+1)×(ℓ+1)≐z∏1≤m<n≤ℓ(xm−xn)(xm−xn−z), |
which yields
det(αℓ(ψ(ℓ)i,j)ℓ×(ℓ−1))≐∏1≤m<n≤ℓ(xm−xn)(xm−xn−z). | (1.1) |
A graph G=(V,E) is defined as an ordered pair, where the set V={1,2,…,ℓ} represents the vertex set, and E is a collection of two-element subsets of V. If {i,j}∈E for some i,j∈V, then {i,j} is referred to as an edge. Writing {i,j}∈G implies {i,j}∈E. Let U⊆V, and define E(U)={{i,j}∣i,j∈U,{i,j}∈E}. We say U induces a subgraph GU=(U,E(U)). Specifically, we use the symbol KU for the induced subgraph of the complete graph Kℓ. For i<j, the interval notation [i,j] represents {i,i+1,…,j}.
For a graph G on the vertex set {1,2,…,ℓ}, the arrangement Shi(G) was defined in [14] by
Shi(G):={{xm−xn=0}|{m,n}∈G}∪{{xm−xn=1}|1≤m<n≤ℓ}. |
Then, Shi(G) is an arrangement between the Linial arrangement
{{xm−xn=1}∣1≤m<n≤ℓ}, |
and the Shi arrangement Shi(ℓ). Write A(G):=cShi(G). It was classified to be free according to the following theorem.
Theorem 1.3. [14, Theorem 3] The arrangement A(G) is free if and only if G consists of all edges of three complete induced subgraphs G[1,s],G[t,ℓ],G[2,ℓ−1], where 1≤s≤ℓ, t≤s+1. The free arrangement A(G) has exponents (0,1,(ℓ−1)ℓ+t−s−2,ℓs−t+1) for s<ℓ and t>1, and (0,1,ℓℓ−1) for s=ℓ or t=1.
For s,t∈Z+, we may define the arrangement
A[s,t]:=A(K[2,ℓ−1])∪{{x1−xn=0}∣2≤n≤s≤ℓ}∪{{xm−xℓ=0}∣1≤t≤m≤ℓ−1}. |
By Theorem 1.3, for 1≤s≤ℓ and t≤s+1, the arrangement A[s,t] is free with exponents (0,1,(ℓ−1)ℓ+t−s−2,ℓs−t+1) for s<ℓ and t>1, and (0,1,ℓℓ−1) for s=ℓ or t=1.
For 0≤q≤ℓ−2, we write A[q]:=A[ℓ−1,ℓ−q], then A[q] is free with exponents (0,1,(ℓ−1)ℓ−q−1,ℓq).
In this section, based on the conclusions of Suyama and Terao, we provide an explicit construction of the basis for D(A[q]), 0≤q≤ℓ−2. First, we shall establish a basis for D(A[0]), which is the ingredient of the basis for D(A[q]).
Theorem 2.1. For 1≤j≤ℓ−2, define homogeneous derivations
φ(0)j:=(xj−xj+1−z)ℓ∑i=1∑0≤k1≤j−10≤k2≤ℓ−j−2(−1)k1+k2Ij−k1−1[1,j−1]Iℓ−j−k2−2[j+2,ℓ−1]¯Bk1,k2(yi,z)∂i,φ(0)ℓ−1:=ℓ−1∏s=1(xs−xℓ−z)∂ℓ∈D(A[0]), |
where
yi={xi,1≤i≤ℓ−1,xℓ+z,i=ℓ. |
Then, the derivations η1,η2,φ(0)1,…,φ(0)ℓ−1 form a basis for D(A[0]).
Proof. Write φ(0)i,j:=φ(0)j(xi),1≤i≤ℓ,1≤j≤ℓ−1, and from the definitions of φ(0)j and ψ(ℓ)j, we can get
φ(0)i,j=ψ(ℓ−1)i,j, | (2.1) |
for 1≤i≤ℓ−1, 1≤j≤ℓ−2. Consequently, for 1≤m<n≤ℓ−1, it follows that φ(0)j(xm−xn) is divisible by xm−xn, and φ(0)j(xm−xn−z) is divisible by xm−xn−z. For 1≤m≤ℓ−1, let the congruence notation (m,k)≡ in the subsequent calculation denote modulo the ideal (xm−xℓ−kz). We derive
φ(0)j(xm−xℓ−z)=(xj−xj+1−z)∑0≤k1≤j−10≤k2≤ℓ−j−2(−1)k1+k2Ij−k1−1[1,j−1]Iℓ−j−k2−2[j+2,ℓ−1][¯Bk1,k2(xm,z)−¯Bk1,k2(xℓ+z,z)](m,1)≡0, |
which implies that φ(0)j(xm−xℓ−z) is divisible by xm−xℓ−z. Thus, φ(0)j∈D(A[0]) for 1≤j≤ℓ−2. Therefore, we have η1,η2,φ(0)1,…,φ(0)ℓ−1∈D(A[0]). Additionally, we obtain
detM(η1,η2,φ(0)1,…,φ(0)ℓ−1)=(−1)ℓ+1zdet(1φ(0)1,1⋯φ(0)1,ℓ−20⋮⋮⋱⋮⋮1φ(0)ℓ−1,1⋯φ(0)ℓ−1,ℓ−201φ(0)ℓ,1⋯φ(0)ℓ,ℓ−2ℓ−1∏s=1(xs−xℓ−z))ℓ×ℓ=(−1)ℓ+1zℓ−1∏s=1(xs−xℓ−z)det(1φ(0)1,1⋯φ(0)1,ℓ−2⋮⋮⋱⋮1φ(0)ℓ−1,1⋯φ(0)ℓ−1,ℓ−2)(ℓ−1)×(ℓ−1)=(−1)ℓ+1zℓ−1∏s=1(xs−xℓ−z)det(αℓ−1(φ(0)i,j)(1≤i≤ℓ−1,1≤j≤ℓ−2))(ℓ−1)×(ℓ−1). |
According to the equalities (1.1) and (2.1), we have
det(αℓ−1(φ(0)i,j)(1≤i≤ℓ−1,1≤j≤ℓ−2))≐∏1≤m<n≤ℓ−1(xm−xn)(xm−xn−z). |
Hence, we obtain
detM(η1,η2,φ(0)1,…,φ(0)ℓ−1).=zℓ−1∏s=1(xs−xℓ−z)∏1≤m<n≤ℓ−1(xm−xn)(xm−xn−z)=z∏1≤m<n≤ℓ−1(xm−xn)∏1≤m<n≤ℓ(xm−xn−z)=Q(A[0]). |
By applying Theorem 1.2, we conclude that the derivations η1,η2,φ(0)1,…,φ(0)ℓ−1 form a basis for D(A[0]).
Definition 2.1. For 1≤q≤ℓ−2, 1≤j≤ℓ−1, define the homogeneous derivations
φ(q)j:={φ(0)j,1≤j≤ℓ−q−2,(xj+1−xℓ)φ(0)j−(xj−xj+1−z)j−1∑a=ℓ−q−1φ(0)a,ℓ−q−1≤j≤ℓ−3,φ(0)j+1+j∑a=ℓ−q−1(ℓ−a−1)φ(0)a,j=ℓ−2,(xj−xℓ)φ(0)j+(xj−xj+1−z)j−2∑a=ℓ−q−1(ℓ−a−2)φ(0)a,j=ℓ−1. |
To prove the derivations η1,η2,φ(q)1,…,φ(q)ℓ−1 form a basis for D(A[q]), first we prove all such derivations belong to the module D(A[q]).
Theorem 2.2. For 1≤m≤ℓ−1, 1≤j≤ℓ−2, we have
φ(0)j(xm−xℓ)(m,0)≡(−z)(xj−xj+1−z)j−1∏s=1(xs−xm−z)ℓ−1∏s=j+2(xs−xm). | (2.2) |
Proof. We have the following congruence relation of polynomials modulo the ideal (xm−xℓ).
φ(0)j(xm−xℓ)=(xj−xj+1−z)∑0≤k1≤j−10≤k2≤ℓ−j−2(−1)k1+k2Ij−k1−1[1,j−1]Iℓ−j−k2−2[j+2,ℓ−1][¯Bk1,k2(xm,z)−¯Bk1,k2(xℓ+z,z)](m,0)≡(xj−xj+1−z)∑0≤k1≤j−10≤k2≤ℓ−j−2(−1)k1+k2+1Ij−k1−1[1,j−1]Iℓ−j−k2−2[j+2,ℓ−1][¯Bk1,k2(xm+z,z)−¯Bk1,k2(xm,z)]=(xj−xj+1−z)∑0≤k1≤j−10≤k2≤ℓ−j−2(−1)k1+k2+1Ij−k1−1[1,j−1]Iℓ−j−k2−2[j+2,ℓ−1]zk1+k2+1(xm+zz)k1(xmz)k2=(−z)(xj−xj+1−z)j−1∑k1=0Ij−k1−1[1,j−1][−(xm+z)]k1ℓ−j−2∑k2=0Iℓ−j−k2−2[j+2,ℓ−1](−xm)k2=(−z)(xj−xj+1−z)j−1∏s=1(xs−xm−z)ℓ−1∏s=j+2(xs−xm). |
We complete the proof.
Remark 2.1 In equality (2.2), we observe that ℓ−1∏s=j+2(xs−xm)=0 for j+2≤m≤ℓ−1. This implies that φ(0)j(xm−xℓ) is divisible by xm−xℓ for 1≤j≤ℓ−3 and j+2≤m≤ℓ−1.
According to Remark 2.1, for 1≤j≤ℓ−q−2, we have φ(0)j(xm−xℓ) is divisible by xm−xℓ for ℓ−q≤m≤ℓ−1, which implies that φ(q)j=φ(0)j∈D(A[q]). Therefore, to prove the derivations belong to the module D(A[q]), it suffices to prove φ(q)j∈D(A[q]) for ℓ−q−1≤j≤ℓ−1.
For the sake of convenience in the proof, let us introduce the notations for f,g,h∈Z+,
A[g,h]f:=h∏s=g(xs−xf),B[g,h]f:=h∏s=g(xs−xf−z). |
If g>h, we agree that A[g,h]f=B[g,h]f=1.
Lemma 2.1. For any u,v,w∈Z+ that satisfy 4≤ℓ−j+1≤u≤ℓ−2, 3≤v≤ℓ−2, and 3≤w≤ℓ−2, we have the following three equalities:
B[ℓ−u,j−1]ℓ−u−1=A[ℓ−u+1,j]ℓ−u−1+j−1∑a=ℓ−u(xa−xa+1−z)A[a+2,j]ℓ−u−1B[ℓ−u,a−1]ℓ−u−1. | (2.3) |
B[ℓ−v,ℓ−1]ℓ−v−1=(v+1)A[ℓ−v,ℓ−1]ℓ−v−1+ℓ−2∑a=ℓ−v−1(ℓ−a−1)(xa−xa+1−z)A[a+2,ℓ−1]ℓ−v−1B[ℓ−v−1,a−1]ℓ−v−1. | (2.4) |
B[ℓ−w,ℓ−2]ℓ−w−1=wA[ℓ−w,ℓ−2]ℓ−w−1+ℓ−3∑a=ℓ−w−1(ℓ−a−2)(xa−xa+1−z)A[a+2,ℓ−2]ℓ−w−1B[ℓ−w−1,a−1]ℓ−w−1. | (2.5) |
Proof. We will only prove equality (2.5) by induction on w. The proofs of equalities (2.3) and (2.4) are similar. For w=3,
3A[ℓ−3,ℓ−2]ℓ−4+ℓ−3∑a=ℓ−4(ℓ−a−2)(xa−xa+1−z)A[a+2,ℓ−2]ℓ−4B[ℓ−4,a−1]ℓ−4=3(xℓ−3−xℓ−4)(xℓ−2−xℓ−4)+2(xℓ−4−xℓ−3−z)(xℓ−2−xℓ−4)+(xℓ−3−xℓ−2−z)(−z)=(xℓ−3−xℓ−4−z)(xℓ−2−xℓ−4−z)=B[ℓ−3,ℓ−2]ℓ−4, |
and the equality holds. Assume that for w=k≤ℓ−3, the equality holds. Then, we replace xℓ−k−1 with xℓ−k−2, and multiply both sides of the equality by (xℓ−k−1−xℓ−k−2−z) to get
B[ℓ−k−1,ℓ−2]ℓ−k−2=(k−1)(xℓ−k−2−xℓ−k−z)(xℓ−k−1−xℓ−k−2−z)A[ℓ−k+1,ℓ−2]ℓ−k−2+k(xℓ−k−1−xℓ−k−2−z)A[ℓ−k,ℓ−2]ℓ−k−2+ℓ−3∑a=ℓ−k(ℓ−a−2)(xa−xa+1−z)A[a+2,ℓ−2]ℓ−k−2B[ℓ−k−2,a−1]ℓ−k−2=(k+1)A[ℓ−k−1,ℓ−2]ℓ−k−2+ℓ−3∑a=ℓ−k−2(ℓ−a−2)(xa−xa+1−z)A[a+2,ℓ−2]ℓ−k−2B[ℓ−k−2,a−1]ℓ−k−2. |
We have completed the induction.
Lemma 2.2. The derivation φ(q)j belongs to the module D(A[q]) for 2≤q≤ℓ−2 and ℓ−q−1≤j≤ℓ−3.
Proof. For 2≤q≤ℓ−2 and j=ℓ−q−1, it is evident from Remark 2.1 that
φ(q)ℓ−q−1=(xℓ−q−xℓ)φ(0)ℓ−q−1∈D(A[q]). |
For 3≤q≤ℓ−2 and ℓ−q≤j≤ℓ−3, we will establish this by induction on q. From Theorem 2.2, for ℓ−q≤m≤ℓ−1, we have
φ(q)j(xm−xℓ)=(xj+1−xℓ)φ(0)j(xm−xℓ)−(xj−xj+1−z)j−1∑a=ℓ−q−1φ(0)a(xm−xℓ)(m,0)≡(−z)(xj−xj+1−z)A[j+1,ℓ−1]mB[1,ℓ−q−2]m[B[ℓ−q−1,j−1]m−j−1∑a=ℓ−q−1(xa−xa+1−z)A[a+2,j]mB[ℓ−q−1,a−1]m]. |
(1) For q=3, we get
φ(3)ℓ−3(xm−xℓ)(m,0)≡(−z)(xℓ−3−xℓ−2−z)A[ℓ−2,ℓ−1]mB[1,ℓ−5]m(xℓ−3−xm). |
If m=ℓ−3,ℓ−2,ℓ−1, then we have φ(3)ℓ−3(xm−xℓ)(m,0)≡0, which indicates that φ(3)ℓ−3(xm−xℓ) is divisible by xm−xℓ for m=ℓ−3,ℓ−2,ℓ−1. Therefore, φ(3)j∈D(A[3]).
(2) For q=k≤ℓ−3, assume that φ(k)j∈D(A[k]), which implies that φ(k)j(xm−xℓ) is divisible by xm−xℓ for ℓ−k≤m≤ℓ−1.
For q=k+1, we observe that
φ(k+1)j=φ(k)j−(xj−xj+1−z)φ(0)ℓ−k−2. |
According to the induction hypothesis and Remark 2.1, it is sufficient to prove that φ(k+1)j(xℓ−k−1−xℓ) is divisible by xℓ−k−1−xℓ. By using the equality (2.3), we obtain
φ(k+1)j(xℓ−k−1−xℓ)(ℓ−k−1,0)≡(−z)(xj−xj+1−z)A[j+1,ℓ−1]ℓ−k−1B[1,ℓ−k−3]ℓ−k−1[B[ℓ−k−2,j−1]ℓ−k−1−j−1∑a=ℓ−k−2(xa−xa+1−z)A[a+2,j]ℓ−k−1B[ℓ−k−2,a−1]ℓ−k−1]=(−z)2(xj−xj+1−z)A[j+1,ℓ−1]ℓ−k−1B[1,ℓ−k−2]ℓ−k−1[B[ℓ−k,j−1]ℓ−k−1−A[ℓ−k+1,j]ℓ−k−1−j−1∑a=ℓ−k(xa−xa+1−z)A[a+2,j]ℓ−k−1B[ℓ−k,a−1]ℓ−k−1]=0. |
Therefore, φ(k+1)j(xℓ−k−1−xℓ) is divisible by xℓ−k−1−xℓ, and it follows that φ(k+1)j∈D(A[k+1]). Consequently, we can conclude that for any 3≤q≤ℓ−2 and ℓ−q≤j≤ℓ−3, φ(q)j∈D(A[q]).
Lemma 2.3. The derivation φ(q)ℓ−2 belongs to the module D(A[q]) for 1≤q≤ℓ−2.
Proof. From Theorem 2.2, we can get the following equality for ℓ−q≤m≤ℓ−1,
φ(q)ℓ−2(xm−xℓ)=φ(0)ℓ−1(xm−xℓ)+ℓ−2∑a=ℓ−q−1(ℓ−a−1)φ(0)a(xm−xℓ)(m,0)≡−B[1,ℓ−1]m+(−z)ℓ−2∑a=ℓ−q−1(ℓ−a−1)(xa−xa+1−z)A[a+2,ℓ−1]mB[1,a−1]m. |
(1) For q=1,2, this conclusion is straightforward to verify.
(2) For q=k≤ℓ−3, assume that φ(k)ℓ−2∈D(A[k]), which implies that φ(k)ℓ−2(xm−xℓ) is divisible by xm−xℓ for ℓ−k≤m≤ℓ−1.
For q=k+1, we have
φ(k+1)ℓ−2=φ(k)ℓ−2+(k+1)φ(0)ℓ−k−2. |
By using the equality (2.4), we have
φ(k+1)ℓ−2(xℓ−k−1−xℓ)(ℓ−k−1,0)≡−B[1,ℓ−1]ℓ−k−1+(−z)ℓ−2∑a=ℓ−k−2(ℓ−a−1)(xa−xa+1−z)A[a+2,ℓ−1]ℓ−k−1B[1,a−1]ℓ−k−1=B[1,ℓ−k−1]ℓ−k−1[−B[ℓ−k,ℓ−1]ℓ−k−1+(k+1)A[ℓ−k,ℓ−1]ℓ−k−1+ℓ−2∑a=ℓ−k−1(ℓ−a−1)(xa−xa+1−z)A[a+2,ℓ−1]ℓ−k−1B[ℓ−k−1,a−1]ℓ−k−1]=0. |
Therefore, φ(k+1)ℓ−2(xℓ−k−1−xℓ) is divisible by xℓ−k−1−xℓ. According to the induction hypothesis and Remark 2.1, we have φ(k+1)ℓ−2∈D(A[k+1]). Hence, we may conclude that for any 1≤q≤ℓ−2, φ(q)ℓ−2∈D(A[q]).
Lemma 2.4. The derivation φ(q)ℓ−1 belongs to the module D(A[q]) for 1≤q≤ℓ−2.
Proof. First, from Theorem 2.2, for ℓ−q≤m≤ℓ−1, we can get
φ(q)ℓ−1(xm−xℓ)=(xℓ−1−xℓ)φ(0)ℓ−1(xm−xℓ)+(xℓ−1−xℓ−z)ℓ−3∑a=ℓ−q−1(ℓ−a−2)φ(0)a(xm−xℓ)(m,0)≡−(xℓ−1−xm)B[1,ℓ−1]m+(−z)(xℓ−1−xm−z)ℓ−3∑a=ℓ−q−1(ℓ−a−2)(xa−xa+1−z)A[a+2,ℓ−1]mB[1,a−1]m. |
(1) For q=1,2, it is obvious that φ(q)ℓ−1∈D(A[q]).
(2) For q=k≤ℓ−3, assume that φ(k)ℓ−1∈D(A[k]), which implies that φ(k)ℓ−1(xm−xℓ) is divisible by xm−xℓ for ℓ−k≤m≤ℓ−1.
For q=k+1, we can see
φ(k+1)ℓ−1=φ(k)ℓ−1+k(xℓ−1−xℓ−z)φ(0)ℓ−k−2. |
By using the equality (2.5), we have
φ(k+1)ℓ−1(xℓ−k−1−xℓ)(ℓ−k−1,0)≡−(xℓ−1−xℓ−k−1)B[1,ℓ−1]ℓ−k−1+(−z)(xℓ−1−xℓ−k−1−z)ℓ−3∑a=ℓ−k−2(ℓ−a−2)(xa−xa+1−z)A[a+2,ℓ−1]ℓ−k−1B[1,a−1]ℓ−k−1=(xℓ−1−xℓ−k−1)(xℓ−1−xℓ−k−1−z)B[1,ℓ−k−1]ℓ−k−1[−B[ℓ−k,ℓ−2]ℓ−k−1+kA[ℓ−k,ℓ−2]ℓ−k−1+ℓ−3∑a=ℓ−k−1(ℓ−a−2)(xa−xa+1−z)A[a+2,ℓ−2]ℓ−k−1B[ℓ−k−1,a−1]ℓ−k−1]=0. |
Therefore, φ(k+1)ℓ−1(xℓ−k−1−xℓ) is divisible by xℓ−k−1−xℓ. According to the induction hypothesis and Remark 2.1, we have φ(k+1)ℓ−1∈D(A[k+1]). Hence, we may conclude that for any 1≤q≤ℓ−2, φ(q)ℓ−1∈D(A[q]).
From the above proof, we finally conclude that φ(q)1,…,φ(q)ℓ−1 belong to the module D(A[q]).
Theorem 2.3. For 1≤q≤ℓ−2, the derivations η1,η2,φ(q)1,…,φ(q)ℓ−1 form a basis for D(A[q]).
Proof. According to Lemmas 2.2–2.4, it suffices to prove that
detM(η1,η2,φ(q)1,⋯,φ(q)ℓ−1)≐Q(A[q]). |
Let
γ1=(q,q−1,…,1,1)T |
and
γ2=((q−1)(xℓ−1−xℓ−z),(q−2)(xℓ−1−xℓ−z),…,xℓ−1−xℓ−z,0,xℓ−1−xℓ)T |
be the (q+1)×1 column vectors, and define a matrix
M(q+1)×(q−1):=(xℓ−q−xℓ−(xℓ−q−xℓ−q+1−z)⋯−(xℓ−3−xℓ−2−z)0xℓ−q+1−xℓ⋯−(xℓ−3−xℓ−2−z)⋮⋮⋱⋮00⋯xℓ−2−xℓ00⋯000⋯0). |
Write ˜M(q+1)×(q+1):=(M(q+1)×(q−1),γ1,γ2), then
det˜M(q+1)×(q+1)=ℓ−1∏s=ℓ−q(xs−xℓ). |
Thus, we obtain the following equality
(η1,η2,φ(q)1,⋯,φ(q)ℓ−1)(ℓ+1)×(ℓ+1)=(η1,η2,φ(0)1,⋯,φ(0)ℓ−1)(Eℓ−q0(ℓ−q)×(q+1)0(q+1)×(ℓ−q)˜M(q+1)×(q+1)). |
Hence,
detM(η1,η2,φ(q)1,⋯,φ(q)ℓ−1)=detM(η1,η2,φ(0)1,⋯,φ(0)ℓ−1)det˜M(q+1)×(q+1).=z∏1≤m<n≤ℓ−1(xm−xn)∏1≤m<n≤ℓ(xm−xn−z)ℓ−1∏s=ℓ−q(xs−xℓ)=Q(A[q]). |
We complete the proof.
Meihui Jiang: writing-original draft; Ruimei Gao: writing-review and editing, methodology and supervision. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have used Artificial Intelligence (AI) tools in the creation of this article.
The work was partially supported by Science and Technology Development Plan Project of Jilin Province, China (No. 20230101186JC) and NSF of China (No. 11501051).
The authors declare no conflict of interest in this paper.
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