# Integer Solutions of Integral Inequalities andH-Invariant

Authors: G. Ortenzi; V. Rubtsov; S. R. Tagne Pelap
Journal: Advances in Mathematical Physics
 (2011)

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## Abstract

We study the Jacobian Poisson structures in any dimension invariant with respect
to the discrete Heisenberg group. The classification problem is related to the discrete
volume of suitable solids. Particular attention is given to dimension 3 whose simplest
example is the Artin-Schelter-Tate Poisson tensors.


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## Body

## 1. Introduction

This paper continues the authors' program of studies of the Heisenberg invariance properties of polynomial Poisson algebras which were started in [1] and extended in [2, 3]. Formally speaking, we consider the polynomials in n variables ℂ[x0,x1,…,xn-1] over ℂ and the action of some subgroup Hn of GLn(ℂ) generated by the shifts operators xi→xi+1(modℤn) and by the operators x→ɛixi, where ɛn=1. We are interested in the polynomial Poisson brackets on ℂ[x0,x1,…,xn-1] which are “stable” under this actions (we will give more precise definition below).The most famous examples of the Heisenberg invariant polynomial Poisson structures are the Sklyanin-Odesskii-Feigin-Artin-Tate quadratic Poisson brackets known also as the elliptic Poisson structures. One can also think about these algebras like the “quasiclassical limits” of elliptic Sklyanin associative algebras. These is a class of Noetherian graded associative algebras which are Koszul, Cohen-Macaulay, and have the same Hilbert function as a polynomial ring withn variables. The above-mentioned Heisenberg group action provides the automorphisms of Sklyanin algebras which are compatible with the grading and defines an Hn-action on the elliptic quadratic Poisson structures on ℙn. The latter are identified with Poisson structures on some moduli spaces of the degree n and rank k+1 vector bundles with parabolic structure (= the flag 0⊂F⊂ℂk+1 on the elliptic curve ℰ). We will denote this elliptic Poisson algebras by qn;k(ℰ). The algebras qn;k(E) arise in the Feigin-Odesskii “deformational” approach and form a subclass of polynomial Poisson structures. A comprehensive review of elliptic algebras can be found in [4] to which we refer for all additional information. We will mention only that as we have proved in [3] all elliptic Poisson algebras (being in particular Heisenberg-invariant) are unimodular.Another interesting class of polynomial Poisson structures consists of so-called Jacobian Poisson structures (JPS). These structures are a special case of Nambu-Poisson structures. Their rank is two, and the Jacobian Poisson bracket{P,Q} of two polynomials P and Q is given by the determinant of Jacobi matrix of functions (P,Q,P1,…,Pn-2). The polynomials Pi, 1≤i≤n-2 are Casimirs of the bracket and under some mild condition of independence are generators of the centrum for the Jacobian Poisson algebra structure on ℂ[x0,…,xn-1]. This type of Poisson algebras was intensively studied (due to their natural origin and relative simplicity) in a huge number of publications among which we should mention [1, 5–9].There are some beautiful intersections between two described types of polynomial Poisson structures: when we are restricting ourselves to the class of quadratic Poisson brackets then there are only Artin-Schelter-Tate (n=3) and Sklyanin (n=4) algebras which are both elliptic and Jacobian. It is no longer true for n>4. The relations between the Sklyanin Poisson algebras qn,k(ℰ) whose centrum has dimension 1 (for n odd) and 2 (for n even) in the case k=1 and is generated by l=𝔤cd(n,k+1) Casimirs for qn,k(ℰ) for k>1 are in general quite obscure. We can easily found that sometimes the JPS structures correspond to some degenerations of the Sklyanin elliptic algebras. One example of such JPS for n=5 was remarked in [8] and was attributed to so-called Briesckorn-Pham polynomials for n=5(1.1)P1=∑i=04αixi;P2=∑i=04βixi2;P3=∑i=04γixi2.
It is easy to check that the homogeneous quintic P=P1P2P3  (see Section 4.2)  defines a Casimir for some rational degeneration of (one of) elliptic algebras q5,1(ℰ) and q5,2(ℰ) if it satisfies the H-invariance condition.In this paper, we will study the Jacobian Poisson structures in any number of variables which are Heisenberg-invariant and we relate all such structures to some graded subvector spaceℋ of polynomial algebra. This vector space is completely determined by some enumerative problem of a number-theoretic type. More precisely, the homogeneous subspace ℋi of ℋ of degree i is in bijection with integer solutions of a system of Diophant inequalities. Geometric interpretation of the dimension of ℋi is described in terms of integer points in a convex polytope given by this Diophant system. In the special case of dimension 3, ℋ is a subalgebra of polynomial algebra with 3 variables and all JPS are given by this space. We solve explicitly the enumerative problem in this case and obtain a complete classification of the H-invariant not necessarily quadratic Jacobian Poisson algebras with three generators. As a byproduct, we explicitly compute the Poincaré series of ℋ. In this dimension, we observe that the H-invariant JPS of degree 5 is given by the Casimir sextic (1.2)P∨=16a(x06+x16+x26)+13b(x03x13+x03x23+x13x23)+c(x04x1x2+x0x14x2+x0x1x24)+12dx02x12x22,a,b,c,d∈ℂ. This structure is a “projectively dual” to the Artin-Schelter-Tate elliptic Poisson structure which is the H-invariant JPS given by the cubic (1.3)P=(x03+x13+x23)+γx0x1x2,
where γ∈ℂ. In fact, the algebraic variety ℰ∨:{P∨=0}∈ℙ2 is the (generically) projectively dual to the elliptic curve ℰ:{P=0}⊂ℙ2.The paper is organized as follows: in Section2, we remind a definition of the Heisenberg group in the Schroedinger representation and describe its action on Poisson polynomial tensors and also the definition of JPS. In Section 3, we treat the above mentioned enumerative problem in dimension 3. The last section concerns the case of any dimension. Here, we discuss some possible approaches to the general enumerative question.


## 2. Preliminary Facts

Throughout of this paper,K is a field of characteristic zero. Let us start by remembering some elementary notions of the Poisson geometry.


### 2.1. Poisson Algebras and Poisson Manifold

Letℛ be a commutative K-algebra. One says that ℛ is a Poisson algebra if ℛ is endowed with a Lie bracket, indicated with {·,·}, which is also a biderivation. One can also say that ℛ is endowed with a Poisson structure, and therefore, the bracket {·,·} is called the Poisson bracket. Elements of the center are called Casimirs: a∈ℛ is a Casimir if {a,b}=0 for all b∈ℛ.A Poisson manifoldM (smooth, algebraic, etc.) is a manifold whose function algebra 𝒜 (C∞(M), regular, etc.) is endowed with a Poisson bracket.As examples of Poisson structures let us consider a particular subclass of Poisson structures which are uniquely characterized by their Casimirs. In the dimension 4, let(2.1)q1=12(x02+x22)+kx1x3,q2=12(x12+x32)+kx0x2
be two elements of ℂ[x0,x1,x2,x3], where k∈ℂ.Onℂ[x0,x1,x2,x3], a Poisson structure π is defined by (2.2){f,g}π:=df∧dg∧dq1∧dq2dx0∧dx1∧dx2∧dx3,
or more explicitly (mod ℤ4) (2.3){xi,xi+1}=k2xixi+1-xi+2xi+3,{xi,xi+2}=k(xi+32-xi+12),i=0,1,2,3.
Sklyanin had introduced this Poisson algebra which carries today his name in a Hamiltonian approach to the continuous and discrete integrable Landau-Lifshitz models [10, 11]. He showed that the Hamiltonian structure of the classical model is completely determined by two quadratic “Casimirs”. The Sklyanin Poisson algebra is also called elliptic due to its relations with an elliptic curve. The elliptic curve enters in the game from the geometric side. The symplectic foliation of Sklyanin's structure is too complicated. This is because the structure is degenerated and looks quite different from a symplectic one. But the intersection locus of two Casimirs in the affine space of dimension four (one can consider also the projective situation) is an elliptic curve ℰ given by two quadrics q1,2. We can think about this curve ℰ as a complete intersection of the couple q1=0, q2=0 embedded in 𝒞P3 (as it was observed in Sklyanin's initial paper).A possible generalization one can be obtained consideringn-2 polynomials Qi in Kn with coordinates xi, i=0,…,n-1. We can define a bilinear differential operation (2.4){⋅,⋅}:K[x1,…,xn]⊗K[x1,…,xn]⟶K[x1,…,xn],
by(2.5){f,g}=df∧dg∧dQ1∧⋯∧dQn-2dx1∧dx2∧⋯∧dxn,f,g∈K[x1,…,xn].
This operation, which gives a Poisson algebra structure on K[x1,…,xn], is called a Jacobian Poisson structure (JPS), and it is a partial case of more general n-m-ary Nambu operation given by an antisymmetric n-m-polyvector field introduced by Nambu [6] and was extensively studied by Takhtajan [5].The polynomialsQi,  i=1,…,n-2 are Casimir functions for the brackets (2.5).There exists a second generalization of the Sklyanin algebra that we will describe briefly in the next subsection (see, for details, [4]).


### 2.2. Elliptic Poisson Algebrasqn(ℰ,η) and qn,k(ℰ,η)

(We report here this subsection from [2] for sake of self-consistency).These algebras, defined by Feĭgin and Odesskiĭ, arise as quasiclassical limits of elliptic associative algebrasQn(ℰ,η) and Qn,k(ℰ,η) [12, 13].LetΓ=ℤ+τℤ⊂ℂ be an integral lattice generated by 1 and τ∈ℂ, with Imτ>0. Consider the elliptic curve ℰ=ℂ/Γ and a point η on this curve.In their article [13], given k<n, mutually prime, Odesskiĭ and Feĭgin construct an algebra, called elliptic, Qn,k(ℰ,η), as an algebra defined by n generators {xi,i∈ℤ/nℤ} and the following relations:(2.6)∑r∈Z/nZθj-i+r(k-1)(0)θkr(η)θj-i-r(-η)xj-rxi+r=0,i≠j,i,j∈ZnZ,
where θα are theta functions [13].These family of algebras has the following properties:(1)
the center of the algebraQn,k(ℰ,η), for generic ℰ and η, is the algebra of polynomial of m=pgcd(n,k+1) variables of degree n/m,(2)
Qn,k(ℰ,0)=ℂ[x1,…,xn] is commutative,(3)
Qn,n-1(ℰ,η)=ℂ[x1,…,xn] is commutative for all η,(4)
Qn,k(ℰ,η)≃Qn,k′(ℰ,η), if kk′≡1 (mod n),(5)
the mapsxi↦xi+1 et xi↦ɛixi, where ɛn=1, define automorphisms of the algebra Qn,k(ℰ,η),(6)
the algebrasQn,k(ℰ,η) are deformations of polynomial algebras. The associated Poisson structure is denoted by qn,k(ℰ,η),(7)
among the algebrasqn,k(ℰ,η), only q3(ℰ,η) (the Artin-Schelter-Tate algebra) and the Sklyanin algebra q4(ℰ,η) are Jacobian Poisson structures.


### 2.3. The Heisenberg Invariant Poisson Structures

#### 2.3.1. The G-Invariant Poisson Structures

LetG be a group acting on a Poisson algebra ℛ.Definition 2.1.
A Poisson bracket{·,·} on ℛ is said to be a G-invariant if G acts on ℛ by Poisson automorphisms.
In other words, for everyg∈G, the morphism φg:ℛ→ℛ, a↦g·a is an automorphism and the following diagram is a commutative:(2.7)


#### 2.3.2. The H-Invariant Poisson Structures

In their paper [2], the authors introduced the notion of H-invariant Poisson structures. That is, a special case of a G-invariant structure when G in the finite Heisenberg group and ℛ is the polynomial algebra. Let us remember this notion.LetV be a complex vector space of dimension n and e0,…,en-1 a basis of V. Take the nth primitive root of unity ɛ=e2πi/n.Considerσ,τ of GL(V) defined by (2.8)σ(em)=em-1,τ(em)=ɛmem.
The Heisenberg of dimension n is nothing else that the subspace Hn⊂GL(V) generated by σ and τ.From now on, we assume thatV=ℂn, with x0,x1,…,xn-1 as a basis and consider the coordinate ring ℂ[x0,x1,…,xn-1].Naturally,σ and τ act by automorphisms on the algebra ℂ[x0,x1,…,xn-1] as follows: (2.9)σ⋅(αx0α0x1α1⋯xn-1αn-1)=αx0αn-1x1α0⋯xn-1αn-2,τ⋅(αx0α0x1α1⋯xn-1αn-1)=ɛα1+2α2+⋯+(n-1)αn-1αx0α0x1α1⋯xn-1αn-1.
We introduced in [2] the notion of τ-degree on the polynomial algebra ℂ[x0,x1,…,xn-1]. The τ-degree of a monomial M=αx0α0x1α1⋯xn-1αn-1 is the positive integer α1+2α2+⋯+(n-1)αn-1∈ℤ/nℤ if α≠0 and -∞ if not. The τ-degree of M is denoted τϖ(M). A τ-degree of a polynomial is the highest τ-degree of its monomials.For simplicity, theHn-invariance condition will be referred from now on just as H-invariance. An H-invariant Poisson bracket on 𝒜=ℂ[x0,x1,…,xn-1] is nothing but a bracket on 𝒜 which satisfy the following: (2.10){xi+1,xj+1}=σ⋅{xi,xj},τ⋅{xi,xj}=ɛi+j{xi,xj},
for all i,j∈ℤ/nℤ.Theτ invariance is, in some sense, a “discrete” homogeneity.Proposition 2.2 (see [2]).
The Sklyanin-Odesskii-Feigin Poisson algebrasqn,k(ℰ) are H-invariant Poisson algebras.Therefore, anH-invariant Poisson structures on the polynomial algebra ℛ includes as the Sklyanin Poisson algebra or more generally of the Odesskii-Feigin Poisson algebras.In this paper, we are interested in the intersection of the two classes of generalizations of Artin-Shelter-Tate-Sklyanin Poisson algebras: JPS andH-invariant Poisson structures.Proposition 2.3 (see [2]).
If{·,·} is an H-invariant polynomial Poisson bracket, the usual polynomial degree of the monomial of {xi,xj} equals to 2+sn,  s∈ℕ.Proposition 2.4 (see [2]).
LetP∈ℛ=ℂ[x0,…,xn-1].
For alli∈{0,…,n-1},
(2.11)σ⋅∂F∂xi=∂(σ⋅F)∂(σ⋅xi).


## 2.1. Poisson Algebras and Poisson Manifold

Letℛ be a commutative K-algebra. One says that ℛ is a Poisson algebra if ℛ is endowed with a Lie bracket, indicated with {·,·}, which is also a biderivation. One can also say that ℛ is endowed with a Poisson structure, and therefore, the bracket {·,·} is called the Poisson bracket. Elements of the center are called Casimirs: a∈ℛ is a Casimir if {a,b}=0 for all b∈ℛ.A Poisson manifoldM (smooth, algebraic, etc.) is a manifold whose function algebra 𝒜 (C∞(M), regular, etc.) is endowed with a Poisson bracket.As examples of Poisson structures let us consider a particular subclass of Poisson structures which are uniquely characterized by their Casimirs. In the dimension 4, let(2.1)q1=12(x02+x22)+kx1x3,q2=12(x12+x32)+kx0x2
be two elements of ℂ[x0,x1,x2,x3], where k∈ℂ.Onℂ[x0,x1,x2,x3], a Poisson structure π is defined by (2.2){f,g}π:=df∧dg∧dq1∧dq2dx0∧dx1∧dx2∧dx3,
or more explicitly (mod ℤ4) (2.3){xi,xi+1}=k2xixi+1-xi+2xi+3,{xi,xi+2}=k(xi+32-xi+12),i=0,1,2,3.
Sklyanin had introduced this Poisson algebra which carries today his name in a Hamiltonian approach to the continuous and discrete integrable Landau-Lifshitz models [10, 11]. He showed that the Hamiltonian structure of the classical model is completely determined by two quadratic “Casimirs”. The Sklyanin Poisson algebra is also called elliptic due to its relations with an elliptic curve. The elliptic curve enters in the game from the geometric side. The symplectic foliation of Sklyanin's structure is too complicated. This is because the structure is degenerated and looks quite different from a symplectic one. But the intersection locus of two Casimirs in the affine space of dimension four (one can consider also the projective situation) is an elliptic curve ℰ given by two quadrics q1,2. We can think about this curve ℰ as a complete intersection of the couple q1=0, q2=0 embedded in 𝒞P3 (as it was observed in Sklyanin's initial paper).A possible generalization one can be obtained consideringn-2 polynomials Qi in Kn with coordinates xi, i=0,…,n-1. We can define a bilinear differential operation (2.4){⋅,⋅}:K[x1,…,xn]⊗K[x1,…,xn]⟶K[x1,…,xn],
by(2.5){f,g}=df∧dg∧dQ1∧⋯∧dQn-2dx1∧dx2∧⋯∧dxn,f,g∈K[x1,…,xn].
This operation, which gives a Poisson algebra structure on K[x1,…,xn], is called a Jacobian Poisson structure (JPS), and it is a partial case of more general n-m-ary Nambu operation given by an antisymmetric n-m-polyvector field introduced by Nambu [6] and was extensively studied by Takhtajan [5].The polynomialsQi,  i=1,…,n-2 are Casimir functions for the brackets (2.5).There exists a second generalization of the Sklyanin algebra that we will describe briefly in the next subsection (see, for details, [4]).


## 2.2. Elliptic Poisson Algebrasqn(ℰ,η) and qn,k(ℰ,η)

(We report here this subsection from [2] for sake of self-consistency).These algebras, defined by Feĭgin and Odesskiĭ, arise as quasiclassical limits of elliptic associative algebrasQn(ℰ,η) and Qn,k(ℰ,η) [12, 13].LetΓ=ℤ+τℤ⊂ℂ be an integral lattice generated by 1 and τ∈ℂ, with Imτ>0. Consider the elliptic curve ℰ=ℂ/Γ and a point η on this curve.In their article [13], given k<n, mutually prime, Odesskiĭ and Feĭgin construct an algebra, called elliptic, Qn,k(ℰ,η), as an algebra defined by n generators {xi,i∈ℤ/nℤ} and the following relations:(2.6)∑r∈Z/nZθj-i+r(k-1)(0)θkr(η)θj-i-r(-η)xj-rxi+r=0,i≠j,i,j∈ZnZ,
where θα are theta functions [13].These family of algebras has the following properties:(1)
the center of the algebraQn,k(ℰ,η), for generic ℰ and η, is the algebra of polynomial of m=pgcd(n,k+1) variables of degree n/m,(2)
Qn,k(ℰ,0)=ℂ[x1,…,xn] is commutative,(3)
Qn,n-1(ℰ,η)=ℂ[x1,…,xn] is commutative for all η,(4)
Qn,k(ℰ,η)≃Qn,k′(ℰ,η), if kk′≡1 (mod n),(5)
the mapsxi↦xi+1 et xi↦ɛixi, where ɛn=1, define automorphisms of the algebra Qn,k(ℰ,η),(6)
the algebrasQn,k(ℰ,η) are deformations of polynomial algebras. The associated Poisson structure is denoted by qn,k(ℰ,η),(7)
among the algebrasqn,k(ℰ,η), only q3(ℰ,η) (the Artin-Schelter-Tate algebra) and the Sklyanin algebra q4(ℰ,η) are Jacobian Poisson structures.


## 2.3. The Heisenberg Invariant Poisson Structures

### 2.3.1. The G-Invariant Poisson Structures

LetG be a group acting on a Poisson algebra ℛ.Definition 2.1.
A Poisson bracket{·,·} on ℛ is said to be a G-invariant if G acts on ℛ by Poisson automorphisms.
In other words, for everyg∈G, the morphism φg:ℛ→ℛ, a↦g·a is an automorphism and the following diagram is a commutative:(2.7)


### 2.3.2. The H-Invariant Poisson Structures

In their paper [2], the authors introduced the notion of H-invariant Poisson structures. That is, a special case of a G-invariant structure when G in the finite Heisenberg group and ℛ is the polynomial algebra. Let us remember this notion.LetV be a complex vector space of dimension n and e0,…,en-1 a basis of V. Take the nth primitive root of unity ɛ=e2πi/n.Considerσ,τ of GL(V) defined by (2.8)σ(em)=em-1,τ(em)=ɛmem.
The Heisenberg of dimension n is nothing else that the subspace Hn⊂GL(V) generated by σ and τ.From now on, we assume thatV=ℂn, with x0,x1,…,xn-1 as a basis and consider the coordinate ring ℂ[x0,x1,…,xn-1].Naturally,σ and τ act by automorphisms on the algebra ℂ[x0,x1,…,xn-1] as follows: (2.9)σ⋅(αx0α0x1α1⋯xn-1αn-1)=αx0αn-1x1α0⋯xn-1αn-2,τ⋅(αx0α0x1α1⋯xn-1αn-1)=ɛα1+2α2+⋯+(n-1)αn-1αx0α0x1α1⋯xn-1αn-1.
We introduced in [2] the notion of τ-degree on the polynomial algebra ℂ[x0,x1,…,xn-1]. The τ-degree of a monomial M=αx0α0x1α1⋯xn-1αn-1 is the positive integer α1+2α2+⋯+(n-1)αn-1∈ℤ/nℤ if α≠0 and -∞ if not. The τ-degree of M is denoted τϖ(M). A τ-degree of a polynomial is the highest τ-degree of its monomials.For simplicity, theHn-invariance condition will be referred from now on just as H-invariance. An H-invariant Poisson bracket on 𝒜=ℂ[x0,x1,…,xn-1] is nothing but a bracket on 𝒜 which satisfy the following: (2.10){xi+1,xj+1}=σ⋅{xi,xj},τ⋅{xi,xj}=ɛi+j{xi,xj},
for all i,j∈ℤ/nℤ.Theτ invariance is, in some sense, a “discrete” homogeneity.Proposition 2.2 (see [2]).
The Sklyanin-Odesskii-Feigin Poisson algebrasqn,k(ℰ) are H-invariant Poisson algebras.Therefore, anH-invariant Poisson structures on the polynomial algebra ℛ includes as the Sklyanin Poisson algebra or more generally of the Odesskii-Feigin Poisson algebras.In this paper, we are interested in the intersection of the two classes of generalizations of Artin-Shelter-Tate-Sklyanin Poisson algebras: JPS andH-invariant Poisson structures.Proposition 2.3 (see [2]).
If{·,·} is an H-invariant polynomial Poisson bracket, the usual polynomial degree of the monomial of {xi,xj} equals to 2+sn,  s∈ℕ.Proposition 2.4 (see [2]).
LetP∈ℛ=ℂ[x0,…,xn-1].
For alli∈{0,…,n-1},
(2.11)σ⋅∂F∂xi=∂(σ⋅F)∂(σ⋅xi).


## 2.3.1. The G-Invariant Poisson Structures

LetG be a group acting on a Poisson algebra ℛ.Definition 2.1.
A Poisson bracket{·,·} on ℛ is said to be a G-invariant if G acts on ℛ by Poisson automorphisms.
In other words, for everyg∈G, the morphism φg:ℛ→ℛ, a↦g·a is an automorphism and the following diagram is a commutative:(2.7)


## 2.3.2. The H-Invariant Poisson Structures

In their paper [2], the authors introduced the notion of H-invariant Poisson structures. That is, a special case of a G-invariant structure when G in the finite Heisenberg group and ℛ is the polynomial algebra. Let us remember this notion.LetV be a complex vector space of dimension n and e0,…,en-1 a basis of V. Take the nth primitive root of unity ɛ=e2πi/n.Considerσ,τ of GL(V) defined by (2.8)σ(em)=em-1,τ(em)=ɛmem.
The Heisenberg of dimension n is nothing else that the subspace Hn⊂GL(V) generated by σ and τ.From now on, we assume thatV=ℂn, with x0,x1,…,xn-1 as a basis and consider the coordinate ring ℂ[x0,x1,…,xn-1].Naturally,σ and τ act by automorphisms on the algebra ℂ[x0,x1,…,xn-1] as follows: (2.9)σ⋅(αx0α0x1α1⋯xn-1αn-1)=αx0αn-1x1α0⋯xn-1αn-2,τ⋅(αx0α0x1α1⋯xn-1αn-1)=ɛα1+2α2+⋯+(n-1)αn-1αx0α0x1α1⋯xn-1αn-1.
We introduced in [2] the notion of τ-degree on the polynomial algebra ℂ[x0,x1,…,xn-1]. The τ-degree of a monomial M=αx0α0x1α1⋯xn-1αn-1 is the positive integer α1+2α2+⋯+(n-1)αn-1∈ℤ/nℤ if α≠0 and -∞ if not. The τ-degree of M is denoted τϖ(M). A τ-degree of a polynomial is the highest τ-degree of its monomials.For simplicity, theHn-invariance condition will be referred from now on just as H-invariance. An H-invariant Poisson bracket on 𝒜=ℂ[x0,x1,…,xn-1] is nothing but a bracket on 𝒜 which satisfy the following: (2.10){xi+1,xj+1}=σ⋅{xi,xj},τ⋅{xi,xj}=ɛi+j{xi,xj},
for all i,j∈ℤ/nℤ.Theτ invariance is, in some sense, a “discrete” homogeneity.Proposition 2.2 (see [2]).
The Sklyanin-Odesskii-Feigin Poisson algebrasqn,k(ℰ) are H-invariant Poisson algebras.Therefore, anH-invariant Poisson structures on the polynomial algebra ℛ includes as the Sklyanin Poisson algebra or more generally of the Odesskii-Feigin Poisson algebras.In this paper, we are interested in the intersection of the two classes of generalizations of Artin-Shelter-Tate-Sklyanin Poisson algebras: JPS andH-invariant Poisson structures.Proposition 2.3 (see [2]).
If{·,·} is an H-invariant polynomial Poisson bracket, the usual polynomial degree of the monomial of {xi,xj} equals to 2+sn,  s∈ℕ.Proposition 2.4 (see [2]).
LetP∈ℛ=ℂ[x0,…,xn-1].
For alli∈{0,…,n-1},
(2.11)σ⋅∂F∂xi=∂(σ⋅F)∂(σ⋅xi).


## 3.H-Invariant JPS in Dimension 3

We consider first a generalization of Artin-Schelter-Tate quadratic Poisson algebras. Letℛ=ℂ[x0,x1,x2] be the polynomial algebra with 3 generators. For every P∈𝒜, we have a JPS π(P) on ℛ given by (3.1){xi,xj}=∂P∂xk,
where (i,j,k)∈ℤ/3ℤ is a cyclic permutation of (0,1,2). Let ℋ be the set of all P∈𝒜 such that π(P) is an H-invariant Poisson structure.Proposition 3.1.
IfP∈ℋ is a homogeneous polynomial, then σ·P=P and τϖ(P)=0.Proof.
Let(i,j,k)∈(ℤ/3ℤ)3 be a cyclic permutation of (0,1,2). One has
(3.2){xi,xj}=∂P∂xk,(3.3){xi+1,xj+1}=∂P∂xk+1.
Using Proposition 2.4, we conclude that for all m∈ℤ/3ℤ,  ∂σP/∂xm=∂P/∂xm.
It gives thatσ·P=P.
On the other hand, from (3.2), one has τ-ϖ(P)≡i+j+kmod3. And we get the second half of the proposition.Proposition 3.2.
ℋ is a subalgebra of ℛ.Proof.
LetP,Q∈ℋ. It is clear that for all α,β∈ℂ,  αP+βQ belongs to ℋ.
Let us denote by{·,·}F the JPS associated with the polynomial F∈ℛ. It is easy to verify that {xi,xj}PQ=P{xi,xj}Q+Q{xi,xj}P. Therefore, it is clear that the H-invariance condition is verified for the JPS associated to the polynomial PQ.We endowℋ with the usual grading of the polynomial algebra ℛ. For F, an element of ℛ, we denote by ϖ(F) its usual weight degree. We denote by ℋi the homogeneous subspace of ℋ of degree i.Proposition 3.3.
If 3 does not dividei∈ℕ (in other words i≠3k), then ℋi=0.Proof.
First of allℋ0=ℂ. We suppose now that i≠0. Let P∈ℋi,P≠0. Then, ϖ(P)=i. It follows from Proposition 2.3 and the definition of the Poisson brackets that there exists s∈ℕ such that ϖ(xi,xj)=2+3s. The result follows from the fact that ϖ(xi,xj)=ϖ(P)-1.SetP=∑cα¯x0α0x1α1x2α2, where α¯=(α0,α1,α2). We suppose that ϖ(P)=3(1+s). We want to find all α0,α1,α2 such that P∈ℋ and, therefore, the dimension ℋ3(1+s) as ℂ-vector space.Proposition 3.4.
There exists′, s′′, and s′′′ such that
(3.4)α0+α1+α2=3(1+s),0α0+α1+2α2=3s′,α0+2α1+0α2=3s′′,2α0+0α1+1α2=3s′′′.Proof.
This is a direct consequence of Proposition3.3.Proposition 3.5.
The system equation (3.4) has as solutions the following set:
(3.5)α0=4r-2s′-s′′,α1=-2r+s′+2s′′,α2=r+s′-s′′,
where r=1+s,  s′ and s′′ live in the polygon given by the following inequalities in ℝ2:(3.6)x+y⩽3r,2x+y⩽4r,-x-2y⩽-2r,-x+y⩽r.Remark 3.6.
Forr=1, one obtains the Artin-Schelter-Tate Poisson algebra which is the JPS given by the Casimir P=α(x03+x13+x23)+βx0x1x2,  α,β∈ℂ. Suppose that α≠0, then it can take the form
(3.7)P=(x03+x13+x23)+γx0x1x2,
where γ∈ℂ. The interesting feature of this Poisson algebra is that their polynomial character is preserved even after the following nonalgebraic changes of variables. Let
(3.8)y0=x0;y1=x1x2-1/2;y2=x23/2.
The polynomial P in the coordinates (y0;y1;y2) has the form
(3.9)P̃=(y03+y13y2+y22)+γy0y1y2.
The Poisson bracket is also polynomial (which is not evident at all!) and has the same form
(3.10){xi,xj}=∂P̃∂xk,
where (i,j,k) is the cyclic permutation of (0,1,2). This JPS structure is no longer satisfying the Heisenberg invariance condition. But it is invariant with respect the following toric action: (ℂ*)3×ℙ2→ℙ given by
(3.11)λ⋅(x0:x1:x2)=(λ2x0:λx1:λ3x2).
Putdegy0=2;degy2=1;degy2=3. Then, the polynomial P̃ is also homogeneous in (y0;y1;y2) and defines an elliptic curve P̃=0 in the weighted projective space 𝕎P2;1;3.
The similar change of variables(3.12)z0=x0-3/4x13/2;z1=x01/4x1-1/2x2;z2=x03/2
defines the JPS structure invariant with respect to the torus action (ℂ*)3×ℙ2→ℙ given by
(3.13)λ⋅(x0:x1:x2)=(λx0:λx1:λ2x2),
and related to the elliptic curve 1/3(z22+z02z2+z0z13)+kz0z1z2=0 in the weighted projective space 𝕎P1;1;2.
These structures had appeared in [1], their Poisson cohomology was studied by Pichereau [14], and their relation to the noncommutative del Pezzo surfaces and Calabi-Yau algebras were discussed in [15].Proposition 3.7.
The subset ofℝ2 given by the system (3.6) is a triangle 𝒯r with (0,r),  (r,2r), and (2r,0) as vertices.  Then, dimℋ3r=Card(𝒯r∩ℕ2).Remark 3.8.
Forr=2, the case of  Figure 1, the generic Heisenberg-invariant JPS is given by the sextic polynomial
(3.14)P∨=16a(p06+p16+p26)+13b(p03p13+p03p23+p13p23)+c(p04p1p2+p0p14p2+p0p1p24)+12dp02p12p22,a,b,c,d∈C.
The corresponding Poisson bracket takes the form
(3.15){xi,xj}=axk5+b(xi3+xj3)xk2+cxixj(xi3+xj3+4xk3)+dxi2xj2xk,
where i,j,k are the cyclic permutations of 0,1,2.
This new JPS should be considered as the “projectively dual” to the Artin-Schelter-Tate JPS, since the algebraic varietyℰ∨:P∨=0 is generically the projective dual curve in ℙ2 to the elliptic curve
(3.16)E:P=(x03+x13+x23)+γx0x1x2=0.
To establish the exact duality and the explicit values of the coefficients, we should use (see [16, chapter 1]) Schläfli's formula for the dual of a smooth plane cubic ℰ=0⊂ℙ2. The coordinates (p0:p1:p2)∈ℙ2* of a point p∈ℙ2* satisfies to the sextic relation ℰ∨=0 if and only if the line x0p0+x1p1+x2p2=0 is tangent to the conic locus 𝒞(x,p)=0, where
(3.17)C(x,p)=|0p0p1p2p0∂2P∂x02∂2P∂x0∂x1∂2P∂x0∂x2p1∂2P∂x1∂x0∂2P∂x1∂x1∂2P∂x1∂x2p2∂2P∂x2∂x0∂2P∂x2∂x1∂2P∂x2∂x2|.
Set𝒮r=𝒯r∩ℕ2.  𝒮r=𝒮r1∪𝒮r2, 𝒮r1={(x,y)∈𝒮r:0≤x≤r} and 𝒮r1={(x,y)∈𝒮r:r<x≤2r}.  dimℋ3r=Card(𝒮r1)+Card(𝒮r2).Figure 1
An example of the triangle𝒯2 in the case r=2.Proposition 3.9.
(3.18)Card(Sr1)={3r2+6r+44ifris even,3r2+6r+34ifris odd.Proof.
Letα∈{0,…,r}, and set 𝒟rα={(x,y)∈𝒮r1:x=α}. Therefore,
(3.19)Card(Sr1)=∑α=0rCard(Drα).
Let
(3.20)βmax⁡α=max⁡⁡{β:(α,β)∈Drα},βmin⁡α=min⁡⁡{β:(α,β)∈Drα}.Card(𝒟rα)=βmaxα-βminα+1.
It is easy to prove that(3.21)Card(Drα)=(α+1)+⌊α2⌋={3α+22ifαis even,3α+12ifαisodd.
The result follows from the summation of all Card(𝒟rα),α∈{0,…,r}.Proposition 3.10.
(3.22)Card(Sr2)={3r24ifris even,3r2+14ifris odd.Proof.
Letα∈{r+1,…,2r}, and set 𝒟rα={(x,y)∈𝒮r2:x=α}. Therefore,
(3.23)Card(Sr2)=∑α=r+12rCard(Drα).
Let
(3.24)βmax⁡α=max⁡⁡{β:(α,β)∈Drα},βmin⁡α=min⁡⁡{β:(α,β)∈Drα}.Card(𝒟rα)=βmaxα-βminα+1.
It is easy to prove that(3.25)Card(Drα)=(3r+1)-2α+⌊α2⌋={6r+2-3α2ifαis even,6r+1-3α2ifαisodd.
The result follows from the summation of all Card(𝒟rα),  α∈{r+1,…,2r}.Theorem 3.11.
(3.26)dim⁡H3(1+s)=32s2+92s+4.Proof.
This result is a direct consequence of Propositions3.9 and 3.10.Corollary 3.12.
The Poincaré series of the algebrasℋ is
(3.27)P(H,t)=∑s≥-1dim⁡⁡(H3(1+s))t3(s+1)=1+t3+t6(1-t3)3.Remark 3.13.
Forr=2, the case of Figure 1, our formula gives the same answer like the classical Pick's formula for integer points in a convex polygon Π with integer vertices on the plane ([17, chapter 10])
(3.28)Card(Π∩Z2)=Area(Π)+12Card(∂Π∩Z2)+1.
Here, s=1 and dimℋ3(1+1)=10. In other hand the Pick's formula ingredients are
(3.29)Area(Π)=12|2-44-2|=6,12Card(∂Π∩Z2)=3,
hence 6+3+1=10.This remark gives a good hint how one can use the developed machinery of integer points computations in rational polytopes to our problems.


## 4.H-Invariant JPS in Any Dimension

In order to formulate the problem in any dimension, let us remember some number theoretic notions concerning the enumeration of nonnegative integer points in a polytope or more generally discrete volume of a polytope.


### 4.1. Enumeration of Integer Solutions to Linear Inequalities

In their papers [18, 19], the authors study the problem of nonnegative integer solutions to linear inequalities as well as their relation with the enumeration of integer partitions and compositions.Define the weight of a sequenceλ=(λ0,λ2,…,λn-1) of integers to be |λ|=λ0+⋯+λn-1. If sequence λ of weight N has all parts nonnegative, it is called a composition of N; if, in addition, λ is a nonincreasing sequence, we call it a partition of N.Given anr×n integer matrix C=[ci,j],  (i,j)∈({-1}∪ℤ/rℤ)×ℤ/nℤ, consider the set SC of nonnegative integer sequences λ=(λ1,λ2,…,λn) satisfying the constraints(4.1)ci,-1+ci,0λ0+ci,1λ1+⋯+ci,n-1λn-1≧0,0≤r≤n-1.
The associated full generating function is defined as follows:(4.2)FC(x0,x2,…,xn-1)=∑λ∈SCx0λ0x1λ1⋯xn-1λn-1.
This function “encapsulates” the solution set SC: the coefficient of qN in FC(qx0, qx1,…,qxn-1) is a “listing” (as the terms of a polynomial) of all nonnegative integer solutions to (4.1) of weight N, and the number of such solutions is the coefficient of qN in FC(q,q,…,q).


### 4.2. Formulation of the Problem in Any Dimension

Letℛ=ℂ[x0,x1,…,xn-1] be the polynomial algebra with n generators. For given n-2 polynomials P1,P2,…,Pn-2∈ℛ, one can associate the JPS π(P1,…,Pn-2) on ℛ given by (4.3){f,g}=df∧dg∧dP1∧⋯∧dPn-2dx0∧dx1∧⋯∧dxn-1,
for f,g∈ℛ.We will denote byP the particular Casimir P=∏i=1n-2Pi of the Poisson structure π(P1,…,Pn-2). We suppose that each Pi is homogeneous in the sense of τ-degree.Proposition 4.1.
Consider a JPSπ(P1,…,Pn-2) given by homogeneous (in the sense of τ-degree) polynomials P1,…,Pn-2. If π(P1,…,Pn-2) is H-invariant, then
(4.4)τ-ϖ(σ⋅P)=τ-ϖ(P)={n2ifnis even,0ifnis odd,
where P=P1P2⋯Pn-2.Proof.
Leti<j∈ℤ/nℤ, and consider the set Ii,j, formed by the integers i1<i2<⋯<in-2∈ℤ/nℤ∖{i,j}. We denote by Si,j the set of all permutation of elements of Ii,j. We have
(4.5){xi,xj}=(-1)i+j-1dxi∧dxj∧dP1∧dP2∧⋯∧dPn-2dx0∧dx1∧dP2∧⋯∧dxn-1=(-1)i+j-1∑α∈Si,j(-1)|α|∂P1∂xα(i1)⋯∂Pn-2∂xα(in-2).
From the τ-degree condition,
(4.6)i+j≡(τ-ϖ(P1)-α(i1))+⋯+(τ-ϖ(Pn-2)-α(in-2))modulon.
We can deduce, therefore, that(4.7)τ-ϖ(P1⋯Pn-2)≡n(n-1)2modulon.
And we obtain the first part of the result. The second part is the direct consequence of facts that
(4.8)σ⋅{xi,xj}={xi+1,xj+1}=(-1)i+j-1∑α∈Si,j(-1)|α|∂(σ⋅P1)∂xα(i1)+1⋯∂(σ⋅Pn-2)∂xα(in-2)+1,α(i1)+1≠⋯≠α(in-2)+1∈Z/nℤ∖{i+1,j+1} and the τ-degree condition.Set(4.9)l={n2ifnis even,0ifnis odd.
Let ℋ be the set of all Q∈ℛ such that τ-ϖ(σ·Q)=τ-ϖ(Q)=l. One can easily check the following result.Proposition 4.2.
ℋ is a subvector space of ℛ. It is subalgebra of ℛ if l=0.We endowℋ with the usual grading of the polynomial algebra ℛ. For Q, an element of ℛ, we denote by ϖ(Q) its usual weight degree. We denote by ℋi the homogeneous subspace of ℋ of degree i.Proposition 4.3.
Ifn is not a divisor of i (in other words, i≠nm) then ℋi=0.Proof.
It is clear theℋ0=ℂ. We suppose now that i≠0. Let Q∈ℋi,  Q≠0. Then,
(4.10)Q=∑k1,…,ki-1ak1,…,ki-1xk1⋯xki-1xl-k1-⋯-ki-1.
Hence,
(4.11)σ⋅Q=∑k1,…,ki-1ak1,…,ki-1xk1+1⋯xki-1+1xl-k1-⋯-ki-1+1.
Since τ-ϖ(σ·Q)=τ-ϖ(Q)=l,  i≡0modulon.SetQ=∑βx0α0x1α1⋯xn-1αn-1. We suppose that ϖ(Q)=n(1+s). We want to find all α0,α1,…,αn-1 such that Q∈ℋ and, therefore, the dimension ℋ3(1+s) as ℂ-vector space.Proposition 4.4.
There exists0,s1,…,sn-1 such that
(4.12)α0+α1+⋯+αn-1=n(1+s),0α0+α1+2α2+⋯(n-1)αn-1=l+ns0,1α0+2α1+3α2+⋯(n-1)αn-2+0αn-1=l+ns1,⋮(n-2)α0+(n-1)α1+0α2+⋯(n-3)αn-4+(n-3)αn-1=l+nsn-2,(n-1)α0+0α1+1α2+⋯(n-3)αn-3+(n-2)αn-1=l+nsn-1.Proof.
That is, the direct consequence of the fact thatτ-ϖ(σ·Q)=τ-ϖ(Q)=l.One can easily obtain the following result.Proposition 4.5.
The system equation (4.12) has as a solution
(4.13)αi=sn-i-1-sn-i+r,i∈ZnZ,
where r=s+1 and the s0,…,sn-1 satisfy the condition
(4.14)s0+s1+⋯sn-1=(n-1)n2r-l.Thereforeα0,α1,…,αn-1 are completely determined by the set of nonnegative integer sequences (s0,s1,…,sn-1) satisfying the constraints(4.15)ci:sn-i-1-sn-i+r≥0,i∈ZnZ,
and such that(4.16)s0+s1+⋯+sn-1=(n-1)n2r-l.
There are two approaches to determine the dimension of ℋnr.The first one is exactly as in the case of dimension 3. The constraint (4.16) is equivalent to say that(4.17)sn-1=-(s0+s1+⋯+sn-2)+(n-1)n2r-l.Therefore, by replacingsn-1 by this value, α0,…,αn-1 are completely determined by the set of nonnegative integer sequences (s0,s1,…,sn-2) satisfying the constraints(4.18)c-1′:s0+s1+⋯+sn-3+sn-2≤(n-1)n2r-l,c0′:2s0+s1+⋯+sn-3+sn-2≤[(n-1)n2+1]r-l,c1′:s0+s1+⋯+sn-3+2+sn-2≥[(n-1)n2k+1]r-l,ci′:sn-i-1-sn-i+r≥0,i∈Z/nZ∖{0,1}.
Hence, the dimension ℋnr is just the number of nonnegative integer points contained in the polytope given by the system (4.18), where r=s+1.In dimension 3, one obtains the triangle inℝ2 given by the vertices A(0,2r), B(r,2r),   and C(2r,0) (see Section 3).In dimension 4, we get the following polytope (see Figure2).Figure 2
An example of the polytopeT4 in the case r=4. The vertices are in (-1/2,r-1/2,2r-1/2), (r-2/3,2r-2/3,3r-2/3), (r,2r-1,3r-1), (r,2r,3r-2), (2r-1/2,3r-1/2,-1/2), and (3r-1/2,-1/2,r-1/2).For the second method, one can observe that the dimension ofℋnr is nothing else that the cardinality of the set SC of all compositions (s0,…,sn-1) of N=((n-1)n/2)r-l subjected to the constraints (4.15). Therefore, if SC is the set of all nonnegative integers (s0,…,sn-1) satisfying the constraints (4.15) and FC is the associated generating function, then the dimension of ℋnr is the coefficient of qN in FC(q,q,…,q).The set SC consists of all nonnegative integers points contained in the polytope of ℝn(4.19)Pn:xn-i-1-xn-i+r≥0,i∈ZnZxi≥0,i∈ZnZ.(See Figure3).Figure 3
An example of the polytope𝒫3 in the case r=2.


## 4.1. Enumeration of Integer Solutions to Linear Inequalities

In their papers [18, 19], the authors study the problem of nonnegative integer solutions to linear inequalities as well as their relation with the enumeration of integer partitions and compositions.Define the weight of a sequenceλ=(λ0,λ2,…,λn-1) of integers to be |λ|=λ0+⋯+λn-1. If sequence λ of weight N has all parts nonnegative, it is called a composition of N; if, in addition, λ is a nonincreasing sequence, we call it a partition of N.Given anr×n integer matrix C=[ci,j],  (i,j)∈({-1}∪ℤ/rℤ)×ℤ/nℤ, consider the set SC of nonnegative integer sequences λ=(λ1,λ2,…,λn) satisfying the constraints(4.1)ci,-1+ci,0λ0+ci,1λ1+⋯+ci,n-1λn-1≧0,0≤r≤n-1.
The associated full generating function is defined as follows:(4.2)FC(x0,x2,…,xn-1)=∑λ∈SCx0λ0x1λ1⋯xn-1λn-1.
This function “encapsulates” the solution set SC: the coefficient of qN in FC(qx0, qx1,…,qxn-1) is a “listing” (as the terms of a polynomial) of all nonnegative integer solutions to (4.1) of weight N, and the number of such solutions is the coefficient of qN in FC(q,q,…,q).


## 4.2. Formulation of the Problem in Any Dimension

Letℛ=ℂ[x0,x1,…,xn-1] be the polynomial algebra with n generators. For given n-2 polynomials P1,P2,…,Pn-2∈ℛ, one can associate the JPS π(P1,…,Pn-2) on ℛ given by (4.3){f,g}=df∧dg∧dP1∧⋯∧dPn-2dx0∧dx1∧⋯∧dxn-1,
for f,g∈ℛ.We will denote byP the particular Casimir P=∏i=1n-2Pi of the Poisson structure π(P1,…,Pn-2). We suppose that each Pi is homogeneous in the sense of τ-degree.Proposition 4.1.
Consider a JPSπ(P1,…,Pn-2) given by homogeneous (in the sense of τ-degree) polynomials P1,…,Pn-2. If π(P1,…,Pn-2) is H-invariant, then
(4.4)τ-ϖ(σ⋅P)=τ-ϖ(P)={n2ifnis even,0ifnis odd,
where P=P1P2⋯Pn-2.Proof.
Leti<j∈ℤ/nℤ, and consider the set Ii,j, formed by the integers i1<i2<⋯<in-2∈ℤ/nℤ∖{i,j}. We denote by Si,j the set of all permutation of elements of Ii,j. We have
(4.5){xi,xj}=(-1)i+j-1dxi∧dxj∧dP1∧dP2∧⋯∧dPn-2dx0∧dx1∧dP2∧⋯∧dxn-1=(-1)i+j-1∑α∈Si,j(-1)|α|∂P1∂xα(i1)⋯∂Pn-2∂xα(in-2).
From the τ-degree condition,
(4.6)i+j≡(τ-ϖ(P1)-α(i1))+⋯+(τ-ϖ(Pn-2)-α(in-2))modulon.
We can deduce, therefore, that(4.7)τ-ϖ(P1⋯Pn-2)≡n(n-1)2modulon.
And we obtain the first part of the result. The second part is the direct consequence of facts that
(4.8)σ⋅{xi,xj}={xi+1,xj+1}=(-1)i+j-1∑α∈Si,j(-1)|α|∂(σ⋅P1)∂xα(i1)+1⋯∂(σ⋅Pn-2)∂xα(in-2)+1,α(i1)+1≠⋯≠α(in-2)+1∈Z/nℤ∖{i+1,j+1} and the τ-degree condition.Set(4.9)l={n2ifnis even,0ifnis odd.
Let ℋ be the set of all Q∈ℛ such that τ-ϖ(σ·Q)=τ-ϖ(Q)=l. One can easily check the following result.Proposition 4.2.
ℋ is a subvector space of ℛ. It is subalgebra of ℛ if l=0.We endowℋ with the usual grading of the polynomial algebra ℛ. For Q, an element of ℛ, we denote by ϖ(Q) its usual weight degree. We denote by ℋi the homogeneous subspace of ℋ of degree i.Proposition 4.3.
Ifn is not a divisor of i (in other words, i≠nm) then ℋi=0.Proof.
It is clear theℋ0=ℂ. We suppose now that i≠0. Let Q∈ℋi,  Q≠0. Then,
(4.10)Q=∑k1,…,ki-1ak1,…,ki-1xk1⋯xki-1xl-k1-⋯-ki-1.
Hence,
(4.11)σ⋅Q=∑k1,…,ki-1ak1,…,ki-1xk1+1⋯xki-1+1xl-k1-⋯-ki-1+1.
Since τ-ϖ(σ·Q)=τ-ϖ(Q)=l,  i≡0modulon.SetQ=∑βx0α0x1α1⋯xn-1αn-1. We suppose that ϖ(Q)=n(1+s). We want to find all α0,α1,…,αn-1 such that Q∈ℋ and, therefore, the dimension ℋ3(1+s) as ℂ-vector space.Proposition 4.4.
There exists0,s1,…,sn-1 such that
(4.12)α0+α1+⋯+αn-1=n(1+s),0α0+α1+2α2+⋯(n-1)αn-1=l+ns0,1α0+2α1+3α2+⋯(n-1)αn-2+0αn-1=l+ns1,⋮(n-2)α0+(n-1)α1+0α2+⋯(n-3)αn-4+(n-3)αn-1=l+nsn-2,(n-1)α0+0α1+1α2+⋯(n-3)αn-3+(n-2)αn-1=l+nsn-1.Proof.
That is, the direct consequence of the fact thatτ-ϖ(σ·Q)=τ-ϖ(Q)=l.One can easily obtain the following result.Proposition 4.5.
The system equation (4.12) has as a solution
(4.13)αi=sn-i-1-sn-i+r,i∈ZnZ,
where r=s+1 and the s0,…,sn-1 satisfy the condition
(4.14)s0+s1+⋯sn-1=(n-1)n2r-l.Thereforeα0,α1,…,αn-1 are completely determined by the set of nonnegative integer sequences (s0,s1,…,sn-1) satisfying the constraints(4.15)ci:sn-i-1-sn-i+r≥0,i∈ZnZ,
and such that(4.16)s0+s1+⋯+sn-1=(n-1)n2r-l.
There are two approaches to determine the dimension of ℋnr.The first one is exactly as in the case of dimension 3. The constraint (4.16) is equivalent to say that(4.17)sn-1=-(s0+s1+⋯+sn-2)+(n-1)n2r-l.Therefore, by replacingsn-1 by this value, α0,…,αn-1 are completely determined by the set of nonnegative integer sequences (s0,s1,…,sn-2) satisfying the constraints(4.18)c-1′:s0+s1+⋯+sn-3+sn-2≤(n-1)n2r-l,c0′:2s0+s1+⋯+sn-3+sn-2≤[(n-1)n2+1]r-l,c1′:s0+s1+⋯+sn-3+2+sn-2≥[(n-1)n2k+1]r-l,ci′:sn-i-1-sn-i+r≥0,i∈Z/nZ∖{0,1}.
Hence, the dimension ℋnr is just the number of nonnegative integer points contained in the polytope given by the system (4.18), where r=s+1.In dimension 3, one obtains the triangle inℝ2 given by the vertices A(0,2r), B(r,2r),   and C(2r,0) (see Section 3).In dimension 4, we get the following polytope (see Figure2).Figure 2
An example of the polytopeT4 in the case r=4. The vertices are in (-1/2,r-1/2,2r-1/2), (r-2/3,2r-2/3,3r-2/3), (r,2r-1,3r-1), (r,2r,3r-2), (2r-1/2,3r-1/2,-1/2), and (3r-1/2,-1/2,r-1/2).For the second method, one can observe that the dimension ofℋnr is nothing else that the cardinality of the set SC of all compositions (s0,…,sn-1) of N=((n-1)n/2)r-l subjected to the constraints (4.15). Therefore, if SC is the set of all nonnegative integers (s0,…,sn-1) satisfying the constraints (4.15) and FC is the associated generating function, then the dimension of ℋnr is the coefficient of qN in FC(q,q,…,q).The set SC consists of all nonnegative integers points contained in the polytope of ℝn(4.19)Pn:xn-i-1-xn-i+r≥0,i∈ZnZxi≥0,i∈ZnZ.(See Figure3).Figure 3
An example of the polytope𝒫3 in the case r=2.



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*Source: 252186-2011-10-03.xml*