Research article

A basis construction for free arrangements between Linial arrangements and Shi arrangements

  • Received: 12 September 2024 Revised: 23 November 2024 Accepted: 02 December 2024 Published: 13 December 2024
  • MSC : 13C10, 32S22

  • 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

    Related Papers:

    [1] Ruimei Gao, Yuyu Wang . The characteristic polynomials of semigeneric threshold arrangements. AIMS Mathematics, 2023, 8(12): 28569-28581. doi: 10.3934/math.20231462
    [2] Ruimei Gao, Jingjing Qiang, Miao Zhang . The Characteristic polynomials of semigeneric graphical arrangements. AIMS Mathematics, 2023, 8(2): 3226-3235. doi: 10.3934/math.2023166
    [3] Rabha W. Ibrahim, Dumitru Baleanu . Global stability of local fractional Hénon-Lozi map using fixed point theory. AIMS Mathematics, 2022, 7(6): 11399-11416. doi: 10.3934/math.2022636
    [4] Peirong Li, Hong Bian, Haizheng Yu, Yan Dou . Clar covering polynomials of polycyclic aromatic hydrocarbons. AIMS Mathematics, 2024, 9(5): 13385-13409. doi: 10.3934/math.2024653
    [5] Qian Cao, Xiaojin Guo . Anti-periodic dynamics on high-order inertial Hopfield neural networks involving time-varying delays. AIMS Mathematics, 2020, 5(6): 5402-5421. doi: 10.3934/math.2020347
    [6] Emad H. M. Zahran, Omar Abu Arqub, Ahmet Bekir, Marwan Abukhaled . New diverse types of soliton solutions to the Radhakrishnan-Kundu-Lakshmanan equation. AIMS Mathematics, 2023, 8(4): 8985-9008. doi: 10.3934/math.2023450
    [7] M. Syed Ali, M. Hymavathi, Bandana Priya, Syeda Asma Kauser, Ganesh Kumar Thakur . Stability analysis of stochastic fractional-order competitive neural networks with leakage delay. AIMS Mathematics, 2021, 6(4): 3205-3241. doi: 10.3934/math.2021193
    [8] Chahn Yong Jung, Muhammad Shoaib Saleem, Shamas Bilal, Waqas Nazeer, Mamoona Ghafoor . Some properties of η-convex stochastic processes. AIMS Mathematics, 2021, 6(1): 726-736. doi: 10.3934/math.2021044
    [9] Mohammed A. Almalahi, Satish K. Panchal, Fahd Jarad, Mohammed S. Abdo, Kamal Shah, Thabet Abdeljawad . Qualitative analysis of a fuzzy Volterra-Fredholm integrodifferential equation with an Atangana-Baleanu fractional derivative. AIMS Mathematics, 2022, 7(9): 15994-16016. doi: 10.3934/math.2022876
    [10] Bundit Unyong, Vediyappan Govindan, S. Bowmiya, G. Rajchakit, Nallappan Gunasekaran, R. Vadivel, Chee Peng Lim, Praveen Agarwal . Generalized linear differential equation using Hyers-Ulam stability approach. AIMS Mathematics, 2021, 6(2): 1607-1623. doi: 10.3934/math.2021096
  • 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 -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 HA 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:SS is called a derivation if for f,gS,

    θ(fg)=fθ(g)+gθ(f).

    Let DerK(S) be the S-module of derivations. When A is central, for each HA, choose αHV with ker(αH)=H. Define an S-submodule of DerK(S), called the module of A-derivations by

    D(A):={θDerK(S)|θ(αH)αHSforallHA}.

    The arrangement A is called free if D(A) is a free S-module. Then, D(A) has a basis comprising of homogeneous elements. For an affine arrangement A in V, cA denotes the cone over A [7], which is a central arrangement in an (+1)-dimensional vector space by adding the new coordinate z.

    Let E=R be an -dimensional Euclidean space with a coordinate system x1,,x, and let Φ be a crystallographic irreducible root system in the dual space E. Let Φ+ be a positive system of Φ. For αΦ+ and kZ, define an affine hyperplane Hα,k by

    Hα,k:={vE(α,v)=k}.

    The Shi arrangement Shi() was introduced by Shi in the study of the Kazhdan-Lusztig representation theory of the affine Weyl groups in [9] by

    Shi():={Hα,kαΦ+,0k1},

    when the root system is of type A1.

    For mZ0, the extended Shi arrangement Shik of the type Φ is an affine arrangement defined by

    Shik:={Hα,kαΦ+,m+1km}.

    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 A1, B, C, and D, 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 A1 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)(Z0)2, the homogenization polynomial of degree k1+k2+1 is defined by

    ¯Bk1,k2(x,z):=zk1+k2+1k1i=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()) was constructed as follows.

    Theorem 1.1. [11, Theorem 3.5] The arrangement cShi() is free with the exponents (0,1,1). The homogeneous derivations

    η1:=1+2++,η2:=x11+x22++x+zz,ψ()j:=(xjxj+1z)i=10k1j10k2j1(1)k1+k2Ijk11[1,j1]Ijk21[j+2,]¯Bk1,k2(xi,z)i,

    form a basis for D(cShi()), where 1j1 and i(1i), z denote xi, z respectively. Ik[u,v] represents the elementary symmetric function in the variables {xu,xu+1,,xv} of degree k for 1uv.

    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,,θD(A), the following two conditions are equivalent:

    (1)detM(θ1,,θ)Q(A),

    (2)θ1,,θ form a basis for D(A) over S,

    where M(θ1,,θ)=(θj(xi))× denotes the coefficient matrix, and AB means that A=cB, cK{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 α=(1,,1)T and β=(x1,,x)T be the ×1 column vectors, and define ψ()i,j:=ψ()j(xi) for 1i, 1j1. Suyuma and Terao in [11] proved the equality

    detM(η1,η2,ψ()1,,ψ()1)=det(αβ(ψ()i,j)×(1)0z01×(1))(+1)×(+1)z1m<n(xmxn)(xmxnz),

    which yields

    det(α(ψ()i,j)×(1))1m<n(xmxn)(xmxnz). (1.1)

    A graph G=(V,E) is defined as an ordered pair, where the set V={1,2,,} represents the vertex set, and E is a collection of two-element subsets of V. If {i,j}E for some i,jV, then {i,j} is referred to as an edge. Writing {i,j}G implies {i,j}E. Let UV, and define E(U)={{i,j}i,jU,{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 K. For i<j, the interval notation [i,j] represents {i,i+1,,j}.

    For a graph G on the vertex set {1,2,,}, the arrangement Shi(G) was defined in [14] by

    Shi(G):={{xmxn=0}|{m,n}G}{{xmxn=1}|1m<n}.

    Then, Shi(G) is an arrangement between the Linial arrangement

    {{xmxn=1}1m<n},

    and the Shi arrangement Shi(). 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,],G[2,1], where 1s, ts+1. The free arrangement A(G) has exponents (0,1,(1)+ts2,st+1) for s< and t>1, and (0,1,1) for s= or t=1.

    For s,tZ+, we may define the arrangement

    A[s,t]:=A(K[2,1]){{x1xn=0}2ns}{{xmx=0}1tm1}.

    By Theorem 1.3, for 1s and ts+1, the arrangement A[s,t] is free with exponents (0,1,(1)+ts2,st+1) for s< and t>1, and (0,1,1) for s= or t=1.

    For 0q2, we write A[q]:=A[1,q], then A[q] is free with exponents (0,1,(1)q1,q).

    In this section, based on the conclusions of Suyama and Terao, we provide an explicit construction of the basis for D(A[q]), 0q2. 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 1j2, define homogeneous derivations

    φ(0)j:=(xjxj+1z)i=10k1j10k2j2(1)k1+k2Ijk11[1,j1]Ijk22[j+2,1]¯Bk1,k2(yi,z)i,φ(0)1:=1s=1(xsxz)D(A[0]),

    where

    yi={xi,1i1,x+z,i=.

    Then, the derivations η1,η2,φ(0)1,,φ(0)1 form a basis for D(A[0]).

    Proof. Write φ(0)i,j:=φ(0)j(xi),1i,1j1, and from the definitions of φ(0)j and ψ()j, we can get

    φ(0)i,j=ψ(1)i,j, (2.1)

    for 1i1, 1j2. Consequently, for 1m<n1, it follows that φ(0)j(xmxn) is divisible by xmxn, and φ(0)j(xmxnz) is divisible by xmxnz. For 1m1, let the congruence notation (m,k) in the subsequent calculation denote modulo the ideal (xmxkz). We derive

    φ(0)j(xmxz)=(xjxj+1z)0k1j10k2j2(1)k1+k2Ijk11[1,j1]Ijk22[j+2,1][¯Bk1,k2(xm,z)¯Bk1,k2(x+z,z)](m,1)0,

    which implies that φ(0)j(xmxz) is divisible by xmxz. Thus, φ(0)jD(A[0]) for 1j2. Therefore, we have η1,η2,φ(0)1,,φ(0)1D(A[0]). Additionally, we obtain

    detM(η1,η2,φ(0)1,,φ(0)1)=(1)+1zdet(1φ(0)1,1φ(0)1,201φ(0)1,1φ(0)1,201φ(0),1φ(0),21s=1(xsxz))×=(1)+1z1s=1(xsxz)det(1φ(0)1,1φ(0)1,21φ(0)1,1φ(0)1,2)(1)×(1)=(1)+1z1s=1(xsxz)det(α1(φ(0)i,j)(1i1,1j2))(1)×(1).

    According to the equalities (1.1) and (2.1), we have

    det(α1(φ(0)i,j)(1i1,1j2))1m<n1(xmxn)(xmxnz).

    Hence, we obtain

    detM(η1,η2,φ(0)1,,φ(0)1).=z1s=1(xsxz)1m<n1(xmxn)(xmxnz)=z1m<n1(xmxn)1m<n(xmxnz)=Q(A[0]).

    By applying Theorem 1.2, we conclude that the derivations η1,η2,φ(0)1,,φ(0)1 form a basis for D(A[0]).

    Definition 2.1. For 1q2, 1j1, define the homogeneous derivations

    φ(q)j:={φ(0)j,1jq2,(xj+1x)φ(0)j(xjxj+1z)j1a=q1φ(0)a,q1j3,φ(0)j+1+ja=q1(a1)φ(0)a,j=2,(xjx)φ(0)j+(xjxj+1z)j2a=q1(a2)φ(0)a,j=1.

    To prove the derivations η1,η2,φ(q)1,,φ(q)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 1m1, 1j2, we have

    φ(0)j(xmx)(m,0)(z)(xjxj+1z)j1s=1(xsxmz)1s=j+2(xsxm). (2.2)

    Proof. We have the following congruence relation of polynomials modulo the ideal (xmx).

    φ(0)j(xmx)=(xjxj+1z)0k1j10k2j2(1)k1+k2Ijk11[1,j1]Ijk22[j+2,1][¯Bk1,k2(xm,z)¯Bk1,k2(x+z,z)](m,0)(xjxj+1z)0k1j10k2j2(1)k1+k2+1Ijk11[1,j1]Ijk22[j+2,1][¯Bk1,k2(xm+z,z)¯Bk1,k2(xm,z)]=(xjxj+1z)0k1j10k2j2(1)k1+k2+1Ijk11[1,j1]Ijk22[j+2,1]zk1+k2+1(xm+zz)k1(xmz)k2=(z)(xjxj+1z)j1k1=0Ijk11[1,j1][(xm+z)]k1j2k2=0Ijk22[j+2,1](xm)k2=(z)(xjxj+1z)j1s=1(xsxmz)1s=j+2(xsxm).

    We complete the proof.

    Remark 2.1 In equality (2.2), we observe that 1s=j+2(xsxm)=0 for j+2m1. This implies that φ(0)j(xmx) is divisible by xmx for 1j3 and j+2m1.

    According to Remark 2.1, for 1jq2, we have φ(0)j(xmx) is divisible by xmx for qm1, which implies that φ(q)j=φ(0)jD(A[q]). Therefore, to prove the derivations belong to the module D(A[q]), it suffices to prove φ(q)jD(A[q]) for q1j1.

    For the sake of convenience in the proof, let us introduce the notations for f,g,hZ+,

    A[g,h]f:=hs=g(xsxf),B[g,h]f:=hs=g(xsxfz).

    If g>h, we agree that A[g,h]f=B[g,h]f=1.

    Lemma 2.1. For any u,v,wZ+ that satisfy 4j+1u2, 3v2, and 3w2, we have the following three equalities:

    B[u,j1]u1=A[u+1,j]u1+j1a=u(xaxa+1z)A[a+2,j]u1B[u,a1]u1. (2.3)
    B[v,1]v1=(v+1)A[v,1]v1+2a=v1(a1)(xaxa+1z)A[a+2,1]v1B[v1,a1]v1. (2.4)
    B[w,2]w1=wA[w,2]w1+3a=w1(a2)(xaxa+1z)A[a+2,2]w1B[w1,a1]w1. (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[3,2]4+3a=4(a2)(xaxa+1z)A[a+2,2]4B[4,a1]4=3(x3x4)(x2x4)+2(x4x3z)(x2x4)+(x3x2z)(z)=(x3x4z)(x2x4z)=B[3,2]4,

    and the equality holds. Assume that for w=k3, the equality holds. Then, we replace xk1 with xk2, and multiply both sides of the equality by (xk1xk2z) to get

    B[k1,2]k2=(k1)(xk2xkz)(xk1xk2z)A[k+1,2]k2+k(xk1xk2z)A[k,2]k2+3a=k(a2)(xaxa+1z)A[a+2,2]k2B[k2,a1]k2=(k+1)A[k1,2]k2+3a=k2(a2)(xaxa+1z)A[a+2,2]k2B[k2,a1]k2.

    We have completed the induction.

    Lemma 2.2. The derivation φ(q)j belongs to the module D(A[q]) for 2q2 and q1j3.

    Proof. For 2q2 and j=q1, it is evident from Remark 2.1 that

    φ(q)q1=(xqx)φ(0)q1D(A[q]).

    For 3q2 and qj3, we will establish this by induction on q. From Theorem 2.2, for qm1, we have

    φ(q)j(xmx)=(xj+1x)φ(0)j(xmx)(xjxj+1z)j1a=q1φ(0)a(xmx)(m,0)(z)(xjxj+1z)A[j+1,1]mB[1,q2]m[B[q1,j1]mj1a=q1(xaxa+1z)A[a+2,j]mB[q1,a1]m].

    (1) For q=3, we get

    φ(3)3(xmx)(m,0)(z)(x3x2z)A[2,1]mB[1,5]m(x3xm).

    If m=3,2,1, then we have φ(3)3(xmx)(m,0)0, which indicates that φ(3)3(xmx) is divisible by xmx for m=3,2,1. Therefore, φ(3)jD(A[3]).

    (2) For q=k3, assume that φ(k)jD(A[k]), which implies that φ(k)j(xmx) is divisible by xmx for km1.

    For q=k+1, we observe that

    φ(k+1)j=φ(k)j(xjxj+1z)φ(0)k2.

    According to the induction hypothesis and Remark 2.1, it is sufficient to prove that φ(k+1)j(xk1x) is divisible by xk1x. By using the equality (2.3), we obtain

    φ(k+1)j(xk1x)(k1,0)(z)(xjxj+1z)A[j+1,1]k1B[1,k3]k1[B[k2,j1]k1j1a=k2(xaxa+1z)A[a+2,j]k1B[k2,a1]k1]=(z)2(xjxj+1z)A[j+1,1]k1B[1,k2]k1[B[k,j1]k1A[k+1,j]k1j1a=k(xaxa+1z)A[a+2,j]k1B[k,a1]k1]=0.

    Therefore, φ(k+1)j(xk1x) is divisible by xk1x, and it follows that φ(k+1)jD(A[k+1]). Consequently, we can conclude that for any 3q2 and qj3, φ(q)jD(A[q]).

    Lemma 2.3. The derivation φ(q)2 belongs to the module D(A[q]) for 1q2.

    Proof. From Theorem 2.2, we can get the following equality for qm1,

    φ(q)2(xmx)=φ(0)1(xmx)+2a=q1(a1)φ(0)a(xmx)(m,0)B[1,1]m+(z)2a=q1(a1)(xaxa+1z)A[a+2,1]mB[1,a1]m.

    (1) For q=1,2, this conclusion is straightforward to verify.

    (2) For q=k3, assume that φ(k)2D(A[k]), which implies that φ(k)2(xmx) is divisible by xmx for km1.

    For q=k+1, we have

    φ(k+1)2=φ(k)2+(k+1)φ(0)k2.

    By using the equality (2.4), we have

    φ(k+1)2(xk1x)(k1,0)B[1,1]k1+(z)2a=k2(a1)(xaxa+1z)A[a+2,1]k1B[1,a1]k1=B[1,k1]k1[B[k,1]k1+(k+1)A[k,1]k1+2a=k1(a1)(xaxa+1z)A[a+2,1]k1B[k1,a1]k1]=0.

    Therefore, φ(k+1)2(xk1x) is divisible by xk1x. According to the induction hypothesis and Remark 2.1, we have φ(k+1)2D(A[k+1]). Hence, we may conclude that for any 1q2, φ(q)2D(A[q]).

    Lemma 2.4. The derivation φ(q)1 belongs to the module D(A[q]) for 1q2.

    Proof. First, from Theorem 2.2, for qm1, we can get

    φ(q)1(xmx)=(x1x)φ(0)1(xmx)+(x1xz)3a=q1(a2)φ(0)a(xmx)(m,0)(x1xm)B[1,1]m+(z)(x1xmz)3a=q1(a2)(xaxa+1z)A[a+2,1]mB[1,a1]m.

    (1) For q=1,2, it is obvious that φ(q)1D(A[q]).

    (2) For q=k3, assume that φ(k)1D(A[k]), which implies that φ(k)1(xmx) is divisible by xmx for km1.

    For q=k+1, we can see

    φ(k+1)1=φ(k)1+k(x1xz)φ(0)k2.

    By using the equality (2.5), we have

    φ(k+1)1(xk1x)(k1,0)(x1xk1)B[1,1]k1+(z)(x1xk1z)3a=k2(a2)(xaxa+1z)A[a+2,1]k1B[1,a1]k1=(x1xk1)(x1xk1z)B[1,k1]k1[B[k,2]k1+kA[k,2]k1+3a=k1(a2)(xaxa+1z)A[a+2,2]k1B[k1,a1]k1]=0.

    Therefore, φ(k+1)1(xk1x) is divisible by xk1x. According to the induction hypothesis and Remark 2.1, we have φ(k+1)1D(A[k+1]). Hence, we may conclude that for any 1q2, φ(q)1D(A[q]).

    From the above proof, we finally conclude that φ(q)1,,φ(q)1 belong to the module D(A[q]).

    Theorem 2.3. For 1q2, the derivations η1,η2,φ(q)1,,φ(q)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)1)Q(A[q]).

    Let

    γ1=(q,q1,,1,1)T

    and

    γ2=((q1)(x1xz),(q2)(x1xz),,x1xz,0,x1x)T

    be the (q+1)×1 column vectors, and define a matrix

    M(q+1)×(q1):=(xqx(xqxq+1z)(x3x2z)0xq+1x(x3x2z)00x2x000000).

    Write ˜M(q+1)×(q+1):=(M(q+1)×(q1),γ1,γ2), then

    det˜M(q+1)×(q+1)=1s=q(xsx).

    Thus, we obtain the following equality

    (η1,η2,φ(q)1,,φ(q)1)(+1)×(+1)=(η1,η2,φ(0)1,,φ(0)1)(Eq0(q)×(q+1)0(q+1)×(q)˜M(q+1)×(q+1)).

    Hence,

    detM(η1,η2,φ(q)1,,φ(q)1)=detM(η1,η2,φ(0)1,,φ(0)1)det˜M(q+1)×(q+1).=z1m<n1(xmxn)1m<n(xmxnz)1s=q(xsx)=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.



    [1] T. Abe, Divisionally free arrangements of hyperplanes, Invent. Math., 204 (2016), 317–346. http://dx.doi.org/10.1007/s00222-015-0615-7 doi: 10.1007/s00222-015-0615-7
    [2] T. Abe, D. Suyama, A basis construction of the extended Catalan and Shi arrangements of the type A2, J. Algebra, 493 (2018), 20–35. http://dx.doi.org/10.1016/j.jalgebra.2017.09.024 doi: 10.1016/j.jalgebra.2017.09.024
    [3] T. Abe, H. Terao, The freeness of Shi-Catalan arrangements, Eur. J. Combin., 32 (2011), 1191–1198. http://dx.doi.org/10.1016/j.ejc.2011.06.005 doi: 10.1016/j.ejc.2011.06.005
    [4] T. Abe, H. Terao, Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements, J. Algebr. Comb., 43 (2016), 33–44. http://dx.doi.org/10.1007/s10801-015-0624-z doi: 10.1007/s10801-015-0624-z
    [5] M. Feigin, Z. Wang, M. Yoshinaga, Integral expressions for derivations of multiarrangements, arXiv: 2309.01287. http://dx.doi.org/10.48550/arXiv.2309.01287
    [6] R. Gao, D. Pei, H. Terao, The Shi arrangement of the type D, Proc. Japan Acad. Ser. A Math. Sci., 88 (2012), 41–45. http://dx.doi.org/10.3792/pjaa.88.41 doi: 10.3792/pjaa.88.41
    [7] P. Orlik, H. Terao, Arrangements of hyperplanes, Berlin: Springer-Verlag, 1992. http://dx.doi.org/10.1007/978-3-662-02772-1
    [8] K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 265–291.
    [9] J. Shi, The Kazhdan-Lusztig cells in certain affine Weyl groups, Berlin: Springer-Verlag, 2006. http://dx.doi.org/10.1007/BFb0074968
    [10] D. Suyama, A basis construction for the Shi arrangement of the type B or C, Commun. Algebra, 43 (2015), 1435–1448. http://dx.doi.org/10.1080/00927872.2013.865051 doi: 10.1080/00927872.2013.865051
    [11] D. Suyama, H. Terao, The Shi arrangements and the Bernoulli polynomials, Bull. Lond. Math. Soc., 44 (2012), 563–570. http://dx.doi.org/10.1112/blms/bdr118 doi: 10.1112/blms/bdr118
    [12] D. Suyama, M. Yoshinaga, The primitive derivation and discrete integrals, SIGMA, 17 (2021), 038. http://dx.doi.org/10.3842/sigma.2021.038 doi: 10.3842/sigma.2021.038
    [13] H. Terao, Multiderivations of Coxeter arrangements, Invent. Math., 148 (2002), 659–674. http://dx.doi.org/10.1007/s002220100209 doi: 10.1007/s002220100209
    [14] Z. Wang, G. Jiang, Free subarrangements of Shi arrangements, Graph. Combinator., 38 (2022), 59. http://dx.doi.org/10.1007/s00373-021-02399-2 doi: 10.1007/s00373-021-02399-2
    [15] M. Yoshinaga, Characterization of a free arrangement and conjecture of Edelman and Reiner, Invent. Math., 157 (2004), 449–454. http://dx.doi.org/10.1007/s00222-004-0359-2 doi: 10.1007/s00222-004-0359-2
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(456) PDF downloads(41) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog